Theory and Applications of Categories, Vol. 8, No. 7, pp. 114–243.

DUALITY FOR SIMPLE ω-CATEGORIES AND DISKS

MIHALY MAKKAI AND MAREK ZAWADOWSKI

A young schizophrenic named Struther When told of the death of his mother,

Said, ’Yes, it’s too bad, But I can’t feel too sad.

After all, I still have each other.’ The Pan Book of Limericks

ABSTRACT. A. Joyal [J] has introduced the category D of the so-called finite disks, and used it to define the concept of θ-category, a notion of weak ω-category. We introduce the notion of an ω-graph being composable (meaning roughly that ’it has a unique composite’), and call an ω-category simple if it is freely generated by a composable ωgraph. The category S of simple ω-categories is a full subcategory of the category, with strict ω-functors as morphisms, of all ω-categories. The category S is a key ingredient in another concept of weak ω-category, called protocategory [MM1], [MZ]. We prove that D and S are contravariantly equivalent, by a duality induced by a suitable schizophrenic object living in both categories. In [MZ], this result is one of the tools used to show that the concept of θ-category and that of protocategory are equivalent in a suitable sense. We also prove that composable ω-graphs coincide with the ω-graphs of the form T^{∗} considered by M.Batanin [B], which were characterized by R. Street (as announced in [S]) and called ‘globular cardinals’. Batanin’s construction, using globular cardinals, of the free ω-category on a globular set plays an important role in our paper. We give a self-contained presentation of Batanin’s construction that suits our purposes.

Contents

1 Introduction 115 2 Preliminaries 122 3 Disks 145 4 Simple ω-categories 175 5 The duality 191 6 Appendices 208 7 Notation 235

The first author wants to thank the NSERC of Canada and the second author wants to express his thanks to KBN for financial support (grant 2 P03A 014 11)

Received by the editors 2000 May 3 and, in revised form, 2001 February 27. Transmitted by Michael Barr. Published on 2001 March 26. 2000 Mathematics Subject Classification: 18D05, 18D10, 18D35. Key words and phrases: omega-category, globular set, omega-graph, disk, schizophrenic object, du-

ality, theta-category.

\begin{matrix}c \\ ◯ \end{matrix} Mihaly Makkai and Marek Zawadowski, 2001. Permission to copy for private use granted.

1. Introduction

This paper is a contribution to recent studies aimed at clarifying the concept of weak higher dimensional category.

In each of certain recent proposals of a precise notion of weak n-category, for n ≤ ω, a specific small category of ’shapes of cells’ (briefly, a shape-category) is introduced, and a weak n-category is defined as a presheaf on the shape-category having certain additional properties (let us emphasize: not as a presheaf with additional structure). Thus, in particular, a weak n-category W of the given kind has cells of various shapes: for an object A of the shape-category, the elements of the set W(A) are the cells of shape A. The various arrows of the shape category are interpreted in W as face and degeneracy operators, by extending the terminology used for simplicial sets.

In [BD], the opetopic weak n-categories (for finite n) are in fact defined without first describing the shape-category, but the latter is implicit: it is the category of opetopes. In [HMP], the shape-category is made explicit: it is the category of multitopes. In [MM2], where the definition of multitopic ω-category is completed, and, also, the ambient structure comprising all multitopic ω-categories is clarified, the shape-category (the category of multitopes) plays an active role.

In Joyal’s concept of θ-category [J], the shape-category is the opposite of a certain category D, called the category of finite disks. Joyal denotes D^{op} by Θ, and calls it the category of Batanin cells. Cellular sets are set-valued functors on D; a θ-category is a cellular set satisfying certain conditions that are analogs of Daniel Kan’s horn-filling conditions (see [K], and many later sources). In fact, the opposite of the (’non-augmented’) simplicial category: the category Δ^{+} of non-empty finite linear orders is equivalent to a full subcategory of D, and thus, every cellular set has, as a part, an underlying simplicial set ’in it’. The underlying simplicial set of an ω-category satisfies the so-called restricted Kan-condition [BV]. We may regard the passage from simplicial sets to cellular sets as the result of extending the range of ’shapes of cells’ under consideration.

The concept of protocategory was arrived at, by the first author of this paper, independently of Joyal’s work on θ-categories; however, the two concepts are closely related. In the talk [MM1], the first author described only a certain part, the subcategory L, of the shape-category for protocategories; the whole shape-category will be spelled out in [MZ]. The description of L will bring us to one of the two main concepts treated in this paper: simple categories.

By an ω-graph, we mean the same as, for instance, [B] means by a globular set. Consider S_{ωgr}, the following category-sketch (graph with composition relations; categorypresentation could be another name):

[figure: d; G_{1}; G_{0}; c; d; G_{2}; c; d; G_{n}; G_{n−1}; c; …; …]

subject to d ∘ d = d ∘ c and c ∘ d = c ∘ c, (of course, the various d’s and c’s have suitable subscripts distinguishing them; and the ’globular’ identities are understood as all the meaningful ones of the forms given). A model of S_{ωgr} (a graph-map, that is, diagram on S_{ωgr}, obeying the identities) is an ω-graph.

The category of small ω-graphs, with morphism all the natural transformations (of diagrams), is denoted by ωGr.

It is well-known how to present the notion of ω-category equationally over ωGr. Writing ωCat for the category, with ordinary ω-functors as morphisms, of all small ωcategories, we have the forgetful functor ωCat ⟶ ωGr, and its left adjoint [−] : ωGr ⟶ ωCat; for an ω-graph G, [G] is freely generated by G.

It is easy to see that if m : G → H is a monomorphism of ω-graphs, then the induced arrow [m] : [G] → [H] is also a monomorphism.

Let G be an ω-graph. Let us call an element (cell) a of [G] maximal if it is proper, that is, not an identity cell, and if the only monomorphisms m : H → G for which a belongs to the image of [m] are isomorphisms. Intuitively, an element is maximal if it is proper, and the whole graph G is needed to generate it. We call G composable if [G] has a unique maximal element; in that case, the maximal element may be called the composite of the graph.

(Some remarks. It is easy to see that the proper arrows in any ω-category of the form [G] form a sub-ω-graph of [G]: in other words, the domain and the codomain of a proper cell is proper. Moreover, the generating ω-graph G is uniquely recoverable from [G] as consisting of those proper cells that are indecomposable in the sense that they are not composites of two proper cells.)

The graphs

[figure: •; •; •; •; •]

are not composable; for the first as G, [G] has no maximal element, for the second, [G] has infinitely many. (As we will see, this is the general situation: there are either 0, or exactly 1, or else infinitely many maximal elements.) The examples that the reader would think are composable, in an intuitive sense, are indeed such, as experimentation shows. Indeed, the definition is a rigorous formulation of the idea of having a well-defined composite, the latter being the unique maximal element. Note that the definition immediately implies that a composable ω-graph is finite.

An ω-category is simple if it is of the form [G] for a composable ω-graph. The category S is defined as the full subcategory of ωCat on the simple ω-categories as objects.

The category L, mentioned above as part of the shape-category for protocategories, is the skeleton of the subcategory of S with the same objects as S, but with only the monomorphisms as arrows. L has the distinguishing property of being one-way: the endo-monoids of all the objects are trivial. This still holds for the full shape-category for protocategories, and indeed, this is a basic fact about it, making the specification of the concept of protocategory to be one in FOLDS (First Order Logic with Dependent Sorts); see [MM3], [MZ]. We are not going to have to do anything with FOLDS here; however, let us remark that the ’one-way’ condition on the shape-category amounts to the fact that, speaking in reference to the second paragraph of this Introduction, the cells of a protocategory do not have degeneracy operators on them, only face operators. Note that S is not a one-way category, and the concept of θ-category is not specified within FOLDS.

As a part of our work, we give an explicit combinatorial description of the composable ω-graphs. More precisely, we introduce the combinatorial concept of ‘simple’ ω-graph, and prove that ‘composable’ coincides with ‘simple’. The concept of ‘simple’ ω-graph is due to R. Street [S], where he called the concept ‘globular cardinal’. Our description of the notion differs only inessentially from his.

Let G be an ω-graph. We call two cells a and b in G parallel if either they are both of dimension 0, or d(a) = d(b) and c(a) = c(b). By hom(a, b), we mean the set of all cells e for which d(e) = a and c(e) = b. Let us fix the parallel cells a and b, and define the binary relation F = F_{a,b} on the set hom(a, b) by saying that eF f (‘f follows e’) holds iff there is g (of dimension exactly 1 higher than e and f) such that d(g) = e and c(e) = f. The ω-graph G is called simple if for any parallel pair (a, b) the transitive closure of F = F_{a,b} is an irreflexive total order R on hom(a, b), and eF f iff f is the immediate successor of e in R: eRf and there is no h such that eRhRf. We will prove that an ω-graph is composable iff it is simple. Let us remark here that the interesting direction of this equivalence is that ‘composable’ implies ‘simple’; the other direction is easy.

Here is an example of a simple ω-graph:

[figure: •; •; ⇓; ⇒ −; ⇓; ⇒ −; ⇓; ⇓; •; •; ⇓; ⇓; ⇒ −; ⇓; ⇓]

Note that the 1-dimensional simple (composable) ω-graphs are the chains of arrows. A disk, according to [J], is a sequence

[figure: ⋯; D^{n}; p; D^{1}; D^{0}; p; D^{n−1}; p; …]

of sets and functions, with D^{0} being a singleton, together with, for any n ∈ ω and x ∈ D^{n} a specified interval structure, that is, a (nonempty) linear order with a bottom and a top element, on the set p^{−1}(x), subject to the following condition: for any n ∈ ω, and for the functions

[figure: t^{n}; D^{n+1}; D^{n}; b^{n}; t^{n−1}; D^{n−1}; b^{n−1}]

assigning to x the top and bottom elements in p^{−1}(x), for x in D^{n−1} or D^{n}, we have that the set

Equ(t^{n}, b^{n}) = {x ∈ D^{n} : t^{n}(x) = b^{n}(x)}

equals the set Im(t^{n−1}) ∪ Im(b^{n−1}) (here, D^{−1} is understood to be the empty set).

Thus, in a disk, in dimension 0, we have exactly one element. In each positive dimension n = 1, 2, …, we have a bundle of intervals over the previous level, i.e., a disjoint union of intervals, each interval being mapped by the projection to a single element on the previous level. An element x in dimension n > 0 is singular, that is, an endpoint of an interval in the bundle, if and only if the fiber p^{−1}(x), an interval in the bundle in dimension n + 1, is a singleton.

The inner nodes of a disk are the non-singular nodes. Note that the unique node in D^{0} is inner. The planar tree ι(D) of the inner nodes of a disk D completely determines the disk: any isomorphism ι(D) ⟶^{≅} ι(D′) can be uniquely extended to an isomorphism D ⟶^{≅} D′. (In ι(D) the nodes over a given node have a specified linear order on them: hence the adjective ”planar”.)

A disk is finite if it has finitely many inner nodes.

A morphism f : D → E maps D_{n} into E_{n} so that f is compatible with the projections, and for any x the induced map f : p^{−1}(x) → p^{−1}(f(x)) preserves the interval structure: it is order- (that is, ≤-) preserving, and maps end-points to end-points. D, the category of finite disks, is thereby defined.

Here is an example of the first 3 bundles or 4 levels of a disk:

[figure: ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; •; •; ∘; ∘; ∘; ∘; ∘; •; •; ∘; •]

showing inner nodes in black (•), singular ones in white (∘).

The main theorem of the paper is that S^{op} and D are equivalent categories.

The category I of finite strict intervals (non-empty finite linear orders with a first and a last element, which are distinguished in the structure and preserved by the morphisms) and the category Δ of finite non-empty linear orders are dual to one another. The Stonetype adjunction based on a schizophrenic object establishing this duality is described in detail in section 2.7 (except that we deal with Δ^{+} instead of Δ and the corresponding part of T instead of T ). This is the duality Joyal describes in section 1.1 of [J]. Also, the same duality appears in [SGL], p. 455. Our treatment of this duality serves a purely expository purpose. We display the ingredients of the schizophrenic object in detail to make the generalization to the ω-dimensional case easier to follow.

In the remaining parts of the paper we show that the higher-dimensional analogs of these notions are related to each other in a similar manner. On the way to this result we study the structure of D, the category of finite disks, and that of S, the category of simple ω-categories. Among other things, we define an internal disk D in S and an internal ω-category C in D. In a sense, D is a transposition of C and this structure is a schizophrenic object for those categories, defining via hom functors a Stone-type adjunction which establishes an equivalence of the categories S and D, in a way similar to the duality presented in section 2.7 for I and Δ. Although the full proof of this fact is long, the correspondence between a particular simple ω-graph G and the internal tree ι(D) of the disk D dual to the simple ω-category generated by G is easy to understand in particular cases. For example, the simple ω-graph and the (planar) tree drawn below correspond in this sense to each other. An ω-graph:

[figure: •; •; ⇓; ⇒ −; 1; ⇓; ⇒ −; 2; ⇓; ⇓; 3; •; 4; •; ⇓; 5; ⇓; ⇒ −; 6; ⇓; ⇓; 7]

and the corresponding tree:

[figure: 1; 2; 6; •; 3; 5; •; 7; •; 4; •; •]

In the ω-graph above, we numbered those cells that are neither domains nor codomains of other cells. The nodes of the tree correspond to those cells in the ω-graph that are not codomains of any other cell. The leaves correspond the cells which are neither domains nor codomains of any other cell in the ω-graph. To indicate the correspondence, we marked the leaves and corresponding cells with the same numbers.

The correspondence indicated above between planar trees and simple ω-graphs is due to M. Batanin [B]. The correspondence described here is the same as his mapping from T to T^{∗}, for T a planar tree, and T^{∗} the corresponding specific globular set given in [B].

We use extensively the technical notion of a ud-vector, (‘up-and-down-vector’) a vector n⃗ = n_{0}, n_{1}, …, n_{2k} of natural numbers in which elements with odd subscripts are smaller than the neighboring elements with even ones. The ud-vectors completely characterize both finite disks and simple ω-categories up to isomorphism, constituting a simple and useful invariant for objects of both categories. Therefore, we are able to define and prove many things concerning these categories by induction on ud-vectors. A finite disk with a given invariant corresponds via duality to a simple ω-category with the same invariant.

We note that there are other ways, cosidered in the literature, to use finite sequences of natural numbers to describe the same combinatorial objects, cf. [L], [Ma].

To form an idea how the ud-vector associated with a tree is formed, consider the example of a tree given above. Now, the ud-vector is 3, 2, 3, 1, 2, 0, 1, 0, 2, 1, 3, 1, 2. This is a sequence of (non-negative) integers; we start numbering the positions with 0. In the seven even positions, we find the heights of the seven leaves, in the order from left to right. In each odd position, we find the height where the leaves corresponding to the neighboring even positions ‘meet’.

Let us comment on the connections of our work to other people’s work.

It goes without saying that one of the starting points for the present paper is A.Joyal’s preprint [J], produced in September 1997. In addition, Joyal conjectured the exact statement of our main result, the equivalence of S^{op} and D, with the small difference that in his version of S, the ingredient ‘composable ω-graph’ is replaced by ‘globular cardinal’ (which, as we said, we proved to be equivalent concepts). We learned about this fact from an e-mail message by Joyal on June 23, 1999, when our work had been completed, and a version of the present paper had been written, and was ready for electronic dissemination. Upon receiving a description of our result, in the e-mail message mentioned above, Joyal wrote, among others: ‘I am happy you have proved this duality. I had suspected it shortly after writing my notes ‘disk, ...’ [which are [J]], but had no proof. I like your description of composable ω-graph’.

Recently we realized that the paper [BS] (that appeared in the year 2000, but which had been put on the internet already in November 1997) contains, in essence, a statement of Joyal’s conjecture. As we had been unaware of [BS], this source did not influence our work. As we indicated at the beginning of this Introduction, the idea of the category S came to the first author in the Spring of 1998 independently of considerations involving disks.

As we mentioned above, A. Joyal called his category Θ the category of Batanin cells, indicating connections of his work to Michael Batanin’s work. In [B], Batanin introduced , and used extensively, the planar trees mentioned above. His construction of the ω-graph T^{∗} out of the tree T is the same as what we described above as the correspondence between simple ω-graphs and trees; in particular, simple ω-graphs are the same as the ones of the form T^{∗}, for T a planar tree. In Proposition 4.2 in [B], Batanin gives a construction of the free ω-category generated by an arbitrary ω-graph, one that uses trees and, ultimately, simple ω-graphs; we will reprove his result in this paper (in sections 4.1 and 6.4).

At the end of the paper [BS], we find a statement concerning Θ, Joyal’s category mentioned above, the formal dual of the category of finite disks. When one compares Joyal’s paper [J] with the description of Θ given in [BS], the coincidence of the two descriptions turns out to be nothing but the main theorem of our paper! We have recently learned from a private communication by Professors Street and Batanin that the description of Θ given in [BS] resulted from conversations of theirs and Andre Joyal’s in which Joyal described his conjecture, ”identifying” the original description of Θ and the one in [BS], which is essentially the same as that of S.

In [BS], there is no proof of our theorem, neither is Joyal’s original description of Θ presented. Therefore the reader who does not know Joyal’s paper gets the misleading impression that the description of [BS] is merely a reformulation of Joyal’s definition. Joyal’s purely combinatorial definition of Θ (already given above in this introduction) is very different from that of the category of simple ω-categories; the equivalence of the two categories is far from obvious as Joyal’s statement ‘[I] had no proof’ quoted above also indicates. Let us point out that we got the idea of the duality theorem of this paper in August 1998, and its proof shortly thereafter.

Since A. Joyal’s preprint [J] is unpublished, we find it appropriate to point out that, despite the appearence of the word ‘duality’ in the title, the paper does not contain an indication of the possibility of the statement of our duality theorem. In fact, [J] makes no reference to (strict) ω-categories at all.

A theorem related to our main result is Theorem 1.13 of [Be]. Note, however, that the expression Θ(S, T) used in the statement of the theorem is defined in a way that is not directly related to hom-sets in Joyal’s category Θ.

Our paper is organized as follows. In chapter 2 we introduce the notions and some notation concerning the main notions used in the paper: disks, simple ω-categories, and ud-vectors. In the last section, 2.7, we present the well known duality for finite linear orders and finite intervals in a way that can be generalized to the case of simple ωcategories and of finite disks. This presentation is more involved than it could be, but we think that doing this exercise will help the reader to understand the main result of the paper.

In chapter 3 we investigate the category of finite disks. In section 3.1, we introduce some notation and state some basic facts concerning disks. In section 3.11, we study certain special pullbacks of disks, and for this purpose we use some simple results discussed in section 3.6, concerning similar limits in some related categories of posets. We obtain a presentation of any finite disk as a multi-pullback of very simple disks, and we associate with every disk a ud-vector which describes it up to isomorphism. In section 3.21, we define three special kinds of morphisms in D and we show that every morphism can be, essentially uniquely, presented as a composition of such morphisms. In section 3.25, we define the internal ω-category C in D and we show that homming into it defines a contravariant functor from D to S.

In chapter 4 the simple ω-graphs and simple ω-categories are investigated. In section 4.1, some notation concerning simple ω-graphs is introduced and the construction of a free ω-category on an ω-graph is presented. The construction is based on simple ω-graphs.

We verify that the construction is correct by relating it to a more general one presented in Appendix 6.8. In section 4.7, we prove that simple ω-graphs are exactly those that are composable. In section 4.9 , we introduce some notation for simple ω-categories, prove some of their properties, define an internal disk D in S. We show that homming into it defines a contravariant functor from S to D.

In chapter 5 we state and prove the main result of the paper. We show that the contravariant functors mentioned above form a Stone adjunction between S and D, which is an equivalence of categories. In section at the end of the section 5.5, we indicate the correspondence of objects and some morphisms in categories D and S via the established duality.

The final section of this chapter contains some applications of our work. Among other things we define a nerve functor for ω-categories, i.e. a full and faithful functor

N_{ω} : ωCat ⟶ Set^{D}

In this way, we identify ω-categories as special pullbacks preserving cellular sets, i.e. a special kind of θ-categories.

The chapter 6 contains four appendices. We spell out the full (elementary) definitions of an internal disk and an internal ω-category. In Appendix 6.3 we prove some facts concerning internal ω-categories. Among other things we prove some general form of the associativity law for ω-categories. In Appendix 6.8 we give a construction of a free internal ω-category over an internal ω-graph and we prove that the free ω-category functor preserves pullbacks. This construction in based on ud-vectors.

For the convenience of the reader, all the notation introduced in the paper is collected in chapter 7.

In the whole paper, ω denotes the set of (von Neuman) natural numbers, i.e. if n ∈ ω then n = {0, …, n − 1}, and ω^{+} the set of positive natural numbers. Set is the category of (small) sets.

Acknowledgements

We thank M.Batanin and R.Street for the valuable informations that they provided for us concerning the historical background to our work.

The first autor thanks the participants of the Montreal Category Seminar who in the Winter and Spring 1999, listened to several talks by him on the subject of the present paper.

The diagrams for this paper were prepared with a help of catmac of Michael Barr.

2. Preliminaries

2.1. The category D of finite disks. The category of finite disks D was introduced by A. Joyal in [J]. In this section, we repeat this definition using the original terminology.

In order to introduce the category of finite disks we need to introduce the category of finite trees.

A bundle of linear orders over the set B is a linear order in Set/B, i.e. a map p : E ⟶ B with each fiber linearly ordered. A (planar) tree T is a sequence of bundles of linear orders

[figure: ⋯; T^{s+1}; p^{s+1}; T^{1}; T^{0} ≅ 1; p^{0}; T^{s}; p^{s}; …]

We often omit the superscript s of projection p^{s}, when it does not lead to confusion.

A morphism of trees f : T ⟶ T′ is a set of functions {f^{s} : T^{s} ⟶ T^{′s}}_{s∈ω} preserving projections and order in fibers. A tree is finite if all T_{n}’s are finite and almost all are empty. Let T ree and T denotes the categories of trees and finite trees, respectively.

If n > s, by p^{(s)} : T^{n} ⟶ T^{s} we denote the composition of n − s projections. By convention, if n = s, then p^{(s)}(x) = x.

We introduce notation for some finite trees. For n ∈ ω, θ_{n} is a tree such that, for s ∈ ω

θ_{n}^{s} = \{ \begin{matrix}\{s\} & if s ≤ n \\ ∅ & if s > n \end{matrix}

and the projections are the obvious ones. Thus, for example, θ_{3} can be drawn as

0 1 2 3

A leaf x of a tree T is a node of T, such that p^{−1}(x) = ∅. Clearly, any tree morphism f : T ⟶ T′ in uniquely determined by the function f restricted to the leaves of T.

A bundle of intervals over set B is an interval in Set/B, i.e. a diagram of sets and functions

[figure: b; E; B; p; t]

with each fiber p^{−1}(x), for x ∈ B being an interval, i.e. a linear order with endpoints b(x) and t(x). The equalizer of b and t, a subset of B, is the singular set of the bundle. A disk D is a sequence of bundles of intervals

[figure: ⋯; D^{s+1}; p^{s+1}; D^{1}; D^{0} ≅ 1; p^{0}; D^{s}; p^{s}; …]

such that the singular set eq(b, t) of p : D^{n+1} ⟶ D^{n} is equal to b(D^{n−1})∪t(D^{n−1}). (Here, by convention, D^{−1} = ∅, and the functions b, t : D^{−1} ⟶ D^{0} are thereby defined.) We call this property the disk condition.

As a consequence of the definition, we have bb = tb and bt = tt. We define the boundary ∂(D^{n}) to be b(D^{n−1}) ∪ t(D^{n−1}) and the interior ı(D^{n}) to be D^{n} \ ∂(D^{n}). By the previous convention ∂(D^{0}) = ∅. The nodes in ı(D^{n}) are called inner and the nodes in ∂(D^{n}) are called outer. Because of the ’disk condition’, the projections p send inner nodes to inner nodes and hence, restricting the projections to the interiors, we obtain the internal tree ı(D) of the disk D

[figure: ⋯; ı(D^{s+1}); p; ı(D^{1}); ı(D^{0}) ≅ 1; p; ı(D^{s}); p; …]

We say that the disk D is finite if ı(D) is.

A morphism of disks f : D ⟶ E is a set of functions {f^{s} : D^{s} ⟶ E^{s}}_{s∈ω} preserving projections, order and endpoints in fibers. Let Dk and D denotes the categories of disks and finite disks, respectively. From now on, by a disk we mean a finite disk, unless explicitly stated otherwise.

Similarly as for trees, if n ≥ s, by p^{(s)} : D^{n} ⟶ D^{s} we denote the composition of n − s projections.

There is an obvious forgetful functor |−| from disks to trees, forgetting the endpoints, which has a left adjoint \overline{(−)}:

[figure: \overline{(−)}; T ree; Dk; | − |]

For a tree T, \overline{T} is the unique disk D, such that ı(D) is isomorphic to T. In fact, Dk is both Kleisli and Eilenberg-Moore category for the monad induced by this adjunction. The above adjunction restricts to finite disks and finite trees:

[figure: \overline{(−)}; T; D; | − |]

It follows that, in order to define a disk morphism f : D ⟶ E, it is enough to define a tree morphism f′ : ı(D) ⟶ E. In the Appendix 6.1, we give an internal version of the notion of a disk in an arbitrary category, so that, disks (not necessarily finite) defined above become internal disks in Set.

2.2. The category S of simple ω-categories . An ω-graph G in a category C has for each n ∈ ω an object G_{n} of n-cells in C and operations

[figure: d_{n}; G_{n+1}; G_{n}; c_{n}]

of domain and codomain. We usually omit the subscripts of the morphisms d_{n} and c_{n}. Furthermore, in the diagram

[figure: d; G_{n+1}; G_{n}; c; d; G_{n−1}; c; …; …]

we have d ∘ d = d ∘ c and c ∘ d = c ∘ c. A morphism between ω-graphs G and G′ is a family of arrows {φ_{n} : G_{n} ⟶ G′_{n}}_{n∈ω} in C commuting with operations of domain and codomain. By ωGr we denote the category of ω-graphs in Set.

A simple ω-graph is an ω-graph G in Set, such that

1. G is non-empty and finite, i.e. each G_{n} in finite G_{0} ≠ ∅ and almost all are empty; the height of G is ht(G) = max{n : G_{n} ≠ ∅};

2. for n ∈ ω, G_{n} is partially ordered; for any x, y ∈ G_{n} the subset of G_{n+1}

G_{n+1}(x, y) = {u ∈ G_{n+1} : d(u) = x and c(u) = y}

is linearly ordered by ≥; as well as G_{0}; let > denote the immediate predecessor relation: u > v means that u, v ∈ G_{0} and v is an immediate predecessor of u or u, v ∈ G_{n+1}(x, y) for some x, y ∈ G_{n} such that x >y, and moreover v is a predecessor of u in that order;

3. for n ∈ ω, if x, y ∈ G_{n} then

x > y iff G_{n+1}(x, y) ≠ ∅

Let sωGr denote the full subcategory of ωGr, whose objects are simple ω-graphs.

The full definition of an ω-category in a category C is given in the Appendix 6.2. Below, we briefly sketch the definition.

An ω-category A is an ω-graph together with operations of identity, for n, l ∈ ω, n ≤ l,

ι_{(n,l)}^{A} = ι_{(l)}^{A} : A_{n} ⟶ A_{l}

and compositions

m_{n_{0},n_{1},n_{2}}^{A} : A_{n_{0},n_{1},n_{2}} ⟶ A_{max(n_{0},n_{2})}

where for a 3-tuple ⟨n_{0}, n_{1}, n_{2}⟩ such that n_{1} < n_{0}, n_{2}, the diagram

[figure: A_{n_{0}}; A_{n_{1}}; c^{A}; A_{n_{0},n_{1},n_{2}}; A_{n_{2}}; π_{1}; π_{0}; d^{A}]

is a pullback. This data is subject to conditions concerning domains and codomains of identities and compositions, neutrality of identities, associativity of compositions, and middle exchange law. The morphisms of ω-categories are defined as the morphisms of the underlying graphs preserving additionally compositions and identities. ωCat denotes the category of ω-categories in Set.

The forgetful functor

U : ωCat ⟶ ωGr

has a left adjoint,

[−] : ωGr ⟶ ωCat

associating to a graph G, the free ω-category [G] generated by G. A specific construction of this functor using simple ω-graphs is given in section 4.1.

If G′ is a sub-ω-graph of G, then [G′] is a sub-ω-category of [G]. Moreover the ωcategory [G] determines uniquely (up to an isomorphism) the ω-graph G, as the ω-graph of those cells in [G] that are not compositions of two other non-identity cells in [G]. We have that if an ω-functor φ : [G] ⟶ [G′] is an isomorphism then there is a unique isomorphism of ω-graphs ψ : G ⟶ G′ such that [ψ] is φ. All this can be easily deduced from the description of the functor [−] : ωGr ⟶ ωCat given in the section 4.1.

Let G be an ω-graph. A cell e in [G], is said to be maximal if it is not an identity cell and it is not contained in [G′], for any proper sub-ω-graph G′ of G. An ω-category S is simple iff for some graph G, it is isomorphic to the category [G] containing exactly one maximal cell. The unique maximal cell in [G] and S, if exists, will be denoted by mac_{G} and mac_{S}, respectively. An ω-graph is composable if [G] is a simple category. The name composable comes from the intuition that if a graph G is composable then it has well define composition in [G] which is the maximal cell mac_{G}.

Note, that a 1-category (i.e. ω-category without non-identity cells of dimension bigger then 1) is simple iff it is generated by a composable string of arrows, e.g. the category [G_{1}], generated by the graph G_{1}:

[figure: •; •; k; •; •; f; g; h]

has no maximal arrow, and in the category [G_{2}], generated by the graph G_{2}:

[figure: •; f]

all non-identity arrows are maximal.

Later (Theorem 4.8) we shall prove that simple ω-categories are exactly those which are free ω-categories generated by a simple ω-graphs, i.e. that a ω-graph is composable iff it is simple.

We shall define a functor

T r : sωGr ⟶ T

Let G be a simple ω-graph. Let max(G_{n}) be the set of maximal elements in G_{n} and max(G_{n}, x, y) the maximal element in G_{n}(x, y), provided x, y ∈ G_{n−1} and x > y. We put

T r(G)^{n} = max(G_{n})

Thus T r(G)^{0} contains only max(G_{0}), the maximal element of G_{0} and for n > 0 and x ∈ G_{n} we have that x ∈ T r(G)^{n} iff x = max(G_{n}, d(x), c(x)).

The projection p_{n} : T r(G)^{n+1} ⟶ T r(G)^{n} is defined as follows. For x ∈ G_{1}, p^{1}(x) = max(G_{0}). For n > 1 and x ∈ G_{n},

p^{n}(x) = max(G_{n−1}, d_{(n−2)}(x), c_{(n−2)}(x))

The order in the fibers of T r(G) is given as follows. If u, v ∈ T r(G)^{1} then u ≥ v iff d(u) ≥ d(v) in G_{0}, and if u, v ∈ T r(G)^{n} for n > 1, then u ≥ v iff d(u) ≥ d(v) in G_{n−1}(d_{(n−2)}(x), c_{(n−2)}(x)).

Let f : G ⟶ G′ be a morphism of simple ω-graphs. Then for x = max(G_{0}) ∈ T r(G)^{0} we put T r(f)^{0}(x) = max(G′_{0}) and for x ∈ T r(G)^{n} for n > 0, we put T r(f)^{n}(x) = max(G′_{n}, d(f_{n}(x)), c(f_{n}(x))).

The functor T r is neither full nor faithful, but it is conservative (i.e. reflects isomorphisms) and essentially surjective. We shall sketch why the last property holds. Let T be a tree. We shall define a simple ω-graph G such that T r(G) is isomorphic to T. To this end, we define a predecessor function

pre_{T} : ⋃_{s=1}^{∞} T^{s} ⟶ ⋃_{s=0}^{∞} T^{s}

such that, for x ∈ T^{n},

pre_{T} (x) = \{ \begin{matrix}next element in the fiber & if exists \\ p^{n−1}(x) & otherwise \end{matrix}

for n ≥ 1. We put

G_{n} = T^{n} + T^{n+1}

for x ∈ T^{n} ↪ G_{n}

d_{n}(x) = pre_{T} (x) ∈ G_{n−1}
c_{n}(x) = x ∈ T^{n} ↪ G_{n−1}

for x ∈ T^{n+1} ↪ G_{n}

d_{n}(x) = pre_{T} (p^{n}(x)) ∈ G_{n−1}
c_{n}(x) = p^{n}(x) ∈ T^{n} ↪ G_{n−1}

Now, one can check that T r(G) is isomorphic to T.

Putting together the functors that we have mentioned so far, we get the following diagram of categories and functors

[figure: T; T r; D; sωGr; ωGr; S; ωCat; [−]; \overline{(−)}; | − |; [ −]; U]

where horizontal arrow going right are inclusions.

2.3. The ud-vectors. In this section we introduce ud-vectors, some vectors of natural numbers. They characterize up to an isomorphism both disks and simple ω-categories, being much simpler then either of them.

The reason, the notion of a ud-vector is rather technical as opposed the other two is that there is no reasonable easy notion of a morphism of ud-vectors. However, it is very convenient to describe domains and codomains of disk morphisms and ω-functors between simple ω-categories using ud-vectors. Some (important) pullbacks in D can be described in terms of operation of amalgamation of ud-vectors. For l ∈ ω, we introduce an l-size of ud-vectors. This will allow us to show easily many properties of disks and simple categories, by induction on l-size.

By an up-and-down vector, ud-vector for short, u⃗, we mean a sequence of natural numbers u⃗ = ⟨u_{0}, …, u_{2k}⟩ with k ∈ ω, such that u_{2i+1} < u_{2i}, u_{2i+2}, for i ∈ k. By the length lh(u⃗) of a ud-vector u⃗, we mean the number of even-numbered elements in it. (To help the intuition of the reader, let us note that, with T the tree corresponding to u⃗ in the sense indicated in the Introduction, we have that lh(u⃗) is the number of the leaves in T. This, and the similar parenthetical remarks that follow, are not needed for the technical development.) Thus, in the above case, lh(u⃗) = k + 1. The height ht(u⃗) of a ud-vector u⃗, we mean the maximum number in u⃗, i.e. max(u⃗). (ht(u⃗) is the maximal dimension of a cell occuring in G, the simple graph corresponding to u⃗.) If we write u⃗ = u⃗′, z, u⃗′′ then we mean that the ud-vector u⃗ is a concatenation of ud-vector u⃗′ followed by a single term z, followed by ud-vector u⃗′′.

Let l, k ∈ ω, u⃗ a ud-vector of length k + 1 . We say that u⃗ is l-primitive iff min(u⃗) ≥ l and max(u⃗) > l. The l-size of a ud-vector n⃗ we define as follows

size_{(l)}(u⃗) = \{ \begin{matrix}1 & if u⃗ = u_{0} ≤ l \\ 1 & if u⃗ is l-primitive \\ size_{(l)}(u⃗′) + size_{(l)}(u⃗′′) & if u⃗ = u⃗′, z, u⃗′′, z = min(u⃗) < l \end{matrix}

We shall prove many statements involving ud-vectors by induction on l-size. (With G as above size_{(l)}(u⃗) is the number of equivalence classes of the relation ‘a parallel to b’ for l-cells a, b, where the l-cells a, b are said to be parallel if either l = 0, or d(a) = d(b) and c(a) = c(b).)

The ud-vector tr_{(l)}(u⃗), the l-truncation of u⃗ is defined, by induction on l-size of u⃗, as follows

tr_{(l)}(u⃗) = \{ \begin{matrix}u_{0} & if u⃗ = u_{0} ≤ l \\ l & if u⃗ is l-primitive \\ tr_{(l)}(u⃗′), z, tr_{(l)}(u⃗′′) & if u⃗ = u⃗′, z, u⃗′′ and z = min(u⃗) < l \end{matrix}

(With G as above, tr_{(l)}(u⃗) is the ud-vector associated with the ‘l-truncation’ G′ of G, where G′ is obtained from G by deleting every cell of dimension greater then l, and replacing each parallelism class of l-cells by just one l-cell.)

For l ∈ ω and ud-vectors u⃗, v⃗, such that tr_{(l)}(u⃗) = tr_{(l)}(v⃗) = w⃗, i.e. u⃗ and v⃗ are l-compatible, we define recursively a ud-vector [u⃗, l, v⃗], an l-amalgam of ud-vectors u⃗ and v⃗, by induction on l-size of w⃗, as follows

[u⃗, l, v⃗] = \{ \begin{matrix}u⃗ & if v⃗ = v_{0} ≤ l, \\ v⃗ & if u⃗ = u_{0} ≤ l, \\ u⃗, l, v⃗ & if both u⃗ and v⃗ are l-primitive, \\ [u⃗′, l, v⃗′], z, [u⃗′′, l, v⃗′′] & if u⃗ = u⃗′, z, u⃗′′, v⃗ = v⃗′, z, v⃗′′, \\ & tr_{(l)}(u⃗′) = tr_{(l)}(v⃗′), \\ & tr_{(l)}(u⃗′′) = tr_{(l)}(v⃗′′) \\ & and z = min(u⃗) < l \end{matrix}

If we write [u⃗, l, v⃗], we always presuppose that u⃗ and v⃗ are l-compatible. (The l-amalgam [u⃗, l, v⃗] can be related to simple graphs as folows. Let G, H and I be the simple graphs corresponding to u⃗, v⃗ and w⃗ = tr_{(l)}(u⃗) = tr_{(l)}(v⃗), respectively. Consider the map c : I ⟶ G, d : I ⟶ H defined as follows. On cells of dimensions less that l, c and d are ‘inclusions’. For a in I of dimension l, c(a) is the last element in the parallelism class of elements of G corresponding to a, d(a) is the last element of the corresponding class in H. Finally, let J be the pushout (in the category of ω-graphs) of the maps c and d. J is the simple graph corresponding to [u⃗, l, v⃗].)

The (n_{1}, n_{3})-amalgam [u⃗, n_{1}, v⃗, n_{3}, w⃗] of three ud-vectors u⃗, v⃗, w⃗, such that u⃗ and v⃗ are n_{1}-compatible and v⃗ and w⃗ are n_{3}-compatible is defined as follows:

[u⃗, n_{1}, v⃗, n_{3}, w⃗] = \{ \begin{matrix}[[u⃗, n_{1}, v⃗], n_{3}, w⃗] & if n_{1} ≥ n_{3} \\ [u⃗, n_{1}, [v⃗, n_{3}, w⃗]] & if n_{1} < n_{3} \end{matrix}

We list the following easy relations between the notions introduced above.

Theory and Applications of Categories, Vol. 8, No. 7 130

2.4. Lemma. Let l, n_{1}, n_{3} ∈ ω, u⃗, v⃗, w⃗ ud-vectors, such that u⃗ and v⃗ are n_{1}-complatible and v⃗ and w⃗ are n_{3}-complatible. Then

1. if ht(u⃗) ≤ l then tr_{(l)}(u⃗) = u⃗;

2. if l ≤ n_{1} then tr_{(l)}(tr_{(n_{1})}(u⃗)) = tr_{(l)}(u⃗);

3. ht(tr_{(l)}(u⃗)) = min(l, ht(u⃗));

4. size_{(l)}(u⃗) = size_{(l)}(tr_{(l)}(u⃗)) = lh(tr_{(l)}(u⃗));

5. ht([u⃗, l, v⃗]) = max(ht(u⃗), ht(v⃗));

6. if l ≤ n_{1} then tr_{(l)}(u⃗) = tr_{(l)}([u⃗, n_{1}, v⃗]) = tr_{(l)}(v⃗);

7. if n_{1} < l then tr_{(l)}([u⃗, n_{1}, v⃗]) = [tr_{(l)}(u⃗), n_{1}, tr_{(l)}(v⃗)];

8. if n_{1} = n_{3} then [[u⃗, n_{1}, v⃗], n_{3}, w⃗] = [u⃗, n_{1}, [v⃗, n_{3}, w⃗]];

9. if n_{1} < n_{3} then [u⃗, n_{1}, v⃗, n_{3}, w⃗] = [[u⃗, n_{1}, v⃗], n_{3}, [tr_{(n_{3})}(u⃗), n_{1}, w⃗]];

10. if n_{1} > n_{3} then

[u⃗, n_{1}, v⃗, n_{3}, w⃗] = [[u⃗, n_{3}, tr_{(n_{1})}(w⃗)], n_{1}, [u⃗, n_{3}, w⃗]];

11. [tr_{(n_{1})}(u⃗), n_{1}, u⃗] = u⃗ = [u⃗, n_{1}, tr_{(n_{1})}(u⃗)];

12. if n_{1} < l then

tr_{(l)}([u⃗, n_{1}, v⃗]) = tr_{(l)}([tr_{(l)}(u⃗), n_{1}, v⃗]) = tr_{(l)}([u⃗, n_{1}, tr_{(l)}(v⃗)]);

13. if n_{1} < l then

[u⃗, n_{1}, v⃗] = [[u⃗, n_{1}, tr_{(l)}(v⃗)], l, [tr_{(l)}(u⃗), n_{1}, v⃗]] =
[[tr_{(l)}(u⃗), n_{1}, v⃗], l, [u⃗, n_{1}, tr_{(l)}(v⃗)]].

Proof. Exercise.

The free ω-category on the terminal ω-graph can be conveniently described in terms of ud-vectors and the operations introduced above. We denote this ω-category by UD.

Intuitively, UD is constructed by formally composing ud-vectors of length 1 (the generating cells of UD) in all possible ways. The cells in UD are diagrams of (generating) cells with some prescribed compatibility (the domains of some cells match the codomains of some other cells, in such a way that they ’compose’ altogether to a single cell). In ud-vector u⃗ the i-th generating cell u_{2i} matches i+1-st generating cell u_{2i+2} at level u_{2i+1}.

The set of n-cells UD_{n} consists of the ud-vectors of height at most n. The domain and the codomain operations:

d_{(l)} = d_{(l)}^{UD}, c_{(l)} = c_{(l)}^{UD} : UD_{n} ⟶ UD_{l}

for l ≤ n, are given by the l-truncation, i.e. for a u⃗ ∈ UD_{n}, we have

d_{(l)}(u⃗) = c_{(l)}(u⃗) = tr_{(l)}(u⃗)

The operations are well defined by Lemma 2.4 3.

The identity operations:

ι_{(n)} = ι_{(n)}^{UD} : UD_{l} ⟶ UD_{n}

are inclusions, for l ≤ n.

The compositions in UD are given by the operations of amalgamation. The set UD_{n_{0},n_{1},n_{2}} is the set of n_{1}-compatible pairs of ud-vectors (u⃗, v⃗), such that ht(u⃗) ≤ n_{0} and ht(v⃗) ≤ n_{2}. The composition

m_{n_{0},n_{1},n_{2}} : UD_{n_{0},n_{1},n_{2}} ⟶ UD_{max(n_{0},n_{2})}

is given, for the pair of ud-vectors (u⃗, v⃗) ∈ UD_{n_{0},n_{1},n_{2}} by

m_{n_{0},n_{1},n_{2}}(u⃗, v⃗) = [u⃗, n_{1}, v⃗]

The composition is well defined by Lemma 2.4 5.

2.5. Proposition. UD defined above is the free ω-category over the terminal ω-graph 1.

Proof. First we shall verify that UD is an ω-category and then we shall show that it is free over 1. To prove this we are going to use Lemma 2.4.

By Lemma 2.4 2, we have,

d ∘ d = d ∘ c = c ∘ d = c ∘ c

so UD is an ω-graph. We shall indicate, why conditions (vi)-(xi) in 6.2 of the definition of an ω-category are satisfied.

(vi) .1: let (u⃗, v⃗) ∈ UD_{n_{0},n_{1},n_{2}}, l ≤ n_{1}. Then, using Lemma 2.4 6, we get

d_{(l)} ∘ π_{0}(u⃗, v⃗) = d_{(l)}(v⃗)
= tr_{(l)}(v⃗) = tr_{(l)}([u⃗, n_{1}, v⃗])
= d_{(l)}([u⃗, n_{1}, v⃗]) = d_{(l)} ∘ m(u⃗, v⃗)

(vi) .2: use Lemma 2.4 6.

(vi) .3 and (vi).4 are similar.

(vii) : use Lemma 2.4 1.

(viii) : use Lemma 2.4 1, 8, 9, 10.

(ix) : says that the compositions, of inclusions is an inclusion.

(x) : use Lemma 2.4 11.

(xi) : use Lemma 2.4 12, 13.

This shows that UD is indeed an ω-category.

Now we shall show that UD free on 1. We define

η : 1 ⟶ UD

so that η_{n}(∗) = n ∈ UD_{n}, for n ∈ ω. Let F : 1 ⟶ A be an ω-graph morphism into an ω-category A. We define an ω-graph functor \overline{F} : UD ⟶ A, as follows:

\overline{F}_{n}(u⃗) = \{ \begin{matrix}ι_{(n)}(F (u⃗_{0})) & if u⃗ = u_{0} \\ m_{n,l,n}^{A}(\overline{F} (u⃗′), \overline{F} (u⃗′′)) & if u⃗ = u⃗′, l, u⃗′′, and = min(u⃗) = l < min(u⃗′) \end{matrix}

It is easy to see that, more generaly, the equality which results by droping the condition l < min(u⃗′) is also true.

The verification that \overline{F} ∘ η = F is trivial. Since every cell in UD is obtained by applaying the ι and the m operations repeatedly starting with ud-vectors of length 1, if follows that \overline{F} must be unique, if it exists. It remains to show that \overline{F} is indeed an ω-functor. \overline{F} preserves domains. Let l < n and u⃗ ∈ UD_{n}. We need to show that d_{(l)}(\overline{F}_{n}(u⃗)) = \overline{F}_{l}((d_{(l)}(u⃗)).

The argument is by induction on l-size of u⃗.

If size_{(l)}(u⃗) = 1, we argue by induction on lenght of u⃗. If lh(u⃗) = 1, i.e. u⃗ = u_{0}, we have for l ≤ u_{0}:

\overline{F}_{l}(d_{(l)}(u⃗)) = \overline{F}_{l}(d_{(l)}(u_{0})) = F_{l}(∗)) =
= F_{l}(d_{(l)}(∗)) = d_{(l)}(F_{l}(∗)) = d_{(l)} ∘ ι_{(n)}(F_{u_{0}}(∗)) =

= d_{(l)}(\overline{F} (u_{0})) = d_{(l)}(\overline{F} (u⃗))

and for l > u_{0}:

\overline{F}_{l}(d_{(l)}(u⃗)) = \overline{F}_{l}(u_{0}) =

= ι_{(l)}(F_{u_{0}}(∗)) = d_{(l)} ∘ ι_{(n)}(F_{u_{0}}(∗)) =

= d_{(l)}(\overline{F}_{n}(u_{0})) = d_{(l)}(\overline{F} (u⃗))

If lh(u⃗) > 1 then u⃗ = u⃗′, k, u⃗′′, where n > k = min(u⃗) ≥ l. Using the inductive hypothesis, we get

\overline{F}_{l}(d_{(l)}(u⃗)) = \overline{F}_{l}(l) = \overline{F}_{l}(d_{(l)}(∗)) = d_{(l)}(F_{u_{0}}(∗) =

= d_{(l)}(\overline{F} (u⃗′) = d_{(l)}(m_{n,k,n}(\overline{F} (u⃗′), \overline{F} (u⃗′′)) = d_{(l)}(\overline{F} (u⃗))

Now assume that size_{(l)}(u⃗) > 1, u⃗ = u⃗′, k, u⃗′′, and k = min(u⃗) < l. Then, using the inductive assumption and the axioms of ω-categories, we have:

\overline{F}_{l}(d_{(l)}(u⃗)) = \overline{F}_{l}(tr_{(l)}(u⃗)) =

= \overline{F}_{l}(tr_{(l)}(u⃗′), k, tr_{(l)}(u⃗′′)) = m_{l,k,l}(\overline{F}_{l}(tr_{(l)}(u⃗′)), \overline{F}_{l}(tr_{(l)}(u⃗′′))) =

= m_{l,k,l}(\overline{F}_{l}(d_{(l)}(u⃗′)), \overline{F}_{l}(d_{(l)}(u⃗′′))) = m_{l,k,l}(d_{(l)}(\overline{F}_{n}(u⃗′)), d_{(l)}(\overline{F}_{n}(u⃗′′))) =

d_{(l)}(m_{n,k,n}(\overline{F}_{n}(u⃗′), d_{(l)}(\overline{F}_{n}(u⃗′′))) = d_{(l)}(\overline{F}_{n}(u⃗))

For the codomains, the argument is the same.

\overline{F} preserves identities. Let n < l and u⃗ ∈ UD_{n}. We need to show that ι_{(l)}(\overline{F}_{n}(u⃗)) = \overline{F}_{l}((ι_{(l)}(u⃗)).

The argument is by induction on lenght of u⃗. If u⃗ = u_{0} then

ι_{(l)}(\overline{F}_{n}(u⃗)) = ι_{(l)}(ι_{(n)}(\overline{F}_{u_{0}}(∗))) =

= ι_{(l)}(\overline{F}_{u_{0}}(∗)) = \overline{F}_{l}(∗) = \overline{F}_{l}(ι_{(l)}(∗))

If u⃗ = u⃗′, k, u⃗′′ where k = min(u⃗), we have

ι_{(l)}(\overline{F}_{n}(u⃗)) = ι_{(l)}(m_{l,k,l}(\overline{F}_{n}(u⃗′), \overline{F}_{n}(u⃗′′))) =

= m_{l,k,l}(ι_{(l)}(\overline{F}_{n}(u⃗′)), ι_{(l)}(\overline{F}_{n}(u⃗′′))) = m_{l,k,l}(\overline{F}_{l}(ι_{(l)}(u⃗′)), \overline{F}_{l}(ι_{(l)}(u⃗′′))) =

= m_{l,k,l}(\overline{F}_{l}(u⃗′), \overline{F}_{l}(u⃗′′)) = \overline{F}_{l}(u⃗) = \overline{F}_{l}(ι_{(l)}(u⃗))

\overline{F} preserves compositions. Let (u⃗, v⃗) ∈ UD_{n_{0},n_{1},n_{2}}, n = max(n_{0}, n_{2}). We need to show that m_{n_{0},n_{1},n_{2}}(\overline{F}_{n_{0}}(u⃗), \overline{F}_{n_{2}}(v⃗)) = \overline{F}_{n}(m_{n_{0},n_{1},n_{2}}(u⃗, v⃗)).

The argument is by induction on n_{1}-size of w⃗ = tr_{n_{1}}(u⃗). If u⃗ = u_{0} ≤ n_{1} then u⃗ = tr_{(n_{1})}(v⃗) = ι_{(n_{0})}(d_{(n_{1})}(v⃗), n = n_{1}, and we have

m_{n_{1}}(\overline{F}_{n_{0}}(u⃗), \overline{F}_{n_{2}}(v⃗)) = m_{n_{1}}(\overline{F}_{n_{0}}(ι_{(n_{0})}(d_{(n_{1})}(v⃗)), \overline{F}_{n_{2}}(v⃗)) =

= m_{n_{1}}(ι_{(n_{0})}(d_{(n_{1})}(\overline{F}_{n_{2}}(v⃗)), \overline{F}_{n_{2}}(v⃗)) = \overline{F}_{n}(v⃗) =

\overline{F}_{n}([tr_{(n_{1})}(v⃗), v⃗]) = \overline{F}_{n}(m_{n_{1}}(u⃗, v⃗))

The case v⃗ = v_{0} ≤ n_{1} is similar. If both u⃗ and v⃗ are n_{1}-primitive then

m_{n_{1}}(\overline{F}_{n_{0}}(u⃗), \overline{F}_{n_{2}}(v⃗)) == \overline{F}_{n}(u⃗, n_{1}, v⃗) =

= \overline{F}_{n}([u⃗, n_{1}, v⃗]) = \overline{F}_{n}(m_{n_{1}}(u⃗, v⃗))

Now, let u⃗ = u⃗′, l, u⃗′′, v⃗ = v⃗′, l, v⃗′′, l = min(u⃗) = min(v⃗) < n_{1}, tr_{n_{1}}(u⃗′) = tr_{n_{1}}(v⃗′), and tr_{n_{1}}(u⃗′′) = tr_{n_{1}}(v⃗′′). Then, using inductive hypothesis, axioms (vi), (viii), and (xi) of the definiton of ω-category, and the fact that

d_{(n_{1})}(\overline{F}_{n_{2}}(v⃗′)) = c_{(n_{1})}(\overline{F}_{n_{0}}(u⃗′)), c_{(n_{1})}(\overline{F}_{n_{0}}(u⃗′′)) = d_{(n_{1})}(\overline{F}_{n_{2}}(v⃗′′))  (1)

we have (we drop indices in \overline{F})

\overline{F} (m_{n_{1}}(u⃗, v⃗)) = \overline{F} ([u⃗, n_{1}, v⃗]) =

(assumption on u⃗ and v⃗)

= \overline{F} ([[u⃗′, l, u⃗′′], n_{1}, [v⃗′, l, v⃗′′]]) =

(definition of n_{1}-amalgam)

= \overline{F} ([u⃗′, n_{1}, v⃗′], l, [u⃗′′, n_{1}, v⃗′′]]) =

Theory and Applications of Categories, Vol. 8, No. 7 135

(definition of \overline{F})

= m_{l}(\overline{F} ([u⃗′, n_{1}, v⃗′]), l, \overline{F} ([u⃗′′, n_{1}, v⃗′′])) =

(inductive hypothesis)

= m_{l}(m_{n_{1}}(\overline{F} (u⃗′), \overline{F} (v⃗′)), m_{n_{1}}(\overline{F} (u⃗′′), \overline{F} (v⃗′′))) =

(axiom viii.4)

= m_{n_{1}}(m_{l}(\overline{F} (u⃗′), d_{(n_{1})}(m_{n_{1}}(\overline{F} (u⃗′′), \overline{F} (v⃗′′)))),
m_{l}(\overline{F} (v⃗′), m_{n_{1}}(\overline{F} (u⃗′′), \overline{F} (v⃗′′)))) =

(axiom viii.2)

= m_{n_{1}}(m_{l}(\overline{F} (u⃗′), d_{(n_{1})}(m_{n_{1}}(\overline{F} (u⃗′′), \overline{F} (v⃗′′)))),
m_{n_{1}}(m_{l}(\overline{F} (v⃗′), \overline{F} (u⃗′′)), m_{l}(c_{(n_{1})}(\overline{F} (v⃗′)), \overline{F} (v⃗′′)))) =

(axiom vi.1)

= m_{n_{1}}(m_{l}(\overline{F} (u⃗′), d_{(n_{1})}(\overline{F} (u⃗′′))),

m_{n_{1}}(m_{l}(\overline{F} (v⃗′), \overline{F} (u⃗′′)), m_{l}(c_{(n_{1})}(\overline{F} (v⃗′)), \overline{F} (v⃗′′)))) =

(axiom ix, middle exchange law)

= m_{n_{1}}(m_{l}(\overline{F} (u⃗′), d_{(n_{1})}(\overline{F} (u⃗′′))),

m_{n_{1}}(m_{n_{1}}(m_{l}(d_{(n_{1})}(\overline{F} (v⃗′)), \overline{F} (u⃗′′)), m_{l}(\overline{F} (v⃗′), c_{(n_{1})}(\overline{F} (u⃗′′)))),

m_{l}(c_{(n_{1})}(\overline{F} (v⃗′)), \overline{F} (v⃗′′)))) =

(axiom viii.1)

= m_{n_{1}}(m_{n_{1}}(m_{l}(\overline{F} (u⃗′), d_{(n_{1})}(\overline{F} (u⃗′′))), m_{l}(d_{(n_{1})}(\overline{F} (v⃗′)), \overline{F} (u⃗′′))),

m_{n_{1}}(m_{l}(\overline{F} (v⃗′), c_{(n_{1})}(\overline{F} (u⃗′′))), m_{l}(c_{(n_{1})}(\overline{F} (v⃗′)), \overline{F} (v⃗′′)))) =

(equations (1))

= m_{n_{1}}(m_{n_{1}}(m_{l}(\overline{F} (u⃗′), d_{(n_{1})}(\overline{F} (u⃗′′))), m_{l}(c_{(n_{1})}(\overline{F} (u⃗′)), \overline{F} (u⃗′′))),

m_{n_{1}}(m_{l}(\overline{F} (v⃗′), d_{(n_{1})}(\overline{F} (v⃗′′))), m_{l}(c_{(n_{1})}(\overline{F} (v⃗′)), \overline{F} (v⃗′′)))) =

(axiom xi, middle exchange law)

= m_{n_{1}}(m_{l}(\overline{F} (u⃗′), \overline{F} (u⃗′′)), m_{l}(\overline{F} (v⃗′), \overline{F} (v⃗′′))) =

(definition of \overline{F})

= m_{n_{1}}(\overline{F} (u⃗), \overline{F} (v⃗))

This ends the proof of the proposition.

2.6. Some notation concerning ω-graphs and ω-categories. In this section we introduce some notation concerning ω-graphs and ω-categories. For any ω-graph G and ud-vector u⃗, we define the pullbacks Gu⃗ of lh(u⃗)-tuples of compatible cells and between such objects we define the ’multi’-versions of the operations of domain d_{u⃗;l}^{G}, codomain c_{u⃗;l}^{G}, and some projections. If additionally, G is an ω-category we define the ’multi’-versions of the operations of composition mu⃗. These ’multi’-operations satisfy some generalizations of the laws defining ω-categories. The explicit statements and proofs of these laws are in Appendix 6.3.

Since we will use these definitions not only when the ambient category is Set but also in the category D which is not even finitely complete, we assume that we have a fixed ω-graph G and ω-category A in a category C with all (finite) limits that are explicitely taken.

We assume that in C, for any ud-vector u⃗, we can form the following limit

[figure: G_{u_{0}}; G_{u_{2}}; G_{u_{1}}; c; d; G_{u_{2k−2}}; G_{u_{2k}}; G_{u_{2k−1}}; c; d; …; Gu⃗; π_{2}^{G}; π_{2k−2}^{G}; π_{2k}^{G}; π_{0}^{G}]

If 0 ≤ i ≤ j < lh(u⃗) we denote by

π_{i..j}^{G} : Gu⃗ ⟶ G_{u_{2i},...,u_{2j}}

the obvious projection.

Let l ∈ ω and u⃗ be a ud-vector, lh(u⃗) = k + 1. The morphisms of multi-domain and multi-codomain in an ω-graph G

d_{u⃗;l}^{G}, c_{u⃗;l}^{G} : Gu⃗ ⟶ G_{tr_{(l)}(u⃗)}

are defined recursively, as follows

d_{u⃗;l}^{G} = \{ \begin{matrix}1_{G_{u_{0}}} & if u⃗ = u_{0} ≤ l; \\ d_{(l)}^{G} ∘ π_{0} & if u⃗ is l-primitive; \\ d_{u⃗′;l}^{G} × d_{u⃗′′;l}^{G} & if u⃗ = u⃗′, w, u⃗′′ and w = min(u⃗) < l. \\ & min(u⃗′) > l. \end{matrix}

and

c_{u⃗;l}^{G} = \{ \begin{matrix}1_{G_{u_{0}}} & if u⃗ = u_{0} ≤ l; \\ c_{(l)}^{G} ∘ π_{k} & if u⃗ is l-primitive; \\ c_{u⃗′;l}^{G} × c_{u⃗′′;l}^{G} & if u⃗ = u⃗′, w, u⃗′′ and w = min(u⃗) < l. \\ & min(u⃗′) > l. \end{matrix}

It is convenient to define morphisms d_{(l)}^{G}, c_{(l)}^{G} : G_{n} ⟶ G_{min(l,n)} for any l, n ∈ ω, by putting d_{(l)}^{G} = c_{(l)}^{G} = id_{G_{n}}, if l ≥ n.

For u⃗, v⃗ ∈ UD_{n_{0},n_{1},n_{2}} the projection morphisms in C

π_{0;u⃗}^{G} : G_{[u⃗,n_{1},v⃗]} ⟶ Gu⃗ π_{1;v⃗}^{G} : G_{[u⃗,n_{1},v⃗]} ⟶ Gv⃗

are defined as follows

π_{0;u⃗}^{G} = \{ \begin{matrix}d_{v⃗;u_{0}}^{G} & if u⃗ = u_{0} ≤ n_{1}; \\ π_{0..k}^{G} & if u⃗ is n_{1}-primitive, and lh(u⃗) = k + 1; \\ π_{0;u⃗′}^{G} × π_{0;u⃗′′}^{G} & if u⃗ = u⃗′, z, u⃗′′, v⃗ = v⃗′, z, v⃗′′, \\ & tr_{(n_{1})}(u⃗′) = tr_{(n_{1})}(v⃗′), tr_{(n_{1})}(u⃗′′) = tr_{(n_{1})}(v⃗′′), \\ & and z = min(u⃗) < n_{1}. \end{matrix}

and

π_{1;v⃗}^{G} = \{ \begin{matrix}c_{u⃗;v_{0}}^{G} & if v⃗ = v_{0} ≤ n_{1}; \\ π_{k+1..k′}^{G} & if v⃗ is n_{1}-primitive, lh(u⃗) = k + 1, \\ & and lh(u⃗, n_{1}, v⃗) = k′ + 1; \\ π_{1;v⃗′}^{G} × π_{1;v⃗′′}^{G} & if u⃗ = u⃗′, z, u⃗′′, v⃗ = v⃗′, z, v⃗′′, \\ & tr_{(n_{1})}(u⃗′) = tr_{(n_{1})}(v⃗′), tr_{(n_{1})}(u⃗′′) = tr_{(n_{1})}(v⃗′′), \\ & and z = min(u⃗) < n_{1}. \end{matrix}

In Lemma 6.5 we show that, with the above definitions, the square

[figure: Gu⃗; Gw⃗; c_{u⃗;n_{1}}^{G}; G_{[u⃗,n_{1},v⃗]}; Gv⃗; π_{1;v⃗}^{G}; π_{0;u⃗}^{G}; d_{v⃗;n_{1}}^{G}]

is a pullback.

Let u⃗ a ud-vector, lh(u⃗) = k + 1 > 2. We put

u = max{u_{2i+1} : i ∈ k} and j = min{i ∈ k : u_{2i+1} = u}

and then u⃗ = u⃗′, u_{2j−1}, u_{2j}, u, u_{2j+2}, u_{2j+3}, u⃗′′, for some ud-vectors u⃗′ and u⃗′′. The morphisms

Au⃗ 1_{Au⃗′} × m_{u_{2j},u,u_{2j+2}}^{A} × 1_{Au⃗′′} \overline{}A_{u⃗′,u_{2j−1},max(u_{2j},u_{2j+2}),u_{2j+3},u⃗′′}  (2)

is the one-step (u⃗-compatible) composition morphism, and the morphisms

Au⃗  mA ; n⃗  \overline{}A_{ht(u⃗)}  (3)

which is a composition of one-step composition morphisms is the canonical composition morphism. By convention, if u⃗ = u_{0} then mA ; u⃗ = id_{A_{u_{0}}}.

2.7. The small duality for S_{1} and D^{1}. For n ∈ ω^{+}, by S_{n} we denote the category of simple n-categories i.e. the simple ω-categories truncated to the first n levels, and by D^{n} we denote the category of finite n-disks, i.e. disks truncated to the first n levels. In this section we sketch the well know duality between S_{1} and D^{1} in the way we will prove duality between S and D. In fact, for any positive n, the duality between S_{n} and D^{n} holds and it is a restriction of the duality between S and D.

S_{1} is equivalent to the category of finite non-empty linear orders and monotone maps and D^{1} is equivalent to the category of finite linear orders with (necessarily different) endpoints and monotone functions preserving endpoints. Thus, it is well know that they are dually equivalent. An easy explanation of this fact can be found in [SGL]. Below, we will exhibit a more involved but, as we believe, more instructive explanation of this fact, that can be generalized to the higher dimensional categories, of which it is a special case.

Before we prove the duality theorem we need to develop some notation. Both elements on level 1 in a disk in D^{1} and objects in categories in S_{1} comes with an order. However it is convenient to draw these orders in reverse directions, in disk as increasing orders, and in categories as decreasing orders. At the end of this section we will return to this point.

The 1-disks. An object of D^{1}, i.e. a 1-disk in Set, is a finite linear order with different endpoints, i.e. it is isomorphic to one of the form

[figure: π_{1}; {⟨k, l⟩ : 0 ≤ k ≤ l ≤ n}; {k : 0 ≤ k ≤ n}; π_{2}; b; {⋆}; p; t]

for some n ∈ ω^{+}, where b(⋆) = 0 and t(⋆) = n. Such a 1-disk will be denoted by ⌊n⌋. We introduce a notation for maps into ⌊1⌋ in D^{1}. For l ∈ n we have disk maps

\overline{l; l + 1} : ⌊n⌋ ⟶ ⌊1⌋

defined by

\overline{l; l + 1}(i) = \{ \begin{matrix}0 & if i ≤ l \\ 1 & otherwise \end{matrix}

We write \overline{l;} or \overline{; l + 1} for \overline{l; l + 1}, as well. These are all the maps from ⌊n⌋ to ⌊1⌋. A map g : ⌊n′⌋ ⟶ ⌊n⌋ induces a dual map by composition

D^{1}(g, ⌊1⌋) : D^{1}(⌊n⌋, ⌊1⌋) ⟶ D^{1}(⌊n′⌋, ⌊1⌋)

D^{1}(g, ⌊1⌋)(\overline{l;}) = \overline{l;} ∘ g

for l ∈ n.

The category D^{1} does not have pullbacks of arbitrary pairs of morphisms with common codomain. But it has some pullbacks. In fact, the morphisms f : ⌊n⌋ ⟶ ⌊l⌋ and f′ : ⌊n′⌋ ⟶ ⌊l⌋ have a pullback in D^{1} iff for any x ∈ ⌊l⌋ either f^{−1}(x) or f^{′−1}(x) has at most one element. If this is the case, the pullback is computed as in the category of posets. Let d, c : ⌊2⌋ ⟶ ⌊1⌋ be morphism in D^{1}, such that d = \overline{0; 1} and c = \overline{1; 2}. Then d and c satisfy the above condition and we have a pullback

[figure: ⌊2⌋; ⌊1⌋; c; ⌊3⌋; ⌊2⌋; π_{1}; π_{0}; d]

Similarly, we have a pullback

[figure: ⌊3⌋; ⌊2⌋; π_{2}; ⌊4⌋; ⌊3⌋; π_{23}; π_{12}; π_{1}]

Let

m : ⌊3⌋ ⟶ ⌊2⌋ ι : ⌊1⌋ ⟶ ⌊2⌋

be morphisms in D^{1} such that m(1) = m(2) = 1. In this way, we have defined an internal category C_{1} in D^{1} given by the diagram

[figure: d; ⌊1⌋; ⌊2⌋; ι; c; π_{1}; ⌊3⌋; m; π_{2}; π_{12}; 1 × m; ⌊4⌋; m × 1; π_{23}]

The simple 1-categories. The objects of S_{1} are free categories generated by finite (possibly empty) strings of arrows. Let for n ∈ ω, ⌈n⌉ denote the free category generated by the string

n ⟶ (n − 1) ⟶ … ⟶ 1 ⟶ 0

Any object in S_{1} is isomorphic to ⌈n⌉ for some n ∈ ω.

Similarly, we introduce a notation for functors into ⌈1⌉ in S_{1}. Since in S_{1} the functors are uniquely determined by their values on objects, we will define only their object functions. We define functors

0↓, (k + 1↓k), ↓n : ⌈n⌉ ⟶ ⌈1⌉

with k ∈ n, so that for i ∈ {0, …, n}

(0↓)(i) = 1
(↓n)(i) = 0

(k + 1↓k)(i) = \{ \begin{matrix}1 & if i > k \\ 0 & if k ≥ i \end{matrix}

We abbreviate (k + 1↓k) to either k + 1↓ or ↓k.

A functor φ : ⌈n′⌉ ⟶ ⌈n⌉ induces a dual map by composition

S_{1}(φ, ⌈2⌉) : S_{1}(⌈n⌉, ⌈2⌉) ⟶ S_{1}(⌈n′⌉, ⌈2⌉)
S_{1}(φ, ⌈2⌉)(h) = h ∘ φ

In S_{1} we have an internal 1-disk, denoted by D_{1}

[figure: ρ^{1}; ⌈2⌉; ⌈1⌉; ρ^{2}; b; ⌈0⌉; p; t]

with ρ^{1} = 2↓1, ρ^{2} = 1↓0, b = ↓n, and t = 0↓.

The schizophrenic object. The category C_{1} in D^{1} and the disk D^{1} in S_{1} form together a schizophrenic object. By this, we mean, that we can form the following diagram in Set

[figure: \begin{matrix}X \\ f \\ Y \end{matrix}; \begin{aligned}X & f & Y \\ & gf & g \\ & & Z \end{aligned}; \begin{matrix}X \\ Y \end{matrix}; \begin{aligned}X & Y \\ & Z \end{aligned}; \begin{matrix}(X, X, X) \\ (X, X, f) \\ (X, f, Y) \\ (f, Y, Y) \\ (Y, Y, Y) \end{matrix}; \begin{aligned}(X, X, X) & (X, X, f) & (X, f, Y) & (f, Y, Y) & (Y, Y, Y) \\ & (X, X, gf) & (X, f, g) & (f, Y, g) & (Y, Y, g) \\ & & (X, gf, Z) & (f, g, Z) & (Y, g, Z) \\ & & & (gf, Z, Z) & (g, Z, Z) \\ & & & & (Z, Z, Z) \end{aligned}; \begin{matrix}(X, X) \\ (X, f) \\ (f, Y) \\ (Y, Y) \end{matrix}; \begin{aligned}(X, X) & (X, f) & (f, Y) & (Y, Y) \\ & (X, gf) & (f, g) & (Y, g) \\ & & (gf, Z) & (g, Z) \\ & & & (Z, Z \end{aligned}; ⋆; ⋆; ⋆; ⋆; ⋮; ⋮; ⋮; b; p; t; b; p; t; b; p; t; b; p; t; ρ^{′0}; ρ^{′1}; ρ^{0}; ρ^{1}; ρ^{00}; ρ^{11}; ρ^{000}; ρ^{111}; ι; c; d; ι; c; d; m; π_{2}; π_{1}; m; π_{2}; π_{1}; m × 1; 1 × m; m × 1; 1 × m]

In this diagram the rows are disks isomorphic to the internal category C_{1} in D, and the columns are categories isomorphic to the internal disk D^{1} in S. Before we will see this we shall describe the data in the above diagram, in details.

The sets in this diagram are given by lists or arrays of its elements, so that the projections were easy to understand, e.g. in the second row of the right column array

\begin{aligned}X & f & Y \\ & gf & g \\ & & Z \end{aligned}

represents the the set {id_{X}, f, id_{Y} , gf, g, id_{Z}}.

As we said, the columns are diagrams of free categories on simple 1-graphs, on one object, on one arrow X ⟶^{f} Y and, on a string of arrows X ⟶^{f} Y ⟶^{g} Z, respectively. Thus they are equivalent to ⌈0⌉, ⌈1⌉, and ⌈2⌉. In the first row there are objects, in the second there are morphisms, in the third there are composable pairs of morphism, and in the fourth there are composable triples of morphism. The object X, Y, Z in 2nd, 3rd and 4th rows stand for identities on them, i.e. for 1_{X}, 1_{Y}, 1_{Z}, respectively.

The rows are disks isomorphic to ⌊1⌋, ⌊2⌋ ,⌊3⌋, and ⌊4⌋, respectively. In the first column there are one-element sets, i.e. level 0 parts of the disks, in the second column there are universes of disks, and in the third row there are linear orders on these universes.

The morphism ρ^{i} and ρ^{′i}, for i = 0, 1 are 1st and 2nd projections, e.g. ρ^{′0}(X) = ρ^{′1}(Y) = X, ρ^{′1}(Y) = ρ^{′1}(Z) = Y, ρ^{0}(g) = ρ^{0}(gf) = f, ρ^{1}(g) = ρ^{1}(Z) = Y, etc. The morphisms ρ^{ii} and ρ^{iii}, for i = 0, 1, denote ρ^{i} × ρ^{i} and ρ^{i} × ρ^{i} × ρ^{i}, respectively. The morphisms p are uniquely determined, b and t picks the least and the largest elements in the order, respectively, e.g. in the forth row b(⋆) = (X, X, X). The way the morphism in the columns are defined is easy to guess: d, c, ι, and m are morphisms of domain, codomain, identity and composition, respectively.

Note that respective morphisms in rows and columns commute, e.g.

m ∘ ρ^{ii} = ρ^{i} ∘ m d ∘ ρ^{i} = ρ^{′i} ∘ d

and so on. This is equivalent to saying, that if we pick the corresponding morphisms in three columns, then they form a disk morphism between disks in the rows, e.g. m’s form a disk morphism m form the disk in third row to the disk in the second row. The same thing remains true if we exchange the role of the rows and columns. This is the essence of being a schizophrenic object.

The duality. We claim

2.8. Theorem. The contravariant functors of homming into C_{1} and into D^{1} give rise to a Stone adjunction

[figure: D^{1}(−, C_{1}); D^{1}; S_{1}; S_{1}(−, D^{1})]

which is an equivalence of categories. Thus the categories S_{1} and D^{1} are dually equivalent.

Proof. To fix the notation, in the following we consider the category C_{1} in D^{1} as being formed of the rows in the diagram describing the schizophrenic object, e.g. the disk universe of object of objects ⌊1⌋ will be {X, Y } rather than {0, 1}. Similarly, we consider the disk D^{1} in S_{1} as being formed from columns in the same diagram, e.g. ⌊2⌋ in D^{1} stands for the second row. This give us certain convenient identifications, e.g. the universe of the disk ⌊2⌋ is equal to the set of maps of the category ⌈1⌉.

For n ∈ ω^{+}, the objects of D^{1}(⌊n⌋, C_{1}) form the set

D^{1}(⌊n⌋, ⌊1⌋) = \{\overline{0; 1}, \overline{1; 2}, …, \overline{n − 1; n}\}

and the category D^{1}(⌊n⌋, C_{1}) isomorphic to ⌈n − 1⌉.

For n ∈ ω, the universe of the disk S_{1}(⌈n⌉, D^{1}) is the set

S_{1}(⌈n⌉, ⌈1⌉) = {↓n, n↓n − 1, …, 1↓0, 0↓}

and with the increasing order so that S_{1}(⌈n⌉, D^{1}) is isomorphic to ⌊n + 1⌋. Thus the functors have the appropriate codomains.

The statement of the theorem says that we have two natural isomorphisms η and ε, satisfying the triangular equalities.

We shall define these natural transformations. The component of η at ⌊n⌋ in D^{1} is a disk map

η_{⌊n⌋} : ⌊n⌋ ⟶ S_{1}(D^{1}(⌊n⌋, C_{1}), D^{1})

such that on universes it is a function

η_{⌊n⌋} : {0, …, n} ⟶ S_{1}(D^{1}(⌊n⌋, C_{1}), ⌈1⌉)

so that, for i ∈ {0, …, n},

η_{⌊n⌋}(i) : D^{1}(⌊n⌋, C_{1}) ⟶ ⌈1⌉

is a functor with the object function

η_{⌊n⌋}(i)_{0} : D^{1}(⌊n⌋, ⌊1⌋) ⟶ {X, Y }

given by

h ⟼ h(i)

Identifying D^{1}(⌊n⌋, C_{1}) with ⌈n − 1⌉ we have, for i ∈ {0, …, n}

η_{⌊n⌋}(i) : ⌈n − 1⌉ ⟶ ⌈1⌉ = \{ \begin{matrix}i↓ & if i < n \\ ↓n − 1 & if i = n \end{matrix}

So η_{⌊n⌋} is iso.

The component of ε at ⌈n⌉ in S_{1} is a functor

ε_{⌈n⌉} : ⌈n⌉ ⟶ D^{1}(S_{1}(⌈n⌉, D^{1}), C_{1})

with object function

ε_{⌈n⌉,0} : {0, …, n} ⟶ D^{1}(S_{1}(⌈n⌉, D^{1}), ⌊1⌋)

such that, for i ∈ {0, …, n},

ε_{⌈n⌉,0}(i) : S_{1}(⌈n⌉, D^{1}) ⟶ ⌊1⌋

is a disk map, given by the function on universes

ε_{⌈n⌉,0}(i) : S_{1}(⌈n⌉, ⌈1⌉) ⟶ {X, Y }

by

h ⟼ h(i)

Identifying S_{1}(⌈n⌉, D^{1}) with ⌊n + 1⌋ we have, for i ∈ {0, …, n}, that

ε_{⌈n⌉,0}(i) = \overline{i; i + 1} : ⌊n + 1⌋ ⟶ ⌊1⌋

So ε_{⌈n⌉} is iso.

We leave to the reader the verification that ε_{⌈n⌉} and η_{⌊n⌋} are well defined, η and ε are natural, and that the following triangles

[figure: S_{1}(⌈n⌉, D^{1}); S_{1}(⌈n⌉, D^{1}); Id; S_{1}(D^{1}(S_{1}(⌈n⌉, D^{1}), C_{1}), D^{1}); η_{S_{1}(⌈n⌉,D^{1})}; S_{1}(ε_{⌈n⌉}, D^{1})]

[figure: D^{1}(⌊n⌋, C_{1}); D^{1}(⌊n⌋, C_{1}); Id; D^{1}(S_{1}(D^{1}(⌊n⌋, C_{1}), D^{1}), C_{1}); ε_{D^{1}(⌊n⌋,C_{1})}; D^{1}(η_{⌊n⌋}, C_{1})]

commute. This is end the proof.

We will make a comment on the orders in 1-disks and simple 1-categories or rather finite intervals (strict with endpoints) and finite non-empty linear orders. In [J] there is a very suggestive picture explaining this duality:

| ∗ | ∗ | ∗ | ∗ |

showing that the interval

| | | | |

is the set of Dedekind cuts of the order

∗ ∗ ∗ ∗

The trick is that the orders on ∗’s and |’s are going in the different directions i.e. to have increasing order on cuts

| ≤ | ≤ | ≤ | ≤ |

we need to have decreasing order on stars

∗ ≥ ∗ ≥ ∗ ≥ ∗

3. Disks

3.1. Some notation concerning disks. We introduce some notation concerning disks. Let x, y be nodes in a disk D. We call x a left endpoint (right endpoint, bi-endpoint) iff it is the least (largest, both) element in the linear order of the fiber over p(x). A node x is a leaf iff the fiber p^{−1}(x) contain exactly two elements. Note that, x is a leaf iff it is the image of a leaf of the tree ı(D) under the unit morphism η : ı(D) ⟶ D. Since our disks are finite, over every inner node there is a leaf. We write x < y to mean that both x and y are nodes in the same fiber and y is the immediate successor of x. Since p^{(0)}(x) = p^{(0)}(y) we can define the number

μ_{x,y}^{D} = max{l : p^{(l)}(x) = p^{(l)}(y)}

We usually omit the superscript D. Note that, if x ∈ D^{l}, y ∈ D^{k} and μ_{x,y} < min{l, k} then the node p^{(μ_{x,y})}(x) = p^{(μ_{x,y})}(y) must be inner (there are at least two elements in the fiber over it), moreover the nodes p^{(μ_{x,y}+1)}(x), p^{(μ_{x,y}+1)}(y) are comparable. This last observation allows us to introduce a natural linear order on the whole set D^{l} extending the existing orders on fibers of p : D^{l} ⟶ D^{l−1}. Namely, if x, y ∈ D^{l} are different nodes then we put

x < y iff p^{(μ_{x,y}+1)}(x) < p^{(μ_{x,y}+1)}(y)

But this order is not preserved by the disk morphisms, in general.

If x, y are different leaves of D then we always have μ_{x,y} < min{l, k}. Thus we can also introduce a linear order on leaves by the above formula. We denote this order by ⪯ and by ≪ we denote the immediate successor relation given by this order.

If x, y are elements of a poset, we write x ⊥ y if x and y are comparable and x ⊥̸ y if they are not.

The category BL is a full subcategory of Poset containing finite sums of finite linear orders, i.e. a poset X is in BL iff ⊥ is an equivalence relation on X (i.e. ⊥ is transitive on X). BI is a subcategory of BL containing the same objects, and morphisms in BI preserve additionally minimal and maximal elements.

Thus the category BL is equivalent to the category of finite bundles of finite linear orders and the category BI is equivalent to the category of finite bundles of finite intervals.

Note that the inclusion functor BI ⟶ BL has a right adjoint \overline{(−)} : BL ⟶ BI which adds to a non-empty object X of BL new minimal and maximal elements at the ends of each maximal linear order in X; \overline{(∅)} is the one-element (non-strict) interval.

The following easy lemma provides an equivalent elementary description of disks.

3.2. Lemma. The sequence of functions

[figure: ⋯; D^{s+1}; p; b; t; D^{s}; p; b; t; D_{1}; D^{0} ≅ 1; p; b; t; …]

between objects of BI is a disk iff for s ∈ ω, 1. for x ∈ D^{s}, bp(x) ≤ x ≤ tp(x);

2. p maps D^{s+1} onto D^{s}, so that, for x, y ∈ D^{s+1}, p(x) = p(y) iff x ⊥ y;

3. for x ∈ D^{s}, b(x) = t(x) iff x = bp(x) or x = tp(x). Proof. The conditions 1. and 2. say that p : D^{s+1} ⟶ D^{s} is a bundle of intervals and the condition 3. expresses the disk condition.

For n ∈ ω, we give a description of the disk γ_{n} which is the image of θ_{n} under the functor \overline{(−)}. The universe of γ_{n} is given by

γ_{n}^{l} = \{ \begin{matrix}2l + 1 & if 0 ≤ l ≤ n \\ 2n + 2 & if n < l \end{matrix}

the projections p : γ_{n}^{l+1} ⟶ γ_{n}^{l} are defined as follows: for 0 ≤ l < n

p(i) = \{ \begin{matrix}i & if 0 ≤ i ≤ l \\ i − 1 & if i = l + 1 \\ i − 2 & if l + 1 < i ≤ 2l \end{matrix}

for l = n

p(i) = \{ \begin{matrix}i & if 0 ≤ i ≤ n \\ i − 1 & if n < i ≤ 2n + 1 \end{matrix}

and for l > n

p(i) = i

The orders in fibers of γ_{n} are defined as follows: if i, j ∈ γ_{n}^{l}, then

i < j iff \{ \begin{matrix}l ≤ n and l − 1 ≤ i < j ≤ l + 1 \\ or \\ l = n + 1 and i = n and j = n + 1 \end{matrix}

In order to make this simple but tedious definition easier to grasp we present below the first six levels of γ_{3}. The inner nodes are marked bold.

[figure: 0; 0; 1; 2; 0; 1; 2; 3; 4; 0; 1; 2; 3; 4; 5; 6; 0; 1; 2; 3; 4; 5; 6; 7; 0; 1; 2; 3; 4; 5; 6; 7]

Thus in γ_{3} at level l = 0, 1, 2, 3 there is exactly one inner node l, and l − 1 < l < l + 1, and there are no inner nodes at higher levels. Moreover, at level 4 we have 3 < 4.

3.3. Lemma. Let l, n_{1}, n_{2} ∈ ω, n_{1} < n_{2}, l. Then

1. γ_{n_{2}}^{l} = γ_{n_{1}}^{l} + γ_{n_{2}−n_{1}−1}^{l−n_{1}−1}

2. min{x ∈ γ_{n_{2}}^{l} : p^{(n_{1})}(x) = n_{1}} = n_{1}

3. max{x ∈ γ_{n_{2}}^{l} : p^{(n_{1})}(x) = n_{1}} = n_{1} + 1 + γ_{n_{2}−n_{1}−1}^{l−n_{1}−1} = γ_{n_{2}}^{l} − n_{1} − 1

Proof. Exercise.

We introduce some notions and notation concerning disks γ_{n}. Let l, n ∈ ω and x ∈ γ_{n}^{l}. We say that x is in the left side of γ_{n} iff either l ≤ n and x < l or n < l and x ≤ n. We say that x is in the right side of γ_{n} iff either l ≤ n and x > l or n < l and x > n. We say that x is in the far left side of γ_{n} iff x is in the left side of γ_{n} and it is a right endpoint. We say that x is in the far right side of γ_{n} iff x is in the right side of γ_{n} and it is a left endpoint. By ls(γ_{n}), fls(γ_{n}), rs(γ_{n}), frs(γ_{n}), we denote left, far left, right, far right sides of γ_{n}, respectively. Note that a node in the left (right) side is left (right) endpoints and a node is far left or far right side iff it is a bi-endpoint.

For later reference, we list below the numeric conditions for a node of to be in the above parts of the disk γ_{n}.

3.4. Lemma. Let l, n ∈ ω and x ∈ γ_{n}. Then

1. x ∈ ls(γ_{n}) iff ( l ≤ n and x < l ) or ( l > n and x ≤ n )

2. x ∈ fls(γ_{n}) iff ( l ≤ n + 1 and x < l − 1 ) or ( l > n + 1 and x ≤ n )

3. x ∈ rs(γ_{n}) iff ( l ≤ n and x > l ) or ( l > n and x > n )

4. x ∈ frs(γ_{n}) iff ( l ≤ n and x > l + 1 ) or ( l = n + 1 and x > n + 1 ) or ( n + 1 < l and x ≥ n + 1 ).

Before defining some specific morphisms in D, we shall make some general observations concerning them.

3.5. Lemma.

1. Any disk morphism f : D ⟶ E is uniquely determined by its values on leaves of D.

2. Let n ∈ ω and D be a disk. Then the disk morphism f : γ_{n} ⟶ D is uniquely determined by the value f^{n}(n) ∈ D^{n}. Moreover, for any x ∈ D^{n} there is a unique morphism f : γ_{n} ⟶ D, such that f^{n}(n) = x. Thus, we have a bijection

D(γ_{n}, D) ⟶ D^{n}

which is natural in D.

3. Let n ∈ ω and D be a disk. Then the disk morphism f : D ⟶ γ_{n} is uniquely determined by the value {f^{l}}_{l≤n+1}.

Proof.

Ad 1 We already noted that any disk morphisms f : D ⟶ E is uniquely determined by its values on inner nodes of D. But since over each inner node there is a leaf and disk morphisms preserves projections, once we fix values of a morphisms on leaves the other values on inner nodes are uniquely determined.

Ad 2 The node n ∈ γ_{n}^{n} is the unique leaf of γ_{n}. Thus the value f^{n}(n) ∈ D^{n} indeed uniquely determines the morphism f. On the other hand, for any x ∈ D^{n} we can define a tree morphism f′ : θ_{n} ⟶ D by the formula f^{′l}(l) = p^{(l)}(x) for 0 ≤ l ≤ n and f′ extends uniquely to a disk morphism f : γ_{n} ⟶ D, such that f^{n}(n) = x.

Ad 3 This follows from the fact that at levels l greater than n + 1 all the intervals contains one element. So, for x ∈ D^{l}, we have f^{l}(x) = f^{n+1}(p^{(n+1)}(x)).

Let n_{0}, n_{1}, n_{2} ∈ ω and n_{1} < n_{0}, n_{2}. We define the codomain morphism

c_{(n_{1})} = c : γ_{n_{0}} ⟶ γ_{n_{1}}

by the condition c^{n_{0}}(n_{0}) = n_{1}, and the domain morphism

d_{(n_{1})} = d : γ_{n_{2}} ⟶ γ_{n_{1}}

by the condition d^{n_{2}}(n_{2}) = n_{1} + 1. Since these morphisms play very important role in this paper we shall give their full descriptions. For x ∈ γ_{n_{0}}^{l} we have

c^{l}(x) = \{ \begin{matrix}x & if (0 ≤ l ≤ n_{1}) or (x < n_{1}) \\ n_{1} & if n_{1} < l and n_{1} ≤ x < γ_{n_{0}}^{l} − n_{1} − 1 \\ n_{1} + 1 & if n_{1} < l and x = γ_{n_{0}}^{l} − n_{1} − 1 \\ x − γ_{n_{0}}^{l} + 2n_{1} + 2 & if n_{1} < l and γ_{n_{0}}^{l} − n_{1} − 1 < x < γ_{n_{0}}^{l} \end{matrix}

and for y ∈ γ_{n_{2}}^{l} we have

d^{l}(y) = \{ \begin{matrix}y & if (0 ≤ l ≤ n_{1}) or (n_{1} < l and y < n_{1}) \\ n_{1} & if n_{1} < l and y = n_{1} \\ n_{1} + 1 & if n_{1} < l and n_{1} < y ≤ γ_{n_{2}}^{l} − n_{1} − 1 \\ y − γ_{n_{2}}^{l} + 2n_{1} + 2 & if n_{1} < l and γ_{n_{2}}^{l} − n_{1} − 1 < y < γ_{n_{2}}^{l} \end{matrix}

3.6. The limits in BI. We want to study the limits in D. In particular, we want to show that all the disk are the multi-pullbacks constructed using objects γ_{n} and morphisms c and d. Before we do this, we shall characterize, in this section, the multi-pullbacks in BI.

For k ∈ ω, by a zigzag of morphisms of length k + 1 in a category we mean a diagram

[figure: C_{0}; C_{2}; C_{1}; f_{1}; f_{2}; C_{4}; C_{3}; f_{3}; f_{4}; C_{2k−2}; C_{2k}; C_{2k−1}; f_{2k−1}; f_{2k}; …]  (4)

The limit of such a zigzag, if exists, is called multi-pullback.

Given a zigzag (4) in Poset. By a compatible tuple for this zigzag, we mean, a k+1-tuple ⟨x_{0}, …, x_{k}⟩ such, that x_{i} ∈ C_{i} for i ∈ k + 1 and f_{2i+1}(x_{i}) = f_{2i+2}(x_{i+1}) for i ∈ k.

Thus, the limit of (4) in Poset is the set of compatible tuples of the zigzag (4) with pointwise order.

The limits in the category of linear orders, it they exists, are computed in the category of partial orders Poset. This is mainly due to the fact that, one- and two-element linear orders are objects of this category. The same is true for the category of (non-strict) intervals I, since if all the maps in a diagram of linear orders preserves endpoints, then so do all the projections. Thus in I the limit of a diagram exists if its limit in Poset is a linear order. For similar reasons the limits in both BL and BI, if exists, are computed in Poset.

We say that a zigzag (4) in I is thin iff for any compatible tuples x⃗, y⃗ if there is 0 ≤ i_{0} ≤ k such, that x_{i_{0}} < y_{i_{0}} and f_{2i_{0}+1}(x_{i_{0}}) = f_{2i_{0}+1}(y_{i_{0}}) than x_{i} = y_{i}, for all i_{0} < i ≤ k.

We have

3.7. Lemma. The limit of a zigzag of morphisms (4) in I exists in I iff (4) is thin.

We say that the zigzag (4) in Poset is thin iff for any compatible tuples x⃗, y⃗ such that, x_{i} ⊥ y_{i}, for i ∈ k + 1, x_{i_{0}} < y_{i_{0}} and f_{2i_{0}+1}(x_{i_{0}}) = f_{2i_{0}+1}(y_{i_{0}}) for some i_{0} ∈ k, we have x_{i} = y_{i}, for all i > i_{0}. We state some equivalent conditions for a zigzag in BI to be thin.

3.8. Lemma. Let (4) be a zigzag of morphisms in Poset. Then, the following are equivalent

1. the zigzag (4) is thin; 2. for any compatible tuples x⃗, y⃗ such that, x_{i} ⊥ y_{i}, for i ∈ k + 1, x_{i_{0}} < y_{i_{0}} and

f_{2i_{0}}(x_{i_{0}}) = f_{2i_{0}}(y_{i_{0}}), for some 0 < i_{0} ≤ k, we have x_{i} = y_{i}, for all i < i_{0};

3. there are no compatible tuples x⃗, y⃗ such that, x_{i} ⊥ y_{i}, for i ∈ k + 1, x_{i_{0}} < y_{i_{0}} and x_{i_{1}} > y_{i_{1}}, for some i_{0}, i_{1} ∈ k + 1.

Proof. We shall prove that 1. is equivalent to 3. The condition 2. is ’symmetric’ to 1. and its equivalence with 3. can be proved similarly.

Suppose 3. does not hold. Let x⃗, y⃗ be compatible tuples for (4) such, that x_{i} ⊥ y_{i} for all i ∈ k + 1, and x_{i_{0}} < y_{i_{0}}, x_{i_{1}} > y_{i_{1}} for some i_{0}, i_{1} ∈ k + 1. Without loss of generality, we can assume that i_{0} < i_{1}. Let

i_{2} = max{i < i_{1} : x_{i} < y_{i}}

Then x_{i_{2}} < y_{i_{2}}, x_{i_{2}+1} ≥ y_{i_{2}+1}, f_{2i_{2}+1}(x_{i_{2}}) ≤ f_{2i_{2}+1}(y_{i_{2}}) and

f_{2i_{2}+1}(x_{i_{2}}) = f_{2i_{2}+2}(x_{i_{2}+1}) ≥ f_{2i_{2}+2}(y_{i_{2}+1}) = f_{2i_{2}+1}(y_{i_{2}})

So, we have f_{2i_{2}+1}(x_{i_{2}}) = f_{2i_{2}+1}(y_{i_{2}}), i_{1} > i_{2} and x_{i_{1}} ≠ y_{i_{1}}, i.e. the zigzag (4) is not thin.

To prove the converse, suppose that 1. does not hold. Let x⃗, y⃗ be compatible tuples for (4) such, that x_{i} ⊥ y_{i} for all i ∈ k + 1, and for some 0 ≤ i_{0} < i_{1} ≤ k, x_{i_{0}} < y_{i_{0}}, f_{2i_{0}+1}(x_{i_{0}}) = f_{2i_{0}+1}(y_{i_{0}}), x_{i_{1}} ≠ y_{i_{1}}. If x_{i_{1}} > y_{i_{1}}, then this contradicts 3. immediately. If x_{i_{1}} < y_{i_{1}} then the tuples

x⃗′ = ⟨x_{0}, …, x_{i_{0}}, y_{i_{0}+1}, …, y_{k}⟩ y⃗′ = < y_{0}, …, y_{i_{0}}, x_{i_{0}+1}, …, x_{k} >

are compatible and contradicts 3.

3.9. Lemma. Let (4) be a zigzag of morphisms in Poset, x⃗, y⃗ compatible tuples such that, x_{i} ⊥ y_{i}, for i ∈ k + 1. Then the tuple z⃗ = ⟨z_{i}⟩_{i∈k+1}, where z_{i} = min{x_{i}, y_{i}} is compatible and z⃗ ≤ x⃗, y⃗.

Proof. We shall prove, for i ∈ k, that

f_{2i+1}(z_{i}) = f_{2i+2}(z_{i+1})  (5)

Fix i ∈ k. If z_{i} and z_{i+1} comes from the same tuple x⃗ or y⃗, then (5) holds, since both tuples are compatible.

Suppose that z_{i} = x_{i} < y_{i} and z_{i+1} = y_{i+1} < x_{i+1}. The other case is similar. We have

f_{2i+1}(x_{i}) = f_{2i+2}(x_{i+1}) ≥ f_{2i+2}(y_{i+1})

and

f_{2i+1}(x_{i}) ≤ f_{2i+1}(y_{i}) = f_{2i+2}(y_{i+1})

i.e. (5) holds.

We have

3.10. Proposition. The zigzag (4) in BI has a limit iff it is thin. Moreover, if this is the case the limit is computed in Poset.

Proof. Assume that (4) is thin. Let x⃗, y⃗, z⃗ be compatible tuples, such that x⃗ ⊥ y⃗ ⊥ z⃗. We shall show that x⃗ ⊥ z⃗.

Since the relation ⊥ is transitive in C_{2i} we have x⃗_{i} ⊥ z⃗_{i}, for i ∈ k + 1. If x⃗ = z⃗ then clearly x⃗ ⊥ z⃗. So, assume that x⃗ ≠ z⃗. Let

i_{0} = min{i : x_{i} ≠ z_{i}}

With out loss of generality, we can assume that x_{i_{0}} < z_{i_{0}}. Let

i_{1} = max{i : x_{i} ≤ z_{i}}

If i_{1} < k then we have x_{i_{0}} < z_{i_{0}} and x_{i_{1}+1} > z_{i_{1}+1} and the zigzag (4) is not thin, by Lemma 3.8. Thus i_{1} = k and x⃗ ≤ z⃗, so x⃗ ⊥ z⃗, as required.

Thus, we have shown that the limit of (4) in Poset is in BI. Hence this zigzag has a limit in BI.

To prove the converse, assume that the zigzag (4) is not thin and let (L, π) be a limiting cone of (4) in Poset. Then, by Lemma 3.8, we have compatible tuples x⃗, y⃗ such that x_{i} ⊥ y_{i}, for i ∈ k + 1, x_{i_{0}} < y_{i_{0}} and x_{i_{1}} > y_{i_{1}}, for some i_{0}, i_{1} ∈ k + 1. So, in particular x⃗ ⊥̸ y⃗. By Lemma 5 the tuple z⃗ = ⟨z_{i}⟩_{i∈k+1}, where z_{i} = min{x_{i}, y_{i}}, for i ∈ k + 1 is compatible and z⃗ < x⃗, y⃗. So ⊥ is not transitive on L, L is not in BI, and hence the zigzag (4) does not have a limit in BI.

3.11. The limits in D. In this section we study the limits in the category D. We show that each disk has a canonical presentation as a multi-pullback of disks of form γ_{n}. Such a presentation describes the correspondence between the disks and the ud-vectors, the numerical invariants for disks, i.e. ud-vectors determine the disks up to isomorphisms.

Note that, for l ∈ ω, we have functors

(−)^{l} : D ⟶ BI

sending disk D to its l-th level D^{l} (not forgetting the order).

3.12. Proposition. The limits in D, if exists, are computed pointwise in Poset, i.e. for a cone in D to be a limiting cone it is necessary and sufficient that, for each l ∈ ω, its image under every functor (−)^{l} be a limiting cone in Poset.

Proof. Fix l ∈ ω.

Recall from Lemma 3.5 that, for any disk D, we have a bijective correspondence between nodes x ∈ D^{l} and disk morphisms f_{x} : γ_{l} ⟶ D, such that f_{x}^{l}(l) = x. Let γ = \overline{θ} be a disk, such that θ is a tree with one node at levels smaller than l, two nodes l ≤ l + 1, at level l, and no other nodes. It easy to see that, we have a bijective correspondence between pairs of nodes x, y ∈ D^{l}, such that x ≤ y and morphisms f_{x,y} : γ ⟶ D, such that f_{x,y}^{l}(l) = x and f_{x,y}^{l}(l + 1) = y.

Let (D, π), be a limit in D of the functor F : J ⟶ D. Then, for l ∈ ω, we have a bijective correspondences

x ∈ D^{l}
f_{x} : γ_{l} ⟶ D
π ∘ f_{x} : γ_{l} ⟶ F
π^{l} ∘ f_{x}^{l}⌈_{{l}}: {l} ⟶ F^{l}

and

x, y ∈ D^{l}, x ≤ y
f_{x,y} : γ ⟶ D
π ∘ f_{x,y} : γ ⟶ F
π^{l} ∘ f_{x,y}^{l}⌈_{{l,l+1}}: {l ≤ l + 1} ⟶ F^{l}
π^{l} ∘ f_{x,y}^{l}⌈_{{l}}: {l} ⟶ F^{l} ≤ π^{l} ∘ f_{x,y}^{l}⌈_{{l+1}}: {l + 1} ⟶ F^{l}

where the second bijection, in both cases, follows from the fact that (D, π) is a limit of F. These correspondences show that, the universe of D^{l} is in bijective correspondence, via π^{l}, with with cones from one-element set to F^{l} and moreover the order of elements in D^{l} corresponds to the pointwise order of cones from one-element set to F^{l}, i.e. (D^{l}, π^{l}) is a limit of F^{l} in Poset.

To finish the proof, note that, the morphisms p, b, t in disk D are morphisms into limits (D^{l}, π^{l}) from cones induced by morphisms p, b, t in disks F (j), for j ∈ Ob(J). Thus they are unique.

We call a zigzag of morphisms in D thin iff its image under every functor (−)^{l}, for l ∈ ω, is thin in BI. We have

3.13. Lemma. The limit of a zigzag of morphisms in D

[figure: D_{0}; D_{2}; D_{1}; f_{1}; f_{2}; D_{4}; D_{3}; f_{3}; f_{4}; D_{2k−2}; D_{2k}; D_{2k−1}; f_{2k−1}; f_{2k}; …]  (6)

exists iff the zigzag (6) is thin.

Proof. By Propositions 3.12, 3.10, the condition is clearly necessary.

Assume that, the zigzag (6) is thin. We shall construct a limit (D, π) of (6) in D. Since (6) is thin, for l ∈ ω, we have a limit (D^{l}, π^{l}) in BI of the zigzag (6)^{l}, the image

of (6) under the functor (−)^{l}. Thus x = ⟨x_{i}⟩_{i∈k+1} ∈ D^{l} iff f_{2i+1}(x_{i}) = f_{2i+2}(x_{i+1}), for i ∈ k.

Let p : D^{l+1} ⟶ D^{l} be the unique morphism from the cone (D^{l+1}, p ∘ π^{l+1}) into the limiting cone (D^{l}, π^{l}), so that p∘π^{l+1} = π^{l} ∘p. Similarly, b, t : D^{l} ⟶ D^{l+1}, are the unique morphisms, into the limiting cone (D^{l+1}, π^{l+1}), such that

b ∘ π^{l} = π^{l+1} ∘ b and b ∘ π^{l} = π^{l+1} ∘ b

This ends the construction of (D, π).

Note that π preserves order, and commutes with p, b, t by construction. Now, by Lemma 3.12, to show that (D, π) is a limit of (6) in D, it is enough to prove that D is a disk. To this end, we shall verify that it satisfy the conditions from Lemma 3.2.

Fix l ∈ ω. Clearly, D^{l} is an object of BI.

If x ∈ D^{l+1}, then

b(p(x)) ≤ x ≤ t(p(x)) iff b(p(x_{i})) ≤ x_{i} ≤ t(p(x_{i})) for i ∈ k + 1

and the latter condition holds, since D_{2i} is a disk, for i ∈ k + 1.

Let x, y ∈ D^{l+1}. If p(x) = p(y) then b(p(x)) ≤ x, y, and since D^{l+1} is in BI, we have x ⊥ y. On the other hand, if x ⊥ y, then x_{i} ⊥ y_{i}, for i ∈ k + 1, and then

p(x) = ⟨p(x_{i})⟩_{i∈k+1} = ⟨p(y_{i})⟩_{i∈k+1} = p(y)

Thus, it remains to show that, for x ∈ D^{l+1}

b(x) = t(x) iff x = b(p(x)) or x = t(p(x))  (7)

If x = b(p(x)), then x_{i} = b(p(x_{i})), and hence b(x_{i}) = t(x_{i}), for i ∈ k + 1. Thus b(x) = t(x).

If x = t(p(x)) the argument is similar. This proves one side of (7).

For the converse, we assume that b(x) = t(x). Then y = b(p(x)) ≤ x ≤ t(p(x)) = z. We need to show, that one of those inequalities is, in fact, an equality.

Assume contrary, that y < x < z. Note that, for i ∈ k + 1, since b(x_{i}) = t(x_{i}), the node x_{i} is outer and then either x_{i} = y_{i} or x_{i} = z_{i}. There are i_{1}, i_{2} ∈ k + 1, such that

x_{i_{1}} = y_{i_{1}} < z_{i_{1}} and y_{i_{2}} < z_{i_{2}} = x_{i_{2}}

Suppose that i_{1} < i_{2}. The case i_{2} < i_{1} can be treated similarly. Let

i_{0} = max{i < i_{2} : x_{i} = y_{i} < z_{i}}

Then i_{0} < i_{2} and

x_{i_{0}} = y_{i_{0}} < z_{i_{0}} and y_{i_{0}+1} ≤ z_{i_{0}+1} = x_{i_{0}+1}

Thus, we have

f_{2i_{0}+1}(y_{i_{0}}) = f_{2i_{0}+1}(x_{i_{0}}) = f_{2i_{0}+2}(x_{i_{0}+1}) = f_{2i_{0}+2}(z_{i_{0}+1}) = f_{2i_{0}+1}(z_{i_{0}})

i_{2} > i_{0}, z_{i_{0}} ≠ y_{i_{0}} and y_{i} ⊥ z_{i}, for i ∈ k + 1. But this contradicts the fact, that 6^{l} is thin. Thus x = b(p(x)) or x = t(p(x)), and (7) holds.

Let n_{0}, n_{1}, n_{2}, c, d be as above, l ∈ ω, x ∈ γ_{n_{0}}^{l} and y ∈ γ_{n_{2}}^{l}. If c^{l}(x) = d^{l}(y), we say that x and y are connected, and more specifically, x is connected to the right to y and y is connected to the left to x. In order to study limits of some diagrams involving c and d, we need to analyze properties of connected nodes. When two elements are connected they may be from different sides of the disks and if one of the elements is inner the other may be not. However, the connected elements inherit some properties one from another. Since the following lemma contains many conditions we add some slogans to help understand and remember them.

3.14. Lemma. Let l, n_{0}, n_{1}, n_{2} ∈ ω, n_{1} < n_{0}, n_{2}, x ∈ γ_{n_{0}}^{l} and y ∈ γ_{n_{2}}^{l}, and c^{l}(x) = d^{l}(y). Then

1. if x is in the (far) left side of γ_{n_{0}} then if y is in the (far) left side of γ_{n_{2}} and y is the unique node of γ_{n_{2}}^{l} with the property c^{l}(x) = d^{l}(y)

(the slogan: elements from the (far) left side are connected to the right to elements on the (far) left side and they are connected in a unique way);

2. if y is in the (far) right side of γ_{n_{2}} then if x is in the (far) right side of γ_{n_{0}} and x is the unique node of γ_{n_{0}}^{l} with the property c^{l}(x) = d^{l}(y)

(the slogan: elements from the (far) right side are connected to the left to elements on the (far) right side and they are connected in a unique way);

3. if y is in the left side of γ_{n_{2}} and x is in the right side of γ_{n_{0}} then either x is in the far right side of γ_{n_{0}} or y is in the far left side of γ_{n_{2}}

(the slogan: if an element in the left side is connected to the left to an element on an element on the right side one of the elements must be in the far side);

4. l ≤ n_{1} then x is inner iff y is inner;

5. if x is inner and n_{1} < l then y = n_{1}, i.e. y is in the left side;

6. if y is inner and n_{1} < l then x = γ_{n_{0}}^{l} − n_{1} − 1, i.e. x is in the right side;

(the slogan the last three conditions: the inner nodes are connected to unique elements; moreover they are connected either to inner nodes or to the left to elements in the right side or to the right to elements in the left side).

Proof. Recall the numeric conditions from Lemma 3.4 for elements to be in special parts of the disk γ_{n}. For the whole proof we assume that l, n_{0}, n_{1}, n_{2} ∈ ω, n_{1} < n_{0}, n_{2}, and x ∈ γ_{n_{0}}^{l} y ∈ γ_{n_{2}}^{l}, and c^{l}(x) = d^{l}(y).

Note that, if l ≤ n_{1} then, if one of the nodes x, y is in either side or far side or is inner then so is the other node. Thus for l ≤ n_{1}, all the statements of the lemma are obvious. Therefore, we assume further that l > n_{1}.

Ad 1. Let x ∈ fls(γ_{n_{0}}). If x < n_{1} then

y = x < n_{1} ≤ min{l − 1, n_{2} − 1}

and hence y ∈ fls(γ_{n_{2}}). If x ≥ n_{1} then l > n_{1} + 1 and then

min{l, n_{2}} > n_{1} = c^{l}(x) = d^{l}(y) = y

and y ∈ fls(γ_{n_{2}}), as well.

Now assume that, x ∈ (ls(γ_{n_{0}}) \ fls(γ_{n_{0}})). Then l ≤ n_{0} + 1, x = l − 1, and we have again

min{l, n_{2}} > n_{1} = c^{l}(x) = d^{l}(y) = y

i.e. y ∈ ls(γ_{n_{2}}). Note that in all the above cases the node y connected to x is unique.

Ad 2. Let y ∈ frs(γ_{n_{2}}). If y > γ_{n_{2}}^{l} − n_{1} + 1 then d^{l}(y) > n_{1} + 1. It follows that, c^{l}(x) > n_{1} + 1, and we get

x = c^{l}(x) + γ_{n_{0}}^{l} − 2n_{1} − 2 > γ_{n_{0}}^{l} − n_{1} − 1 > min{l, n_{0}}

This means that x ∈ frs(γ_{n_{0}}). If y ≤ γ_{n_{2}}^{l} − n_{1} − 1 then l > n_{1} + 1, n_{1} + 1 = d^{l}(y) = c^{l}(x) and

x = γ_{n_{0}}^{l} − n_{1} − 1 > min{l, n_{0} + 1}

Hence x ∈ frs(γ_{n_{0}}).

Now assume that, y ∈ (rs(γ_{n_{2}}) \ frs(γ_{n_{2}})). Then, either l ≤ n_{2} and y = l + 1 or l = n_{2} + 1 and y = n_{2} + 2. In either case, c^{l}(x) = d^{l}(y) = n_{1} + 1. So x = γ_{n_{0}}^{l} − n_{1} − 1. If l > n_{1} + 1, then we already know from the above argument that x ∈ frs(γ_{n_{0}}) ⊆ rs(γ_{n_{0}}). If l = n_{1} + 1 then x = n_{1} + 2 and x ∈ frs(γ_{n_{0}}). As before, it is easy to see that in all the cases the node x connected to y is unique.

Ad 3. Let x ∈ frs(γ_{n_{0}}) and y ∈ fls(γ_{n_{2}}). Then, either c^{l}(x) = n_{1} or c^{l}(x) = n_{1} + 1. In the former case, y = n_{1} and it is a bi-endpoint, unless l = n_{1} + 1. But then

x ∈ {n_{1}, n_{1} + 1} i.e. x ∉ rs(γ_{n_{0}}).

In the latter case, x = γ_{n_{0}}^{l} − n_{1} − 1 and it is a bi-endpoint, unless l = n_{1} + 1. But then y ∈ {n_{1} + 1, n_{1} + 2} i.e. y ∉ ls(γ_{n_{2}}).

The remaining cases are left for the reader as an exercise.

3.15. Lemma. Let u⃗ be a ud-vector of length k + 1. Then the zigzag

[figure: γ_{u_{0}}; γ_{u_{2}}; γ_{u_{1}}; c; d; γ_{u_{4}}; γ_{u_{3}}; c; d; γ_{u_{2k−2}}; γ_{u_{2k}}; γ_{u_{2k−1}}; c; d; …]  (8)

is thin.

Proof. Fix l ∈ ω, and let x⃗, y⃗ be compatible tuples in the zigzag (8)^{l}, the image of (8) under the functor (−)^{l}, such that for some i_{0} ∈ k + 1, x_{i_{0}} < y_{i_{0}} and c(x_{i_{0}}) = c(y_{i_{0}}). By Lemma 3.14, l > u_{2i_{0}+1} and the node x_{i_{0}+1} = y_{i_{0}+1} = u_{2i_{0}+1} must be in the left side of γ_{u_{2i_{0}+2}}, and then for all i ≥ i_{0} we have x_{i} = y_{i}. Hence (8)^{l} is thin.

Thus, by Proposition 3.12 and Lemmas 3.13, 3.15, for any ud-vector u⃗, the zigzag of form (8) has a limit in D. We shall describe a specific limit (γu⃗, π) of the diagram (8). We have a limiting cone in D

[figure: γ_{u_{0}}; γ_{u_{2}}; γ_{u_{1}}; c; d; γ_{u_{2k−2}}; γ_{u_{2k}}; γ_{u_{2k−1}}; c; d; …; γu⃗; π_{2}; π_{2k−2}; π_{2k}; π_{0}]  (9)

where for l ∈ ω, γl ; u⃗ is a set of compatible tuples for the zigzag (8)^{l}, i.e. x_{i} ∈ γ_{u_{2i}}^{l}, for i ∈ k + 1, and c^{l}(x_{i}) = d^{l}(x_{i+1}), for i ∈ k. The order on γl ; u⃗ is defined pointwise, as well, as the operations p, b, t on γu⃗, e.g. p(x⃗) = ⟨p(x_{i})⟩_{i∈k+1}. The projections π_{2i} : γu⃗ ⟶ γ_{u_{2i}}, for i ∈ k + 1, are the usual projections, i.e. π_{2i}^{l}(x⃗) = x_{i}, for x⃗ ∈ γl ; u⃗.

The following lemma gives more informations about γu⃗.

3.16. Lemma. All the unexplained notation refers to (9). Let l ∈ ω, x⃗ ∈ γl ; u⃗. Then

1. | γl ; u⃗ |= ∑_{i=0}^{2k}(−1)^{i}γ_{u_{i}}^{l};

2. there are 0 ≤ i_{0} ≤ i_{1} ≤ k + 1, such that

x_{i} ∈ rs(γ_{2u_{i}}) for i < i_{0}
x_{i} ∈ ls(γ_{2u_{i}}) for i ≥ i_{1}
x_{i} is inner (x_{i} = l) for i_{0} ≤ i < i_{1};

3. x⃗ is inner iff x_{i_{0}} ∈ γ_{u_{2i_{0}}}^{l} is inner, for some i_{0} ∈ k + 1;

4. there are k + 1 leaves: x^{0}, …, x^{k} in γu⃗, such that x^{j} ∈ γu_{2j} ; u⃗, for j ∈ k + 1 and x^{j} ≪ x^{j+1}, for j ∈ k;

5. x⃗ is a j-th leaf iff x_{j} = u_{2j}, moreover x_{j}^{j} ∈ γ_{u_{2j}} is the only inner node in x^{j};

6. for j ∈ k, μ_{x^{j},x^{j+1}} = u_{2j+1}.

Proof. 1. is left as an exercise.

2. follows from Lemma 3.14, since elements in the right sides are connected to the left to elements in the right side, elements in the left sides are connected to the right to elements in the left side, and inner nodes are either connected to inner nodes or to the left to elements in the right side, and to the right to elements in the left side.

Fix l ∈ ω and x⃗ ∈ γl ; u⃗. If, for some i_{0} ∈ k + 1, x_{i_{0}} ∈ γ_{u_{2i_{0}}}^{l} is inner then b(x_{i_{0}}) ≠ t(x_{i_{0}}) and hence b(x⃗) ≠ t(x⃗). So, x⃗ is inner, as well. If x_{i} is outer then b(x_{i}) = t(x_{i}). So, if all x_{i}’s are outer then x⃗ is outer, as well. This shows 3.

If x_{i_{0}} ∈ γ_{u_{2i_{0}}}^{l} is a leaf, then x_{i_{0}} = l = u_{2i_{0}} and x_{i_{0}−1} ∈ rs(γ_{2u_{0}−2}), x_{i_{0}+1} ∈ ls(γ_{2u_{0}+2}). Hence, by Lemma 3.14, for i ≠ i_{0}, x_{i} is an outer node. It follows, that b(x⃗) and t(x⃗) are the only compatible tuples over x⃗, i.e. x⃗ is a leaf.

On the other hand, if x⃗ ∈ γl ; u⃗ is a leaf, then b(x⃗) ≠ t(x⃗) are the only compatible tuples over x⃗. Then, there is i_{0} ∈ k + 1, such that b(x_{i_{0}}) < t(x_{i_{0}}).

In case x_{i_{0}} is not a leaf, then we have an inner node y ∈ γ_{u_{2i_{0}}}^{l+1}, such that b(x_{i_{0}}) < y < t(x_{i_{0}}).

In case, either x_{i_{0}−1} or x_{i_{0}+1} is inner, by Lemma 3.14, l < u_{2i_{0}+1} < u_{2i_{0}} and we have, as in previous case, an inner node y ∈ γ_{u_{2i_{0}}}^{l+1}, such that b(x_{i_{0}}) < y < t(x_{i_{0}}).

Having y as in either case, we show that x⃗ is not inner. As, y is inner, there is a unique tuple y⃗ ∈ γl ; u⃗, such that y_{i_{0}} = y. Clearly, b(x⃗) ≠ y⃗ ≠ t(x⃗). Since x_{i_{0}} is inner, x⃗ is the unique compatible tuple in γl ; u⃗ with x_{i_{0}} on i_{0}-th place and p(y⃗) has x_{i_{0}} = p(y) on i-th place, as well. Therefore p(y⃗) = x⃗, i.e. there are at least three elements over x⃗ and x⃗ is not a leaf.

Thus, we have shown that x⃗ ∈ γl ; u⃗ is a leaf iff there is a unique i_{0}, such that x_{i_{0}} is an inner node and moreover this x_{i_{0}} is a leaf in γ_{u_{2i_{0}}}^{l}. Then, we must have x_{i_{0}} = l = u_{2i_{0}}.

For j ∈ k + 1, we define compatible tuples

x^{j} = ⟨x_{i}^{j}⟩_{i∈k+1} ∈ γu_{2j} ; u⃗

by the condition x_{j}^{j} = u_{2j}. Since n_{2j} ∈ γ_{u_{2j}}^{u_{2j}} is the unique leaf of γ_{u_{2j}}, x^{j}’s are well defined by the above condition. Then, by the above considerations, they are all and only leaves of γu⃗.

Since x_{j+1}^{j+1} is inner, for l ≤ u_{2j}, u_{2j+2}, we have

p^{(l)}(x^{j}) = p^{(l)}(x^{j+1}) iff p^{(l)}(x_{j+1}^{j}) = p^{(l)}(x_{j+1}^{j+1})

We can easily check, that

x_{j+1}^{j} = u_{2j+1} ∈ γ_{u_{2j+2}}^{u_{2j}}

and, for l ≥ u_{2j+1}

p^{(l)}(x_{j+1}^{j}) = p^{(l)}(u_{2j+1}) = u_{2j+1}
p^{(l)}(x_{j+1}^{j+1}) = p^{(l)}(u_{2j+2}) = l

Thus, we have

p^{(u_{2j+1})}(x_{j+1}^{j}) = p^{(u_{2j+1})}(x_{j+1}^{j+1})
p^{(u_{2j+1}+1)}(x_{j+1}^{j}) = u_{2j+1} < u_{2j+1} + 1 = p^{(u_{2j+1}+1)}(x_{j+1}^{j+1})

This shows, that μ_{x^{j},x^{j+1}} = u_{2j+1} and that x^{j} ⪯ x^{j+1}, for j ∈ k. So, we also have x^{j} ≪ x^{j+1}, for j ∈ k.

Using the above Lemma we can prove the following Proposition.

3.17. Proposition. Let u⃗, v⃗ be ud-vectors, such that the disks γu⃗ and γv⃗ are isomorphic. Then u⃗ = v⃗.

Proof. Let u⃗, v⃗ be ud-vectors, and let f : γu⃗ ⟶ γv⃗ be a disk isomorphism. The leaves, as well, as order on leaves are defined in terms of operations p, b, t, and order in disks. Thus, any disk isomorphism must send leaves to leaves bijectively. Hence, γu⃗ and γv⃗ have the same number of leaves, say k + 1, i.e. the ud-vectors u⃗ and v⃗ have equal length. Let x^{0}, …, x^{k} be the leaves of γu⃗ and y^{0}, …, y^{k} be the leaves of γv⃗, listed in order. Then f(x^{j}) = y^{j}, for j ∈ k + 1. Thus u_{2j} = v_{2j}, for j ∈ k + 1. For l ≤ u_{2j}, u_{2j+2}, we also have

p^{(l)}(x^{j}) = p^{(l)}(x^{j+1}) p^{(l)}(y^{j}) = p^{(l)}(y^{j+1})

Thus

u_{2j+1} = μ_{x^{j},x^{j+1}} = μ_{y^{j},y^{j+1}} = v_{2j+1}

i.e. u⃗ = v⃗.

We have shown that the inner nodes of the disk γu⃗, form a tree θu⃗, which have lh(u⃗) leaves. The j-th leaf is at the level u_{2j} and it ’meets’ j + 1-st leaf at the level u_{2j+1}. This describes θu⃗ and hence γu⃗ uniquely. For example the tree θ_{3,2,3,1,2,0,1,0,4,1,3} looks like that:

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •]

We shall show, that any disk is isomorphic to a disk γu⃗, for some ud-vector u⃗. To this end we introduces some ’projection’ morphisms from any disk into disks of form γ_{n}, and gives some connections between them.

3.18. Proposition. Let D be a disk, n_{0}, n_{1}, n_{2}, n ∈ ω. Then

1. Let x = x_{n} ∈ D^{n} be a leaf, and x_{l} = p^{(l)}(x_{n}) for 0 ≤ l < n. Then, there is a unique

disk morphism

\overline{x} : D ⟶ γ_{n}

such that, for 0 ≤ l ≤ n and y ∈ D^{l}, we have

\overline{x}^{l}(y) = l iff y = x_{l}  (10)

2. Let x ∈ D^{n_{0}}, y ∈ D^{n_{2}} be leaves, such that x ≪ y, and n_{1} = μ_{x,y}. Then the diagram

[figure: c; D; γ_{n_{0}}; \overline{x}; γ_{n_{1}}; d; γ_{n_{2}}; \overline{y}]  (11)

commutes.

3. Let x ∈ D^{n_{0}}, y ∈ D^{n_{2}} and n_{1} = μ_{x,y}, such that p^{n_{1}+1}(x) < p^{n_{1}+1}(y). Then there is

a unique morphism

\overline{x; y} : D ⟶ γ_{n_{1}}

such that, for 0 ≤ l ≤ n_{1} and z ∈ D^{l}

\overline{x; y}^{l}(z) = l iff z = p^{(l)}(x) (= p^{(l)}(y))  (12)

and

\overline{x; y}^{n_{0}}(x) = n_{1} and \overline{x; y}^{n_{2}}(y) = n_{1} + 1  (13)

Proof. Ad 1. Let D, n, x, x_{l} be as in the statement. We put for l ≤ n and y ∈ D^{l}

\overline{x}^{l}(y) = \{ \begin{matrix}μ_{x,y} & if p^{(μ_{x,y}+1)}(y) < x_{μ_{x,y}+1} \\ l & if y = x_{l} \\ 2l − μ_{x,y} & if p^{(μ_{x,y}+1)}(y) > x_{μ_{x,y}+1} \end{matrix}

and for y ∈ D^{n+1}

\overline{x}^{n+1}(y) = \{ \begin{matrix}μ_{x,y} & if p^{(μ_{x,y}+1)}(y) < x_{μ_{x,y}+1} \\ n & if y = b(x) \\ n + 1 & if y = t(x) \\ 2n + 1 − μ_{x,y} & if p^{(μ_{x,y}+1)}(y) > x_{μ_{x,y}+1} \end{matrix}

Since any node y ∈ D^{l} is either equal to x_{l} or p^{(μ_{x,y}+1)}(y) is comparable but different than p^{(μ_{x,y}+1)}(x), it follows, that for l ≤ n, the above formulas defines functions \overline{x}^{l} : D^{l} ⟶ γ_{n}^{l}. As x is a leaf, if y ∈ D^{l} and p(y) = x then y ∈ {b(x), t(x)}. Moreover, if y ∈ D^{l} and p(y) ≠ x then x_{μ_{x,y}+1} and p^{(μ_{x,y}+1)}(y) are comparable but different. Hence \overline{x}^{n+1} : D^{n+1} ⟶ γ_{n}^{n+1} is a well defined function, as well.

We need to verify that the functions \{\overline{x}^{l}\}_{l∈n+2} preserves projections, order in fibers, and endpoints. Then, by Lemma 3.5, we will get a unique morphism \overline{x} : D ⟶ γ_{n} extending the above set of functions.

First, we shall show that the orders are preserved.

Note that, if l ≤ n, y, z ∈ γ_{n}^{l} and y < z, then p(y) = p(z) and either μ_{x,y} = μ_{x,z} < l −1 or p(y) = x_{l−1} ∈ D^{l−1}. In the former case, we simply have that \overline{x}^{l}(y) = \overline{x}^{l}(z), by first or third close of the definition of \overline{x}^{l}. In the latter case, y, z ∈ p^{−1}(x_{l−1}). If u ∈ p^{−1}(x_{l−1}) then u = x_{l−1} or μ_{x,u} = l − 1. Hence

\overline{x}^{l}(y) = \{ \begin{matrix}l − 1 & if u < x_{l} \\ l & if u = x_{l} \\ l + 1 & if u > x_{l} \end{matrix}

Since, y < z, and l − 1 < l < l + 1 in order of γ_{n}^{l}, we have \overline{x}^{l}(y) ≤ \overline{x}^{l}(z).

If y, z ∈ D^{n+1} and y < z, then either y = b(x) and z = t(x) and hence

\overline{x}^{n+1}(y) = n < \overline{x}^{n+1}(z)

or μ_{x,y} = μ_{x,z} < n and then \overline{x}^{n+1}(y) = \overline{x}^{n+1}(z). Thus, for l ≤ n + 1, the functions \overline{x}^{n+1} preserves the orders.

To see, that projections are preserved, note that, if l ≤ n + 1 and y ∈ D^{l}, then either y ∈ p^{−1}(l − 1) or μ_{x,y} < l − 1. In both cases, we have that

p ∘ \overline{x}^{l}(y) = \overline{x}^{l}(p(y))

i.e. projections are preserved.

For the preservation of endpoints, note that, if l ≤ n + 1 and y ∉ p^{−1}(x_{l−1}), then \overline{x}^{l}(y) is the unique element in the fiber over p ∘ \overline{x}^{l}(y), i.e. it is a bi-endpoint. If y ∈ p^{−1}(x_{l−1}), then

\overline{x}^{l}(y) = \{ \begin{matrix}l − 1 & if l ≤ n and y < x_{l} \\ n & if y = b(x_{l−1}) \\ n + 1 & if y = t(x_{l−1}) \\ l + 1 & if l ≤ n and y > x_{l} \end{matrix}

Since all the elements are endpoints of th appropriate kind, the endpoints in p^{−1}(x) are also preserved by \overline{x}^{l}, for l ≤ n + 1.

It remains to show that, the morphism \overline{x} is a unique disk morphism satisfying (10). Suppose contrary, that π : D ⟶ γ_{n} is a different morphism satisfying (10). Let

l = min\{l′ : π^{l′}(y) ≠ \overline{x}_{x}^{l′}(y) for some y ∈ D^{l′}\}

and fix y ∈ D^{l}, such that π^{l}(y) ≠ \overline{x}^{l}(y).

Since l is minimal we have

p ∘ π^{l}(y) = π^{l−1}(p(y)) = \overline{x}^{l−1}(p(y)) = p ∘ \overline{x}^{l}(y)

Thus p(y) = l − 1, as l − 1 is the only node in γ_{n} having more than one element in the fiber over it. So, \overline{x}^{l}(y), π^{l}(y) ∈ \{l − 1, l, l + 1\}. Then one of the following

x_{l} < y y = x_{l} y < x_{l}

holds. If x_{l} < y, then by (10) for \overline{x} we have

\overline{x}^{l}(y) ≠ \overline{x}^{l}(x_{l}) = l and \overline{x}^{l}(x_{l}) ≤ \overline{x}^{l}(y)

Thus \overline{x}^{l}(y) = l + 1. Since (10) holds for π, as well, we also have, that \overline{x}^{l}(y) = l + 1, contrary to the choice of y.

The case y < x_{l} is similar.

If y = x_{l}, then by (10) for both \overline{x} and π, we have

π^{l}(y) = π^{l}(x_{l}) = \overline{x}^{l}(x_{l}) = \overline{x}^{l}(y)

again contrary to the choice of y. This contradiction shows, that \overline{x} satisfying (10) is unique. Ad 2. Let x_{0} ∈ D^{n_{0}}, x_{2} ∈ D^{n_{2}} be leaves, such that x ≪ y, and n_{1} = μ_{x_{0},x_{2}}. By Lemma 3.5, it is enough to verify that, for l ≤ n_{1} + 1 and y ∈ D^{l}

c^{l} ∘ \overline{x_{0}}^{l}(y) = d^{l} ∘ \overline{x_{2}}^{l}(y)  (14)

If l ≤ n_{1}, then μ_{x_{0},y} = μ_{x_{2},y}. Moreover, p^{(l)}(x_{0}) = p^{(l)}(x_{2}). Thus \overline{x_{0}}^{l}(y) = \overline{x_{2}}^{l}(y). Since, for l ≤ n_{1}, c^{l} = d^{l}, as well, (14) holds for l ≤ n_{1}.

If l = n_{1} + 1 and y ∉ {p^{(n_{1}+1)}(x_{0}), p^{(n_{1}+1)}(x_{2})}, then μ_{x_{0},y} = μ_{x_{2},y} and, since p^{(n_{1}+1)}(x_{0}) < p^{(n_{1}+1)}(x_{2}), we have

y < p^{(n_{1}+1)}(x_{0}) iff y < p^{(n_{1}+1)}(x_{2})

Thus \overline{x_{0}}^{n_{1}+1}(y) = \overline{x_{2}}^{n_{1}+1}(y) ≠ n_{1} + 1. But, for n_{1} + 1 ≠ x ∈ γ_{n_{0}}^{n_{1}+1} = γ_{n_{2}}^{n_{1}+1}, we have c^{n_{1}+1}(x) = d^{n_{1}+1}(x), i.e. (14) holds for y ∉ {p^{(n_{1}+1)}(x_{0}), p^{(n_{1}+1)}(x_{2})}.

Since p^{(n_{1}+1)}(x_{0}) < p^{(n_{1}+1)}(x_{2}), we have

\overline{x_{0}}^{n_{1}+1}(p^{(n_{1}+1)}(x_{0})) = n_{1} + 1 \overline{x_{0}}^{n_{1}+1}(p^{(n_{1}+1)}(x_{2})) = n_{1} + 2
\overline{x_{2}}^{n_{1}+1}(p^{(n_{1}+1)}(x_{0})) = n_{1} \overline{x_{2}}^{n_{1}+1}(p^{(n_{1}+1)}(x_{2})) = n_{1} + 1
c^{n_{1}+1}(n_{1} + 1) = n_{1} c^{n_{1}+2}(n_{1} + 2) = n_{1} + 1
d^{n_{1}+1}(n_{1}) = n_{1} d^{n_{1}+1}(n_{1} + 1) = n_{1} + 1

i.e. (14) holds for y ∈ {p^{(n_{1}+1)}(x_{0}), p^{(n_{1}+1)}(x_{2})}. Thus (14) holds for l = n_{1} + 1, as well. This proves that the diagram (11) commutes.

Ad 3. Let x ∈ D^{n_{0}}, y ∈ D^{n_{2}} and n_{1} = μ_{x,y}, such that x′ = p^{n_{1}+1}(x) < p^{n_{1}+1}(y) = y′. We shall give the definition of \overline{x; y} and leave to the reader the verification that, they satisfy the conditions (12) and (13).

By x′′ and y′′ we denote the largest leaf over x′ and the least leaf over y′, respectively, if they exists, i.e. if x′ and y′ are inner, respectively. We put

\overline{x; y} : D ⟶ γ_{n_{1}} = \{ \begin{matrix}c_{(n_{1})} ∘ \overline{x′′} & if x′ is inner \\ d_{(n_{1})} ∘ \overline{y′′} & if y′ is inner \\ \overline{p^{μ_{x,y}}(x)} & otherwise \end{matrix}

For further reference we give an explicit formula for \overline{x; y}^{l} : D^{l} ⟶ γ_{n}^{l}, where x, y ∈ D^{n+1} and x < y. For z ∈ D^{l} we have

\overline{x; y}^{l}(z) = \{ \begin{matrix}n & if l > n and p^{(n+1)}(z) = x \\ n + 1 & if l > n and p^{(n+1)}(z) = y \\ l & if l ≤ n and p^{(l)}(x) = z \\ μ_{x,z} & if μ_{x,z} < min(l, n + 1) \\ & and p^{(μ_{x,z}+1)}(z) < p^{(μ_{x,z}+1)}(x) \\ γ_{n}^{l} − μ_{x,z} − 1 & if μ_{x,z} < min(l, n + 1) \\ & and p^{(μ_{x,z}+1)}(x) < p^{(μ_{x,z}+1)}(z) \end{matrix}  (15)

3.19. Theorem. Let D be a disk, and let {x_{i}}_{i∈k+1} be the set of leaves of D such that x_{i} ∈ D^{u_{2i}} for i ∈ k + 1, x_{i} ≪ x_{i+1} and u_{2i+1} = μ_{x_{i},x_{i+1}} for i ∈ k. Then the diagram

[figure: γ_{u_{0}}; γ_{u_{2}}; γ_{u_{1}}; c; d; γ_{u_{2k−2}}; γ_{u_{2k}}; γ_{u_{2k−1}}; c; d; …; D; \overline{x_{2}}; \overline{x_{2k−2}}; \overline{x_{2k}}; \overline{x_{0}}]  (16)

is a limiting cone in D.

Proof. By Lemma 3.12, we need to verify that for l ∈ ω, the image (16)^{l} of cone (16) under the functor (−)^{l} is a limiting cone in Poset, i.e. we need to verify, that (16)^{l} is a limit in Set and that for l ∈ ω and y, z ∈ D^{l}, we have

y ≤ z iff \overline{x_{i}}^{l}(y) ≤ \overline{x_{i}}^{l}(z) for i ∈ k + 1

First, we shall show that the diagram (16)^{l} is a limit in Set. We have Claim 1. If y, z ∈ D^{l} and y < z, then there is i_{0} ∈ k + 1 such that \overline{x_{i_{0}}}^{l}(y) < \overline{x_{i_{0}}}^{l}(z).

If y is inner, then for some i_{0} ∈ k + 1 there is a leaf x_{i_{0}} over y, and we have

\overline{x_{i_{0}}}^{l}(y) = l < l + 1 = \overline{x_{i_{0}}}^{l}(z).

If z is inner, then for some i_{0} ∈ k + 1 there is a leaf x_{i_{0}} over z, and we have

\overline{x_{i_{0}}}^{l}(y) = l − 1 < l = \overline{x_{i_{0}}}^{l}(z).

If both y and z are outer, then p(y) is a leaf, i.e. for some i_{0} ∈ k + 1, p(y) = x_{i_{0}}. Then

\overline{x_{i_{0}}}^{l}(y) = l − 1 < l =≤ \overline{x_{i_{0}}}^{l}(z).

This ends the proof of Claim 1.

Now, let y, z ∈ D^{l} be arbitrary different nodes, l′ = μ_{y,z} + 1, y′ = p^{(l′)}(y), z′ = p^{(l′)}(z). Then the nodes y′, z′ ∈ D^{l′} are comparable and different, say y′ < z′. Thus, we have a node y′′ ∈ D^{l′}, such that y′ ≤ y′′ < z′. By Claim 1, there is i_{0} ∈ k + 1, such that \overline{x_{i_{0}}}^{l′}(y′′) < \overline{x_{i_{0}}}^{l′}(z′), and then

\overline{x_{i_{0}}}^{l}(y) = \overline{x_{i_{0}}}^{l′}(y′) ≤ \overline{x_{i_{0}}}^{l}(y′′) < \overline{x_{i_{0}}}^{l′}(z′) < \overline{x_{i_{0}}}^{l′}(z)

Hence \overline{x_{i_{0}}}^{l}(y) < \overline{x_{i_{0}}}^{l}(z).

We shall show by induction on l ∈ ω

Claim 2 For each compatible tuple {y_{i}}_{i∈k+1} at level l, there is y ∈ D^{l} such that

\overline{x_{i}}^{l}(y) = y_{i} for i ∈ k + 1

For l = 0, Claim 2 is obvious, since at level 0 all disks have exactly one element.

Now, suppose that Claim 2 holds for l −1. Let {y_{i}}_{i∈k+1} be a compatible tuple at level l, i_{0} and i_{1} as in Lemma 3.16.2 for that tuple. Since c and d commutes with projection the tuple {z_{i}}_{i∈k+1} at level l − 1, such that

z_{i} = p(y_{i}) ∈ γ_{n_{2i}}^{l−1} for i ∈ k + 1

is compatible at the level l − 1. By induction hypothesis there is z ∈ D^{l−1}, such that

\overline{x_{i}}^{l−1}(z) = z_{i} for i ∈ k + 1

We shall consider four cases

1. i_{0} = k + 1;

2. i_{1} = 0;

3. i_{0} < i_{1};

4. 0 < i_{0} = i_{1} ≤ k.

Case 1. If i_{0} = k+1 then all y_{i}’s are right endpoints. Then, since \overline{x_{i}}^{l}’s preserves endpoints, the node y = t(z) satisfy the Claim 2. Case 2. If i_{1} = 0 then all y_{i}’s are left endpoints, and similarly as before, y = b(z) satisfy the Claim 2. Case 3. If i_{0} < i_{1} then y_{i_{0}} = l is an inner node in γ_{2i_{0}}. We put y = p^{(l)}(x_{i_{0}}). Then

\overline{x_{i_{0}}}^{l}(y) = l = y_{i_{0}}

By Lemma 3.14, {y_{i}}_{i∈k+1} is a unique compatible tuple {u_{i}}_{i∈k+1} at level l, such that u_{i_{0}} = y_{i_{0}}. But, \{\overline{x_{i}}^{l}(y)\}_{i∈k+1} is a compatible tuple with the same property, i.e. y is as in the Claim 2. Case 4. If 0 < i_{0} = i_{1} ≤ k then y_{i_{0}} is in the right side of γ_{2i_{0}} and is connected to the right to y_{i_{1}} which is in the left side of γ_{2i_{1}}. Thus by Lemma 3.14 either y_{i_{0}} or y_{i_{1}} is in the far side.

If y_{i_{0}} is in far side then, again by Lemma 3.14, all y_{i} for i ≤ i_{0} are in the far right side. Thus y_{i} is a left endpoint for all i ∈ k + 1. Thus as in Case 2, y = b(z) satisfy the Claim 2.

If y_{i_{1}} is in far side then, for similar reasons as previously, all y_{i} for i ≤ i_{0} are in the far left side. Thus y_{i} is a right endpoint for all i ∈ k + 1. Thus as in Case 1, y = t(z) satisfy the Claim 2.

This shows that for l ∈ ω, (16)^{l} is a limiting cone in Set.

In order to show that (16)^{l} is a limiting cone in Poset we need to verify, that if y, z ∈ D^{l} and

\overline{x_{i}}^{l}(y) ≤ \overline{x_{i}}^{l}(z) in γ_{n_{2i}} for i ∈ k + 1 then y ≤ z D^{l}  (17)

Since, for i ∈ k + 1

\overline{x_{i}}^{l−1}(p(y)) = p ∘ \overline{x_{i}}^{l}(y) = p ∘ \overline{x_{i}}^{l}(z) = \overline{x_{i}}^{l−1}(p(z))

it follows, form previous considerations, that p(y) = p(z). Therefore y and z are comparable. If y > z then, again from previous considerations, there is i_{0} ∈ k + 1, such that \overline{x_{i_{0}}}^{l}(y) ≠ \overline{x_{i_{0}}}^{l}(z). But \overline{x_{i_{0}}}^{l}(y) ≤ \overline{x_{i_{0}}}^{l}(z), so \overline{x_{i_{0}}}^{l}(y) < \overline{x_{i_{0}}}^{l}(z), and we get a contradiction with the fact that \overline{x_{i_{0}}}^{l} preserves order. Thus, we have y ≤ z, i.e. (17) holds, and (16)^{l} is a limiting cone in Poset, as was to be shown.

Thus we have

3.20. Corollary. For every disk D, there is a unique ud-vector u⃗ such that D is isomorphic to γu⃗.

Proof. By Lemmas 3.13, 3.15, for any ud-vector u⃗, we have a disk γu⃗. By Theorem 3.19, for any disk D, there is a ud-vector u⃗ such, that D is isomorphic to γu⃗, and by Proposition 3.17 such a ud-vector u⃗ is unique.

3.21. Factorizations in D. In this section we shall study morphisms of disks.

It is easy to see that the epi in D are onto and mono are one-to-one and that these classes of morphisms form a factorization system in D. However in D there is another factorization system.

Let f : D ⟶ E be a disk morphism. We say that f is inner iff f sends inner nodes to inner nodes, i.e. f(ı(D)) ⊆ ı(E). We say that f is outer iff f is epi and for any inner nodes x, y in D, if f(x) = f(y) then f(x) is an outer node.

The inner morphisms are exactly those that are induced by the maps of underlying trees. In fact, the category of finite disks with the inner maps as morphisms is equivalent to the category of finite trees.

If there is an inner morphism from γu⃗ to γ_{l} then it is unique, and it is the case iff ht(u⃗) ≤ l. We denote by

mu⃗ : γu⃗ ⟶ γ_{ht(u⃗)} ι_{(l)} : γ_{n} ⟶ γ_{l}

the unique inner morphisms between these disks, where n ≤ l. Note that this notaton agree with the one in section 2.6 i.e. if we define for n_{1} < n_{0}, n_{2} the morphisms

m_{n_{0},n_{1},n_{2}} : γ_{n_{0},n_{1},n_{2}} ⟶ γ_{max(n_{0},n_{2})}

as the unique inner morphisms, then since the composition of inner morphisms is inner, mu⃗ is indeed the canonical internal composition morphism. This will be important in the next section.

We have

3.22. Lemma.

1. Outer morphisms are split epimorphisms with a unique splitting;

2. Let u⃗ be a ud-vector. Then, a disk morphism f : D ⟶ γu⃗ is outer iff π_{i} ∘f is outer for i ∈ lh(u⃗);

3. For any ud-vector u⃗ the projection π_{i} : γu⃗ ⟶ γ_{u_{2i}} is outer, for i ∈ lh(u⃗);

4. The following are equivalent

(a) A disk morphism f : D ⟶ γ_{n} is outer;

(b) there are x, y ∈ D^{n+1}, such that x < y and f = \overline{x; y};

(c) for l ≤ n there is a unique z ∈ ı(D)^{l} such that f^{l}(z) = l ∈ γ_{n}^{l}.

Proof. Ad 1. If f : D ⟶ E is an outer morphism of disks, then the inverse image of each inner node of E contains one element. Thus f^{−1}⌈ı(E) : ı(E) ⟶ D is a tree morphisms. Therefore, it extends to a disk morphism \overline{f} : E ⟶ D. Since f ∘ (f^{−1}⌈ı(E) is identity on inner nodes f ∘ \overline{f} = id_{E}. Since there are no choices involved in construction of \overline{f} it is clear that \overline{f} with the above property is unique.

Ad 2. Let u⃗ be a ud-vector and f : D ⟶ γu⃗ a disk morphism. For any i ∈ lh(u⃗), π_{i} : γu⃗ ⟶ γ_{u_{2}i} is outer. Hence if f is outer so is π_{i} ∘ f for i ∈ lh(u⃗).

Suppose now, that π_{i} ∘ f for i ∈ lh(u⃗). Let x, y be inner nodes in D such that f(x) = f(y) and f(x) is inner. Then, by Proposition 3.16.3, for some i_{0} ∈ lh(u⃗), π_{i_{0}} ∘f(x) is inner. Since π_{i_{0}} ∘ f is outer, we have that x = y.

Ad 3. Follows from 2.

Ad 4. (b) ⇒ (a). From the description in Proposition 3.18 easily follows, that \overline{x; y} is outer.

(a) ⇒ (c). If f : D ⟶ γ_{n} is outer then, it is epi and there is a a node z ∈ D^{n} such that f^{n}(z) = n ∈ γ_{n}^{n}. Since n ∈ γ_{n}^{n} is inner z must be inner. And since f is outer z must be unique.

(c) ⇒ (b). Having such z as above, in the fiber p^{−1}(z) = {z_{1}, …, z_{k}} ⊆ D^{n+1}, there are nodes z_{i} and z_{i+1} such that f(b(z)) = f(z_{i}) and f(b(z)) = f(z_{i+1}). Putting x = z_{i} and y = z_{i+1}, we have f = \overline{x; y}.

3.23. Proposition. Outer and inner morphisms of disks form a factorization system.

Proof. For a disk morphism f : D ⟶ E we define an equivalence relation ∼ on D so that for x, y ∈ D,

x ∼ y iff x ⊥ y and f(x) = f(y) and f(x) is outer

On D/_{∼} we have an obvious disk structure inherited from D and the obvious morphisms

[figure: D; E; f; D/_{∼}; f^{∘}; f^{•}]

form the outer-inner factorization of f. If in the commuting diagram in D

[figure: g_{2}; f_{2}; f_{1}; g_{1}]

f_{1} is outer and g_{1} is inner then h = f_{2} ∘ \overline{f_{1}} is the unique lifting. It exists, by Lemma 3.22, since f_{1} is outer, and the commutations of the triangles easily follows given the fact that g_{1} is inner.

From the proposition we get

3.24. Corollary. For any n ∈ ω and any disk morphism f : D ⟶ γ_{n} there is a ud-vector u⃗, ht(u⃗) = u, such that f factorizes as follows

[figure: γu⃗; γ_{u}; mu⃗; D; γ_{n}; f; f^{∘}; ι_{(n)}]

with f^{∘} outer part of f, and the morphisms ι_{(n)}, mu⃗ defined above.

Proof. In order to get this factorization of a disk morphism f we need to take the factorization (f^{∘}, f^{•}) from Proposition 3.23 and furthermore take epi-mono factorization of

its inner part f^{•}.

We call the factorization of a disk morphism f described in the above Corollary the canonical factorization of f.

3.25. An internal ω-category C in D. In this section we define an internal ω-category C in D, and we show that by homming into C we get a functor into the category of simple ω-categories S.

For n ∈ ω, γ_{n} is the object of n-cells of C, for n ≥ l

d_{(l)}, c_{(l)} : γ_{n} ⟶ γ_{l}

are the domain and codomain morphisms in C, for n ≤ l

ι_{(l)} : γ_{n} ⟶ γ_{l}

is the identity morphism, and for any ud-vector n⃗ of length 2

mn⃗ : γn⃗ ⟶ γ_{ht(n⃗)}

is the composition morphism in C.

We have

3.26. Proposition. C defined above is an internal ω-category in D.

Proof. We need to check, that the data specified above verify the conditions (vi)-(xi) of the definition of an ω-category given in Appendix 6.2. Thus, we need to check that many diagrams commutes. We shall only verify some chosen ones, and comment on the other leaving to the reader to calculate the rest.

Note that by Lemma 3.5 it is enough to verify the equality of morphisms at leaves only. Let for a ud-vector u⃗, λi ; u⃗ ∈ γu_{2i} ; u⃗ be the i-th leaf of γu⃗, for i ∈ lh(u⃗). Usually we drop subscript u⃗ in λi ; u⃗.

The condition (vi).1 holds, i.e. for l ≤ n_{1} < n_{0}, n_{2} the square

[figure: γ_{n_{0}}; γ_{l}; d_{(l)}; γ_{n_{0},n_{1},n_{2}}; γ_{max(n_{0},n_{2})}; m; π_{0}; d_{(l)}]

commutes, as the following calculations show

d_{(l)} ∘ m(λ^{0}) = d_{(l)}(n_{0}) = l + 1 = d_{(l)}(n_{0}) = d_{(l)} ∘ π_{0}(λ^{0})
d_{(l)} ∘ m(λ^{1}) = d_{(l)}(n_{2}) = l + 1 = d_{(l)}(γ_{max(n_{0},n_{2})}^{n_{2}} − n_{1} − 1) = d_{(l)} ∘ π_{0}(λ^{0})

The condition (vii) since, for l ≥ n, we have

d_{(l)} ∘ ι_{(l)}(λ^{0}) = c_{(l)}(n) = n = d_{(n)} ∘ ι_{(l)}(λ^{0})

The conditions (vii)-(x) holds, since all the morphisms involved are inner and inner the morphism into a disk of form γ_{n} from a given disk is unique, if exists.

The condition (xi) holds as well, i.e. for n_{1} < l < n_{0}, n_{2}, the diagram

[figure: γ_{n_{0},l,n_{2}}; γ_{max(n_{0},n_{2})}; m_{n_{0},l,n_{2}}; γ_{n_{0},n_{1},n_{2}}; γ_{n_{2},l,n_{0}}; ⟨m_{l,n_{1},n_{2}}(d_{(l)} × 1), m_{n_{0},n_{1},l}(1 × c_{(l)})⟩; ⟨m_{n_{0},n_{1},l}(1 × d_{(l)}), m_{l,n_{1},n_{2}}(c_{(l)} × 1)⟩; m_{n_{2},l,n_{0}}; m]

commutes. The morphism ⟨m_{l,n_{1},n_{2}}(d_{(l)} × 1), m_{n_{0},n_{1},l}(1 × c_{(l)})⟩ glue the two branches up to the level l and switch the remaining parts, whereas the morphism ⟨m_{n_{0},n_{1},l}(1 × d_{(l)}), m_{l,n_{1},n_{2}}(c_{(l)}×1)⟩ just glue together the branches up to level l without switching them. Thus both morphisms are inner and composed with m_{n_{2},l,n_{0}} and m_{n_{0},l,n_{2}}, respectively must be equal one to another.

From the proposition we get immediately

3.27. Corollary. For any disk D, D(D, C) is an ω-category in Set.

By an l-D-cut (or l-cut, D-cut) we mean an outer morphism D ⟶ γ_{l}.

Since both d and c are outer morphisms, on D-cuts we can define operations of domain and codomain simply by composing them with d and c, respectively.

Let Cut(D)_{n} be the set of n-D-cuts. We have a sequence of sets and functions, denoted by Cut(D)

[figure: d; Cut(D)_{n}; c; d; c; d; Cut(D)_{1}; Cut(D)_{0}; c; …; …]

Note that if f : D ⟶ γ_{n} is outer then there is a unique inner node z ∈ D^{n}, such that f^{n}(z) = n ∈ γ_{n}^{n}. Thus b(z) ≠ t(z) and there is a unique pair x, y ∈ p^{−1}(z) ⊆ D^{n+1}, such that x < y, f^{n+1}(x) = b(z) and f^{n+1}(y) = t(z). By Proposition 3.18 and Lemma 3.5, we have f = \overline{x; y}. Clearly, for any pair x, y ∈ D^{n+1}, such that x < y, \overline{x; y} is an n-D-cut. Moreover, both first and the second component determines the pair uniquely. We write \overline{x;} (\overline{; x}) for the outer morphisms D ⟶ γ_{n} determined by the pair whose first (second) component is x. In this notation, we have

d ∘ \overline{x; y} = \overline{; p(x)} c ∘ \overline{x; y} = \overline{p(x);}

We define an order on n-D-cuts by putting for f = \overline{x; x′} and g = \overline{y; y′} such that d∘f = d∘g and c ∘ f = c ∘ g (i.e. p(x) = p(y)) that

f ≥ g iff x ≤ y in D^{n+1}

3.28. Proposition. Cut(D) is a simple ω-graph and the inclusion

η_{D} : Cut(D) ⟶ D(D, C)

is an ω-graph morphism.

Proof. Since d ∘ d = d ∘ c and c ∘ d = c ∘ c, Cut(D) is indeed an ω-graph. Moreover, the domain and codomain operations in D(D, C) are defined by composition with d and c, as well. Thus η_{D} is an inclusion of ω-graphs. It remains to show that Cut(D) is simple.

Clearly, for n ∈ ω, Cut(D)_{n} is finite.

Since fibers of p are linearly ordered, the above condition defines a linear order on those n-D-cuts whose domains and codomains coincide. Moreover, for f, g as above, we have

f > g iff x′ = y

But, then y must be an inner node, and b(y) ≠ t(y). So \overline{b(y);} : D ⟶ γ_{n+1} is an n + 1-D-cut for which

d ∘ \overline{b(y);} = f and c ∘ \overline{b(y);} = g

We have as well that

d ∘ f = \overline{; p(x)} > \overline{p(x);} = c ∘ f

Thus Cut(D) is a simple ω-graph.

From the proof we get a Corollary.

3.29. Corollary. If n ∈ ω and f, g are n-D-cuts then f > g iff there is x ∈ D^{n+1} such that f = \overline{; x} and g = \overline{x;}.

3.30. Proposition. For any disk D, the inclusion

η_{D} : Cut(D) ⟶ D(D, C)

is the universal morphism from graph Cut(D) to the forgetful functor ωCat ⟶ ωGr.

Proof. Let D be a disk, A an ω-category, and F : Cut(D) ⟶ A. We need to show that there is a unique ω-functor G : D(D, C) ⟶ A such that the triangle

[figure: Cut(D); D(D, C); η_{D}; F; A; G]  (18)

commutes. First we extend F to all outer morphisms. For any ud-vector u⃗, we shall define a function

F⃗ : D_{outer}(D, γu⃗) ⟶ Au⃗

by the formula

F⃗ (e) = ⟨F (π_{i}^{A} ∘ e)⟩_{i∈lh(u⃗)} ∈ Au⃗

Since F preserves domains and codomains, F⃗ is well defined. Let f : D ⟶ γ_{n} be a disk morphism, i.e. n-cell in D(D, C). By Corollary 3.24, we have a factorization

[figure: γu⃗; γ_{u}; mu⃗; D; γ_{n}; f; f^{∘}; ι_{(n)}]  (19)

with f^{∘} outer. We put

G(f) = ι_{(n)}^{A} ∘ mA ; u⃗(F⃗ (f^{∘})) ∈ A_{n}

First note that (19) shows that any n-cell is D(D, C) is a result of application of composition and identity operations to a compatible tuple of D-cuts. Thus G with the required properties, if exists is, unique. Secondly, for a D-cut f : D ⟶ γ_{n} we have

G(η_{D}(f)) = G(f) = ι_{(n)}^{A} ∘ m_{n}^{A}(F (f)) = F (f)

i.e. (18) commutes. It remains to show that G defined above preserve domains, codomains, identities and compositions.

Before we show that G preserves compositions we shall prove Claim. For any l ∈ ω, ud-vector u⃗ and outer morphism e : D ⟶ γu⃗ we have

d_{(l)}^{A} ∘ F⃗ (e) = F⃗ (d_{u⃗;l} ∘ e)  (20)
c_{(l)}^{A} ∘ F⃗ (e) = F⃗ (c_{u⃗;l} ∘ e).  (21)

We prove the Claim by induction on l-size of u⃗.

If u⃗ = u_{0} < l then e is an u_{0}-cut, F⃗ (e) = F (e), d^{(l)} = 1_{γ_{u_{0}}} and d_{(l)}^{A} = id_{A_{u_{0}}}. Hence (20) holds in this case.

If u⃗ is l-primitive, then we have

d_{u⃗;l}^{A}(F⃗ (e)) = d_{(l)}^{A} ∘ π_{0}^{A}(F⃗ (e)) = d_{(l)}^{A}(F (π_{0} ∘ e)) =
= F (d_{(l)} ∘ π_{0} ∘ e) = F⃗ (d_{u⃗;l} ∘ e) = F⃗ (d_{u⃗;l} ∘ e)

If u⃗ = u⃗′, z, u⃗′′, lh(u⃗) = k + 1, lh(u⃗′) = k′ + 1, and z = min(u⃗) < l, then using the

Theory and Applications of Categories, Vol. 8, No. 7 171

inductive assumption on ud-vectors u⃗′ and u⃗′′ of smaller l-size we have

dA,(l) ; u⃗(F⃗ (e)) = d_{u⃗′;l}^{A} ∘ π_{0..k′}^{A} ∘ (F⃗ (e)) × d_{u⃗′′;l}^{A} ∘ π_{k′+1..k}^{A} ∘ (F⃗ (e))

= d_{u⃗′;l}^{A}(F⃗ (π_{0..k′} ∘ e)) × d_{u⃗′′;l}^{A}(F⃗ (π_{k′+1..k} ∘ e)) = F⃗ (d_{u⃗′;l} ∘ π_{0..k′} ∘ e) × F⃗ (d_{u⃗′′;l} ∘ π_{k′+1..k} ∘ e)

=^{∗} F⃗ (⟨d_{u⃗′;l} ∘ π_{0..k′} ∘ e, d_{u⃗′′;l} ∘ π_{k′+1..k} ∘ e⟩)

= F⃗ ((d_{u⃗′;l} × d(l) ; u⃗′′) ∘ e)

= F⃗ (d_{u⃗;l} ∘ e)

where =^{∗} follows from the fact that z = min(u⃗) and hence

c_{(z)} ∘ d_{u⃗′;l} ∘ π_{0..k′} ∘ e = d_{(z)} ∘ d_{u⃗′′;l} ∘ π_{k′+1..k} ∘ e

The proof of (21) is similar. This ends the proof of the Claim.

We can show now, that G preserves domains and codomains. Let l, n ∈ ω, and f : D ⟶ γ_{n} be a disk map. By Lemma 6.4 we have the following commutative diagram

[figure: γ_{tr_{(l)}(u⃗)}; γ_{min(l,u)}; m_{tr_{(l)}(u⃗)}; γu⃗; γ_{u}; mu⃗; d_{u⃗;l}; d_{(l)}; γ_{l}; ι_{(l)}; γ_{n}; ι^{(n)}; d_{(l)}; D; f^{∘}; f]

with f^{∘} outer. Then using the Claim we obtain

G(d_{(l)} ∘ f) = ι_{(l)}^{A} ∘ m_{tr_{(l)}(u⃗)}^{A}(F⃗ (d_{u⃗;l}∘ f^{∘})

= ι_{(l)}^{A} ∘ m_{tr_{(l)}(u⃗)}^{A} ∘ d_{u⃗;l}^{A}(F⃗ (f^{∘})

= ι_{(l)}^{A} ∘ d_{(l)}^{A} ∘ mA ; u⃗(F⃗ (f^{∘})

= d_{(l)}^{A} ∘ ι_{(n)}^{A} ∘ mA ; u⃗(F⃗ (f^{∘})

= d_{(l)}^{A}(G(f))

Thus G preserves domains. The preservation of codomains can be proved similarly. To see that G preserves identities, let l, n ∈ ω, l > n, and f : D ⟶ γ_{n} be a disk map.

Then, we can form a commuting diagram

[figure: γu⃗; γ_{u}; mu⃗; D; γ_{n}; f; f^{∘}; ι_{(n)}; γ_{l}; ι_{n}^{(l)}; ι_{(l)}]

with f^{∘} outer. Thus, we have

\begin{aligned}G(ι_{(l)} ∘ f) & = G(ι_{(l)} ∘ mu⃗∘ f^{∘}) = ι_{(l)}^{A} ∘ mA ; u⃗(G(f^{∘})) \\ & = ι_{(l)}^{A} ∘ ι_{(n)}^{A} ∘ mA ; u⃗(G(f^{∘})) = ι_{(l)}^{A}(G(f)) \end{aligned}

i.e. G preserves identities.

It remains to show that G preserves compositions. Let u′, v′, l ∈ ω, l < u′, v′ and f : D ⟶ γ_{u′}, f : D ⟶ γ_{v′} be disk morphisms, such that c_{(l)} ∘ f = d_{(l)} ∘ g. We have the following commutative diagrams displaying the canonical factorizations of f and c_{(l)} ∘ f,

[figure: γu⃗; γ_{u}; mu⃗; D; γ_{u′}; f; f^{∘}; ι_{(u′)}; γ_{min(u,l)}; c_{(l)}; γ_{l}; c_{(l)}; ι_{(l)}; γ_{tr_{(l)}(u⃗)}; c_{u⃗;l}; m_{(tr_{(l)}(u⃗)}]

and the canonical factorizations of g and d_{(l)} ∘ g

[figure: γv⃗; γ_{v}; mv⃗; D; γ_{v′}; g; g^{∘}; ι_{(v′)}; γ_{min(v,l)}; d_{(l)}; γ_{l}; d_{(l)}; ι_{(l)}; γ_{tr_{(l)}(v⃗)}; d_{v⃗;l}; m_{(tr_{(l)}(v⃗)}]

Since c_{(l)} ∘ f = d_{(l)} ∘ g and the canonical factorizations are unique, we have that tr_{(l)}(u⃗) =

tr_{(l)}(v⃗), min(v, l) = min(u, l), and c_{u⃗;l}∘ f^{∘}= d_{v⃗;l}∘ g^{∘}. Thus we can form a commuting diagram

[figure: D; f^{∘}; g^{∘}; e; γu⃗; γw⃗; γ_{[u,l,v]}; γv⃗; π_{0;u⃗}; π_{1;v⃗}; c_{u⃗;l}; d_{v⃗;l}; mu⃗ × mv⃗; mu⃗; mv⃗; mw⃗; γ_{u}; γ_{w}; γ_{[u,w,v]}; γ_{v}; π_{0}; π_{1}; c_{(w)}; d_{(w)}; ι_{(u′)} × ι_{(v′)}; ι_{(u′)}; ι_{(v′)}; ι_{(l)}; γ_{u}; γ_{l}; γ_{u,l,v}; γ_{v}; π_{0}; π_{1}; c^{(l)}; d^{(l)}]

in which the three horizontal squares are pullbacks, and e is the induced map into a pullback. Moreover, by Lemma 6.7, for z = min(u, v) and z′ = min(u′, v′) we have a commuting diagram

[figure: γ_{u′,l,v′}; γ_{z′}; m_{u′,l,v′}; γ_{[u,l,v]}; γ_{z}; m_{[u,l,v]}; ι_{(u′)} × ι_{(v′)}; ι_{(z′)}; m_{[u⃗,l,v⃗]}; γ_{[u⃗,l,v⃗]}; mu⃗ × mv⃗]

Thus we have

G(f) = ι_{(u′)}^{A} ∘ mu⃗(F⃗ (f^{∘}))

and

G(g) = ι_{(v′)}^{A} ∘ mv⃗(F⃗ (g^{∘}))

Now using Lemma 6.7 we have

G(m_{u′,l,v′})(f, g)) = ι_{(z′)}^{A} ∘ m_{[u⃗,l,v⃗]}^{A}(F⃗ (e))

= ι_{(z′)}^{A} ∘ m_{[u,l,v]}^{A} ∘ (mA ; u⃗ × mA ; v⃗)(F⃗ (e))

= m_{u′,l,v′}^{A} ∘ (ι_{(u′)}^{A} × ι_{(v′)}^{A}) ∘ (mA ; u⃗ × mA ; v⃗)(⟨F⃗ (π_{0;u⃗} ∘ e), F⃗ (π_{1;v⃗} ∘ e)⟩)

= m_{u′,l,v′}^{A} ∘ ⟨ι_{(u′)}^{A} ∘ mA ; u⃗(F⃗ (f^{∘}), ι_{(v′)}^{A} ∘ mA ; v⃗(F⃗ (g^{∘})⟩

= m_{u′,l,v′}^{A}(G(f), G(g))

i.e. the compositions are preserved as well.

We get

3.31. Corollary. The essential image of the hom-functor D(−, C) into ωCat is contained in the category of simple ω-categories S, and hence we have a contravariant homfunctor

D(−, C) : D^{op} ⟶ S

Proof. By Proposition 4.8 an ω-category is simple iff it is a free ω-category on a simple ωgraph. By Proposition 3.28 Cut(D) is a simple ω-graph and by Proposition 3.30 D(D, C) is a free ω-category on Cut(D), for any disk D. Thus D(D, C) is a simple ω-category.

4. Simple ω-categories

4.1. Simple ω-graphs and free ω-categories. This section begins the investigation of simple ω-categories. We shall introduce some notation and prove some basic facts concerning simple ω-graphs. In Appendix 6.8, we give an internal construction of a free ω-category over an ω-graph, in an ambient category satisfying some mild exactness properties. The construction uses essentially ud-vectors. When the ambient category is Set the same construction can be described more conveniently, using simple ω-graphs. We shall present the construction at the end of the section.

We shall define some specific ω-graphs αu⃗, for any ud-vector u⃗. For u⃗ = n, we put

α_{l}^{n} = \{ \begin{matrix}∅ & if l > n \\ \{2n\} & if l = n \\ \{2l + 1, 2l\} & if 0 ≤ l < n \end{matrix}

d, c : α_{l}^{n} ⟶ α_{l−1}^{n}

d(x) = 2l − 1 c(x) = 2l − 2

for x ∈ α_{l}^{n}, and 1 ≤ l ≤ n.

For example α^{4} can be pictured as follows:

[figure: 1; 0; 3; 2; 5; 4; 7; 6; 8]

For any two cells e, e′ in a simple ω-graph G we define a number

ν_{e,e′}^{G} = max{l : d_{(l)}(e) ⊥ d_{(l)}(e′)}

Since G_{0} is linearly ordered ν_{e,e′}^{G} is well defined. We have

4.2. Lemma. Let f : G ⟶ H be a morphisms of simple ω-graphs. Then f is one-to-one and the image of f is a sub-ω-graph of H. In particular, if m ∈ ω and e, e′, e′′ ∈ G′_{m} such that e > e′ > e′′ and e, e′′ are in the image of f, so is e′.

Proof. First note that if a function between finite linear orders preserves the successor relation then it reflects the order and is one-to-one.

Let f : G ⟶ H be an ω-graph morphism between simple ω-graphs. f maps G_{0} to H_{0} and, for n ∈ ω, x, y ∈ G_{n} f maps G_{n+1}(x, y) to H_{n+1}(f(x), f(y)). In particular, if G_{n+1}(x, y) is not empty, so is H_{n+1}(f(x), f(y)). Since G and H are simple f preserves >. Thus, by the remark above, it also reflects it and is one-to-one when restricted to these sets. But if x, y ∈ G_{n} and x ⊥̸ y then l = ν_{x,y} < n. Hence d_{(l)}(x) ≠ d_{(l)}(y) and d_{(l)}(x) ⊥ d_{(l)}(y). Thus by the above d_{(l)}(f(x)) ≠ d_{(l)}(f(y)), and then f(x) ≠ f(y). Therefore f is one-to-one.

For l < n we define, the bottom and the top ω-graph morphisms

b^{n} = b , t^{n} = t : α^{l} ⟶ α^{n}

by the conditions b_{n}(2l) = 2l and t_{n}(2l) = 2l + 1.

Let u⃗ be a ud-vector. The set of nodes in the simple ω-graph αu⃗ is the set of pairs ⟨i, x⟩ such that i ∈ lh(u⃗) and 0 ≤ x ≤ 2u_{2i} divided by the equivalence relation ∼ generated by the relation ∼_{0} such that

⟨i, x⟩ ∼_{0} ⟨i′, x′⟩ iff \{ \begin{matrix}i = i′ and x = x′ \\ or \\ i + 1 = i′ and x = x′ and x < 2u_{2i+1} \\ or \\ i + 1 = i′ and x = 2u_{2i+1} and x′ = 2u_{2i+1} + 1 \end{matrix}

We write [i, x] for the equivalence class of the pair ⟨i, x⟩. We put, for l ∈ ω,

αu⃗ ; l = {[i, x] ∈ αu⃗ : ⌊x/2⌋ = l}

and domain and codomain functions

d^{αu⃗}, c^{αu⃗} : αu⃗ ; l+1 ⟶ αu⃗ ; l

are given by

d^{αu⃗}([i, x]) = [i, d(x)] c^{αu⃗}([i, x]) = [i, c(x)]

for [i, x] ∈ αu⃗ ; l+1.

The embedding morphisms

κ^{i} : α^{u_{2i}} ⟶ αu⃗

are defined by the condition κ^{i}(2u_{2i}) = [i, 2u_{2i}], for i ∈ lh(u⃗).

We have 4.3. Proposition. The diagram

[figure: α^{u_{0}}; α^{u_{2}}; α^{u_{1}}; b; t; α^{u_{2k−2}}; α^{u_{2k}}; α^{u_{2k−1}}; b; t; …; αu⃗; κ^{2}; κ^{2k−2}; κ^{2k}; κ^{0}]  (22)

in sωGr is a colimiting cone in ωGr. Proof. αu⃗ described above is clearly a colimit of (22) in ωGr. The verification that αu⃗ is a simple ω-graph is left for the reader.

For l ∈ ω, and ud-vector u⃗ the multi-bottom and the multi-top ω-graph morphisms

b^{u⃗;l}, t^{u⃗;l} : α^{tr_{(l)}(u⃗)} ⟶ αu⃗

are defined as follows

b^{u⃗;l} = \{ \begin{matrix}1_{α_{u_{0}}} & if u⃗ = u_{0} ≤ l; \\ κ^{lh(u⃗)−1} ∘ b^{(u_{0})} & if u⃗ is l-primitive; \\ b^{u⃗′;l} + b^{u⃗′′;l} & if u⃗ = u⃗′, w, u⃗′′ and w = min(u⃗) < l. \end{matrix}

t^{u⃗;l} = \{ \begin{matrix}1_{α_{u_{0}}} & if u⃗ = u_{0} ≤ l; \\ κ^{0} ∘ t^{(u_{0})} & if u⃗ is l-primitive; \\ t^{u⃗′;l} + t^{u⃗′′;l} & if u⃗ = u⃗′, w, u⃗′′ and w = min(u⃗) < l. \end{matrix}

For l ∈ ω and ud-vectors u⃗ and v⃗ such that tr_{(l)}(u⃗) = tr_{(l)}(v⃗) = w⃗ the embedding morphisms in sωGr

κ^{0;u⃗} : αu⃗ ⟶ α^{[u⃗,l,v⃗]} κ^{1;v⃗} : αv⃗ ⟶ α^{[u⃗,l,v⃗]}

are as follows.

κ^{0;u⃗} = \{ \begin{matrix}t^{v⃗;u_{0}} & if u⃗ = u_{0} ≤ n_{1}; \\ κ^{0..k} & if u⃗ is n_{1}-primitive, and lh(u⃗) = k + 1; \\ κ^{0;u⃗′} + κ^{0;u⃗′′} & if u⃗ = u⃗′, z, u⃗′′, v⃗ = v⃗′, z, v⃗′′, \\ & tr_{(n_{1})}(u⃗′) = tr_{(n_{1})}(v⃗′), tr_{(n_{1})}(u⃗′′) = tr_{(n_{1})}(v⃗′′), \\ & and z = min(u⃗) < n_{1}. \end{matrix}

and

κ^{1;v⃗} = \{ \begin{matrix}b^{u⃗;u_{0}} & if u⃗ = u_{0} ≤ n_{1}; \\ κ^{k+1..k′} & if v⃗ is n_{1}-primitive, lh(u⃗) = k + 1, \\ & and lh(u⃗, n_{1}, v⃗) = k′ + 1; \\ κ^{1;v⃗′} + κ^{1;v⃗′′} & if u⃗ = u⃗′, z, u⃗′′, v⃗ = v⃗′, z, v⃗′′, \\ & tr_{(n_{1})}(u⃗′) = tr_{(n_{1})}(v⃗′), tr_{(n_{1})}(u⃗′′) = tr_{(n_{1})}(v⃗′′), \\ & and z = min(u⃗) < n_{1}. \end{matrix}

We have 4.4. Proposition. For l ∈ ω and ud-vectors u⃗ and v⃗ such that tr_{(l)}(u⃗) = tr_{(l)}(v⃗) = w⃗ the diagram in sωGr

[figure: αu⃗; αw⃗; b^{u⃗;l}; α^{[u⃗,l,v⃗]}; αv⃗; κ^{1;v⃗}; κ^{0;u⃗}; t^{v⃗;l}]  (23)

is a pushout in ωGr.

Proof. Let u⃗, v⃗, and l be as in the proposition. We prove the Proposition by induction on l-size of tr_{(l)}(u⃗).

If u⃗ = u_{0} ≤ l then (23) is

[figure: α^{u_{0}}; α^{u_{0}}; 1; αv⃗; αv⃗; 1; t^{u⃗;l}; t^{u⃗;l}]

so it is a pushout.

The case v⃗ = v_{0} ≤ l is similar.

If u⃗, v⃗ are l-primitive and then [u⃗, l, v⃗] = u⃗, l, v⃗ and (23) is

[figure: αu⃗; α^{l}; b^{u⃗;l}; α^{u⃗,l,v⃗}; αv⃗; κ^{k+1..k′}; κ^{0..k}; t^{v⃗;l}]

where k = lh(u⃗) and k′ = ln(u⃗, l, v⃗) + 1. So by Proposition 4.4 it is again pushout.

Finally, if z = min(w⃗) u⃗ = u⃗′, z, u⃗′′, v⃗ = v⃗′, z, v⃗′′ tr_{(l)}(u⃗′) = tr_{(l)}(v⃗′) = w⃗′, tr_{(l)}(u⃗′′) = tr_{(l)}(v⃗′′) = w⃗′′ then [u⃗, l, v⃗] = [u⃗′, l, v⃗′], z, [u⃗′′, l, v⃗′′] and (23) is the following square

[figure: α^{u⃗′,w,u⃗′′}; α^{u⃗′,w,u⃗′′}; b^{u⃗′;l} + b^{u⃗′′;l}; α^{[u⃗′,l,v⃗′],w,[u⃗′′,l,v⃗′′]}; α^{v⃗′,w,v⃗′′}; κ^{1;v⃗′} + κ^{1;v⃗′′}; κ^{0;u⃗′} + κ^{0;u⃗′′}; t^{v⃗′;l} + t^{v⃗′′;l}]

which arises as a pushout over α^{z} of two squares, which are pushouts by inductive assumption for vectors w⃗′ and w⃗′′ with smaller l-size then w⃗′, z, w⃗′′. Thus it is again a pushout.

4.5. Proposition.

1. For any simple ω-graph G, there is a ud-vector u⃗ such that G is isomorphic to αu⃗.

2. Let ud-vectors u⃗, v⃗ be ud-vectors. If f : αu⃗ ⟶ αv⃗ is an isomorphism then u⃗ = v⃗ and f = 1_{αu⃗}.

Proof. 2. follows easily from 1. For 1. we give only a sketch of the proof.

Let G be a simple ω-graph. By T_{G} we denote the set of cells in G which are neither domains nor codomains of other cells in G. We define a linear order on T_{G}, so that for a, b ∈ T_{G}

a ≤ b iff d_{(ν_{a,b})}(a) ≥ d_{(ν_{a,b})}(b)

Let T_{G} = {a_{0}, …, a_{k}} such that a_{i} ≤ a_{i+1} for 0 ≤ i ≤ k. Let u_{2i} be such a number that a_{i} ∈ G_{u_{2i}}, for 0 ≤ i ≤ k, and u_{2i+1} = ν_{a_{i},a_{i+1}} for 0 ≤ i ≤ k − 1. Then, we can prove, by induction on the cardinality of T_{G} that G is isomorphic to αu⃗, where u⃗ = u_{0}, …, u_{2k}.

Let G be an arbitrary ω-graph. In Appendix 6.8 the construction of the free internal ω-category [G] over an internal ω-graph G is given, when the ambient category has finite limits and disjoint and universal coproducts. In Set, this construction can be described more conveniently using simple ω-graphs.

To distinguish (temporarily) between these two constructions by [[G]] we will denote the ω-category defined in terms of simple ω-graphs.

The set of n-cells is

[[G]]_{n} = {f : αu⃗ → G : ht(u⃗) ≤ n}

The set of n_{1}-compatible pairs of n_{0}- and n_{2}-cells is

[[G]]_{n_{0},n_{1},n_{2}} = {⟨f, g⟩ : f : αu⃗ → G, g : αv⃗ → G,
ht(u⃗) ≤ n_{0}, ht(v⃗) ≤ n_{2}, and f ∘ b^{u⃗;n_{1}} = g ∘ t^{v⃗;n_{1}}}

For l ≤ n, the domain and the codomain operations

d_{l}^{[[G]]}, c_{l}^{[[G]]} : [[G]]_{n} ⟶ [[G]]_{l}

for f : αu⃗ → G ∈ [[G]]_{n}, are defined by composition

d_{l}^{[[G]]}(f) = f ∘ t^{u⃗;l} : α^{tr_{(l)}(u⃗)} → G,
c_{l}^{[[G]]}(f) = f ∘ b^{u⃗;l} : α^{tr_{(l)}(u⃗)} → G.

For n ≤ l, the identity operations

ι_{l}^{[[G]]} : [[G]]_{n} ⟶ [[G]]_{l}

are inclusions. For n_{1} < n_{0}, n_{2}, the compositions

m_{n_{0},n_{1},n_{2}}^{[[G]]} : [[G]]_{n_{0},n_{1},n_{2}} ⟶ [[G]]_{max(n_{0},n_{2})}

are defined, via pushouts in sωGr. By Proposition 4.4, for any pair ⟨f, g⟩ ∈ [[G]]_{n_{0},n_{1},n_{2}} there is a unique morphism [f, g] : α^{[u⃗,n_{1},v⃗]} ⟶ G making the diagram

[figure: αu⃗; αw⃗; t^{u⃗;n_{1}}; α^{[u⃗,n_{1},v⃗]}; αv⃗; κ^{1;v⃗}; κ^{0;u⃗}; b^{v⃗;n_{1}}; g; f; G; [f, g]]

commutes, where w⃗ = tr_{(n_{1})}(u⃗)(= tr_{(n_{1})}(v⃗)). We put

m_{n_{0},n_{1},n_{2}}^{[[G]]}(⟨f, g⟩) = [f, g]

We define

η_{G} : G ⟶ [[G]]

such that, for x ∈ G_{n}, η_{G}(x) : α^{n} ⟶ G so that η_{G}(x)(2n) = x.

Thus by Theorem 6.9 we obtain

4.6. Theorem. For any ω-graph G, [[G]] as described above is an ω-category. Moreover, the association G ⟼ [[G]] is functorial in G and the functor

[[−]] : ωGr ⟶ ωCat

is a left adjoint to the forgetful functor ωCat ⟶ ωGr, with η as the unit of adjunction, i.e. [[G]] is the free ω-category on the ω-graph G.

Remark. This is essentially a special case of Proposition 4.2 in [B].

Proof. We shall show that [[G]] defined above is isomorphic to [G] defined in Appendix 6.8.

To this end, we define a functor

ξ^{G} : sωGr^{op} ⟶ Set

such that, for a ud-vector u⃗, we put

ξ^{G}(αu⃗) = ωGr(αu⃗, G).

On morphisms ξ^{G} acts by composition. Since colimits in ωGr are computed pointwise, ξ^{G} sends pointwise colimits in sωGr to limits in Set. In particular ξ^{G} sends the diagrams (22) and (23) in sG to the limiting diagrams in Set.

For n ∈ ω, we have a bijection

φ_{n} : ξ^{G}(α^{n}) ⟶ G_{n}

such that φ_{n}(f) = f(2n), for f ∈ ξ^{G}(α^{n}). Moreover, it is easy to see that the diagram

[figure: ξ^{G}(α^{l}); G_{l}; φ_{l}; ξ^{G}(α^{n}); G_{n}; φ_{n}; ξ^{G}(t); ξ^{G}(b); d_{(l)}^{G}; c_{(l)}^{G}]

commutes serially, i.e. ξ^{G} ’sends’ α^{n} to G_{n}, t to d^{G}, and b to c^{G}. It follows, that for any udvector u⃗, the colimit αu⃗ with the coprojections κ^{j} : α^{u_{2i}} ⟶ αu⃗ is sent by ξ^{G} to Gu⃗ with the projections π_{j} : Gu⃗ ⟶ G^{u_{2i}}. We denote by φu⃗ : ξ^{G}(αu⃗) ⟶ Gu⃗ the induced morphisms commuting with the projections. Thus we have φu⃗(f : αu⃗ → G) = ⟨f ∘ κ^{i}(2u_{2i})⟩_{i∈lh(u⃗)}. Then, for any pair of vectors u⃗, v⃗ ∈ UD_{n_{0},n_{1},n_{2}}, ξ^{G} sends the pushout α^{[u⃗,n_{1},v⃗]} with the coprojections κ^{0;u⃗}, κ^{1;v⃗} to the pullback G_{[u⃗,n_{1},v⃗]} with the projections π_{0;u⃗}, π_{1;v⃗}.

For l ∈ ω, and a ud-vector u⃗ the definition of d_{u⃗;l}^{G} is dual to the definition of t^{u⃗;l}, one is using d^{G}’s, projections, and pullbacks and the other one is using t’s, coprojections, and pushouts. Therefore ξ^{G} is sending t^{u⃗;l} to d_{u⃗;l}^{G}, and for the similar reason ξ^{G} is sending b^{u⃗;l} to c_{u⃗;l}^{G}, i.e. the diagram

[figure: ξ^{G}(α^{tr_{(l)}(u⃗)}); G_{tr_{(l)}(u⃗)}; Φ_{tr_{(l)}(u⃗)}; ξ^{G}(αu⃗); Gu⃗; Φu⃗; ξ^{G}(t^{tr_{(l)}(u⃗)}); ξ^{G}(b^{tr_{(l)}(u⃗)}); d_{u⃗;l}^{G}; c_{u⃗;l}^{G}]  (24)

commutes serially.

Since, for n ∈ ω,

[[G]]_{n} = ∐_{u⃗∈UD_{n}} ξ^{G}(αu⃗)  (25)

we have a bijection

Φ_{n} = ∐_{u⃗∈UD_{n}} φu⃗ : [[G]]_{n} ⟶ [G]_{n}.

It remain to show that Φ commutes with the ω-category operations.

The preservations of the domains and the codomains follows from (25) and the diagram 24. Preservation of identities is trivial. Finally, to show that the compositions are preserved consider the following diagram where n = max(n_{0}, n_{2}), ⟨u⃗, v⃗⟩ ∈ UD_{n_{,}n_{1},n_{2}}, ζ_{u⃗,v⃗} is the coprojection morphism, i.e. the morphism of composing with coprojections κ^{0;u⃗} and κ^{1;v⃗}, and the unnamed morphism is an inclusion. The commutation of the left triangle follows from the universal property of the pushout, the right triangle commutes by definition of m_{n_{0},n_{1},n_{2}}^{G}, and the outer square commutes by definition of Φ_{n}. The upper square commutes, since for ⟨f : αu⃗ → G, g : αv⃗ → G⟩ ∈ [[G]]_{n_{0},n_{1},n_{2}}, we have

[figure: [[G]]_{n}; [G]_{n}; Φ_{n}; [[G]]_{n_{0},n_{1},n_{2}}; [G]_{n_{0},n_{1},n_{2}}; Φ_{n_{0}} × Φ_{n_{2}}; m; m; ξ^{G}(α^{[u⃗,n_{1},v⃗]}); ζ_{u⃗;v⃗}; κ_{u⃗,v⃗}; κ_{[u⃗,n_{1},v⃗]}; G_{[u⃗,n_{1},v⃗]}; φ_{[u⃗,n_{1},v⃗]}]

Φ_{n_{0}} × Φ_{n_{2}}(f, g) = φ_{[u⃗,n_{1},v⃗]}([f, g])

Hence the inner square commutes as well, i.e. Φ preserves compositions and it is an isomorphism of ω-categories [[G]] and [G], as required.

In the following, we shall write [G], rather then [[G]].

4.7. Composable ω-graphs. In this section we make the connection between simple ωgraphs and simple ω-categories. The latter are defined as free ω-categories with a unique maximal cell, i.e. free on composable ω-graphs. We have

4.8. Proposition. An ω-graph is composable iff it is simple.

Proof. Let G be an ω-graph. A cell a : αu⃗ ⟶ G in [G]_{n} is a non-identity cell iff ht(u⃗) = n. We have

Claim.

a is a maximal in [G] iff a is surjective.

Consider the surjective-injective factorization of a in ωGr

[figure: αu⃗; G; a; G′; e; i]

Then a is in the image of [i] : [G′] ⟶ [G]. If a is maximal, then, by definition, i an iso and a is a surjection. On the other hand, if a is a surjection then so is i, and since it is an injection it mast be an iso. Hence a is maximal. This proves the Claim.

Thus if G is a simple ω-graph then there is a unique surjection a : αu⃗ ⟶ G in ωGr, for some ud-vector u⃗, which is in fact an iso, as any morphism of simple ω-graphs is one-to-one. Hence [G] is a simple ω-category, with the unique maximal cell a ∈ G_{ht(u⃗)}.

Now assume that [G] is a simple ω-category. We need to show that G is a simple ωgraph. Let m ∈ [G]_{n} be the unique maximal arrow in [G]. By the Claim m is surjective. To finish the proof it is enough to show that m is injective, as well. Suppose not, let k be the minimal such that m_{k} : αu⃗ ; k ⟶ G_{k} is not one-to-one, and let x, y ∈ αu⃗ ; k, such that m_{k}(x) = m_{k}(y). By minimality of k, we have d(x) = d(y) = x′ and c(x) = c(y) = y′. We can divide the set of those cells o in αu⃗ for which d_{(k−1)}(o) = x′ and c_{(k−1)}(o) = x′ into three classes which we can picture schematically as follows

[figure: BLUE; x; RED; ⋯ x′; y′ ⋯; y; GREEN]

where the BLUE class contains those cells o in αu⃗ for which c_{(}k)(o) > x, the RED class contains those cells o for which x ≥ d_{(}k)(o) and c_{(}k)(o) ≥ y, and the GREEN class contains those cells o for which y > d_{(}k)(o). Clearly x and y are in the RED class. We shall construct another simple ω-graph H by dubbling the RED class in αu⃗. The relevant part of the ω-graph H we can draw schematically as follows

[figure: BLUE; x; RED; ⋯ x′; y′ ⋯; y = x̄; PINK; ȳ; GREEN]

where the PINK class is the second copy of the RED class. A cell in the PINK class corresponding to a cell in the RED class we distinguish by putting a bar over it, i.e. z̄ is the PINK version of the RED cell z. The cell y is identified with x̄. The domains and codomains in H remains as they were in αu⃗ except that for those cells o in the GREEN class for which we had d_{(k)}(o) = y now we put d_{(k)}(o) = ȳ and for the cell ō in the PINK class we put for domains

d_{(l)}(ō) = \{ \begin{matrix}ō′ & if l > k and d_{(l)}(o) = o′ \\ o′ & if l ≤ k and d_{(l)}(o) = o′ \end{matrix}

and similarly for codomains

c_{(l)}(ō) = \{ \begin{matrix}ō′ & if l > k and c_{(l)}(o) = o′ \\ o′ & if l ≤ k and c_{(l)}(o) = o′ \end{matrix}

We have a ω-graph morphism

m′ : H ⟶ G

by m′(o) = m(o) for o which are not in PINK and m′(ō) = m(o) for o in RED class. Since m(x) = m(y) the morphism m′ is well defined. By Proposition 4.5, we can assume that H = αv⃗ for some ud-vector v⃗. Since H in not equal to αu⃗, m ≠ m′. But m′ is surjective (since m was), so by the Claim, it is a maximal cell in [G] as well, i.e. G is not composable contrary to the supposition.

4.9. An internal disk D in S. We introduce below some notation for simple ω-categories and prove some basic facts about them.

For any ud-vector s⃗, δs⃗ is, by definition, the free ω-category on αs⃗. However, for some simple ω-categories we need a more precise description. Below, we give a specific presentation for the simple categories δ^{s} and δ^{s+1,s,s+1}, for s ∈ ω.

It is maybe much simpler to look at the picture of δ^{4} given below to understand what δ^{s} is, however before giving the picture we put the general definition of δ^{s}.

The n-cells of δ^{s} are

δ_{n}^{s} = \{ \begin{matrix}2n + 2 & if n < s \\ 2s + 1 & if n ≥ s \end{matrix}

and the domain, codomain, and identity functions

d_{n}, c_{n} : δ_{n+1}^{s} ⟶ δ_{n}^{s} ι_{n+1} : δ_{n}^{s} ⟶ δ_{n+1}^{s}

are, for n < s − 1

d_{n}(x) = \{ \begin{matrix}x & if x ≤ n \\ x − 2 & if x > n \end{matrix}

c_{n}(x) = \{ \begin{matrix}x & if x < n \\ x − 2 & if x ≥ n \end{matrix}

ι_{n+1}(x) = \{ \begin{matrix}x & if x ≤ n \\ x + 2 & if x > n \end{matrix}

for n = s − 1

d_{s−1}(x) = \{ \begin{matrix}x & if x ≤ s \\ x − 1 & if x > s \end{matrix}

c_{s−1}(x) = \{ \begin{matrix}x & if x < s \\ x − 1 & if x ≥ s \end{matrix}

ι_{s}(x) = \{ \begin{matrix}x & if x < s \\ x + 1 & if x ≥ s \end{matrix}

and for n ≥ s

d_{n} = c_{n} = ι_{n} = id

We don’t give an explicit formula for compositions m_{n_{0},n_{1},n_{2}} since any compositions are compositions with identities.

The first six levels of δ^{4} can be pictured as follows:

[figure: 1; 0; 3; 2; 1; 0; 5; 4; 3; 2; 1; 0; 7; 6; 5; 4; 3; 2; 1; 0; 8; 7; 6; 5; 4; 3; 2; 1; 0; 8; 7; 6; 5; 4; 3; 2; 1; 0]

For s ∈ ω^{+}, the n-cells of δ^{s,s−1,s} are

δ_{n}^{s,s−1,s} = \{ \begin{matrix}2n + 2 & if n < s − 1 \\ 2s + 1 & if n = s − 1 \\ 2s + 4 & if n ≥ s \end{matrix}

and the definitions of domain, codomain, identity and compositions in δ^{s,s−1,s} are left for the reader. For example δ^{4,3,4} can be pictured as follows (no-identity arrows are marked bold):

[figure: 1; 0; 3; 2; 1; 0; 5; 4; 3; 2; 1; 0; 8; 7; 6; 5; 4; 3; 2; 1; 0; 11; 10; 9; 8; 7; 6; 5; 4; 3; 2; 1; 0; 11; 10; 9; 8; 7; 6; 5; 4; 3; 2; 1; 0]

For s < s′, and ud-vector s⃗ and i ∈ lh(s⃗) the ω-functors

b, t : δ^{s} ⟶ δ^{s′} b^{s⃗;l}, t^{s⃗;l} : δ^{tr_{(l)}(s⃗)} ⟶ δs⃗
κ^{i} : δ^{s_{2i}} ⟶ δs⃗

and for l ∈ ω and ud-vectors u⃗ and v⃗ such that tr_{(l)}(u⃗) = tr_{(l)}(v⃗) = w⃗ the ω-functors and

κ^{0;u⃗} : δu⃗ ⟶ δ^{[u⃗,l,v⃗]} κ^{1;v⃗} : δv⃗ ⟶ δ^{[u⃗,l,v⃗]}

are defined to be the images of the ω-graph morphisms with the same name in sωGr under the free ω-category functor [−]. Thus, as an immediate corollary from Propositions 4.3 and 4.4 we obtain 4.10. Corollary.

1. For any ud-vector s⃗, the diagram

[figure: δ^{s_{0}}; δ^{s_{2}}; δ^{s_{1}}; t; b; δ^{s_{2k−2}}; δ^{s_{2k}}; δ^{s_{2k−1}}; t; b; …; δs⃗; κ^{2}; κ^{2k−2}; κ^{2k}; κ^{0}]  (26)

is a colimiting cone in S.

2. For l ∈ ω and ud-vectors u⃗ and v⃗ such that tr_{(l)}(u⃗) = tr_{(l)}(v⃗) = w⃗ the diagram

[figure: δu⃗; δw⃗; t^{u⃗;l}; δ^{[u⃗,l,v⃗]}; δv⃗; κ^{1;v⃗}; κ^{0;u⃗}; b^{v⃗;l}]  (27)

is a pushout in S.

Let G be a simple ω-graph and s ∈ ω. We shall introduce some notation for the morphisms

[G] ⟶ δ^{s} and [G] ⟶ δ^{s+1,s,s+1}

There is a unique ω-functor [G] ⟶ δ^{0}, which we denote by ↓. For s ∈ ω, e, e′ ∈ G_{s} such that, e > e′ we define an ω-functor

e↓e′ : [G] ⟶ δ^{s+1}

so that its composition with the ω-graph morphisms η_{G} : G ⟶ [G], is given for n ≥ 0 and x ∈ G_{n} by

(e↓e′ ∘ η_{G})_{n}(x) = \{ \begin{matrix}s + 1 & if n > s and d_{(s)}(e) = x \\ l & if n ≥ l and c_{(l)}(e′) ≥ d_{(l)}(x) \\ δ_{n}^{s+1} − l − 1 & if n ≥ l and c_{(l)}(x) ≥ d_{(l)}(e) \end{matrix}

The ω-functor e↓e′ sends e and e′ to the only two non-identity s-cells and the remaining cells in [G] to the only cell in δ^{s+1} with the suitable domain and codomain. It is the only ω-functor from [G] to δ^{s+1} distinguishing e and e′. If x ∈ G_{n} then we can define

l = max{l′ : l′ ≤ s, n and d_{(l′)}(e) ⊥ d_{(l′)}(x)}

Then, either

l = s and d_{(l)}(x) = e and c_{(l)}(x) = e′

or

c_{(l)}(e′) ≥ d_{(l)}(x) and d_{(l′)}(e′) = d_{(l′)}(x) forl′ < l

or

c_{(l)}(x) ≥ d_{(l)}(e) and c_{(l′)}(e) = c_{(l′)}(x) forl′ < l

Thus the formula for e↓e′∘η_{G} defines the only graph morphism such that e↓e′∘η_{G}(e) = s+1 and e↓e′ ∘ η_{G}(e′) = s. We subsume the properties of e↓e′ in the following lemma.

4.11. Lemma. Let s ∈ ω and G be a simple ω-graph.

1. Let e, e′ ∈ G_{s}, such that e > e′. Then e↓e′ : [G] ⟶ δ^{s+1} is a well defined ω-functor and it is the unique ω-functor from [G] to δ^{s+1} such that

(e↓e′)_{s}(e) = s + 1 (e↓e′)_{s}(e′) = s

2. The ω-functor f : [G] ⟶ δ^{s+1} is onto iff there is e ∈ G_{s+1} such that f = (d(e)↓c(e)).

Proof. For 1. we argued before the statement of the lemma.

The ω-functor f : [G] ⟶ δ^{s+1} is onto iff there is an s + 1-cell e ∈ G_{s+1} such that f_{s+1}(e) = s + 1. But then f = (d(e)↓c(e)). Thus 2. holds as well.

If e is minimal in G_{s}(d(e), c(e)) then, we define

e↓ = b ∘ (d(e)↓c(e)) : [G] ⟶ δ^{s+1}

and if e is maximal in G_{s}(d(e), c(e)) then, we define

↓e = t ∘ (d(e)↓c(e)) : [G] ⟶ δ^{s+1}

Since in the morphism (e↓e′) we have e > e′, both e and e′ determine the morphism (e↓e′) uniquely. Hence we can write e↓ = e↓e′ = ↓e′, for short. Note that in this way, for any s-cell e the morphisms ↓e and e↓ are defined, no matter e has successor and predecessor or not.

Using this notation we can define the some ω-functors. Let s ∈ ω. We have

p^{s} : δ^{s+1} ⟶ δ^{s}

such that p^{s} = (s↓s − 1), i.e. p^{s} ’glue together the only non-identity s-cells s + 1 and s. Moreover, we define

ρ^{0;s+1}, ρ^{1;s+1} : δ^{s+1,s,s+1} ⟶ δ^{s+1}e

such that ρ^{0;s+1} = (s + 2↓s + 1) and ρ^{1;s+1} = (s + 1↓s) with s + 2, s + 1, s ∈ δ_{s}^{s+1,s,s+1}, and

ρ^{s+1} : δ^{s+1} ⟶ δ^{s+1,s,s+1}

such that ρ_{s+1}^{s+1}(s + 1) = s + 3, where s + 3 ∈ δ_{s+1}^{s+1,s,s+1} is the maximal cell in δ^{s+1,s,s+1}. We shall prove that any s ∈ ω, the diagram

[figure: ρ^{0}; δ^{s+1,s,s+1}; δ^{s+1}; ρ^{1}; b; δ^{s}; p; t]  (28)

is an internal bundle of intervals in S. To this end, we introduce below and in the following lemma more notation and state some properties of the ω-functors that we have defined so far.

For ω-functors z_{0}, z_{1} : [G] ⟶ δ^{s+1}, we write z_{0} ≤ z_{1} iff there is an ω-functors f :

[G] ⟶ δ^{s+1,s,s+1} such that ρ^{i} ∘ f = z − i, for i = 0, 1. 4.12. Lemma. Let s ∈ ω and [G] be a simple ω-category, e, e′ ∈ G_{s}. Then

1. We have

(↓e)_{s}(e) = s (e↓)_{s}(e) = s + 1

Moreover, if e is maximal (minimal) in G_{(}d(e), c(e)) then the equality ↓e_{s}(e) = s (e↓_{s}(e) = s + 1) determines the ω-functor ↓e (e↓) uniquely.

2. If s = 0 then

p ∘ (↓e) = p ∘ (e↓) = ↓;

3. If s > 0 then

p ∘ (↓e) = p ∘ (e↓) = (d(e)↓c(e));

4. If e > e′ then

b ∘ (e↓e′) = (min(G_{s}(e, e′)↓) t ∘ (e↓e′) = (↓max(G_{s}(e, e′));

5. ρ^{i} ∘ ρ = id for i = 0, 1;

6. If e ≥ e′ then there is a unique morphism

↓e..e′↓ : [G] ⟶ δ^{s+1,s,s+1}

such that for e′′ ∈ G_{s}

↓e..e′↓(e′′) = s + 1 iff e ≥ e′′ ≥ e′

If e = e′ we write ↓e↓ for ↓e..e↓;

7. If e ≥ e′ then

ρ^{0} ∘ ↓e..e′↓ = ↓e ρ^{1} ∘ ↓e..e′↓ = e′↓

8. if z_{0}, z_{1} : [G] ⟶ δ^{s+1} are two different ω-functors and z_{0} ≤ z_{1} then there are e_{0}, e_{1} ∈ G_{s} such that e_{0} ≥ e_{1} and z_{0} = ↓e_{0} and z_{1} = e_{1}↓;

9. Let z : [G] ⟶ δ^{s+1} be an ω-functor. If z_{s}(e) = s (z_{s}(e) = s + 1) then there is e_{0} ∈ G_{s} (e_{1} ∈ G_{s}) such that e_{0} ≥ e (e ≥ e_{1}) and z = ↓e_{0} (z = e_{1}↓); Proof. Exercise.

Now we can prove

4.13. Lemma. For any s ∈ ω the diagram (28) is a bundle of intervals S.

Proof. We shall verify the conditions 1. - 4. in Appendix 6.1.

To see 1. we calculate

p_{s} ∘ b_{s}(s) = p_{s}(s + 1) = s = p_{s}(s) = p_{s} ∘ t_{s}(s)

Hence we have p ∘ b = 1_{δ^{s}} = p ∘ t.

2. is easy .

Ad 3. let z_{0}, z_{1}, z_{0} : [G] ⟶ δ^{s+1} be ω-functors with G simple.

Then (a) holds since, by Lemma 4.12 ρ^{i} ∘ ρ ∘ z_{0} = z_{0}, for i = 0, 1.

To show (b), we suppose contrary, that z_{0} ≤ z_{1} and z_{1} ≤ z_{0} but z_{0} ≠ z_{1}. Then by Lemma 4.12 there are cells e_{0}, e_{1}, e′_{0}, e′_{1} ∈ G_{s} such that e_{0} ≥ e_{1}, e′_{0} ≥ e′_{1} and z_{0} = ↓e_{0} = e′_{0}↓, z_{1} = e_{1}↓ = ↓e′_{0}. Thus we get

e′_{0} > e_{0} ≥ e_{1} > e′_{1} ≥ e′_{0}

which is a contradiction.

Ad (c). Suppose that z_{0} ≤ z_{1} and z_{1} ≤ z_{2}. We shall show that z_{0} ≤ z_{2}. We assume that z_{0} ≠ z_{1}, z_{1} ≠ z_{2} since otherwise the thesis is obvious. Thus, by Lemma 4.12, there are e_{0}, e_{1}, e′_{1}, e_{2}, such that e_{0} ≥ e_{1}, e′_{1} ≥ e_{2} and

ρ^{0} ∘ ↓e_{0}..e_{1}↓ = z_{0} ρ^{1} ∘ ↓e_{0}..e_{1}↓ = z_{1}
ρ^{0} ∘ ↓e′_{1}..e_{2}↓ = z_{1} ρ^{1} ∘ ↓e′_{1}..e_{2}↓ = z_{2}

Hence, in particular e_{1} > e′_{1}. But then e_{0} ≥ e_{2} and

ρ^{0} ∘ ↓e_{0}..e_{2}↓ = z_{0} ρ^{1} ∘ ↓e_{0}..e_{2}↓ = z_{2}

i.e. z_{0} ≤ z_{2}, as required.

To show (d), we assume that z_{0} ≠ z_{1}. If p ∘ z_{0} = p ∘ z_{1} then there is e ∈ G_{s} such that either z_{0}(e) = s and z_{1}(e) = s + 1 or z_{0}(e) = s + 1 and z_{1}(e) = s. We suppose that the former condition holds. The latter can be treated similarly. There are s-cells e_{0}, e_{1} ∈ G_{s} such that e_{0} ≥ e ≥ e_{1} and z − 0 = ↓e_{0} and z_{1} = e_{1}↓. So, we have

ρ^{0} ∘ ↓e_{0}..e_{1}↓ = z_{0} ρ^{1} ∘ ↓e_{0}..e_{1}↓ = z_{1}

i.e. z_{0} ≤ z_{1}, as required. On the other hand, if z_{0} ≤ z_{1} then, there are e_{0}, e_{1} ∈ G_{s}, such that e_{0} ≥ e_{1}, z_{0} = ↓e_{0} and z_{1} = e_{1}↓. But then

p ∘ z_{0} = p_{0}↓e_{0} = d(e_{0})↓c(e_{0}) = d(e_{1})↓c(e_{1}) = p ∘ e_{1} = p ∘ z_{1}

as required.

Ad 4. Note that for morphisms κ^{0}, κ^{1} : δ^{s+1} ⟶ δ^{s+1,s,s+1} we have κ^{0} = ↓s↓, κ^{1} = ↓s + 1↓. Therefore, we have

ρ^{0} ∘ κ^{0} = s↓ = b ∘ p ρ^{1} ∘ κ^{0} = ↓s = 1

and

ρ^{0} ∘ κ^{1} = s + 1↓ = t ∘ p ρ^{1} ∘ κ^{0} = s + 1↓ = 1

i.e. b ∘ p ≤ 1 ≤ t ∘ p, as required.

Thus for any m ∈ ω and simple ω-category [G] there is a partial order ≤ on the set S([G], δ^{s+1}). Moreover, since (28) is a bundle of intervals each element has at most one immediate successor. We denote by < the immediate successor relation induced by this partial order. We have

4.14. Lemma. For any s ∈ ω and ω-functors z_{0}, z_{1} : [G] ⟶ δ^{s+1} we have z_{0} < z_{1} iff there is e ∈ G_{s} such that z_{0} = ↓e and z_{1} = e↓.

Proof. We can assume that z_{0} ≠ z_{1}. By Lemma 4.12, z_{0} < z_{1} iff there are e_{0}, e_{1} ∈ G_{s} such that e_{0} ≥ e_{1}, z_{0} = ↓e_{0} = ρ^{0} ∘ ↓e_{0}..e_{1}↓, and z_{1} = e_{1}↓ = ρ^{1} ∘ ↓e_{0}..e_{1}↓. Thus z_{0} < z_{1} iff e_{0} = e_{1}, as required.

4.15. Lemma. Let s > 0 and [G] be a simple ω-category. Then any ω-functor f : [G] ⟶ δ^{s} is either onto and there are e, e′ ∈ G_{s−1} such that, e > e′ and f = (e↓e′) or f admits one of the following factorizations

[figure: [G]; δ^{s}; f; δ^{0}; ↓; b; [G]; δ^{s}; f; δ^{0}; ↓; t]  (29)

or there are 0 < s′ < s and e, e′ ∈ G_{s′−1} such that, e >e′ and f admits one of the following factorizations

[figure: [G]; δ^{s}; f; δ^{s′}; e↓e′; b; [G]; δ^{s}; f; δ^{s′}; e↓e′; t]  (30)

Proof. Let G be a simple ω-graph and f : [G] ⟶ δ^{s} be an ω-graph morphism. If f is onto then the thesis follows from Lemma 4.11. Suppose that f is not onto and let s′ be the the maximal number so that the ω-functor

f′ : [G] ⟶^{f} δ^{s} ⟶^{p} δ^{s′}

is onto. If s′ = 0 then f′ = ↓ and if s > 0 then, by Lemma 4.11 there is e ∈ G′_{s} such that f′ = d(e)↓c(e). Thus in either case there is e ∈ G_{s′} such that f′_{s′}(e) = s′. Since p(s′) = {s′ +1, s′}, we have that either f_{s′}(e) = s′ +1 or f_{s′}(e) = s′. Then from definitions of p, b, and t follows that in the former case f = t ∘ d(e)↓c(e) and in the latter case f = b ∘ d(e)↓c(e), as required.

Thus, we have defined all the objects and morphisms of the following diagram D

[figure: b^{s}; δ^{s}; δ^{s−1}; p^{s−1}; t^{s}; b^{s+1}; δ^{s+1}; δ^{s}; p^{s}; t^{s+1}; b^{1}; δ^{1}; δ^{0} ≅ 1; p^{1}; t^{1}; δ^{s+1,s,s+1}; ρ^{0;s+1}; ρ^{1;s+1}; δ^{s,s−1,s}; ρ^{0;s}; ρ^{1;s}; δ^{s−1,s−2,s−1}; ρ^{0;s−1}; ρ^{1;s−1}; δ^{1,0,1}; ρ^{0;1}; ρ^{1;1}; …; …]

in S.

Now, we are ready to define a contravariant functor from S to D. We have

4.16. Proposition. The diagram D defined above is an internal disk in S.

Proof. We need to verify that the elementary axioms of an internal disk given in Appendix 6.1 are satisfied. The first for axioms holds by Lemma 4.13. Thus, it remains to verify the ’disk condition’. As we remarked earlier δ^{0} is a terminal object in S. From the proof of Theorem 4.6 follows that, for s ∈ ω, both b^{s} and t^{s} equalizes b^{s+1} and t^{s+1}. Moreover, if an ω-functor f : [G] ⟶ δ^{s+1}, equalizes b^{s+1} and t^{s+1} then f is not onto. Therefore by Lemma 4.15, there is f′ : [G] ⟶ δ^{s+1}, such that either f = b ∘ f′ or f = t ∘ f′, i.e. ’disk condition’ holds.

From the proposition we get

4.17. Corollary. The internal disk D in S induces a contravariant hom-functor

S(−, D) : S^{op} ⟶ D.

Proof. Proposition 4.16, is essentially saying that hom functors send the diagram D in S into a disk in Set. To see that it is finite, it is enough to notice that, there is finitely many morphism from any ω-category S into δ^{s}, for s ∈ ω. Moreover, by Lemma 4.15, for any S the numbers of such morphisms is bound by 2 times the cardinality of the simple ω-graph G, such that [G] ≃ S.

5. The duality

5.1. Stone adjunction. In this section we show that the the contravariant hom-functors

[figure: D(−, C); D; S; S(−, D)]

give rise to a Stone adjunction based on a schizophrenic object. Then, in the next section we shall show that it is in fact a duality.

First we make some easy observation, showing the connections between γ’s and δ’s, and between internal operations and and external morphisms on those. This explains, in what sense, the collections of δ^{s}, for s ∈ ω and the collection of γ_{n}, for n ∈ ω, are equal. In fact, it is the essence of being the schizophrenic object for the adjunctions between D and S.

5.2. Lemma.

1. For any n, s ∈ ω we have

γ_{n}^{s} = δ_{n}^{s};

2. for any s ∈ ω and n ∈ ω^{+}, the diagrams in Set

[figure: π_{0,n}^{s}; γ_{n,n−1,n}^{s}; γ_{n}^{s}; m_{n,n−1,n}^{s}; π_{1,n}^{s}; d_{n−1}^{s}; γ_{n−1}^{s}; ι_{n}^{s}; c_{n−1}^{s}]

and

[figure: π_{0,n}^{s}; δ_{n}^{s} ×_{δ_{n−1}^{s}} δ_{n}^{s}; δ_{n}^{s}; m_{n,n−1,n}^{s}; π_{1,n}^{s}; d_{n−1}^{s}; δ_{n−1}^{s}; ι_{n}^{s}; c_{n−1}^{s}]

are equal;

3. for any n ∈ ω and s ∈ ω^{+}, the diagrams in Set

[figure: ρ_{n}^{0;s}; δ_{n}^{s,s−1,s}; δ_{n}^{s}; ρ_{n}^{1;s}; b_{n}^{s}; δ_{n}^{s−1}; p_{n}^{s−1}; t_{n}^{s}]

and

[figure: ρ_{n}^{0;s}; ≤_{n}^{s}; γ_{n}^{s}; ρ_{n}^{1;s}; b_{n}^{s}; γ_{n}^{s−1}; p_{n}^{s−1}; t_{n}^{s}]

are equal, where ≤_{n}^{s} is the order on γ_{n}^{s} in the disk γ_{n}.

This observation allow to define the following morphisms.

Let s ∈ ω, D be a disk and x ∈ D^{s}. Then we can define an evaluation ω-functor

ev_{x} : D(D, C) ⟶ δ^{s}

such that, for n ∈ ω,

ev_{x,n} : D(D, γ_{n}) ⟶ δ_{n}^{s}

is a function

f : D → γ_{n} ⟼ f^{s}(x) ∈ γ_{n}^{s} = δ_{n}^{s}

Let n ∈ ω, S be a simple ω-category and e ∈ S_{n}. Then we can similarly define an evaluation disk map

ev_{e} : S(S, D) ⟶ γ_{n}

such that, for m ∈ ω,

ev_{e}^{s} : S(S, δ^{s}) ⟶ γ_{n}^{s}

is a function

g : S → δ^{s} ⟼ g_{n}(e) ∈ δ_{n}^{s} = γ_{n}^{s}

Having these evaluation maps we can define the following two natural transformations:

η : 1_{D} ⟶ S(D(−, C), D)

such that, for a disk D,

η_{D} : D ⟶ S(D(D, C), D)

is a disk map such that, for s ∈ ω and x ∈ D^{m},

η_{D}^{s}(x) = ev_{x}

ε : 1_{S} ⟶ D(S(−, D), C)

such that, for a simple ω-category S,

ε_{S} : S ⟶ D(S(S, D), C)

is an ω-functor such that, for n ∈ ω and e ∈ S_{n},

ε_{S,n}(e) = ev_{e}

We have

5.3. Lemma. η and ε are well defined natural transformations.

Theory and Applications of Categories, Vol. 8, No. 7 194

Proof. We need to verify that 1. evaluation disk maps are indeed disk maps;

2. evaluation ω-functors are indeed ω-functors;

3. for any disk D, η_{D} is disk map;

4. for any simple ω-category S, ε_{S} is ω-functor;

5. η is a natural transformation;

6. ε is a natural transformation.

In the following, we shall use Lemma 5.2 many times.

Ad 1. Let D be a disk, s ∈ ω, x ∈ D^{s}. We shall show that

ev_{x} : D(D, C) ⟶ δ^{s}

is an ω-functor. Let n ∈ ω, f : D ⟶ γ_{n}. We have

ev_{x,n−1}(d ∘ f) = d_{n−1}^{s} ∘ f^{s}(x) = d_{n−1}^{s}(f^{s}(x)) = d_{n−1}^{s}(ev_{x,n}(f))

i.e. ev_{x} preserves domains. Preservation of codomains can be shown similarly. Moreover, we have

ev_{x,n+1}(ι ∘ f) = ι_{n+1}^{s} ∘ d^{s}(x) = ι_{n+1}^{s}(f^{s}(x)) = ι_{n+1}^{s}(ev_{x,n}(f))

i.e. ev_{x} preserves identities. To show that ev_{x} preserves compositions, let f, g : D ⟶ γ_{n} be disk maps, such that c ∘ f = d ∘ g. Then, we have

ev_{x,n}(m_{n,n−1,n} ∘ ⟨f, g⟩) = m_{n,n−1,n}^{s}(f^{s}(x), g^{s}(x)) =

= m_{n,n−1,n}^{s}(f^{s}(x), g^{s}(x)) = m_{n,n−1,n}^{s}(ev_{x,n}(f), ev_{x,n}(g))

i.e. ev_{x} preserves compositions as well and it is an ω-functor.

Ad 2. Let S be a simple ω-category, n ∈ ω, and e ∈ S_{n}. We shall show that

ev_{e} : S(S, D) ⟶ γ_{n}

is a disk map. Let s ∈ ω^{+}, φ : S ⟶ δ^{s} be an ω-functor. Then

ev_{e}^{s−1}(p^{s−1} ∘ φ) = (p^{s−1} ∘ φ)_{n}(e) = p_{n}^{s−1}(φ_{n}(e)) = p_{n}^{s−1}(ev_{e}^{s}(φ))

i.e. ev_{e} preserves projections. Moreover, for s ∈ ω, and φ as above, we have ev_{e}^{s+1}(b^{s+1} ∘ φ) = (b^{s+1} ∘ φ)_{n}(e) = b_{n}^{s+1}(φ_{n}(e)) = b_{n}^{s+1}(ev_{e}^{s}(φ)) i.e. ev_{e} preserves left end-points. For right end-points the calculations are similar. To see that ev_{e} preserves the order let s ∈ ω^{+} and φ^{0}, φ^{1} : S ⟶ δ^{s} be ω-functors, such that φ^{0} ≤ φ^{1}, i.e. there is φ : S ⟶ δ^{s,s−1,s} and φ^{i} = ρ^{i} ∘ φ, for i = 0, 1. Then, we have

ev_{e}(φ^{0}) = φ_{n}^{0}(e) = ρ_{n}^{0} ∘ φ_{n}(e) = ρ_{n}^{0} ∘ φ_{n}(e)

and

ev_{e}(φ^{1}) = φ_{n}^{1}(e) = ρ_{n}^{1} ∘ φ_{n}(e) = ρ_{n}^{1} ∘ φ_{n}(e)

i.e. ev_{e}(φ^{0}) ≤ ev_{e}(φ^{1}) in γ_{n}^{s} and ev_{e} preserves the order, as well.

Ad 3. Let D be a disk. We shall show that

η_{D} : D ⟶ S(D(D, C), D)

is a disk map. For s ∈ ω^{+}

η_{D}^{s} : D ⟶ S(D(D, C), δ^{s})

Then, for and x ∈ D^{s}, n ∈ ω and f : D ⟶ γ_{n} we have

η_{D}^{s−1}(p(x))_{n}(f) = ev_{p(x),n}(f) = f^{s−1}(p(x)) = p_{n}^{s−1}(f^{s}(x)) =

= p_{n}^{s−1}(ev_{x,n}(f)) = p_{n}^{s−1} ∘ ev_{x,n}(f) = p_{n}^{s−1} ∘ η_{D}^{s}(x)_{n}(f)

i.e. η_{D}^{s−1}(p(x)) = p_{n}^{s−1} ∘ η_{D}^{s}(x) and η_{D} preserves the projections. To see that η_{D} preserves the endpoints, we take s ∈ ω and n and f as above. We have

η_{D}^{s+1}(b(x))_{n}(f) = ev_{b(x),n}(f) = f^{s+1}(b(x)) = b_{n}^{s+1}(f^{s}(x)) =

= b_{n}^{s+1}(ev_{x,n}(f)) = b_{n}^{s+1} ∘ ev_{x,n}(f) = p_{n}^{s+1} ∘ η_{D}^{s}(x)_{n}(f)

i.e. η_{D}^{s+1}(b(x)) = b_{n}^{s+1} ∘η_{D}^{s}(x) and η_{D} preserves the left endpoints. For the right endpoints the argument is similar. Finally, let x_{0}, x_{1} ∈ D^{s}, x_{0} ≤ x_{1}, and f : D ⟶ γ_{n}. Then

η_{D}^{s}(x_{0})_{n}(f) = f^{s}(x_{0}) = f^{s}(x_{1}) = η_{D}^{s}(x_{1})_{n}(f)

i.e. η_{D}^{s}(x_{0}) ≤ η_{D}^{s}(x_{1}), and η_{D} preserves the orders, so it is a disk map.

Ad 4. Let S be a simple ω-category. We shall show that

ε_{S} : S ⟶ D(S(S, D), C)

is an ω-functor. For n ∈ ω,

ε_{S,n} : S_{n} ⟶ D(S(S, D), γ_{n})

Fix s ∈ ω^{+}, e ∈ S_{n}, and φ : S ⟶ δ^{s}. Then, we have

ε_{S,n−1}(d(e))^{s}(φ) = φ_{n−1}(d(e)) = d_{n−1}^{s}(φ_{s}(e)) =

= d_{n−1}^{s}(φ_{n}(e)) = d_{n−1}^{s}(ε_{S,n−1}(e)^{s})(φ)) = (d_{n−1}(ε_{S,n−1}(e))^{s}(φ))

i.e. ε_{S} preserves the domains. The argument for preservation of codomains is similar. To show that ε_{S} preserves the identities we take n ∈ ω and s, e, φ as above. We have

ε_{S,n+1}(ι(e))^{s}(φ) = φ_{n+1}(ι(e)) = ι_{n+1}^{s}(φ_{n}(e)) =

= ι_{n+1}^{s}(ε_{S,n}(e))^{s}(φ)) = (ι_{n+1}^{s} ∘ ε_{S,n}(e))^{s}(φ))

i.e. ε_{S,n+1} ∘ ι = ι_{n+1}^{s} ∘ ε_{S,n} and ε_{S} preserves identities. For preservation of compositions, let e, e′ ∈ S_{n}, such that c(e) = d(e′). Then

ε_{S,n}(m(e, e′))^{s}(φ) = φ_{n}(m(e, e′)) = m_{n,n−1,n}^{s}(φ_{n}(e), φ_{n}(e′)) =

= m_{n,n−1,n}^{s}(φ_{n}(e), φ_{n}(e′)) = m_{n,n−1,n}^{s}(ε_{S,n}(e)^{s}(φ), ε_{S,n}(e′)^{s}(φ)) =
= (m_{n,n−1,n}^{s} ∘ ⟨ε_{S,n}(e)^{s}(φ), ε_{S,n}(e′)⟩)^{s}(φ)

i.e. ε_{S,n}(m(e, e′)) = m_{n,n−1,n}^{s} ∘ ⟨ε_{S,n}(e)^{s}(φ), ε_{S,n}(e′)⟩ and the compositions are preserved as well. Hence ε_{S} is an ω-functor indeed, as required.

Ad 5. Let f : D ⟶ E be a disk map. We need to show that the square

[figure: E; S(D(E, C), D); η_{E}; S; S(D(D, C), D); η_{D}; f; S(D(f, C), D)]

commutes. Let n, s ∈ ω, x ∈ D^{s} and g : E ⟶ γ_{n} a disk map. Then, we have

(S(D(f, C), D) ∘ η_{D})(x)_{n}(g) = (S(D(f, C), D)(η_{D}^{s}(x)))_{n}(g) =

(η_{D}^{s}(x) ∘ D(f, C))_{n}(g) = evx, n ∘ D(f, C)_{n}(g) = ev_{x,n}(g ∘ f) = (g ∘ f)^{s}(x) =

= (g^{s}(f^{s}(x)) = ev_{f(x),n}(g) = η_{E}(f(x))_{n}(g) = ((η_{E} ∘ f)(x))_{n}(g)

i.e. the above square commutes.

Ad 6. Let φ : E ⟶ T be an ω-functor. We need to show that the square

[figure: T; D(S(T, D), C); ε_{T}; S; D(S(S, D), C); ε_{S}; φ; D(S(φ, D), C)]

commutes. Let n, s ∈ ω, e ∈ S_{n} and ψ : T ⟶ δ^{s} an ω-functor. Then, we have

(D(S(φ, D), C) ∘ ε_{S})_{n}(e))^{s}(ψ) = (D(S(φ, D), C)_{n}(ε_{S,n}(e))^{s}(ψ) =
= ev_{e}^{s} ∘ S(φ, D)^{s}(ψ) = ev_{e}^{s}(ψ ∘ φ) = ψ_{n} ∘ φ_{n}(e) =
= ev_{φ_{n}(e)}^{s}(ψ) = ε_{T,n}(φ_{n}(e))^{s}(ψ) = (ε_{T} ∘ φ)_{n}(e)^{s}(ψ)

i.e. the above square commutes.

This ends the proof of the Lemma.

5.4. Proposition. The natural transformations η and ε are the unit and counit of the Stone adjunction

[figure: D(−, C); D; S; S(−, D)]

Proof. We need to verify that the following triangles

[figure: D(−, C); D(−, C); 1_{D(−,C)}; D(S(D(−, D), C), D); ε_{D(−,C)}; D(η, C)]  (31)

and

[figure: S(−, D); S(−, D); 1_{S(−,D)}; S(D(S(−, D), C), D); η_{S(−,D)}; S(ε, D)]  (32)

commute.

For commutation of (31), let D be a disk, n, s ∈ ω, f : D ⟶ γ_{n} a disk map, and x ∈ D^{s}. Then

((D(η_{D}, C) ∘ ε_{D(D,C)})_{n}(f))^{s}(x) =

= (D(η_{D}, C)_{n}(ev_{f}))^{s}(x) = (ev_{f} ∘ η_{D})^{s}(x) =

= ev_{f}^{s}(ev_{x}) = (ev_{x})_{n}(f) = f^{s}(x)

i.e. the triangle (31) commutes.

For commutation of (32), let S be a simple ω-category, n, s ∈ ω, φ : S ⟶ δ^{s} an ω-functor, and e ∈ S_{n}. Then

((S(ε_{S}, D) ∘ η_{S(S,D)})^{s}(φ))_{n}(e) =

= (S(ε_{S}, D)^{s}(ev_{φ}))_{n}(e) = ev_{φ,n} ∘ ε_{D,n}(e) =

ev_{φ,n}(ev_{e}) = (ev_{e})^{s}(φ) = φ(e)

i.e. the triangle (32) commutes.

5.5. The duality theorem. This section is devoted to the proof of the main theorem. We show that the Stone adjunction that we established in the previous section is an equivalence of categories. At the end, we list some specific corresponding objects and morphisms in both categories.

5.6. Lemma. Let s ∈ ω, D be a disk and z ∈ D^{s}. Then the ω functors ev_{z} and \overline{; z}↓\overline{z;}, from D(D, C) to δ^{s}, are equal.

Proof. Fix s ∈ ω and z ∈ D^{s}. Then, we have disk morphisms \overline{; z}, \overline{z;} : D ⟶ γ_{s−1}, such that \overline{; z} > \overline{z;} and an ω-functors

\overline{; z}↓\overline{z;} , ev_{z} : D(D, C) ⟶ δ^{s}

We have

\overline{; z}^{s}(z) = s \overline{z;}^{s}(z) = s + 1

Then

\overline{; z}↓\overline{z;}_{s−1}(\overline{; z}) = s + 1 = \overline{; z}^{s}(z) = (ev_{z})_{s−1}(\overline{; z})

and

\overline{; z}↓\overline{z;}_{s−1}(\overline{z;}) = s = \overline{z;}^{s}(z) = (ev_{z})_{s−1}(\overline{z;})

Hence, by Lemma4.11 we get that \overline{; z}↓\overline{z;} = ev_{z}, as required.

We have

5.7. Proposition. For every disk D, the disk map

η_{D} : D ⟶ S(D(D, C), D)

is an isomorphism.

Proof. Let D be a disk. First we shall show that η_{D} is mono. Fix n ∈ ω and x, y ∈ D^{n} such that x ≠ y. Then l = μ_{x,y} < n and the nodes p^{(l+1)}(x) and p^{(l+1)}(y) are comparable but different. Suppose p^{(l+1)}(x) < p^{(l+1)}(y). Thus, with the disk morphism \overline{x; y} : D ⟶ γ_{l} in D(D, γ_{l}), we have

η_{D}^{n}(x)_{l}(\overline{x; y}) = ev_{x,l}(\overline{x; y}) = \overline{x; y}_{n}(x) = l ≠ l + 1 =
= \overline{x; y}_{n}(y) = ev_{y,l}(\overline{x; y}) = η_{D}^{n}(y)_{l}(\overline{x; y})

so, η_{D}^{n}(x) ≠ η_{D}^{n}(y), i.e. η_{D} mono, as required.

It remains to show that η_{D} is epi. Let s ∈ ω, and f : D(D, C) ⟶ δ^{s} be an ω-functor. If s = 0 then f = ↓ and η_{D}^{0}(0) = ↓ = f. If s > 0 and there are e, e′ ∈ Cut(D)_{s−1} such that e > e′ and f = e↓e′. By, Corollary

3.29, there is z ∈ D^{m} such that e = \overline{; z} and e′ = \overline{z;}. Thus e↓e′ = \overline{; z}↓\overline{z;}. So, by Lemma 5.6,

η_{D}^{s}(z) = ev_{z} = \overline{; z}↓\overline{z;} = e↓e′ = f

Now, if f : D(D, C) ⟶ δ^{s} is an arbitrary ω-functor, by Lemma 4.15, it can be factorized as b or t followed by an ω-functor considered above, i.e. in the image of η_{D}. But η_{D} is a disk morphism so it preserves endpoints i.e. operations b and t. Hence, any morphism f : D(D, C) ⟶ δ^{s} is in the image of η_{D}, so η_{D} is epi, as required.

5.8. Lemma. Let n ∈ ω, G a simple ω-graph, and e ∈ G_{n}.

1. Then the disk map

ev_{e} : S([G], D) ⟶ γ_{n}

is outer;

2. the disk maps ev_{e} and \overline{↓e; e↓} are equal.

Proof. Ad 1. This is an immediate corollary from Lemmas 3.22.4(c) and 4.11.

Ad 2. First note that by 1. and Lemma 3.22 both morphisms ev_{e} and \overline{↓e; e↓} are outer. Moreover, we have

(\overline{↓e; e↓})^{n+1}(↓e) = n = (↓e)_{n}(e) = (ev_{e})^{n+1}(↓e)

and

(\overline{↓e; e↓})^{n+1}(e↓) = n + 1 = (e↓)n(e) = (ev_{e})^{n+1}(e↓)

By, Proposition 3.18.3, \overline{↓e; e↓} is the unique morphism with this property so ev_{e} = \overline{↓e; e↓}, as required.

5.9. Proposition. For every simple ω-category S the ω-functor

ε_{S} : S ⟶ D(S(S, D), C)

is an equivalence of ω-categories.

Proof. By Proposition 4.8, we can assume that S = [G] for some simple ω-graph G. Thus we will show that for any n ∈ ω the function

ε_{[G],n} : [G]_{n} ⟶ D(S([G], D), γ_{n}).

is a bijection.

First we shall show that ε_{[G]} is mono. Let m ∈ ω, σ, σ′ ∈ [G]_{m} and σ ≠ σ′. Then, there is s′ ≤ s and e ∈ G_{s′} such that either e ⊑ σ and e ⋢ σ′ or e ⋢ σ and e ⊑ σ′. We can assume that s′ is minimal such that e ⊑ σ and e ⋢ σ′. By minimality of s′, we have that d(e), c(e) ⊑ σ′. Then, there is e′ ∈ G_{s′} such that d(e) = d(e′) and c(e) = c(e′). Then, either e > e′ or e′ > e. Since both cases are similar, we consider the case e > e′. We have

(e↓)_{s}(σ) = \{ \begin{matrix}s′ + 1 & if there is e′′ ∈ G_{s′+1} \\ & such that d^{s′}(e′′) = e and e′ ⊑ σ \\ δ_{s}^{s′+1} − s′ − 1 & otherwise \end{matrix}

Thus (e↓)_{s}(σ) > s′. By Lemma 4.2, if e′′ ∈ G_{s′}, e′′ ⊑ σ′ and e′′ ⊥ e then e > e′′. Therefore (e↓)_{s}(σ′) = s′ and

(ev_{σ})^{s′+1}(e↓) = (e↓)_{s}(σ) > s′ = (e↓)_{s}(σ′) = (ev_{σ′})^{s′+1}(e↓)

i.e. (ev_{σ}) ≠ (ev_{σ′}) and ε_{[G]} is mono, as required.

Next, we show that ε_{[G]} is onto. Let f : S([G], D) ⟶ γ_{n} be a disk map. In the following diagram

[figure: γ_{u_{2i}}; π_{i}; \overline{x; y}; γu⃗; γ^{u}; mu⃗; S(S, D); γ_{n}; f; f^{∘}; ι_{n}]

the square is the canonical factorization of f. Thus f is obtained by the operation of composition mu⃗ and identity ι_{n} from the compatible tuple f^{∘} of D-cuts, π_{i}∘ f^{∘}, for i ∈ lh(u⃗). Since ε_{[G]} is an ω-functor it preserves compositions and identities it is enough

to show that the tuple f^{∘} is in the image of ε_{[G]}. But ε_{[G]} is mono so it both preserves and reflects domains and codomains operations. Thus it is enough to show that for i ∈ lh(u⃗),

π_{i}∘ f^{∘}: S([G], D) ⟶ γ_{u_{2i}} is in the image of ε_{[G]}.

Fix i ∈ lh(u⃗). Since both f^{∘} and π_{i} are outer so is the composition. Hence there are

ω-functors x, y : [G] ⟶ δ^{u_{2i}+1} such that x < y and \overline{x; y} = π_{i}∘ f^{∘}. By Lemma 4.14 there is e ∈ G_{u_{2i}} such that x = ↓e and y = e↓. Thus, by Proposition 5.8.2, we have

π_{i}∘ f^{∘}= \overline{↓e; e↓} = ev_{e} = ε_{[G],u_{2i}}(e)

i.e. π_{i}∘ f^{∘} is in the image of ε_{[G]}, and hence ε_{[G]} is epi, as required.

5.10. Theorem. The Stone adjunction

[figure: D(−, C); D; S; S(−, D)]

is an equivalence of categories.

Proof. The theorem follows from Propositions 5.4, 5.7, 5.9.

In the table below we show how objects and some morphisms in S and D correspond one another via the above duality. We need two notions concerning simple ω-categories. We say, that an ω-fumctor g : S ⟶ T between simple ω-categories is an immersion iff it sends indecomposable^{1} cells to indecomposable cells. Let ht(S) = u ≥ v = ht(T). We say that g is essential iff g(mac_{S}) = ι_{(u)}(mac_{T}).

The morphisms κ^{i}, κ^{0;u⃗}, κ^{1;v⃗}, b, t, b^{u⃗;l}, t^{u⃗;l} are typical immersions. The morphism p is an essential epi. By ϱu⃗ : δ^{u} ⟶ δu⃗, where u = ht(u⃗), we denote the unique essential epimorphism sending the maximal cell mac_{δ^{u}} = u ∈ δ_{u}^{u} to the maximal cell mac_{δu⃗} of δu⃗. Thus, for n ∈ ω, ϱ^{n+1,n,n+1} = ρ^{n+1}.

In the following, we denote by (−)^{∗} both dualising functors D(−, C) and S(−, D). In the table k, l, n ∈ ω and u⃗, v⃗ are ud-vectors, u = ht(u⃗), and if necessary tr_{(l)}(u⃗) =

tr_{(l)}(u⃗).

1Recall that a cell a is indecomposable if it is not an identity of some other cell and if a = a′ ∘ a′, then either a′ of a′′ is an identity.

in D  in S
γ_{n}  δ^{n}
γu⃗  δu⃗
π_{i} : γu⃗ ⟶ γ_{u_{2i}}  κ^{i} : δ^{u_{2i}} ⟶ δu⃗
π_{0;u⃗} : γ_{[u⃗,l,v⃗]} ⟶ γu⃗  κ^{0;u⃗} : δu⃗ ⟶ δ^{[u⃗,l,v⃗]}
π_{1;v⃗} : γ_{[u⃗,l,v⃗]} ⟶ γv⃗  κ^{1;v⃗} : δv⃗ ⟶ δ^{[u⃗,l,v⃗]}
d : γ_{n} ⟶ γ_{k}  t : δ^{k} ⟶ δ^{n}
c : γ_{n} ⟶ γ_{k}  b : δ^{k} ⟶ δ^{n}
d_{u⃗;l} : γu⃗ ⟶ γ_{tr_{(l)}(u⃗)}  t^{u⃗;l} : δ^{tr_{(l)}(u⃗)} ⟶ δu⃗
c_{u⃗;l} : γu⃗ ⟶ γ_{tr_{(l)}(u⃗)}  b^{u⃗;l} : δ^{tr_{(l)}(u⃗)} ⟶ δu⃗
ι : γ_{k} ⟶ γ_{n}  p : δ^{n} ⟶ δ^{k}
m_{n+1,n,n+1} : γ_{n+1,n,n+1} ⟶ γ_{n+1}  ϱ : δ^{n+1} ⟶ δ^{n+1,n,n+1}
mu⃗ : γu⃗ ⟶ γ_{u}  ϱu⃗ : δ^{u} ⟶ δu⃗
f : D → E is outer  f^{∗} : E^{∗} → D^{∗} is an immersion
f : D → E is inner  f^{∗} : E^{∗} → D^{∗} is essential

We shall make some comments about the above table. It is easy to see that

Cut(γ_{n})_{k} = \{ \begin{matrix}∅ & if k > n \\ 1_{γ_{n}} & if k = n \\ \{d_{k} ≥ c_{k}\} & if 0 ≤ k < n \end{matrix}

i.e. Cut(γ_{n}) is isomorphic to α_{n}, by isomorphism sending 1_{γ_{n}} to 2n. But the ω-categories D(γ_{n}, C) and δ^{n} are free over the ω-graphs Cut(γ_{n}) and α_{n}, respectively. Thus they are isomorphic via the unique isomorphisms sending 1_{γ_{n}} to n. Hence (γ_{n})^{∗} ≅ δ^{n}.

The morphism t : α^{n} ⟶ α^{n+1} is determined by the condition b_{n}(2n) = 2n + 1 so it corresponds via the above isomorphism of the ω-graphs to the morphism Cut(γ_{n}) ⟶ Cut(γ_{n+1}) sending 1_{γ_{n}} to d : γ_{n+1} ⟶ γ_{n}. Thus the square

[figure: D(γ_{n+1}, C); δ^{n+1}; ≅; D(γ_{n}, C); δ^{n}; ≅; .. ∘ d; t]

commute, i.e. d^{∗} ≅ t. Similarly we can prove that c^{∗} ≅ b. The object γu⃗ is a limit of a diagram in which the morphism are of form c and d and δu⃗ is a colimit of the dual of that diagram, see pages 156 and 186. Thus (γu⃗)^{∗} ≅ δu⃗. Since the projections from the limit correspond to coprojections into the colimit, we also have (π_{i})^{∗} ≅ κ_{i}.

The morphism D(ι, C) : D(γ_{n+1}, C) ⟶ D(γ_{n}, C) sends both c, d : γ_{n+1} ⟶ γ_{n} to 1_{γ_{n}} = c ∘ ι = d ∘ ι. So the diagram

[figure: D(γ_{n}, C); δ^{n}; ≅; D(γ_{n+1}, C); δ^{n+1}; ≅; D(ι, C); p]

commutes, i.e. ι^{∗} ≅ p.

The morphism mu⃗ : γu⃗ ⟶ γ_{u}, is the maximal arrow in D(γu⃗, C), so the square

[figure: D(γu⃗, C); δu⃗; ≅; D(γ_{u}, C); δ^{u}; ≅; D(mu⃗, C); ϱu⃗]

commutes as both morphism D(mu⃗, C) and ϱu⃗ sends the maximal arrow to the maximal arrows. Thus (mu⃗)^{∗} ≅ ρu⃗.

Let D be a disk. Then, an n-cell a : D ⟶ γ_{n} in D^{∗} is indecomposable iff a is an outer disk morphism. The maximal cell D^{∗} is the unique inner disk morphism mac_{D^{∗}} : D ⟶ γ_{ht(D)}.

Let f : D ⟶ E be a disk morphism, and ht(D) = v ≤ u = ht(E). Recall that the inner morphism from D to γ_{u} is unique. Moreover, if g ∘ f is an inner morphism so is f. Consider the following diagram

[figure: γ_{v}; γ_{u}; ι_{(u)}; D; E; f; mac_{D^{∗}}; mac_{E^{∗}}]  (33)

We have:

f : D → E is inner

iff the square (33) is commutes

iff

f^{∗}(mac_{E^{∗}}) = ι_{(u)}(mac_{D^{∗}})

iff

f^{∗} : E^{∗} → D^{∗} is essential

Since outer morphisms do compose, f^{∗} is an immersion whenever f is an outer morphism.

Finally, assume that g : S ⟶ T is an immersion of simple ω-categories, n > 0, and x, y : T ⟶ δ^{n} are two nodes of the disk T^{∗}, such that x ∘ g = y ∘ g is an inner node of S^{∗}. Then x ∘ g has a splitting z : δ^{n} ⟶ S, such that x ∘ g ∘ z = id_{δ^{n}} = y ∘ g ∘ z, and moreover z_{n}(n) is an indecomposable cell. Hence

x ∘ g = d(z_{n}(n))↓c(z_{n}(n)) = y ∘ g.

Since f preserves indecomposable cells, we have

x = g(d(z_{n}(n)))↓g(c(z_{n}(n))) = y.

Thus g^{∗} is an outer morphism.

5.11. Closing remarks. In this final section, we shall show that the notion of a disk plays a similar role for ω-categories to the role the notion of an interval plays for categories. In picture

\frac{D}{ω-categories} = \frac{I}{categories}

The syntactic category for ω-categories

Since the theory of ω-categories is a finite limit theory it has an obvious syntactic category, the opposite of the category of the finitely presented ω-categories. However, there is a smaller category which is much easier to describe and contains all the necessary syntactic data. We shall describe below what we mean by that.

Let S_{ω} be the sketch for the notion of ω-category, which is the extension of the one defined in Appendix 6.2 by objects Ou⃗ and cones stating that they are multipullbacks of O_{u_{i}}’s. Clearly, it is still true that, for any category C with finite limits, the category of internal ω-categories in C, ωCat(C) is (equivalent to) the category of models of S_{ω} in C, Mod_{C}(S_{ω}). Such an extension is natural in the sense that any ’reasonable’ morphism definable in the theory of ω-categories has Ou⃗ as its domain, for some ud-vector u⃗. For example, O_{n} × O_{k} is not a ’reasonable’ type since any definable function from O_{n} × O_{k} to O_{l} must be a projection followed by a definable function, i.e. there is no operation involving two (or more) cells without a specification on what level the codomain of the first cell match the domain of the second cell. Ou⃗ is a type of tuples of cells with a prescribed matching condition.

We can set a formal system to derive terms in contexts of form

x⃗ : Ou⃗ ⊢ t⃗ : Ov⃗  (34)

and to derive equations of terms in contexts of form

x⃗ : Ou⃗ ⊢ t⃗ = r⃗ : Ov⃗  (35)

Intuitively (34) means that x⃗ is a sequence of variable, the variable x_{i} has type O_{u_{2i}}, the variables x_{i} and x_{i+1} match in the sense that c_{(u_{2i+1})}(x_{i}) = d_{(u_{2i+1})}(x_{i+1}). Moreover, t⃗ is a sequence of matching terms t_{i} of type O_{v_{2i}} (c_{(v_{2i+1})}(t_{i}) = d_{(v_{2i+1})}(t_{i+1})) over S_{ω}, i.e. it is build from variables in x⃗ and the operations of domains, codomains, composition, and identity. (35) means that the theory of ω-categories proves that the terms t⃗ and r⃗ are equal.

Using this, we can define a syntactic category T_{ω} with objects Ou⃗ for any ud-vector u⃗ and whose morphism from Ou⃗ to Ov⃗ are terms in context (34) divided by equations (35). The identity on Ou⃗ is

x⃗ : Ou⃗ ⊢ x⃗ : Ou⃗

and the composition is defined by substitution in the obvious way.

Each term (34) corresponds to a morphism in S

∥x⃗ ⊢ t⃗∥ : δv⃗ ⟶ δu⃗  (36)

The definition of ∥ − ∥ can be given along with term forming rules. If v⃗ = v_{0} then

∥x⃗ ⊢ t_{0}∥ : δ^{v_{0}} ⟶ δu⃗

sends the only non-identity v_{0}-cell u_{0} in δ_{v_{0}}^{v_{0}} to the value of the term t_{0} in δu⃗, where the variables in x⃗ are interpreted as the obvious cells generating δu⃗. For vectors v⃗ of larger length we can use the fact that δv⃗ is a colimit of δ^{v_{i}}’s.

Using the fact that all the ω-categories δu⃗ are free, we obtain 5.12. Proposition. For any pair of tuples of terms t⃗, r⃗, we have

x⃗ : Ou⃗ ⊢ t⃗ = r⃗ : Ov⃗ iff ∥x⃗ ⊢ t⃗∥ = ∥x⃗ ⊢ r⃗∥ □

The above proposition allow us to define a functor

∥ − ∥ : T_{ω} ⟶ S^{op}  (37)

sending equivalence class of (34) to a morphism (36). It can be shown that the functor ∥ − ∥ is an equivalence of categories. Since the diagram

[figure: γ_{u_{0}}; γ_{u_{2}}; γ_{u_{1}}; c; d; γ_{u_{2k−2}}; γ_{u_{2k}}; γ_{u_{2k−1}}; c; d; …; γu⃗; π_{2}; π_{2k−2}; π_{2k}; π_{0}]  (38)

is a limiting cone in D (we call it a special pullback), the corresponding diagram in T_{ω}

[figure: O_{n_{0}}; O_{u_{2}}; O_{u_{1}}; [x_{0} ⊢ c(x_{0})]; [x_{1} ⊢ d(x_{1})]; O_{u_{2k−2}}; O_{u_{2k}}; O_{u_{2k−1}}; [x_{k} ⊢ d(x_{k})]; …; Ou⃗; [x⃗ ⊢ x_{k}]; [x⃗ ⊢ x_{0}]]  (39)

is a limiting cone, as well. We have an obvious sketch morphism

| − | : S_{ω} ⟶ T_{ω}  (40)

which is identity on objects and sends

m_{n_{0},n_{1},n_{2}} : O_{n_{0},n_{1},n_{2}} ⟶ O_{max(n_{0},n_{2})}

to

[x_{0}, x_{1} ⊢ m_{n_{0},n_{1},n_{2}}(x_{0}, x_{1})] : O_{n_{0},n_{1},n_{2}} ⟶ O_{max(n_{0},n_{2})}

etc. Composing (37) with (40) and the equivalence S(−, D), we get a sketch morphism, also denoted by [−]

[figure: [−] : S_{ω}; T_{ω}; | − |; S^{op}; ∥ − ∥; D; S(−, D)]  (41)

Recall that if S is a sketch and A is a category then a model of S in A is an association of objects and morphisms of S to objects and morphisms of A preserving domains, codomains, sending of identity specifications and commutative diagram specifications in S to identities and commutative diagrams, respectively, moreover sending the specified (co)cones to (co)limiting (co)cones. A realized sketch S is a sketch based on a category so that identity on S is a model of S in S.

Let Dspb be the finite limit realized sketch based on the category D, with cones being the special pullbacks (38) in D. We have

5.13. Proposition.

1. The morphism [−] : S_{ω} ⟶ D is a model of S_{ω} in D. Moreover, for any realized finite limit sketch S and a sketch morphism ψ there is a unique sketch morphism φ such that the diagram

[figure: S_{ω}; Dspb; [−]; ψ; S; φ]

commutes, i.e. Dspb is the realized sketch generated by S_{ω}.

2. For any category C with finite limits, the functor of composing with [−] : S_{ω} ⟶ D

Mod_{C}(Dspb) \overline{}Mod_{C}(S_{ω})

is equivalence of categories, i.e. the category of functors from D to C preserving special pullbacks is equivalent to the category of internal ω-categories in C.

The nerve of ω-categories and geometric realization

The above result is to be compared with the well known fact that the category Cat(C) of internal categories in a category with finite limits C, is equivalent to the category of functor from the category of intervals I to C, preserving analogous special pullbacks. The essential inverse functor is the nerve functor

N : Cat(C) ⟶ CAT(I, C)

associating to any small category its nerve. N is full, faithful and its essential image consists of special pullbacks preserving functors.

A similar ω-nerve functor can be defined for ω-categories

N_{ω} : ωCat(C) ⟶ CAT(D, C)

with similar properties, i.e. N_{ω} is full, faithful and its essential image consists of special pullbacks preserving functors. For C being the category of sets Set, it can be defined by the formula

N_{ω}(A) = ωCat((−, C), A) : D ⟶ Set

for any small ω-category A.

The ω-nerve functor composed with the geometric realization functor

R : Set^{D} ⟶ Top

defined in [J] give rise to a geometric realization functor for arbitrary ω-categories

R : ωCat ⟶ Top

The image of N_{ω} can be identified in a different way, using the notion of a homogenous theories, cf. [Be]. In our terminology this can be explained as follows. The canonical topology J on the category of simple ω-graphs sωGr, consists of the jointly surjective families of morphisms. Moreover, we have two faithful, essentially surjective functors:

[−] : sωGr ⟶ S

the free ω-category functor restricted to the simple ω-graphs, and

D(| − |, C) : T ⟶ S

associating to a given tree T the simple ω-category dual to the disk whose internal tree is T. The morphisms in the image of [−] are immersions, and the morphisms in the image of D(|−|, C) are essential (homogenous in the terminology of [Be]). Since the immersions and the essential ω-functors form a factorization system on S, S together with the above embedding functors [−], D(| − |, C), and the topology J form a homogenous theory. The models of this homogenous theory, i.e. the contravariant functors on S, which are Jsheaves when restricted to sωGr via [−], constitute the essential image of the ω-nerve functor

N′_{ω} : ωCat ⟶ Set^{S^{op}}

given by

N′_{ω}(A) = ωCat(−, A) : S^{op} ⟶ Set.

This is due to the fact, that for any presheaf X : S^{op} ⟶ Set, the functor

[figure: D; S^{op}; D(−, C); Set; X]

preserves the special pullbacks iff the functor

[figure: sG; S^{op}; [−]; Set; X]

is a J-sheaf. ω-categories as θ-categories

N_{ω} is also an embedding of strict ω-categories into θ-categories, i.e. the strict ωcategories are those θ-categories which preserves special pullbacks. To be more precise, recall from [J] that a face of disk γu⃗ is any disk epimorphism with domain γu⃗. The dimension of disk γu⃗ is ∑_{i=0}^{lh(u⃗)−1}(−1)^{i} u_{i}. A θ-category is a cellular set X (i.e. an object in Set^{D}) such that any compatible filling of all but one inner face of γu⃗ in X can be extended to a compatible filling of all faces of γu⃗ in X, for any ud-vector u⃗. Note that this condition can be rephrased as saying that X is a θ-category iff it sends limits of inner horns in D to weak limits. The strict ω-categories satisfy the following much stronger ’horn-filling’ condition: for any ud-vector u⃗ if outer faces of the shape π_{i} : γu⃗ ⟶ γ_{u_{2i}} are filled in X in a compatible way then all other faces of γu⃗, including γu⃗ itself have a unique filling in X extending the given one. However it is not the case that the nerves of ω-categories are necesserily flat functors. For example, they don’t preserve the following equaliser:

[figure: γ_{n}; γ_{n+1}; ι; c; d; γ_{n}]

in general. The disk classifier

In [SGL] it is shown that the topos Set^{I} classifies intervals. Using essentially the same method, adopted for the disk case it can be shown that the topos Set^{D} classifies disks, cf. Theorem 1 of [J].

6. Appendices

6.1. The notion of an internal disk. In this Appendix, we present the concept of disk. Note that this concept is not (generalized) algebraic; we need a category with rich-enough internal logic in which existential quantifier and disjunction can be interpreted. We take a category C, and the only assumption we make on C is that it has a terminal object. Our concept of an internal disk ’in C’ will have all its data located in the category C; however, the conditions imposed on the data will be evaluated in Set^{C^{op}}, a topos, into which C is embedded be the Yoneda functor. e also give a reformulation of the resulting notion that is purely elementary, without reference to Set^{C^{op}}. We do not distinguish between an object in C and its image under Yoneda. A diagram

[figure: r^{0}; R; X; r^{1}; b; B; p; t]

in C is a bundle of intervals over B if it is a linear order with (not necessarily different) endpoints b and t in the topos Set^{C^{op}}/B, cf. [SGL] p. 455.

A disk D in a category C is a diagram

[figure: b^{n}; D^{n}; D^{n−1}; p^{n}; t^{n}; b^{n+1}; D^{n+1}; D^{n}; p^{n+1}; t^{n+1}; b^{1}; D^{1}; D^{0} ≅ 1; p^{1}; t^{1}; R^{n+1}; r^{0;n+1}; r^{1;n+1}; R^{n}; r^{0;n}; r^{n;1}; R^{n−1}; r^{0;n−1}; r^{1;n−1}; R^{1}; r^{0;1}; r^{1;1}; …; …]

in C, such that for n > 0 the diagram

[figure: r^{0;n}; R^{n}; D^{n}; r^{1;n}; b^{n}; D^{n−1}; p^{n}; t^{n}]

is a bundle of intervals over D^{n−1} and moreover, in Set^{C^{op}}, the boundary of D^{n}, i.e. the subobject ∂(D^{n}) = im(b^{n}) ∪ im(t^{n}) of D^{n}, is the singular set for the bundle p^{n+1}, i.e. the equalizer of b^{n+1} and t^{n+1}, for n = 1, 2, . . .. By convention, we put ∂(D^{0}) = ∅. A morphism of disks in C is a family of morphisms {f_{n} : D^{n} ⟶ D′_{n}}_{n∈ω} preserving all the additional structure, i.e. projections, order in fibers and endpoints in the obvious internal sense.

The notion of a disk can be expressed in elementary terms as follows:

1. b^{n} and t^{n} are splittings of p^{n}, i.e. p^{n} ∘ b^{n} = p^{n} ∘ t^{n} = 1_{D^{n−1}};

2. r^{0;n}, r^{1;n} are jointly mono;

3. R^{n} is a linear order in fibers of p^{n}. Explicitely, for z_{0}, z_{1} : Z ⟶ D^{n}, we write z_{0} ≤_{R^{n}} z_{1}, whenever there is z : Z ⟶ R^{n}, such that r^{i;n} ∘ z = z_{i}, for i = 0, 1. Then for any z_{0}, z_{1}, z_{2} : Z ⟶ D^{n} we have

(a) z_{0} ≤_{R^{n}} z_{0} (b) if z_{0} ≤_{R^{n}} z_{1} and z_{1} ≤_{R^{n}} z_{0} then z_{0} = z_{1};

(c) if z_{0} ≤_{R^{n}} z_{1} and z_{1} ≤_{R^{n}} z_{2} then z_{0} ≤_{R^{n}} z_{2};

(d) if p^{n} ∘ z_{0} = p^{n} ∘ z_{1} iff either z_{0} ≤_{R^{n}} z_{1} or z_{1} ≤_{R^{n}} z_{0};

4. b and t are bottom and top endpoints in the fibers, i.e. b^{n} ∘ p^{n} ≤_{R^{n}} 1_{D^{n}} ≤_{R^{n}} t^{n} ∘ p^{n};

5. disk condition: for z : Z ⟶ D^{n} we have b^{n+1}∘z = t^{n+1}∘z iff there is z′ : Z ⟶ D^{n−1} such that either z = b^{n} ∘ z′ or z = t^{n} ∘ z′.

6.2. The notion of an internal ω-category. It is well known that the concept of ω-category is equational over the category of ω-graphs. In this section, give the details of a definition of ω-category that exhibits its equational character over ω-graphs, and which works internally in any category. Although of course the idea of an internal equational description is common place, the details here are crucial for our purposes.

Our procedure here is slightly different from Appendix 6.1. We do not assume that the category C has all finite limits, but our definition implicitly assumes that certain finite limits exists in C. This will be seen e.g. in condition (ii) where the definition requires that we have certain pullbacks in C as part of the data for the internal ω-category. In the application, when the category C will be the category D of finite disks, we will have one particular internal ω-category for which all the required pullbacks do indeed exist in D, despite the fact that D is not a finitely complete category.

Since the notion of ω-category is generalized algebraic, we do not need to refer to the topos Set^{C^{op}}.

An ω-category A in C consists of data (i)-(v) subject to the conditions (vi)-(xi). Data

(i) an ω-graph (or ω-globular object): i.e. the following diagram

[figure: d; A_{1}; A_{0}; c; d; A_{2}; c; d; A_{n}; A_{n−1}; c; …; …]

such that the globularity conditions hold

d ∘ d = d ∘ c c ∘ d = c ∘ c

Notation: If l < n, d_{(l)} = d_{(n;l)} : A_{n} ⟶ A_{l} we denote a compositions of d’s with the codomain A_{l}, and similarly by c_{(l)} = c_{(n;l)} : A_{n} ⟶ A_{l} we denote a compositions of c’s with the codomain A_{l}. By convention, if l ≥ n, d_{(l)} and c_{(l)} denotes the identity on A_{n}. When both domain and codomain of d_{(l)} and c_{(l)} is clear from the context we also write d and c for d_{(l)} and c_{(l)}, respectively.

(ii) for a 3-tuple ⟨n_{0}, n_{1}, n_{2}⟩ such that n_{1} < n_{0}, n_{2} a pullback

[figure: A_{n_{0}}; A_{n_{1}}; c; A_{n_{0},n_{1},n_{2}}; A_{n_{2}}; π_{1}; π_{0}; d]

(iii) for a 5-tuple ⟨n_{0}, n_{1}, n_{2}, n_{3}, n_{4}⟩ such that n_{1} < n_{0}, n_{2}, n_{3} < n_{2}, n_{4} a (triple) pullback

[figure: A_{n_{1}}; d; A_{n_{0}}; π_{0}; c; A_{n_{2}}; A_{n_{3}}; c; A_{n_{0},n_{1},n_{2},n_{3},n_{4}}; A_{n_{4}}; π_{2}; π_{1}; d]

(iv) composition morphisms

m_{n_{0},n_{1},n_{2}} = m_{n_{1}} = m : A_{n_{0},n_{1},n_{2}} ⟶ A_{max(n_{0},n_{2})}

(v) identity morphisms, for n ≤ l

ι_{(l)} = ι_{(n;l)} : A_{n} ⟶ A_{l}

Conditions (vi) Domains and codomains of compositions We have the following commutative squares:

1. for l ≤ n_{1}

[figure: A_{n_{0}}; A_{l}; d_{(l)}; A_{n_{0},n_{1},n_{2}}; A_{max(n_{0},n_{2})}; m; π_{0}; d_{(l)}]

2. for n_{1} < l ≤ max(n_{0}, n_{2})

and similarly for codomains, 3. for l ≤ n_{1}

[figure: A_{min(n_{0},l),n_{1},min(n_{2},l)}; A_{l}; m; A_{n_{0},n_{1},n_{2}}; A_{max(n_{0},n_{2})}; m; d_{(l)} × d_{(l)}; d_{(l)}]

[figure: A_{n_{0}}; A_{l}; c_{(l)}; A_{n_{0},n_{1},n_{2}}; A_{max(n_{0},n_{2})}; m; π_{1}; c_{(l)}]

4. for n_{1} < l ≤ max(n_{0}, n_{2})

[figure: A_{min(n_{0},l),n_{1},min(n_{2},l)}; A_{l}; m; A_{n_{0},n_{1},n_{2}}; A_{n_{0}}; m; c_{(l)} × c_{(l)}; c_{(l)}]

(vii) Domains and codomains of identities for n ≤ l

1.

[figure: A_{n}; A_{l}; ι_{(l)}; 1; A_{n}; c_{(n)}]

2.

[figure: A_{n}; A_{l}; ι_{(l)}; 1; A_{n}; d_{(n)}]

(viii) Associativity of compositions

1. for n_{1} = n_{3}

[figure: A_{n_{0},n_{1},max(n_{2},n_{4})}; A_{max(n_{0},n_{2},n_{4})}; m; A_{n_{0},n_{1},n_{2},n_{3},n_{4}}; A_{max(n_{0},n_{2}),n_{3},n_{4}}; m × 1; 1 × m; m]

2. for n_{1} < n_{3} < n_{0}

[figure: A_{n_{0},n_{1},max(n_{2},n_{4})}; A_{max(n_{0},n_{2},n_{4})}; m; A_{n_{0},n_{1},n_{2},n_{3},n_{4}}; A_{max(n_{0},n_{2}),n_{3},n_{4}}; ⟨m(π_{0}, π_{1}), m(c_{(n_{3})}π_{0}, π_{2})⟩; 1 × m; m]

3. for n_{0} ≤ n_{3}

[figure: A_{n_{0},n_{1},max(n_{2},n_{4})}; A_{max(n_{2},n_{4})}; m; A_{n_{0},n_{1},n_{2},n_{3},n_{4}}; A_{n_{2},n_{3},n_{4}}; ⟨m(π_{0}, π_{1}), m(π_{0}, π_{2})⟩; 1 × m; m]

4. for n_{3} < n_{1} < n_{4}

[figure: A_{n_{0},n_{1},max(n_{2},n_{4})}; A_{max(n_{0},n_{2},n_{4})}; m; A_{n_{0},n_{1},n_{2},n_{3},n_{4}}; A_{max(n_{0},n_{2}),n_{3},n_{4}}; m × 1; ⟨m(π_{0}, d_{(n_{1})}π_{2}), m(π_{1}, π_{2})⟩; m]

5. for n_{4} ≤ n_{1}

[figure: A_{n_{0},n_{1},n_{2}}; A_{max(n_{0},n_{2},n_{4})}; m; A_{n_{0},n_{1},n_{2},n_{3},n_{4}}; A_{max(n_{0},n_{2}),n_{3},n_{4}}; m × 1; ⟨m(π_{0}, π_{2}), m(π_{1}, π_{2})⟩; m]

(ix) Associativity of identities for n_{0} < n_{1} < n_{2}

[figure: A_{n_{0}}; A_{n_{1}}; ι_{(n_{1})}; ι_{(n_{2})}; A_{n_{2}}; ι_{(n_{2})}]

(x) Compositions with identities On the left:

1. for l ≥ n_{0}

[figure: A_{n_{0}}; A_{max(l,n_{2})}; ι; A_{n_{0},n_{1},n_{2}}; A_{l,n_{1},n_{2}}; ι_{(l)} × 1; m; m]

2.

[figure: A_{n_{2}}; A_{n_{0},n_{1},n_{2}}; ⟨ι_{(n_{0})} ∘ d_{(n_{1})}, 1⟩; ι; A_{max(n_{0},n_{2})}; m]

On the right:

3. for l ≥ n_{2}

[figure: A_{n_{2}}; A_{max(n_{0},l)}; ι; A_{n_{0},n_{1},n_{2}}; A_{n_{0},n_{1},l}; 1 × ι_{(l)}; m; m]

4.

[figure: A_{n_{0}}; A_{n_{0},n_{1},n_{2}}; ⟨1, ι_{(n_{2})} ∘ c_{(n_{1})}⟩; ι; A_{max(n_{0},n_{2})}; m]

(xi) Middle Exchange Law

for n_{1} < l < n_{0}, n_{2}

[figure: A_{n_{0},l,n_{2}}; A_{max(n_{0},n_{2})}; m; A_{n_{0},n_{1},n_{2}}; A_{n_{2},l,n_{0}}; ⟨m(d_{(l)} × 1), m(1 × c_{(l)})⟩; ⟨m(1 × d_{(l)}), m(c_{(l)} × 1)⟩; m; m]

6.3. Some properties of ω-categories. In this Appendix we shall prove that the analogs of the axioms of ω-categories, expressed for the ’multi’-version of the operations of ωcategories (d_{u⃗;l}, c_{u⃗;l}, mu⃗, etc.) defined in section 2.6, holds. For example, Lemma 6.7.1, gives some general form of the associativity law for ω-categories. We also prove, a more general form of the Middle Exchange Law. The definition of an ω-category is in Appendix 6.2.

In the whole section A denotes is an internal ω-category in a category C. We assume that for any ud-vector u⃗ the ’multi’-pullback Au⃗ exists.

We have

6.4. Lemma. Let A be an ω-category in a category C, l ∈ ω, u⃗ a ud-vector, l ≤ u = ht(u⃗). Then the following squares

[figure: A_{u}; A_{l}; d_{(l)}^{A}; Au⃗; A_{tr_{(l)}(u⃗)}; d_{u⃗;l}^{A}; mA ; u⃗; m_{tr_{(l)}(u⃗)}^{A}]  (42)

and

[figure: A_{u}; A_{l}; c_{(l)}^{A}; Au⃗; A_{tr_{(l)}(u⃗)}; c_{u⃗;l}^{A}; mA ; u⃗; m_{tr_{(l)}(u⃗)}^{A}]  (43)

commute.

Proof. We shall prove, by induction on l-size of u⃗ that the square (42) commutes. The commutation of the square (43) is similar and is left for the reader.

If u⃗ = u_{0} ≤ l then all morphisms in (42) are identities on A_{u_{0}}.

If u⃗ is l-primitive then tr_{(l)}(u⃗) = l, d_{n⃗;l}^{A} = d_{(l)}^{A} ∘ π_{0}, and m_{tr_{(l)}(u⃗)}^{A} = id_{A_{l}}. Thus the commutativity of (42) reduces to the commutation of the following triangle

[figure: An⃗; A_{l}; d_{u⃗;l}^{A}; A_{u}; mA ; u⃗; d_{(l)}^{A}]  (44)

and the commutation of (44), we can show by induction on the length of the l-primitive vector u⃗.

If lh(u⃗) = 1 then (44) clearly commutes. Assume that (44) commutes for vectors of length smaller then lh(u⃗) > 1. Let z = min(u⃗) ≥ l, u⃗ = u⃗′, z, u⃗′′, k′ = lh(u⃗′), u′ = ht(u⃗′), and u′′ = ht(u⃗′′). Then the triangles

[figure: A_{l}; d_{u⃗′;l}^{A} ∘ π_{0..k′}; Au⃗; A_{u′,z,u′′}; mA ; u⃗′ × mA ; u⃗′′; d_{(l)}^{A} ∘ π_{0}^{A}; d_{(l)}^{A}; A_{u}; m_{u′,z,u′′}^{A}]

commute. The left one by inductive assumption on u⃗′ and the right one by the axioms (vi).1 of the definition of an ω-category. Moreover, since

d_{u⃗′;l}^{A} ∘ π_{0..k′} = d_{(l)}^{A} ∘ π_{0}^{A} ∘ π_{0..k′}^{A} = d_{(l)}^{A} ∘ π_{0}^{A} = d_{u⃗;l}^{A}

and

mA ; u⃗ = m_{u′,z,u′′}^{A} ∘ (mA ; u⃗′ × mA ; u⃗′′)

the outer triangle is (44). Thus (44) commutes for any ud-vector u⃗.

Finally, let size_{l}(u⃗) > 1 and (42) commutes for ud-vectors of l-size smaller than size_{l}(u⃗) . Let u⃗ = u⃗′, z, u⃗′′, where z = min(u⃗) < l, u = ht(u⃗), u′ = ht(u⃗′), u′′ = ht(u⃗′′), l′ = min(l, u′), and l′′ = min(l, u′′). Then in the following diagram

[figure: A_{tr_{(l)}(u⃗′),z,tr_{(l)}(u⃗′′}); A_{l′,z,l′′}; m_{tr_{(l)}(u⃗′)}^{A} × m_{tr_{(l)}(u⃗′′)}^{A}; A_{u⃗′,z,u⃗′′}; A_{u′,z,u′′}; mA ; u⃗′ × mA ; u⃗′′; d_{u⃗′;l}^{A} × d_{u⃗′′;l}^{A}; A_{l}; m_{l′,z,l′′}^{A}; A_{u}; m_{u′,z,u′′}^{A}; d_{(l)}^{A} × d_{(l)}^{A}; d_{(l)}^{A}]

the left square commutes by inductive assumption on ud-vectors u⃗′ and u⃗′′, and and the right one commutes by condition (vi).2 of the definition of an ω-category. Thus 42 commutes. The commutation of 43 is similar.

We have

6.5. Lemma. For any ω-graph A and l ∈ ω and ud-vectors u⃗, v⃗ such that tr_{(l)}(u⃗) = tr_{(l)}(v⃗) = w⃗ the square

[figure: Au⃗; Aw⃗; c_{u⃗;l}^{A}; A_{[u⃗,l,v⃗]}; Av⃗; π_{1;v⃗}^{A}; π_{0;u⃗}^{A}; d_{v⃗;l}^{A}]  (45)

is a pullback.

Proof. We show that (45) is a pullback by induction on l-size(u⃗).

If v⃗ = v_{0} ≤ l then (45) is the following commutative square

[figure: Au⃗; A_{v_{0}}; c_{u⃗;l}^{A}; Au⃗; A_{v_{0}}; c_{u⃗;l}^{A}; id_{Au⃗}; id_{A_{v_{0}}}]

The case u⃗ = u_{0} ≤ l is similar.

If both u⃗ and v⃗ are l-primitive then (45) is

[figure: Au⃗; A_{l}; c_{u⃗;l}^{A}; A_{u⃗,l,v⃗}; Av⃗; π_{k_{1}+1..k_{1}+k_{2}+1}^{A}; π_{0..k_{1}}^{A}; d_{v⃗;l}^{A}]

where k_{1} = lh(u⃗) − 1, and k_{2} = lh(v⃗) − 1. Since c_{u⃗;l}^{A} = c_{(l)}^{A} ∘ π_{k_{1}}^{A} and d_{v⃗;l}^{A} = d_{(l)}^{A} ∘ π_{0}^{A}, it is a pullback, as well.

Assume now that l-size(u⃗) > 1. Let w = min(u⃗) < l, u⃗ = u⃗′, w, u⃗′′, v⃗ = v⃗′, w, v⃗′′, tr_{(l)}(u⃗′) = tr_{(l)}(v⃗′) = w⃗′, and tr_{(l)}(u⃗′′) = tr_{(l)}(v⃗′′) = w⃗′′. Then (45) is

[figure: A_{u⃗′,w,u⃗′′}; A_{w⃗′,w,w⃗′′}; c_{u⃗′;l}^{A} × c_{u⃗′′;l}^{A}; A_{[u⃗′,l,v⃗′],w,[u⃗′′,l,v⃗′′]}; A_{v⃗′,w,v⃗′′}; π_{1;v⃗′}^{A} × π_{1;v⃗′′}^{A}; π_{0;u⃗′}^{A} × π_{0;u⃗′′}^{A}; d_{v⃗′;l}^{A} × d_{u⃗′′;l}^{A}]

and it is a pullback, since by inductive hypothesis on u⃗′ and u⃗′′ it is a pullback of two pullback squares. This ends the proof.

In Lemma 6.7 we shall prove among other things a kind of general associativity law. But for this, we shall need the following form of the Middle Exchange Law.

6.6. Lemma. Let u′, u′′, v′, v′′, l, z ∈ ω, z < u′, u′′, v′, v′′, l. Moreover

min(u′, l) = min(v′, l) = l′, min(u′′, l) = min(v′′, l) = l′′
u = max(u′, u′′), v = max(v′, v′′),
n′ = max(u′, v′), n′′ = max(u′′, v′′), n = max(u, v).

Then the square

[figure: A_{n′,z,n′′}; A_{n}; m_{n′,z,n′′}; A_{[u′,l,v′],z,[u′′,l,v′′]}; A_{[u,l,v]}; ⟨m(π_{0}, π_{2}), m(π_{1}, π_{3})⟩; m_{[u′,l,v′]} × m_{[u′′,l,v′′]}; m_{[u,l,v]}]

commutes.

Proof. We need to show that the outer square in the following the diagram

[figure: A_{[u′,l,v′],z,[u′′,l,v′′]}; m × m; ⟨m(π_{0}, π_{2}), m(π_{1}, π_{3}), ⟩; [1]; [2]; [3]; A_{n′,z,n′′}; •; •; •; •; A_{[u,l,v]}; m; m; [4]; [5]; [6]; A_{n}] (46)

commutes. Below we explain what are the unexplained shapes in this diagram and why they commutes. The square [1] is

[figure: A_{n′,z,n′′}; A_{[n′,l,n′′]}; ⟨m(π_{0}, d_{(l)}π_{1}), m(c_{(l)}π_{0}, π_{1})⟩; A_{[u′,l,v′],z,[u′′,l,v′′]}; A_{[[u′,l,v′],l,[u′′,l,v′′]]}; m(c_{(l)}π_{1}, π_{2}), m(c_{(l)}π_{1}, π_{3}) >; m × m; m × m; < m(π_{0}, d_{(l)}π_{2}), m(π_{1}, d_{(l)}π_{2}),]

and it commutes by the axioms for domain and codomain of compositions (vi) and associativity (viii).3 and (viii).5.

The square [2] is

[figure: A_{[[u′,l,v′],l,[u′′,l,v′′]]}; A_{[u′,l,max(v′,u′′),l,v′′]}; 1 × m × 1; A_{[u′,l,v′],z,[u′′,l,v′′]}; A_{[[u′,l,u′′],l,[v′,l,v′′]]}; m(π_{1}, c_{(l)}π_{2}), m(c_{(l)}π_{1}, π_{3}), >; 1 × m × 1; < m(π_{0}, d_{(l)}π_{2}), m(π_{1}, d_{(l)}π_{2}),; m(c_{(l)}π_{1}, π_{2}), m(c_{(l)}π_{1}, π_{3}) >; < m(π_{0}, d_{(l)}π_{2}), m(d_{(l)}π_{1}, π_{2}),]

and it commutes by middle exchange law (xi).

The triangle [3] is

[figure: A_{[[u′,l,u′′],l,[v′,l,v′′]]}; A_{[u,l,v]}; m × m; ⟨m(π_{0}, π_{2}), m(π_{1}, π_{3})⟩; A_{[u′,l,v′],z,[u′′,l,v′′]}; < m(π_{0}, d_{(l)}π_{2}), m(d_{(l)}π_{1}, π_{2}),; m(π_{1}, c_{(l)}π_{2}), m(c_{(l)}π_{1}, π_{3}) >]

and, again, it commutes by middle exchange law (xi).

The triangle [4] is

[figure: A_{[n′,l,n′′]}; A_{n}; m; m; A_{n′,z,n′′}; ⟨m(π_{0}, d_{(l)}π_{1}), m(c_{(l)}π_{0}, π_{1})⟩]

and, again, it commutes by middle exchange law (xi).

The square [5] is

[figure: A_{[n′,l,n′′]}; A_{n}; m; A_{[[u′,l,v′],l,[u′′,l,v′′]]}; A_{[u′,l,max(v′,u′′),l,v′′]}; 1 × m × 1; m × m; m]

and it commutes by associativity axiom (viii).1.

Finally, the square [6] is

[figure: A_{[u′,l,max(v′,u′′),l,v′′]}; A_{n}; m; A_{[[u′,l,u′′],l,[v′,l,v′′]]}; A_{[u,l,v]}; m × m; 1 × m × 1; m]

and, again, it commutes by associativity axiom (viii).1.

Thus all the shapes including the outer square in the diagram (46) commute.

6.7. Lemma. For any ω-category A.

1. for l ∈ ω and ud-vectors u⃗, v⃗ such that tr_{(l)}(u⃗) = tr_{(l)}(v⃗) = w⃗ the triangle

[figure: A_{[u⃗,l,v⃗]}; A_{max(u,v)}; m_{[u⃗,l,v⃗]}^{A}; A_{[u,l,v]}; mA ; u⃗ × mA ; v⃗; m_{[u,l,v]}^{A}]  (47)

commutes;

2. for any u, v, u′, v′, z, z′, l ∈ ω such that u ≤ u′, v ≤ v′, l < u′, v′, z = max(u, v) and z′ = max(u′, v′), the square

[figure: A_{u′,l,v′}; A_{z′}; m_{u′,l,v′}^{A}; A_{[u,l,v]}; A_{z}; m_{[u,l,v]}^{A}; ι_{(u′)}^{A} × ι_{(v′)}^{A}; ι_{(z′)}^{A}]  (48)

commutes.

Proof. Ad 1. We argue by induction on the l-size of u⃗.

If v⃗ = v_{0}, then (47) is

[figure: Au⃗; A_{u}; mA ; u⃗; A_{u}; mA ; u⃗; 1_{A_{u}}]

so it commutes. The case u⃗ = u_{0} is similar.

If u⃗ and v⃗ are l-primitive then (47) is the following triangle

[figure: A_{u⃗,l,v⃗}; A_{max(u,v)}; m_{u⃗,l,v⃗}^{A}; A_{u,l,v}; mA ; u⃗ × mA ; v⃗; m_{u,l,v}^{A}]

commutes, by definition of m_{[u⃗,l,v⃗]}, since l = min([u⃗, l, v⃗]).

Finally, we assume that size_{l}(u⃗) > 1. Let

z = min(u⃗), u⃗ = u⃗′, z, u⃗′′, v⃗ = v⃗′, z, v⃗′′,

u = ht(u⃗), u′ = ht(u⃗′), u′′ = ht(u⃗′′),
v = ht(v⃗), v′ = ht(v⃗′), v′′ = ht(v⃗′′),
w⃗′ = tr_{(l)}(u⃗′) = tr_{(l)}(v⃗′), w⃗′′ = tr_{(l)}(u⃗′′) = tr_{(l)}(v⃗′′),
w′ = ht(w⃗′), w′′ = ht(w⃗′′),

l′ = min(l, u′) = min(l, v′), l′′ = min(l, u′′) = min(l, v′′), l^{.}= min(l′, l′′),

π_{0}^{.}= π_{0} ∘ π_{0;[u′,l,v′]}, π_{1}^{.}= π_{1} ∘ π_{0;[u′,l,v′]},

π_{2}^{.}= π_{0} ∘ π_{1;[u′′,l,v′′]}, π_{3}^{.}= π_{1} ∘ π_{1;[u′′,l,v′′]}.

Then, the following diagram

[figure: A_{u⃗′,z,u⃗′′}; A_{w⃗′,z,w⃗′′}; A_{[u⃗′,l,v⃗′],z,[u⃗′′,l,v⃗′′]}; A_{v⃗′,z,v⃗′′}; π_{0;u⃗′} × π_{0;u⃗′′}; π_{1;v⃗′} × π_{1;v⃗′′}; c_{u⃗′;l} × c_{u⃗′′;l}; d_{v⃗′;l} × d_{v⃗′′;l}; (mu⃗′ × mv⃗′) × (mu⃗′′ × mv⃗′′); mu⃗′ × mu⃗′′; mv⃗′ × mv⃗′′; mw⃗′ × mw⃗′′; A_{u′,z,u′′}; A_{l′,z,l′′}; A_{[u′,l,v′],z,[u′′,l,v′′]}; A_{v′,z,v′′}; ⟨π_{0}^{.}, π_{2}^{.}⟩; ⟨π_{1}^{.}, π_{3}^{.}⟩; c_{(l′)} × c_{(l′′)}; d_{(l′)} × d_{(l′′)}; ⟨m(π_{0}, π_{2}), m(π_{1}, π_{3})⟩; m; m; m; A_{u}; Al^{.}; A_{[u,l,v]}; A_{v}; π_{0}; π_{1}; c_{(l)}; d_{(l)}]

commutes, in which, the three horizontal squares are pullbacks by Lemma 6.5, the four back squares commute formally, and four front squares commutes by Lemma 6.4. Moreover, by definition of mA ; u⃗, the composition of vertical morphisms on the left is equal to mA ; u⃗ and the composition of vertical morphisms on the right is equal to mA ; v⃗. This shows that the following triangle

[figure: A_{[u⃗′,l,v⃗′],z,[u⃗′′,l,v⃗′′]}; A_{[u,l,v]}; mA ; u⃗ × mA ; v⃗; A_{[u′,l,v′],z,[u′′,l,v′′]}; mA ; u⃗′ × mA ; v⃗′ × mA ; u⃗′′ × mA ; v⃗′′; ⟨m^{A}(π_{0}, π_{2}), m^{A}(π_{1}, π_{3})⟩]  (49)

commutes. Then the triangle (47) commutes by Lemma 6.6.

Ad 2. To proof that (48) commutes we consider four cases according to whether the conditions

u ≤ l v ≤ l

hold true.

If u ≤ l and v ≤ l, then the square (48) is

[figure: A_{u′,l,v′}; A_{z}‘; m; A_{z}; A_{z}; 1_{A_{z}}; ⟨ι_{(u′)}, ι_{(v′)}⟩; ι_{(z′)}]

If u′ ≤ v′, it can be decomposed as a diagram

[figure: A_{u′}; A_{v′}; ι_{(v′)}; A_{z}; A_{z}; 1_{A_{z}}; ι_{(u′)}; ι_{(v′)}; A_{u′,l,v′}; ⟨1, ι_{(v′)}c_{(l)}⟩; m]

in which the square commutes by associativity of identity (ix) and the triangle commutes by axiom (x).4, and if u′ > v′, it can be decomposed as a diagram

[figure: A_{v′}; A_{u′}; ι_{(u′)}; A_{z}; A_{z}; 1_{A_{z}}; ι_{(v′)}; ι_{(u′)}; A_{u′,l,v′}; ⟨ι_{(v′)}d_{(l)}, 1⟩; m]

in which the square commutes by associativity of identity (ix) and the triangle commutes by axiom (x).2.

If u > l and v ≤ l, then the square (48) is the outer square in the diagram

[figure: A_{u,l,v}; A_{u}; m; A_{u}; A_{u}; 1_{A_{u}}; ⟨1, ι_{(v)}c_{(l)}⟩; 1_{A_{u}}; A_{u′,l,v}; A_{u′}; m; ι_{(u′)} × 1; ι_{(u′)}; A_{u′,l,v′}; A_{z′}; m; 1 × ι_{(v′)}; ι_{(z′)}]

in which the top square commutes by (x).4, the middle square commutes by (x).1, and the bottom square commutes by (x).3.

The case u ≤ l and v > l is similar to the previous one.

If u > l and v > l, then the square (48) is the outer square in the diagram

[figure: A_{u′,l,v}; A_{max(u′,v)}; m; A_{u,l,v}; A_{z}; m; ι_{(u)} × 1; ι_{(max(u′,v)}; A_{u′,l,v′}; A_{z′}; m; 1 × ι_{(v′)}; ι_{(z′)}]

in which the top square commutes by (x).3 and bottom square commutes by (x).1. This ends the proof of 2.

6.8. Free internal ω-categories. Let us fix for the whole section a category C with pullbacks and stable disjoint countable coproducts, ωGr(C) is the category of ω-graphs in C and ωCat(C) is the category of ω-categories in C. Then we have a forgetful functor

| − | : ωCat(C) ⟶ ωGr(C).

In this section, we shall construct a left adjoint to | − |:

[−] : ωGr(C) ⟶ ωCat(C)

Let G be an ω-graph in C. We shall construct an ω-category [G], and ω-graph morphisms η_{G} : G ⟶ [G], which is universal from G to | − |.

Since C has pullbacks, for any ud-vector u⃗, we can define, as in section 2.6, the multipullback Gu⃗, for 0 ≤ i ≤ j < lh(u⃗), the projections

π_{i}^{G} : Gu⃗ ⟶ G_{u_{2}i}, π_{i..j}^{G} : Gu⃗ ⟶ G_{u_{2i},...,u_{2j}},

for l ∈ ω, the morphisms of multi-domain and the multi-codomain in an ω-graph G

d_{u⃗;l}^{G}, c_{u⃗;l}^{G} : Gu⃗ ⟶ G_{tr_{(l)}(u⃗)}

and for ⟨u⃗, v⃗⟩ ∈ UD_{n_{0},n_{1},n_{2}} the projection morphisms

π_{0;u⃗}^{G} : G_{[u⃗,n_{1},v⃗]} ⟶ Gu⃗ π_{1;v⃗}^{G} : G_{[u⃗,n_{1},v⃗]} ⟶ Gv⃗.

Then, the object of n-cells of [G] is the following coproduct

[G]_{n} = ∐_{u⃗∈UD_{n}} Gu⃗

with the embedding morphisms κu⃗ : Gu⃗ ⟶ ∐_{u⃗∈UD_{n}} Gu⃗, for any ud-vector u⃗ ∈ UD_{n}. For n ≥ l, the domain and the codomain morphisms

d_{(l)}^{[G]}, c_{(l)}^{[G]}; [G]_{n} ⟶ [G]_{l}

are so defined that, for any u⃗ ∈ UD_{n}, the squares

[figure: Gu⃗; G_{tr_{(l)}(u⃗)}; d_{u⃗;l}; [G]_{n}; [G]_{l}; d_{(l)}^{[G]}; κu⃗; κ_{tr_{(l)}(u⃗)}; Gu⃗; G_{tr_{(l)}(u⃗)}; c_{u⃗;l}; [G]_{n}; [G]_{l}; c_{(l)}^{[G]}; κu⃗; κ_{tr_{(l)}(u⃗)}]

commute. For n ≤ l, the identity morphisms

ι_{(l)}^{[G]} : [G]_{n} ⟶ [G]_{l}

are so defined that, for any u⃗ ∈ UD_{n}, the triangle

[figure: [G]_{n}; [G]_{l}; ι_{(l)}^{[G]}; Gu⃗; κu⃗; κu⃗]

commutes.

In Lemma 6.5 we have shown that the square

[figure: Gu⃗; Gw⃗; c_{u⃗;n_{1}}^{G}; G_{[u⃗,n_{1},v⃗]}; Gv⃗; π_{1,v⃗}^{G}; π_{0,u⃗}^{G}; d_{v⃗;n_{1}}^{G}]

is a pullback. Since the countable coproducts are disjoint and universal in C they commutes with pullbacks and hence [G]_{n_{0},n_{1},n_{2}} in the pullback

[figure: [G]_{n_{0}}; [G]_{n_{1}}; c; [G]_{n_{0},n_{1},n_{2}}; [G]_{n_{2}}; π_{1}; π_{0}; d]

is given by the coproduct

[G]_{n_{0},n_{1},n_{2}} = ∐_{⟨u⃗,v⃗⟩∈UD_{n_{0},n_{1},n_{2}}} G_{[u⃗,n_{1},v⃗]}

with the embedding morphisms κ_{u⃗,v⃗} : G_{[u⃗,n_{1},v⃗]} ⟶ [G]_{n_{0},n_{1},n_{2}}, for any ⟨u⃗, v⃗⟩ ∈ UD_{n_{0},n_{1},n_{2}}. The projections

π_{0} : [G]_{n_{0},n_{1},n_{2}} ⟶ [G]_{n_{0}} π_{1} : [G]_{n_{0},n_{1},n_{2}} ⟶ [G]_{n_{2}}

are so defined that, the squares

[figure: G_{[u⃗,n_{1},v⃗]}; Gu⃗; π_{0,u⃗}; [G]_{n_{0},n_{1},n_{2}}; [G]_{n_{0}}; π_{0}; κ_{u⃗,v⃗}; κu⃗; G_{[u⃗,n_{1},v⃗]}; Gv⃗; π_{1,v⃗}; [G]_{n_{0},n_{1},n_{2}}; [G]_{n_{2}}; π_{1}; κ_{u⃗,v⃗}; κv⃗]

commute. Similarly, for a ud-vector n_{0}, n_{1}, n_{2}, n_{3}, n_{4}, the object of compatible triples of cells in [G] is the coproduct

[G]_{n_{0},n_{1},n_{2},n_{3},n_{4}} = ∐_{⟨u⃗,v⃗,w⃗⟩∈UD_{n_{0},n_{1},n_{2},n_{3},n_{4}}} G_{[u⃗,n_{1},v⃗,n_{3},w⃗]}

with the embedding morphisms κ_{u⃗,v⃗,w⃗} : G_{[u⃗,n_{1},v⃗,n_{3},w⃗]} ⟶ [G]_{n_{0},n_{1},n_{2},n_{3},n_{4}}, for any ⟨u⃗, v⃗, w⃗⟩ ∈ UD_{n_{0},n_{1},n_{2},n_{3},n_{4}}. We have a limiting diagram in C

[figure: [G]_{n_{1}}; d; [G]_{n_{0}}; π_{0}; c; [G]_{n_{2}}; [G]_{n_{3}}; c; [G]_{n_{0},n_{1},n_{2},n_{3},n_{4}}; [G]_{n_{4}}; π_{2}; π_{1}; d]

The composition morphisms

m_{n_{0},n_{1},n_{2}}^{[G]} : [G]_{n_{0},n_{1},n_{2}} ⟶ [G]_{max(n_{0},n_{2})}

are so defined that, for any ⟨u⃗, v⃗⟩ ∈ UD_{n_{0},n_{1},n_{2}}, the triangle

[figure: [G]_{n_{0},n_{1},n_{2}}; [G]_{max(n_{0},n_{2})}; m_{n_{0},n_{1},n_{2}}^{[G]}; G_{[u⃗,n_{1},v⃗]}; κ_{u⃗,v⃗}; κ_{[u⃗,n_{1},v⃗]}]

commutes. This ends the construction of [G]. We have

6.9. Theorem. The functor | − | : ωCat(C) ⟶ ωGr(C) has a left adjoint

[−] : ωGr(C) ⟶ ωCat(C).

Proof. We verify that [G] is indeed an ω-category and that

η_{G} : G ⟶ [G]

is given, for n ∈ ω, by

η_{G,n} = κ_{n} : G_{n} ⟶ [G]_{n}

is a universal morphism from G to | − |.

Note that since [G]_{n}, [G]_{n_{0},n_{1},n_{2}}, and [G]_{n_{0},n_{1},n_{2},n_{3},n_{4}} are defined as coproducts the commutations of diagrams involving such objects reduces to commutations involving objects Gu⃗, G_{u⃗;v⃗}, and G_{u⃗;v⃗;w⃗}, where u⃗ ∈ UD_{n}, ⟨u⃗, v⃗⟩ ∈ UD_{n_{0},n_{1},n_{2}}, and ⟨u⃗, v⃗, w⃗⟩ ∈ UD_{n_{0},n_{1},n_{2},n_{3},n_{4}}, respectively. After such reductions the commutations of the resulting diagrams follows from some equalities concerning ud-vectors and additionally, if these diagrams involve domains and codomains, some facts concerning muti-domains and multi-codomains of ω-graphs (from the previous section) are needed.

First we verify that [G] is and ω-category in C. To show that d ∘ d = d ∘ c, note that, for n ∈ ω and u⃗ ∈ UD, in the diagram

[figure: Gu⃗; G_{tr_{(n+1)}(u⃗)}; [G]_{n+2}; [G]_{n+1}; κu⃗; κ_{tr_{(l)}(u⃗)}; G_{tr_{(n+1)}(u⃗)}; G_{tr_{(n)}(u⃗)}; d_{tr_{(n+1)}(u⃗);n}; [G]_{n}; d; κ_{tr_{(n)}(u⃗)}; c_{u⃗;n+1}; d_{u⃗;n+1}; c; d]

the right square commutes and the left square commutes serially, by definition of d and c. Thus the equality d ∘ d = d ∘ c reduces to

d_{tr_{(l)}(u⃗);n} ∘ d_{u⃗;n+1} = d_{tr_{(l)}(u⃗);n} ∘ c_{u⃗;n+1}  (50)

for any n ∈ ω and u⃗ ∈ UD. We prove (50) by induction on length u⃗.

If lh(u⃗) = 1 then (50) reduces to d_{n} ∘ d_{n+1} = d_{n} ∘ c_{n+1} which holds in any ω-graph. Assume that lh(u⃗) > 1, z = min(u⃗) and that u⃗ = u⃗′, z, u⃗′′. If z > n, using inductive

hypothesis and Lemma 6.5, we have

d_{tr_{(n+1)}(u⃗);n} ∘ d_{u⃗;n+1} = d_{n} ∘ d_{tr_{(n+1)}(u⃗′);n} ∘ π_{0;u⃗′} =
d_{n} ∘ c_{tr_{(n+1)}(u⃗′);n} ∘ π_{0;u⃗′} = d_{n} ∘ d_{tr_{(n+1)}(u⃗′′);n} ∘ π_{1;u⃗′′} =
d_{n} ∘ c_{tr_{(n+1)}(u⃗′′);n} ∘ π_{1;u⃗′′} = d_{tr_{(l)}(u⃗);n} ∘ c_{u⃗;n+1}

If z = n, using inductive hypothesis, we have

d_{tr_{(n+1)}(u⃗);n} ∘ d_{u⃗;n+1} = d_{n} ∘ π_{0} ∘ (d_{u⃗′;n+1} × d_{u⃗′′;n+1}) =

d_{n} ∘ π_{0} ∘ d_{u⃗′;n+1} ∘ π_{0;u⃗′} = d_{n} ∘ π_{0} ∘ c_{u⃗′;n+1} ∘ π_{0;u⃗′} =
d_{n} ∘ π_{0} ∘ (c_{u⃗′;n+1} × c_{u⃗′′;n+1}) = d_{tr_{(n+1)}(u⃗);n} ∘ c_{u⃗;n+1}

If z < n, using inductive hypothesis, we have

d_{tr_{(n+1)}(u⃗);n} ∘ d_{u⃗;n+1} =
(d_{tr_{(n+1)}(u⃗′);n} × d_{tr_{(n+1)}(u⃗′′);n}) ∘ (d_{u⃗′;n+1} × d_{u⃗′′;n+1}) =
(c_{tr_{(n+1)}(u⃗′);n} × c_{tr_{(n+1)}(u⃗′′);n}) ∘ (c_{u⃗′;n+1} × c_{u⃗′′;n+1}) =
d_{tr_{(n+1)}(u⃗);n} ∘ c_{u⃗;n+1}

So (50) holds.

The equality c^{[G]} ∘ d^{[G]} = c^{[G]} ∘ c^{[G]} can be proved similarly.

Ad (vi).1. Let n_{1}⟨n_{0}, n_{2} and < u⃗, v⃗⟩ ∈ UD_{n_{0},n_{1},n_{2}}. Then in the diagram

[figure: [G]_{n_{0}}; [G]_{n_{1}}; d_{(n_{1})}; [G]_{n_{0},n_{1},n_{2}}; [G]_{n}; m; π_{0}; d_{(n_{1})}; Gu⃗; G_{tr_{(n_{1})}(u⃗)}; d_{u⃗;n_{1}}; G_{[u⃗,n_{1},v⃗]}; G_{[u⃗,n_{1},v⃗]}; 1; π_{0;u⃗}; d_{[u⃗,n_{1},v⃗];n_{1}}; κu⃗; κ_{[u⃗,n_{1},v⃗]}; κ_{u⃗;v⃗}; κ_{tr_{(l)}(u⃗)}]

the four side squares commute. Thus the commutation of the inner square reduces to the commutation of the outer square for any pair ⟨u⃗, v⃗⟩ ∈ UD_{n_{0},n_{1},n_{2}}. The commutation of the outer square can be proved by induction on n_{1}-size of u⃗. From this (vi).1 follows.

To show (vi).2, let ⟨u⃗, v⃗⟩ ∈ UD_{n_{0},n_{1},n_{2}}, l ≤ max(n_{0}, n_{2}) = n, n′_{0} = min(l, n_{0}), n′_{1} = min(l, n_{1}), and consider the diagram

[figure: [G]_{n′_{0},n_{1},n′_{2}}; [G]_{l}; m; [G]_{n_{0},n_{1},n_{2}}; [G]_{n}; m; d_{(l)} × d_{(l)}; d_{(l)}; G_{[tr_{(l)}(u⃗),n_{1},v⃗]}; G_{[tr_{(l)}(u⃗),n_{1},v⃗]}; 1; G_{[u⃗,n_{1},v⃗]}; G_{[u⃗,n_{1},v⃗]}; 1; d_{u⃗;l} × d_{v⃗;l}; d_{[u⃗,n_{1},v⃗];l}; κ_{tr_{(l)}(u⃗);tr_{(l)}(v⃗)}; κ_{[u⃗,n_{1},v⃗]}; κ_{u⃗;v⃗}; κ_{tr_{(l)}(u⃗);tr_{(l)}(v⃗)}]

As the four side squares commute, then again the commutation of the inner square reduces to the commutation of the outer square for any pair ⟨u⃗, v⃗⟩ ∈ UD_{n_{0},n_{1},n_{2}}. This is left as an exercise.

The commutation of (vi).3 and (vi).4 can be proved similarly. The fact that [G] satisfies (vii), (viii), (ix), and (x), essentially follows from the equalities on ud-vectors proved in Lemma 2.4.

We end the verification that [G] is an ω-category in C by showing that it satisfies Middle Exchange Law (xi). We need to show that for, l < n_{1} < n_{0}, n_{2}, n = max(n_{0}, n_{2}), the diagram

[figure: [G]_{n_{0},n_{1},n_{2}}; [G]_{n_{2},l,n_{0}}; ⟨m(d_{(l)} × 1), m(1 × d_{(l)})⟩; [G]_{n}; m; m]  (51)

commutes. Let ⟨u⃗, v⃗⟩ ∈ UD_{n_{0},n_{1},n_{2}}. By Lemma 2.4.13, we have [u⃗, n_{1}, v⃗] = [[u⃗, n_{1}, tr_{(l)}(v⃗)], l, [tr_{(l)}(u⃗), n_{1}, v⃗]]. Moreover the square

[figure: G_{[tr_{(l)}(u⃗),n_{1},v⃗]}; G_{[tr_{(l)}(u⃗),n_{1},tr_{(l)}(v⃗)]}; 1 × c_{v⃗;l}; G_{[u⃗,n_{1},v⃗]}; G_{[u⃗,n_{1},tr_{(l)}(v⃗)]}; 1 × c_{v⃗;l}; d_{u⃗;l} × 1; d_{u⃗;l} × 1]

is a pullback. Thus by Lemma 6.5

d_{v⃗;l} × id = π_{0;[tr_{(l)}(u⃗),n_{1},v⃗]} : G_{[[tr_{(l)}(u⃗),n_{1},v⃗],l,[u⃗,n_{1},tr_{(l)}(v⃗)]]} ⟶ G_{[[tr_{(l)}(u⃗),n_{1},v⃗]}  (52)

id × c_{v⃗;l} = π_{1;[u⃗,n_{1},tr_{(l)}(v⃗)]} : G_{[[tr_{(l)}(u⃗),n_{1},v⃗],l,[u⃗,n_{1},tr_{(l)}(v⃗)]]} ⟶ G_{[u⃗,n_{1},tr_{(l)}(v⃗)]}  (53)

Thus to show that (51) commutes , it is enough to show that the following three squares

[figure: G_{n_{0},n_{1},n_{2}}; G_{n_{2},l,n_{0}}; ⟨m(d_{(l)} × 1), m(1 × d_{(l)})⟩; G_{[u⃗,n_{1},v⃗]}; G_{[[tr_{(l)}(u⃗),n_{1},v⃗],l,[u⃗,n_{1},tr_{(l)}(v⃗)]]}; 1; κ_{u⃗;v⃗}; κ_{[tr_{(l)}(u⃗),n_{1},v⃗];[u⃗,n_{1},tr_{(l)}(v⃗)]}]  (54)

[figure: G_{[u⃗,n_{1},v⃗]}; [G]_{n}; κ_{[u⃗),n_{1},v⃗]}; G_{[u⃗,n_{1},v⃗]}; [G]_{n_{0},n_{1},n_{2}}; κ_{u⃗;v⃗}; 1; m_{n_{0},n_{1},n_{2}}]

[figure: [G]_{n}; G_{[u⃗,n_{1},v⃗]}; κ_{u⃗;v⃗}; [G]_{n_{2},l,n_{0}}; G_{[[tr_{(l)}(u⃗),n_{1},v⃗],l,[u⃗,n_{1},tr_{(l)}(v⃗)]]}; κ_{[tr_{(l)}(u⃗),n_{1},v⃗];[u⃗,n_{1},tr_{(l)}(v⃗)]}; m; 1]

commute. The second and the third squares commute by definition of m_{n_{0},n_{1},n_{2}} and m_{n_{2},l,n_{0}}, respectively. Thus it remains to show that (54) commutes, as well. As [G]_{n_{0},n_{1},n_{2}} is a pullback, we shall show that the square (54) commutes, when composed with the projections

π_{0} : [G]_{n_{2},l,n_{0}} ⟶ [G]_{n_{2}} π_{1} : [G]_{n_{2},l,n_{0}} ⟶ [G]_{n_{0}}.

Composing (54) with π_{0}, we get the diagram

[figure: [G]_{n_{0},n_{1},n_{2}}; [G]_{n_{2},l,n_{0}}; ⟨m(d_{(l)} × 1), m(1 × d_{(l)})⟩; G_{[u⃗,n_{1},v⃗]}; G_{[[tr_{(l)}(u⃗),n_{1},v⃗],l,[u⃗,n_{1},tr_{(l)}(v⃗)]]}; 1; κ_{u⃗;v⃗}; [G]_{l,n_{1},n_{2}}; [G]_{n_{2}}; m; d_{(l)} × 1; π_{0}; G_{[[tr_{(l)}(u⃗),n_{1},v⃗]}; d_{u⃗×1}; π_{0;[tr_{(l)}(u⃗),n_{1},v⃗]}]

where the unmarked morphisms are the obvious embeddings into coproducts. The outer triangle commutes as it is (52). The left and right squares commute, by definition of d_{(l)} and π_{0}, respectively. The triangle at the bottom commutes, by definition of m_{l,n_{1},n_{2}}. Hence (54) followed by π_{0} commutes.

Composing (54) with π_{1}, we get the diagram

[figure: [G]_{n_{0},n_{1},n_{2}}; [G]_{n_{2},l,n_{0}}; ⟨m(d_{(l)} × 1), m(1 × d_{(l)})⟩; G_{[u⃗,n_{1},v⃗]}; G_{[[tr_{(l)}(u⃗),n_{1},v⃗],l,[u⃗,n_{1},tr_{(l)}(v⃗)]]}; 1; κ_{u⃗;v⃗}; [G]_{n_{0},n_{1},l}; [G]_{n_{0}}; m; 1 × c_{(l)}; π_{1}; G_{[[tr_{(l)}(u⃗),n_{1},v⃗]}; 1 × cu⃗; π_{1;[u⃗,n_{1},tr_{(l)}(v⃗)]}]

which by similar reasons as before shows that (54) followed by π_{1} commutes. Therefore (54) commutes as well, and [G] satisfy (xi). In this way we have proved that [G] is an ω-category in C.

Now we show that η : G ⟶ [G] is a universal morphism. So assume that \overline{F} :

[G] ⟶ A is an ω-graph morphism into an ω-category A. We construct an ω-functor \overline{F} : [G] ⟶ A, extending F.

Since F is an ω-graph morphism, it preserves both domains and codomains, and hence it induces a morphism between pullbacks

Fu⃗ : Gu⃗ ⟶ Au⃗

for any ud-vector u⃗. Then, for m ∈ ω, we define the morphism

\overline{F}_{k} : [G]_{k} ⟶ A_{k}

so that, for any ud-vector u⃗ ∈ UD_{k}, its composition with the embedding κu⃗ : Gu⃗ ⟶ [G]_{k} is the following morphism:

[figure: Gu⃗; Au⃗; Fu⃗; A_{ht(u⃗)}; mA ; u⃗; A_{k}; ι_{(k)}^{A}]

To see that \overline{F} so defined preserves domains, it is enough to note that, for any l ≤ n, u⃗ ∈ UD_{n}, the following three squares

[figure: G_{tr_{(l)}(u⃗)}; A_{tr_{(l)}(u⃗)}; F_{tr_{(l)}(u⃗)}; Gu⃗; Au⃗; Fu⃗; d_{u⃗;l}; A_{l′}; m; A_{u}; m; d_{u⃗;l}; d_{(l′)}; A_{l}; ι_{(l)}; A_{n}; ι_{(n)}; d_{(l)}]

commute, where u = ht(u⃗) and l′ = min(l, u). The commutations of these squares can be easily shown by induction on l-size of u⃗.

The fact that \overline{F} preserves codomains and identities is left for the reader.

To prove that \overline{F} preserves compositions, we need to show that for n_{1} < n_{0}, n_{2}, the diagram

[figure: [G]_{n}; A_{n}; \overline{F}_{n}; [G]_{n_{0},n_{1},n_{2}}; A_{n_{0},n_{1},n_{2}}; \overline{F}_{n_{0}} × \overline{F}_{n_{2}}; m; m]

commutes. But this reduces to the commutation of the following diagram

[figure: G_{[u⃗,n_{1},v⃗]}; A_{[u⃗,n_{1},v⃗]}; F_{[u⃗,n_{1},v⃗]}; G_{[u⃗,n_{1},v⃗]}; A_{[u⃗,n_{1},v⃗]}; Fu⃗ × Fv⃗; 1; 1; A_{k}; m_{[u⃗,n_{1},v⃗]}; A_{u,n_{1},v}; mu⃗ × mv⃗; m; A_{n}; ι_{(n)}; A_{n_{0},n_{1},n_{2}}; ι_{(n_{0})} × ι_{(n_{2})}; m]

for any ⟨u⃗, v⃗⟩ ∈ UD_{n_{0},n_{1},n_{2}}, where u = ht(u⃗), u = ht(v⃗), k = max(u, v), n = max(n_{0}, n_{2}). The commutation of the left square is trivial, the commutation of the middle square follows from Lemma , and the commutation of the right square follows from axioms (x).1 and (x).3. of the definition of the ω-category. Thus \overline{F} is an ω-functor.

The uniqueness of \overline{F} can be proved by a similar easy inductive argument.

Having the explicit description of the free internal ω-categories we can easily show the following corollary.

6.10. Corollary. The functor

[−] : ωGr(C) ⟶ ωCat(C)

preserves pullbacks.

Remark. This result appears in [S]; see the Proposition in Section 5 in [S].

Proof. Suppose that

[figure: G; K; f; P; H; k; h; g]

is a pullback in ωGr(C). Thus for n ∈ ω, the square

[figure: G_{n}; K_{n}; f_{n}; P_{n}; H_{n}; k_{n}; h_{n}; g_{n}]

is a pullback in C. Since limits commutes with limits and k, h, f, g preserves domains and codomains, for any ud-vector u⃗, the square

[figure: Gu⃗; Ku⃗; fu⃗; Pu⃗; Hu⃗; ku⃗; hu⃗; gu⃗]

is a pullback in C, as well. Then, since coproducts are commutes with pullbacks in C, the coproduct of pullbacks

[figure: ∐_{u⃗∈UD_{n}} Gu⃗; ∐_{u⃗∈UD_{n}} Ku⃗; ∐_{u⃗∈UD_{n}} fu⃗; ∐_{u⃗∈UD_{n}} Pu⃗; ∐_{u⃗∈UD_{n}} Hu⃗; ∐_{u⃗∈UD_{n}} ku⃗; ∐_{u⃗∈UD_{n}} hu⃗; ∐_{u⃗∈UD_{n}} gu⃗]  (55)

is again a pullback in C. But the above square is just

[figure: [G]_{n}; [K]_{n}; [f]_{n}; [P]_{n}; [H]_{n}; [k]_{n}; [h]_{n}; [g]_{n}]

and hence [−] preserves pullbacks.endproof

7. Notation

The numbers in square brackets indicate the page or pages where the notation is first introduced.

Conventions:

1. The ud-vectors are invariants for finite trees, finite disks, simple ω-graphs, and simple ω-categories. For a given ud-vector u⃗ we define a specific finite tree θu⃗ [123, 158], a specific finite disk γu⃗ [146, 156] (ud-vector in subscript), a specific simple ω-graph αu⃗ [175, 176], and a specific simple ω-category δu⃗ [184] (ud-vector in superscript). All the above structures are ’graded’ structures. The levels of trees and of disks are marked in superscripts, and the levels of ω-graphs, and of simple ω-categories are marked in subscripts. Thus, for example, D^{n} is the n-th level of the disk D, whereas A_{n} is the set of n-cells of the ω-category A. Adopting this convention, we have for any n, s ≥ 0 that

γ_{n}^{s} = δ_{n}^{s}

i.e. the s-th level of γ_{n} is equal to the n-th level of δ^{s}.

2. The operations in (specific) structures are not in bold-face, whereas the operations on (specific) structures are in bold-face. Thus, for example, for s ≥ s′ and any

Theory and Applications of Categories, Vol. 8, No. 7 236

natural n, the projection inside a disk γ_{n} from level s to level s′ is marked p : γ_{n}^{s} ⟶ γ_{n}^{s′}, and the projection morphism between the simple categories δ^{s} to δ^{s′} is marked p : δ^{s} ⟶ δ^{s′}. Then the some specific instances of bold and not bold morphisms coincide, e.g.

p_{n}^{s} : γ_{n}^{s+1} ⟶ γ_{n}^{s} = p_{n}^{s} : δ_{n}^{s+1} ⟶ δ_{n}^{s}.

7.1. ud-vectors.

1. u⃗: a ud-vector [128];

2. Some functions defined on ud-vectors: (a) lt(u⃗): the length of a ud-vector u⃗ [128];

(b) ht(u⃗): the height of a ud-vector u⃗ [128];

(c) size_{(l)}(u⃗): the l-size of u⃗ [129];

3. Some sets of ud-vectors:

(a) UD_{n}: the set of ud-vectors of height less or equal n ∈ ω [131];

(b) UD_{n_{0},n_{1},n_{2}}: the set of pairs of n_{1}-compatible ud-vectors of height less or equal

n_{0} and n_{2}, respectively [131];

4. Some operations defined on ud-vectors: (a) tr_{(l)}(u⃗): the l-truncation of u⃗ [129];

(b) [u⃗, l, v⃗]: the l-amalgam of l-compatible pair ud-vectors u⃗ and v⃗ [129]. 7.2. Trees, bundles, and disks.

1. A tree T [123]:

[figure: ⋯; T^{s+1}; p^{s+1}; T^{1}; T^{0} ≅ 1; p^{0}; T^{s}; p^{s}; …]

2. Some special trees:

(a) θu⃗: the tree corresponding to a ud-vector u⃗ [123, 158];

(b) The tree θ_{3} can be drawn as [123]:

0 1 2 3

(c) The tree θ_{3,2,3,1,2,0,1,0,4,1,3} can be drawn as [158]:

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •]

3. A bundle of intervals over B [123]:

[figure: r^{0}; R; X; r^{1}; b; B; p; t]

4. A (finite) disk D [123]:

[figure: ⋯; D^{s+1}; p^{s+1}; t^{s+1}; b^{s+1}; D^{1}; D^{0} ≅ 1; p^{0}; t^{0}; b^{0}; D^{s}; p^{s}; t^{s}; b^{s}; …]

(a) ∂(D^{n}) = b(D^{n−1}) ∪ c(D^{n−1}): the boundary of D^{n} [124];

(b) ı(D^{n}) = D^{n} \ ∂(D^{n}): the interior of D^{n} [124];

(c) ı(D): the internal tree of the disk D [124];

(d) x < y: the successor in a fiber relation, x, y ∈ D_{n}, n ∈ ω [145];

(e) μ_{x,y}^{D} = max{l : p^{(l)}(x) = p^{(l)}(y)}: the level on which the nodes x and y of D do

match [145].

5. Some special disks:

(a) γu⃗: the disk corresponding to a ud-vector u⃗ [156];

(b) The first six levels of γ_{3} (the inner nodes are marked bold) [146]:

[figure: 0; 0; 1; 2; 0; 1; 2; 3; 4; 0; 1; 2; 3; 4; 5; 6; 0; 1; 2; 3; 4; 5; 6; 7; 0; 1; 2; 3; 4; 5; 6; 7]

(c) A bundle in a disk γ_{n} over γ_{n}^{s}, s ≥ 0,n ≥ 0 [192]:

[figure: ρ_{n}^{0;s+1}; ≤_{n}^{s+1}; γ_{n}^{s+1}; ρ_{n}^{1;s+1}; b_{n}^{s+1}; γ_{n}^{s}; p_{n}^{s}; t_{n}^{s+1}]

6. Some special disk morphisms:

(a) d = d_{(s)} = d_{(n;s)}, c = c_{(s)} = c_{(n;s)} : γ_{n} ⟶ γ_{s}: the domain and the codomain morphisms [148];

(b) mu⃗ : γu⃗ ⟶ γ_{ht(u⃗)}: the compositions [164];

(c) ι = ι_{(l)} = ι_{(s;l)} : γ_{s} ⟶ γ_{l}: the identity morphisms, for l ≥ s [164];

(d) in the diagram: l ≤ n

[figure: d_{(l)}^{s}; γ_{n}^{s}; γ_{l}^{s}; ι_{(n)}^{s}; c_{(l)}^{s}]

(e) π_{i} : γu⃗ ⟶ γ_{u_{2i}}: the projections, for i ∈ lh(u⃗) [156]:

[figure: γ_{u_{0}}; γ_{u_{2}}; γ_{u_{1}}; c_{u_{1}}; d_{u_{1}}; γ_{u_{2k−2}}; γ_{u_{2k}}; γ_{u_{2k−1}}; c_{u_{2k−1}}; d_{u_{2k}}; …; γu⃗; π_{2}; π_{2k−1}; π_{2k}; π_{0}]

Theory and Applications of Categories, Vol. 8, No. 7 239

(f) n-cuts of disks (outer morphisms into γ_{n}) [168]: \overline{x} : D ⟶ γ_{n}: if x ∈ D^{n} is a leaf in D [158]; \overline{x; y} : D ⟶ γ_{n_{1}}: if x ≠ y, n_{1} = μ_{x,y}, p^{n_{1}+1}(x) < p^{n_{1}+1}(y) [158];

(g) the canonical factorization of a morphism f in D [166]:

[figure: γu⃗; γ_{u}; mu⃗; D; γ_{n}; f; f^{∘}; ι_{(n)}]

with f^{∘} outer part of f, ι_{(n)} inner mono, mu⃗ inner epi;

(h) Cut(D): the ω-graph of cuts of D [168];

(i) A (fragment of) an internal ω-category C in D at level s, s ≥ 0, n ≥ 0 [167]:

[figure: π_{0,n+1}^{s}; γ_{n+1,n,n+1}^{s}; γ_{n+1}^{s}; m_{n+1,n,n+1}^{s}; π_{1,n+1}^{s}; d_{n}^{s}; γ_{n}^{s}; ι_{n+1}^{s}; c_{n}^{s}]

7.3. ω-graphs and ω-categories.

1. A general ω-graph G [125, 210]:

[figure: d_{n}; G_{n+1}; G_{n}; c_{n}; …; …]

(a) d_{u⃗;l}^{G}, c_{u⃗;l}^{G} : Gu⃗ ⟶ G_{tr_{(l)}(u⃗)}: the multi-domain and the multi-codomain morphisms [136];

(b) G_{n+1}(x, y): hom-set of n + 1-cells with domain x and codomain y in G [125];

(c) x > y: predecessor relation [125];

(d) max(G_{n}): the set of maximal elements in G_{n} [127];

(e) max(G_{n+1}, x, y): the maximal element of G_{n+1}(x, y) [127];

(f) x ⊥ y : x is comparable with y [145];

(g) ν_{e,e′}^{G} = max{l : d_{(l)}(e) ⊥ d_{(l)}(e′)}: the level on which domains (and codomains)

of cells e and e′ are comparable [175].

2. Some special simple ω-graphs:

(a) αu⃗: the ω-graph corresponding to a ud-vector u⃗ [176];

(b) α^{4} can be pictured as follows [175]:

[figure: 1; 0; 3; 2; 5; 4; 7; 6; 8]

3. Some special morphisms of simple ω-graphs: (a) b, t : α^{n} ⟶ α^{n′}: the bottom and the top morphisms [176];

(b) b^{u⃗;l}, t^{u⃗;l} : α^{tr_{(l)}(u⃗)} ⟶ αu⃗: the multi-bottom and the multi-top morphisms [177];

(c) κ^{i} : α^{u_{2i}} ⟶ αu⃗: the coprojections into αu⃗ [176]:

[figure: α^{u_{0}}; α^{u_{2}}; α^{u_{1}}; b^{u_{0}}; t^{u_{2}}; α^{u_{2k−2}}; α^{u_{2k}}; α^{u_{2k−1}}; b^{u_{2k−2}}; t^{u_{2k}}; …; αu⃗; κ^{2}; κ^{2k−2}; κ^{2k}; κ^{0}]

(d) κ^{0;u⃗} : αu⃗ ⟶ α^{[u⃗,l,v⃗]}, κ^{1;v⃗} : αv⃗ ⟶ α^{[u⃗,l,v⃗]}: the coprojections into α^{[u⃗,l,v⃗]} [177].

4. A general ω-category A [125, 210]:

(a) d_{(s)}^{A} = d_{n;s}^{A}, c_{(s)}^{A} = c_{n;s}^{A} : A_{n} ⟶ A_{s}: the domain and the codomain morphisms [210];

(b) d_{u⃗;l}^{A}, c_{u⃗;l}^{A} : Au⃗ ⟶ A_{tr_{(l)}(u⃗)}: the multi-domain and the multi-codomain morphisms [136];

(c) ι_{(n,l)}^{A} = ι_{(l)}^{A} = ι^{A} : A_{n} ⟶ A_{l}: the identities [211];

(d) m_{n_{0},n_{1},n_{2}}^{A} = m_{n_{1}}^{A} = m^{A} : A_{n_{0},n_{1},n_{2}} ⟶ A_{max(n_{0},n_{2})}: the compositions [211];

(e) mA ; u⃗ : Au⃗ ⟶ A_{ht(u⃗)}: the canonical composition morphism [138];

(f) π_{i..j}^{A} : Au⃗ ⟶ A_{u_{2i},...,u_{2j}}: the projection i ≤ j [136];

(g) π_{o;u⃗}^{A} : A_{[u⃗,l,v⃗]} ⟶ Au⃗, π_{1;v⃗}^{A} : A_{[u⃗,l,v⃗]} ⟶ Av⃗: the projections [137].

Theory and Applications of Categories, Vol. 8, No. 7 241

5. The free ω-categories:

(a) [G]: the free ω-category on a ω-graph G [179, 225];

(b) [G]_{n} = ∐_{u⃗∈UD_{n}} Gu⃗: the set/object of n-cells in [G] [225];

(c) κu⃗ = κ_{u⃗;n} : Gu⃗ ⟶ [G]_{n}: the coprojections [225];

(d) [G]_{n_{0},n_{1},n_{2}} = ∐_{(u⃗,v⃗)∈UD_{n_{0},n_{1},n_{2}}} G_{[u⃗,n_{1},v⃗]}: the set/object of n_{1}-compatible pairs of n_{0}- and n_{2}-cells in [G] [226];

(e) κ_{u⃗,v⃗} = κ_{u⃗,v⃗;n_{0},n_{1},n_{2}} : G_{[u⃗,n_{1},v⃗]} ⟶ [G]_{n_{0},n_{1},n_{2}}: the coprojections [226]; ;

(f) δu⃗ = [αu⃗]: the simple ω-category corresponding to a ud-vector u⃗ [184];

(g) A fragment of a simple ω-category δ^{s} in S, s ≥ 0, n ≥ 0 [184]:

[figure: π_{0,n+1}^{s}; δ_{n+1}^{s} ×_{δ_{n}^{s}} δ_{n+1}^{s}; δ_{n+1}^{s}; m_{n+1,n,n+1}^{s}; π_{1,n+1}^{s}; d_{n}^{s}; δ_{n}^{s}; ι_{n+1}^{s}; c_{n}^{s}]

(h) l ≤ n

[figure: d_{(l)}^{s}; δ_{n}^{s}; δ_{l}^{s}; ι_{(n)}^{s}; c_{(l)}^{s}]

the composite morphisms;

(i) mac_{G} and mac_{S}: the maximal cells in [G] and S, respectively (if exist) [126].

6. Some special morphisms of simple ω-categories: (a) b, t : δ^{s} ⟶ δ^{s′}: the bottom and the top morphisms [185];

(b) b^{s⃗;l}, t^{s⃗;l} : δ^{tr_{(l)}(s⃗)} ⟶ δs⃗: the multi-bottom and the multi-top morphisms [185];

(c) κ^{0;u⃗} : δu⃗ ⟶ δ^{[u⃗,l,v⃗]} κ^{1;v⃗} : δv⃗ ⟶ δ^{[u⃗,l,v⃗]} the embeddings into the coproduct [185];

(d) l ≤ s

[figure: t_{n}^{(s)}; δ_{n}^{s}; δ_{n}^{l}; p_{n}^{(l)}; b_{n}^{(s)}]

the composite morphisms;

(e) ↓ : [G] ⟶ δ^{0}, e↓e′ : [G] ⟶ δ^{s+1}: the morphisms into δ^{n}, e > e′, e, e′ ∈ G_{s}

Theory and Applications of Categories, Vol. 8, No. 7 242

(f) p^{s} : δ^{s+1} ⟶ δ^{s}: the projection [187];

(g) ρ^{0;s+1}, ρ^{1;s+1} : δ^{s+1,s,s+1} ⟶ δ^{s+1}, ρ^{s+1} : δ^{s+1} ⟶ δ^{s+1,s,s+1}: the order on δ^{s+1} [187];

(h) The (a fragment of) the internal disk D in S at level n, s ≥ 0, n ≥ 0 [187]; :

[figure: ρ_{n}^{0;s+1}; δ_{n}^{s+1,s,s+1}; δ_{n}^{s+1}; ρ_{n}^{1;s+1}; b_{n}^{s}; δ_{n}^{s}; p_{n}^{s}; t_{n}^{s}]

(i) ϱu⃗ : δ^{u} ⟶ δu⃗: the unique essential morphism from δ^{u} to δu⃗ [201].

References

[BD] J. Baez, J. Dolan, Higher-dimensional algebra II: n-Categories and the algebra of opetopes. Advances in Math. 135 (1998), 145-206.

[HMP] C. Hermida, M. Makkai, J. Power, On weak higher dimensional categories, I. Part 1. J. Pure and Applied Alg. 153 (2000), pp 221-246. Parts 2 and 3: to appear in the same Journal.

[B] M. Batanin, Monoidal globular categories as a natural environment for the theory of weak n-categories. Advances in Math. 136 (1998), 39-103.

[BS] M. Batanin, R.Street The universal property of the multitude of trees. J. Pure and Applied Algebra 154 (2000), 3-13.

[Be] C. Berger, A cellular nerve for higher order categories. Preprint, (1999).

[BV] J. M. Boardman and R. M. Vogt, Homotopy Invariant Algebraic Structures on Topological Spaces. Lecture Notes in Mathematics, vol 347, Springer-Verlag, (1973).

[J] A. Joyal, Disks, Duality and Θ-categories. Preprint, (1997).

[CWM] S. MacLane, Categories for Working Mathematician, Springer-Verlag, New York, (1971).

[K] D. Kan, A combinatorial definition of homotopy groups, Annals of Mathematics 67 (1958), 282-312.

[SGL] S. MacLane, I.Moerdijk, Sheaves in Geometry and Logic: a first introduction to topos theory, Springer-Verlag, New York, (1992).

[MM1] M.Makkai, Protocategories, Talk at the Category Theory Special Session of the Summer Meeting of the CMS in Saint John, NB, June, 1998.

[MM2] M.Makkai, The multitopic omega-category of all multitopic omega-categories. Preprint (1999).

[MM3] M.Makkai, Towards a categorical foundation of mathematics. In Logic Colloquium ’95 (J.A.Makowski and E.V.Ravve, eds.), Lecture Notes in Logic, vol II, Springer-Verlag, New York, (1998), 153-190.

[MZ] M. Makkai and M. Zawadowski, Protocategories: the link between Joyal’s thetacategories, and multitopic categories. In preparation.

[L] T.Leinster, Operads in Higher-Dimensional Category Theory. PH.D. thesis, University of Cambridge, England, (2000).

[Ma] F.Marty, Sur generalisation des 3-categories augmentt d’Olivier Leroy, II. preprint, University of Montpellier II, France, (1998).

[S] R.Street, The petit topos of globular sets. J. Pure and Applied Algebra 154 (2000), 299-315.

Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St., Montréal, PQ, H3A 2K6, Canada Instytut Matematyki, Uniwersytet Warszawski ul. S.Banacha 2, 00-913 Warszawa, Poland

Email: makkai@math.mcgill.ca and zawado@mimuw.edu.pl

This article may be accessed via WWW at http://www.tac.mta.ca/tac/ or by anony-

mous ftp at ftp://ftp.tac.mta.ca/pub/tac/html/volumes/8/n7/n7.{dvi,ps}

THEORY AND APPLICATIONS OF CATEGORIES (ISSN 1201-561X) will disseminate articles that significantly advance the study of categorical algebra or methods, or that make significant new contributions to mathematical science using categorical methods. The scope of the journal includes: all areas of pure category theory, including higher dimensional categories; applications of category theory to algebra, geometry and topology and other areas of mathematics; applications of category theory to computer science, physics and other mathematical sciences; contributions to scientific knowledge that make use of categorical methods.

Articles appearing in the journal have been carefully and critically refereed under the responsibility of members of the Editorial Board. Only papers judged to be both significant and excellent are accepted for publication.

The method of distribution of the journal is via the Internet tools WWW/ftp. The journal is archived electronically and in printed paper format.

Subscription information. Individual subscribers receive (by e-mail) abstracts of articles as they are published. Full text of published articles is available in .dvi, Postscript and PDF. Details will be e-mailed to new subscribers. To subscribe, send e-mail to tac@mta.ca including a full name and postal address. For institutional subscription, send enquiries to the Managing Editor, Robert Rosebrugh,

rrosebrugh@mta.ca.

Information for authors. The typesetting language of the journal is TEX, and LaTEX is the preferred flavour. TEX source of articles for publication should be submitted by e-mail directly to an appropriate Editor. They are listed below. Please obtain detailed information on submission format and style files from the journal’s WWW server at http://www.tac.mta.ca/tac/. You may also write to tac@mta.ca to receive details by e-mail.

Editorial board.

John Baez, University of California, Riverside: baez@math.ucr.edu Michael Barr, McGill University: barr@barrs.org, Associate Managing Editor Lawrence Breen, Université Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: r.brown@bangor.ac.uk Jean-Luc Brylinski, Pennsylvania State University: jlb@math.psu.edu Aurelio Carboni, Università dell Insubria: aurelio.carboni@uninsubria.it P. T. Johnstone, University of Cambridge: ptj@dpmms.cam.ac.uk G. Max Kelly, University of Sydney: maxk@maths.usyd.edu.au Anders Kock, University of Aarhus: kock@imf.au.dk F. William Lawvere, State University of New York at Buffalo: wlawvere@acsu.buffalo.edu Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.uu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Andrew Pitts, University of Cambridge: Andrew.Pitts@cl.cam.ac.uk Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca, Managing Editor Jiri Rosicky, Masaryk University: rosicky@math.muni.cz James Stasheff, University of North Carolina: jds@math.unc.edu Ross Street, Macquarie University: street@math.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Insubria: walters@fis.unico.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca