Theory and Applications of Categories, Vol. 4, No. 5, 1998, pp. 82–136.

A 2-CATEGORICAL APPROACH TO CHANGE OF BASE AND GEOMETRIC MORPHISMS II

A.CARBONI, G.M.KELLY, D.VERITY AND R.J.WOOD

Transmitted by Michael Barr

ABSTRACT. We introduce a notion of equipment which generalizes the earlier notion of pro-arrow equipment and includes such familiar constructs as relK, spnK, parK, and proK for a suitable category K, along with related constructs such as the V-pro arising from a suitable monoidal category V. We further exhibit the equipments as the objects of a 2-category, in such a way that arbitrary functors F : L \overline{}K induce equipment arrows relF : relL \overline{}relK, spnF : spnL \overline{}spnK, and so on, and similarly for arbitrary monoidal functors V \overline{}W. The article I with the title above dealt with those equipments M having each M(A, B) only an ordered set, and contained a detailed analysis of the case M = relK; in the present article we allow the M(A, B) to be general categories, and illustrate our results by a detailed study of the case M = spnK. We show in particular that spn is a locally-fully-faithful 2-functor to the 2-category of equipments, and determine its image on arrows. After analyzing the nature of adjunctions in the 2-category of equipments, we are able to give a simple characterization of those spnG which arise from a geometric morphism G.

1. Introduction

1.1. Given a regular category K we have the bicategory relK whose objects are those of K, whose arrows are the relations in K, and whose transformations (2-cells) are containments. Evidently, a functor G : K \overline{}L between regular categories which is left exact and preserves regular epimorphisms (called a regular functor) gives rise to a homomorphism of bicategories relK \overline{}relL that warrants the name relG. This simple observation does not suffice for a detailed examination of “change of base” problems — by which we mean generally the effect of functors such as G : K \overline{}L on constructs such as relK. Rather, one is led, notably when considering adjunctions, to consider what can be put forward as an arrow relG : relK \overline{}relL when G : K \overline{}L is an arbitrary functor between regular categories. The rationale was discussed fully in [CKW] and an answer provided there. In general, relG is not a morphism of bicategories between those in question (nor any of their duals) but something different in kind. In [CKW] a general theory was explored that is applicable not only to rel but also to other constructions that yield bicategories whose hom categories are only ordered sets (also called locally ordered bicategories), such as those of order ideals and of partial maps. A sequel was promised that would extend

The authors gratefully acknowledge financial support from the Italian CNR, the Australian ARC and the Canadian NSERC. Diagrams typeset using M. Barr’s diagram macros.

Received by the editors 1995 September 15 and, in revised form, 1998 January 15. Published on 1998 July 1. 1991 Mathematics Subject Classification: 18A25. Key words and phrases: equipment, adjunction, span. \begin{matrix}c \\ ◯ \end{matrix} A.Carboni, G.M.Kelly, D.Verity and R.J.Wood 1998. Permission to copy for private use granted.

the considerations of [CKW] to constructions such as those of spans and profunctors, for which the resulting bicategories are usually not locally ordered. Independently, the more general problem had also been considered in [VTY] and the present article builds on both earlier works.

1.2. To continue for a moment with relations and a regular category K, consider (−)_{∗} : K \overline{}relK, defined by sending an arrow f : X \overline{}A of K to its graph f_{∗} : X \overline{}A. A naive analogy with ring-homomorphisms allows us to consider relK as a “K-algebra” via (−)_{∗}. For a regular functor G : K \overline{}L, the homomorphism of bicategories relG : relK \overline{}relL is then seen as a “homomorphism” of “algebras” that takes account of the variation of “scalars” provided by G. The key idea we build on here is that a “Kalgebra” is also a “K, K-bimodule” together with a “multiplication”. It transpires that if we forget “multiplication” and concentrate instead on the “actions” giving the bimodule structure, then an arbitrary functor G : K \overline{}L gives rise to a “morphism” of “bimodules”, compatible with the variation of “scalars”. While this analogy was not explicitly expressed in [CKW] and [VTY], the results there are now easily seen in this context.

1.3. Just as for rings, it turns out be useful to study K, L-(bi)modules and see the K, Kmodules alluded to above, which we call equipments, as a special case. Loosely speaking, a K, L-module M is a CAT-valued profunctor M : L \overline{}K, so our use of the word “module” is in fact well-established. Section 2 provides the precise definition of modules in this context and of the 2-category MOD of all modules. Its goal is a characterization of adjunctions in MOD in terms of the data that arise in change-of-base considerations. Many, but not all, of the K, L-modules M arising in change-of-base problems have the property that for each l : L \overline{}L′ in L, the action functor l(−) = M(K, l) : M(K, L) \overline{}M(K, L′) has a right adjoint l^{∗}(−); while for each k in K, the action (−)k has a left adjoint (−)k^{∗}; and the resulting families of adjunctions satisfy conditions of the Beck-Chevalley type. We speak of starred modules and of starred equipments in these cases; and throughout Section 2 we conduct a parallel study of the simplifications, particularly with respect to adjunctions, that ensue in the starred case. We emphasize that being starred is a property, the “starring” being essentially unique when it exists.

1.4. In this paper we have no need to consider a full “K-algebra” stucture on a K, Kmodule M; yet, by analogy with the identity element of an algebra, the additional stucture of a “base point” for such an equipment provides an important tightening of the ideas that are encountered in change of base. For any category K, the hom-functor provides a simple, canonical K, K-module structure on K; and a pointing for M is but a certain kind of arrow (−)_{∗} : K \overline{}M in the 2-category MOD of all modules. Section 3 is devoted to the study of pointed equipments, and a starred pointed equipment is defined to be a pointed and starred equipment for which the pointing structure is suitably compatible with the starring. Once again, the goal of this section is a characterization of adjunctions, this time in the 2-category EQT_{∗} of pointed equipments, in terms of the data that tend to arise in change of base. Here too, we keep track of the simplifications provided by the starred property; and the result is a theorem, 3.19, that fully extends Theorem 3.5 of [CKW] from the locally-ordered to the general case.

1.5. In [CKW] the theory was illustrated in detail with a section devoted to rel seen as a colax functor from the 2-category REG of regular categories, arbitrary functors, and natural transformations that, in the context of this paper, takes values in the 2category ^{∗}EQT_{∗} of starred pointed equipments. In Section 4 we carry out a similar analysis for spans, beginning with the observation that the span construction, denoted by spn, provides a locally-fully-faithful 2-functor to the 2-category of starred pointed equipments. Here the domain of spn is the 2-category of categories with pullbacks, arbitrary functors, and natural transformations. We determine its image on arrows, and give a simple characterization of those spnG which arise from a geometric morphism G.

2. Modules and Equipments

2.1. Informally, an equipment is a 4-sorted structure having objects ⋯, K, L, ⋯, scalar arrows f : K \overline{}L, vector arrows μ : K \overline{}L and vector transformations Φ : μ \overline{}ν : K \overline{}L. The objects and scalar arrows carry the structure of a category; for each pair of objects K, L, the vector arrows from K to L and the vector transformations between these carry the structure of a category; finally, there are actions of the scalars on the vectors, as suggested by

[figure: K′; k; μ; K; L; ν; Φ; =; μk; K′; L; νk; Φk]

and

[figure: μ; K; L; ν; Φ; L′; l; =; lμ; K; L′; lν; lΦ]

which are functorial in Φ, strictly unitary, and coherently associative in the three possible senses. (We will make this precise in the next subsection.)

To see spnK as such a structure, begin by taking the category of scalars to be K itself and, for each pair of objects K, L in K, taking the category of vectors from K to L to be the usual category spnK(K, L) of spans from K to L, a typical object of which we denote by x : K \overline{}S \overline{}L : a, or just (x, S, a). For k : K′ \overline{}K in K, (x, S, a)k is the span p : K′ \overline{}P \overline{}L : aq where

[figure: K′; K; k; P; S; q; p; x]

is a (chosen) pullback; while for l : L \overline{}L′, l(x, S, a) = (x, S, la).

An equipment may be starred: which is to say that for each scalar k the action (−)k has a left adjoint, denoted by (−)k^{∗}, and for each scalar l the action l(−) has a right adjoint, denoted by l^{∗}(−), with the adjunctions satisfying conditions of the Beck-Chevalley type to ensure that these further actions satisfy associative laws with each other and with the original actions.

The equipments of the form spnK, for K with pullbacks, are starred. For K′ \overline{}K : k in K, (x, S, a)k^{∗} is given by composition; and for L \overline{}L′ : l in K, l^{∗}(x, S, a) is given by a pullback.

An equipment may carry the further structure of a pointing, which provides a distinguished vector arrow ι_{K} : K \overline{}K for each object K, and isomorphisms fι_{K} ^{\overline{}^{≃}} ι_{L}f for each scalar f : K \overline{}L, these satisfying (1ι_{K} ^{\overline{}^{≃}} ι_{K}1) = 1_{ι_{K}} and the familiar hexagonal coherence condition. In the case of spnK, such “identity vectors” are provided by the spans 1 : K \overline{}K \overline{}K : 1.

Finally, a starred pointed equipment is a starred equipment together with a pointing for which the mates (in the sense of [K&S]) ι_{K}f^{∗} \overline{}f^{∗}ι_{L} of the isomorphisms fι_{K} ^{\overline{}^{≃}} ι_{L}f are again isomorphisms. In fact, spnK is a starred pointed equipment.

Our formal definition is the following. (Recall that a homomorphism of bicategories is said to be normal when it preserves identities strictly.)

2.2. Definition. An equipment with scalar category K is a normal homomorphism of bicategories M : K \overline{}Hom(K^{op}, CAT), where Hom(−, −) denotes normal homomorphisms of bicategories, strong transformations, and modifications.

All our homomorphisms of bicategories will be assumed to be normal, so for brevity we drop the word “normal” when speaking of such homomorphisms. It is evident in our displays of actions in 2.1 that the scalars acting from the left need bear no relationship to those acting from the right. We will need this extra generality to explore equipments fully, and so pose:

2.3. Definition. For categories K and L, by a K, L-module M is meant a homomorphism M : L \overline{}Hom(K^{op}, CAT) of bicategories. We also write M : L \overline{}K.

To give a homomorphism K^{op} \overline{}Hom(L, CAT) amounts to the very same thing. Accordingly, we often write such a module M as M : K^{op}, L \overline{}CAT, looking upon it as being compatibly a homomorphism in each variable separately, sometimes called a “bi-homomorphism”.

For each K in K and for each L in L, then, we have a category M(K, L), the vector arrows and vector transformations of our less formal definition in 2.1. (We use this “vector” terminology for general modules as well as equipments.) For each k : K′ \overline{}K in K and each L in L, the functor M(k, L) : M(K, L) \overline{}M(K′, L) provides pre-action by k as in the first display of 2.1 and for each K in K and each l : L \overline{}L′ in L, the functor M(K, l) : M(K, L) \overline{}M(K, L′) provides post-action by l as in the second display. When we need to mention them explicitly we write

ξ = ξ_{l′,l,μ} : (l′l)μ ^{\overline{}^{≃}} l′(lμ)

η = η_{l,μ,k} : l(μk) ^{\overline{}^{≃}} (lμ)k

ζ = ζ_{μ,k,k′} : (μk)k′ ^{\overline{}^{≃}} μ(kk′)

for the structural isomorphisms of M (which are natural in μ). Of course our normality assumptions on homomorphisms are equivalent to the actions being strictly unitary. As for the conditions on ξ, η, ζ arising from the homomorphism assumptions, see 2.5 below.

2.4. Before further consideration of modules we give some simple examples.

(i) spn′ For any category C, we define a module spn′C : C \overline{}C^{op}. Here spn′C(X, A) is the usual category of spans. For an arrow g : A \overline{}B in C we define g(x, S, a) : X \overline{}B to be the span (x, S, ga). In dealing with actions of the opposite of a given category it is convenient to write f^{∗} : Y \overline{}X for the arrow in C^{op} that is determined by Y \overline{}X : f in C, and now for such an f and for (x, S, a) : X \overline{}A we define (x, S, a)f^{∗} : Y \overline{}A to be the span (fx, S, a). In this example the associativities ξ, η and ζ are identities. If C has pullbacks then we can define further actions: for A \overline{}C : h we can define h^{∗}(x, S, a) : X \overline{}C to be (xp, P, q) where

[figure: S; A; a; P; C; q; p; h]

is a (chosen) pullback, and for k : Z \overline{}X we can define (x, S, a)k : Z \overline{}A using a similar pullback. These last two actions on the categories spn′C(X, A) produce a new module spn′′C : C^{op} \overline{}C, with spn′′C(X, A) = spn′C(X, A); for this module, of course, the ξ, η and ζ are not identities.

(ii) rel′′ Write rel′C(X, A) for the full subcategory of spn′C(X, A) determined by the monic spans. The module spn′C does not restrict to a module rel′C; but if C has pullbacks then the module structure of spn′′C : C^{op} \overline{}C restricts to give a module rel′′C : C^{op} \overline{}C.

(iii) par′ For a category C, let C_{mono} be the category whose objects are those of C and whose arrows are the monomorphisms of C. Define par′C(X, A) to be the full subcategory of spn′C(X, A) determined by the spans x : X \overline{}S \overline{}A : a with x in C_{mono}. The module structure of spn′C : C \overline{}C^{op} restricts to give a module par′C : C \overline{}Cmono^{op}. If C has pullbacks of arrows in C_{mono} along arbitrary arrows then the module structure of spn′′C : C^{op} \overline{}C restricts to a module Cmono^{op} \overline{}C with underlying categories the par′C(X, A).

(iv) pro′′ We write cat for the 2-category of small categories, cat_{0} for the underlying ordinary category of small categories and all functors, and set for the category of small sets. For small categories X and A, we set pro′′(X, A) = set^{A^{op}×X} and write Φ : X \overline{}A for a typical object of pro′′(X, A), which we call a profunctor from X to A. For a functor F : Y \overline{}X define ΦF : Y \overline{}A by (ΦF)(A, Y) = Φ(A, F Y). For a functor A \overline{}B : G, define G^{∗}Φ : X \overline{}B by (G^{∗}Φ)(B, X) = Φ(GB, X). In this way we get a module pro′′ : cat_{0}^{op} \overline{}cat_{0}. Observe too that this example can be “parametrized” in at least two ways. For a category K with pullbacks, there is an evident pro′′K : (catK)_{0}^{op} \overline{}(catK)_{0}. For a monoidal category V (more generally a bicategory V) there is V-pro′′ : (V-cat)_{0}^{op} \overline{}(V-cat)_{0}. In the case of cat there are also the actions HΦ for Φ : X \overline{}A and H : A \overline{}B, given by (HΦ)(B, X) = ∫^{A} B(B, HA) × Φ(A, X); which, together with ΦK^{∗} for Y \overline{}X : K given by a similar formula, give rise to a module pro′ : cat_{0} \overline{}cat_{0}^{op}, with pro′(X, A) = pro′′(X, A). With reasonable assumptions on K or V, this last remark also applies to the parametrized versions.

(v) hty Let top denote the category of small topological spaces. For r a non-negative real number, write I_{r} for the closed interval [0, r]. For spaces X and A define hty(X, A) to be the category whose objects are continuous functions f : X \overline{}A and whose arrows are homotopies with duration, as in Moore’s definition of the path space. Explicitly, an arrow from f to g is a pair (r, H) where r is a non-negative real and H : I_{r} × X \overline{}A is a continuous function with H(0, x) = fx and H(r, x) = gx for all x in X. Composition is given by pasting homotopies, the first component of the composite (s, K)(r, H) being s + r. With the evident actions we have a module hty : top \overline{}top. This example is interesting in that there is not an obvious “horizontal” composition of homotopies with duration, so that it is is a genuine example of a “module” rather than a bicategory. Were we to restrict to homotopies in the usual sense (with duration 1), we would gain horizontal composition but at the expense of associativity of “vertical” composition.

(vi) comod′ For R a commutative ring, write coalgR for the category of cocommutative R-coalgebras and coalgebra homomorphisms. For such coalgebras X and A, write comod′R(X, A) for the category of X, A-bicomodules. An object M : X \overline{}A of this is an R-module M together with compatible coactions x : M \overline{}X ⊗_{R} M and a : M \overline{}M ⊗_{R} A. An arrow Φ : M \overline{}N is an R-module homomorphism M \overline{}N which is also a homomorphism with respect to the coactions. For a coalgebra homomorphism g : A \overline{}B, we define gM : X \overline{}B to be M together with x : M \overline{}X ⊗_{R} M and (M ⊗_{R} g)a : M \overline{}M ⊗_{R} B. Similarly, for a coalgebra homomorphism Y \overline{}X : f, we define Mf^{∗} : Y \overline{}A by composition. The result is a module comodR : coalgR \overline{}(coalgR)^{op}. It should be compared with the coalgebra-indexed category of vector spaces in [G&P]. We have given this example independently of example (iv) so as to make it more readily accessible; but observe that (R-mod)^{op} is a symmetric monoidal category for which the one-object enriched categories are the R-coalgebras, so that the bicomodules can be seen as profunctors.

(vii) Any S-indexed category as in [P&S] is a module S^{op} \overline{}1 or, equivalently, a module 1 \overline{}S.

2.5. Given a K, L-module M : L \overline{}K, the action displays in 2.1 suggest that the data for M can also be considered to constitute those of a bicategory obtained via a glueing construction. This is the case. We write glM for the bicategory with objects those of K and L disjointly; with glM(K′, K) = K(K′, K), glM(K, L) = M(K, L), glM(L, L′) = L(L, L′), glM(L, K) = 0 (the empty category); with horizontal composition given by the actions; and with structural isomorphisms given by the ξ, η and ζ above. The coherence conditions on the latter, resulting from the definition of M as a homomorphism, are precisely those required to make glM a bicategory. Of course K and L are literally contained in glM, so that we have the cospan K \overline{}glM \overline{}L. Central to our work are squares in glM of the following kind:

[figure: L; L′; l; K; K′; k; μ; μ′; Φ; ,]

and it is convenient to speak of this as a square in M. The Grothendieck construction applied to the homomorphism M : K^{op}, L \overline{}CAT yields a span

K ^{\overline{}^{∂_{0}}} grM ^{\overline{}^{∂_{1}}} L,

with grM a category whose objects are triples (K, μ, L), where μ : K \overline{}L is a vector arrow of M, and whose arrows are triples (k, Φ, l) : (K, μ, L) \overline{}(K′, μ′, L′), where the data constitute a square in M. Composition is given by (horizontal) pasting of squares. That this composition is well defined and associative follows from Appendix A of Verity’s thesis [VTY], which provides an extension to bicategories of Power’s “pasting theorem”

[PAJ] for 2-categories. (A direct proof of the associativity of this particular pasting is interesting too, because it shows that the result does not depend on the invertibility of the ξ, η, and ζ in 2.3.) Clearly an identity for (K, μ, L) is provided by

[figure: L; L; 1_{L}; K; K; 1_{K}; μ; μ; 1_{μ}; .]

The functors ∂_{0} and ∂_{1} are the evident projections, regarded as domain and codomain with respect to the squares. In the terminology of [ST1], K \overline{}grM \overline{}L is a fibration from L to K. The module M can be recovered from this fibration; in particular, for each K and L we have in CAT a pullback

[figure: 1; K × L; (K, L); M(K, L); grM; .]

Observe that the typical arrow (k, Φ, l) : (K, μ, L) \overline{}(K′, μ′, L′) in grM has a canonical factorization as the composite

[figure: L; L′; l; K; K; 1_{K}; μ; L′; 1_{L′}; K; 1_{K}; lμ; μ′k; L′; 1_{L′}; K′; k; μ′; 1_{lμ}; Φ; 1_{μ′k}; .]

In the terminology of [ST1], (1_{K}, 1_{lμ}, l) is right cartesian and (k, 1_{μ′k}, 1_{L′}) is left cartesian, while (1_{K}, Φ, 1_{L′}) is an arrow in the fibre M(K, L). The arrows of these three basic kinds generate those of grM. Note that the first basic arrow is an identity if l is an identity, the second if Φ is an identity, and the third if k is an identity. The general right cartesian and left cartesian arrows for this fibration are easy to describe. It is convenient to record:

2.6. Lemma. An arrow (k, Φ, l) : (K, μ, L) \overline{}(K′, μ′, L′) in grM is

i) right cartesian if and only if k and Φ are invertible

ii) left cartesian if and only if Φ and l are invertible

iii) invertible if and only if each of k, Φ and l is invertible.

2.7. The starred equipments introduced informally in 2.1 are a special case of starred modules. In discussing these we use the language of liftings and extensions, introduced for 2-categories in [S&W], but equally applicable to bicategories.

2.8. Definition. A module M : L \overline{}K is said to be starred if

i) for each vector arrow μ : K′ \overline{}L′ in M and each scalar arrow l : L \overline{}L′ in L, the bicategory glM admits a right lifting

[figure: L; L′; l; l^{∗}μ; K′; μ; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; l̂μ]

of μ through l, whose right-lifting property moreover is respected by all arrows k : K \overline{}K′ in K; ii) similarly, for each ν : K \overline{}L in M and each k : K \overline{}K′ in K, the bicategory

glM admits a left extension

[figure: K; K′; k; L; ν; νk^{∗}; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; νǩ]

of ν along k, whose left extension property is respected by all arrows l : L \overline{}L′ in L.

In other words, M : L \overline{}K is starred if and only if

(a) each M(K′, l) : M(K′, L) \overline{}M(K′, L′) admits a right adjoint μ ↦ l^{∗}μ, which we may denote by M(K′, l^{∗}) : M(K′, L′) \overline{}M(K′, L);

(b) each M(k, L) : M(K′, L) \overline{}M(K, L) admits a left adjoint ν ↦ νk^{∗}, which we may denote by M(k^{∗}, L) : M(K, L) \overline{}M(K′, L);

(c) the mate

η̃: M(k, L)M(K′, l^{∗}) \overline{}M(K, l^{∗})M(k, L′) : M(K, L) \overline{}M(K′, L′)

of the isomorphism

η : M(K, l)M(k, L) \overline{}M(k, L′)M(K′, l) : M(K′, L) \overline{}M(K, L′)

under the adjunctions M(K, l) ⊣ M(K, l^{∗}) and M(K′, l) ⊣ M(K′, l^{∗}), is itself invertible; and

(d) the mate

η′ : M(k^{∗}, L′)M(K′, l) \overline{}M(K′, l)M(k^{∗}, L) : M(K, L) \overline{}M(K′, L′)

under the adjunctions M(k^{∗}, L) ⊣ M(k, L) and M(k^{∗}, L′) ⊣ M(k, L′) of the isomorphism η above is also invertible. (Recall that conditions such as (c) and (d) are said to be of the Beck-Chevalley type.)

Let us suppose that a choice has been made of the liftings and extensions above; we may always suppose it is so made that 1_{L}^{∗}μ = μ and ν1_{K}^{∗} = ν. Because composites of adjunctions are adjunctions, the structural isomorphisms ξ : (l′l)μ ^{\overline{}^{≃}} l′(lμ), η : l(μk) ^{\overline{}^{≃}} (lμ)k, and ζ : (μk)k′ ^{\overline{}^{≃}} μ(kk′) of M give rise to isomorphisms ξ̄: (l^{∗}l^{′∗})μ ^{\overline{}^{≃}} l^{∗}(l^{′∗}μ), η̄ : l^{∗}(μk^{∗}) ^{\overline{}^{≃}} (l^{∗}μ)k^{∗}, and ζ̄ : (μk^{′∗})k^{∗} ^{\overline{}^{≃}} μ(k^{′∗}k^{∗}); and besides the six isomorphisms ξ, η, ζ, ξ̄, η̄, ζ̄, we have as above the two Beck-Chevalley type isomorphisms η̃^{−1} : l^{∗}(μk) ^{\overline{}^{≃}} (l^{∗}μ)k and η^{′−1} : l(μk^{∗}) ^{\overline{}^{≃}} (lμ)k^{∗}. These eight isomorphisms provide between them the structural isomorphisms for three more modules in addition to M : L \overline{}K, to be denoted by M′ : L \overline{}K^{op}, M′′ : L^{op} \overline{}K, and M^{∗} : L^{op} \overline{}K^{op}, and having M′(K, L) = M′′(K, L) = M^{∗}(K, L) = M(K, L): in the case of M′, for instance, the action of l ∈ L(L, L′) on μ ∈ M(K, L) sends it to lμ, while the action of k^{∗} : K′ \overline{}K in K^{op} (which is just k : K \overline{}K′ in K) on μ ∈ M(K, L) sends it to μk^{∗}; the coherence conditions for M′ and the others follow from those for M by a simple application of the “naturality of mates” established in [K&S, Section 2].

Although we have obtained M′, M′′, and M^{∗} by starting with M : L \overline{}K and requiring M(K′, l) and M(k, L) to have respectively a right adjoint and a left adjoint satisfying Beck-Chevalley type conditions, observe that we might equally have begun with with any of the four modules M, M′,M′′, M^{∗}. For instance, we might begin with a module M′ : L \overline{}K^{op}, writing its actions as l, μ ↦ lμ and μ, k^{∗} ↦ μk^{∗}, where now k^{∗} : K′ \overline{}K again denotes the arrow of K^{op} corresponding to the arrow k : K \overline{}K′ of K; and then require the existence of the right adjoints in l(−) ⊣ l^{∗}(−) and (−)k^{∗} ⊣ (−)k, again subject to Beck-Chevalley conditions: the resulting M : L \overline{}K is a starred module, whose M′ (which is in any case determined only to within isomorphism) is isomorphic to that we began with. In practical examples, it is often M′ that is most simply described: as we saw with spn′C in 2.4, where the corresponding M : C \overline{}C is the starred module — indeed the equipment — spnC : C \overline{}C of 2.1.

To reconcile our approach to these matters with others, suppose that we are given a category K and a bicategory M whose objects are those of K. Any homomorphism of bicategories K \overline{}M which is the identity on objects determines a module M : K \overline{}K — and hence an equipment. In particular, when K is a regular category and M is the bicategory relK, the inclusion K \overline{}relK gives in this way the equipment K \overline{}K which is also called relK. For such a homomorphism to be proarrow equipment in the sense of [WRJ] one requires that K \overline{}M be locally fully faithful and that for each arrow k in K there be given an adjunction k ⊣ k^{∗} in M. Such adjunctions make the resulting equipment M : K \overline{}K starred. The objects of the 3-category F introduced in [CKW] are proarrow equipments with the bicategory M (and hence also K) merely locally ordered. The definition of proarrow equipment did not require that K be locally discrete. Had this condition been imposed in [CKW], a 2-category of starred locally ordered equipments would have resulted, rather than a 3-category. Only minor adjustments to the treatment in [CKW] are necessary to carry the results there into the present context.

2.9. We now define the 2-category MOD of all modules. An object (K, M, L) of MOD consists of categories K and L, along with a K, L-module M; we may often call it M for short. It determines as in 2.5 a span ∂_{0} : K \overline{}grM \overline{}L : ∂_{1} in CAT, called ĝrM. This span is an object of the functor-2-category [Λ, CAT], where Λ denotes the three-object category

0 \overline{}⋆ \overline{}1.

An arrow (K, M, L) \overline{}(R, N , S) is by definition an arrow ĝrM \overline{}ĝrN in [Λ, CAT] and is thus a triple (G, T, H) of functors making commutative

[figure: R; grN; ∂_{0}; K; grM; ∂_{0}; G; T; S; ∂_{1}; L; ∂_{1}; H; ;]

while a transformation (G, T, H) \overline{}(F, S, J) in MOD is by definition a transformation (G, T, H) \overline{}(F, S, J) in [Λ, CAT], consisting therefore of transformations t : G \overline{}F, u : T \overline{}S, and s : H \overline{}J for which ∂_{0}.u = t.∂_{0} and ∂_{1}.u = s.∂_{1}. It is immediate that MOD so defined is a 2-category, with a fully-faithful 2-functor ĝr : MOD \overline{}[Λ, CAT]. When using the one-letter notation M for a module it is convenient to write ĝrM as

M_{0} ^{\overline{}^{∂_{0}}} grM ^{\overline{}^{∂_{1}}} M_{1},

with T = (T_{0}, grT, T_{1}) for a typical arrow and u = (u_{0}, gru, u_{1}) for a typical transformation. Thus we have 2-functors (−)_{0}, (−)_{1}, gr : MOD \overline{}CAT, all of which are in fact representable (since one can easily exhibit modules M for which ĝrM is any of the representables 1 \overline{}1 \overline{}1, 1 \overline{}0 \overline{}0, and 0 \overline{}0 \overline{}1), along with 2-natural transformations ∂_{0} : gr \overline{}(−)_{0} and ∂_{1} : gr \overline{}(−)_{1}. We further define the 2-category ^{∗}MOD of all starred modules: it is just the full sub-2-category of MOD given by those objects (K, M, L) for which M is a starred module. Since an equipment is merely a module for which the scalars acting from the left are the same as those acting from the right, we define the 2-categories of equipments and of starred equipments by the following pullback diagrams (in which Δ denotes the diagonal 2-functor):

[figure: CAT; CAT × CAT; Δ; EQT; MOD; (−)_{#}; ((−)_{0}, (−)_{1}); and; CAT; CAT × CAT; Δ; ^{∗}EQT; ^{∗}MOD; (−)_{#}; ((−)_{0}, (−)_{1}); .]

Both 2-functors named (−)_{#} above are represented by the starred equipment M with M_{#} = 1 and glM = 1, while the restrictions of gr to EQT and to ^{∗}EQT are represented by the starred equipment M with M_{#} equal to the discrete category 2 and glM the arrow category 2.

2.10. For a K, L-module M : L \overline{}K, the bicategory glM has the dual (glM)^{coop} given by reversing both the arrows [(−)^{op}] and the transformations [(−)^{co}]. It is apparent from the first two diagrams in 2.1 that (glM)^{coop} results from glueing an L^{op}, K^{op}-module that we may as well call M^{coop} : K^{op} \overline{}L^{op}, giving (glM)^{coop} = gl(M^{coop}). Thus M^{coop}(L, K) = M(K, L)^{op}, while for the actions we have kμ in M^{coop} given by μk in M, and so on. One might think of defining M^{op} and M^{co} similarly; but it is only M^{coop}, in this generality, that is well-behaved with respect to the Grothendieck construction, in terms of which we defined MOD in 2.9. An arrow (k, Φ, l) : (K, μ, L) \overline{}(K′, μ′, L′) in grM is equally an arrow (l, Φ, k) : (L′, μ′, K′) \overline{}(L, μ, K) in gr(M^{coop}); this gives an isomorphism

(grM)^{op} ^{\overline{}^{≃}} gr(M^{coop}),

which in fact constitutes an isomorphism

(ĝrM)^{op} ^{\overline{}^{≃}} ĝr(M^{coop})

of spans, since it clearly takes ∂_{1}^{op} : (grM)^{op} \overline{}L^{op} to the ∂_{0} : gr(M^{coop}) \overline{}L^{op} for M^{coop}.

It follows that an arrow T : M \overline{}N in MOD, being the same thing as an arrow ĝrM \overline{}ĝrN of spans, is in effect the same thing as an arrow T^{coop} : M^{coop} \overline{}N^{coop} in MOD. Again, a transformation u : T \overline{}S : M \overline{}N in MOD, being a transformation u : T \overline{}S : ĝrM \overline{}ĝrN of spans, and hence a transformation u^{op} : S^{op} \overline{}T^{op} : (ĝrM)^{op} \overline{}(ĝrN)^{op}, is the same thing as a transformation

u^{coop} : S^{coop} \overline{}T^{coop} : M^{coop} \overline{}N^{coop}.

We conclude that (−)^{coop} constitutes an isomorphism

(−)^{coop} : MOD ^{\overline{}^{≃}} (MOD)^{co}

of 2-categories. Moreover, M^{coop} is starred when M is so; and similarly for the property of being an equipment; hence we have also, by restriction, similar dualities ^{∗}MOD ^{\overline{}^{≃}} ∗MOD^{co}, EQT ^{\overline{}^{≃}} EQT^{co}, and ^{∗}EQT ^{\overline{}^{≃}} ^{∗}EQT^{co}.

Since the other operations (−)^{co} and (−)^{op} suggested above by consideration of glM play no significant role for general modules we can henceforth abandon such meanings of (−)^{co} and (−)^{op}, thus freeing these symbols for a new role. The point is that, for starred modules, there are new involutary isomorphisms that it is convenient to denote by

(−)^{op} : ^{∗}MOD ^{\overline{}^{≃}} ^{∗}MOD

and

(−)^{co} : ^{∗}MOD ^{\overline{}^{≃}} (^{∗}MOD)^{co}.

Given the K, L-module M : L \overline{}K, we have an L, K-module M^{op} : K \overline{}L, where M^{op}(L, K) = M(K, L), and we may write μ : K \overline{}L in M as μ^{∗} : L \overline{}K in M^{op}. Then the actions for M^{op} are defined by setting kμ^{∗} = (μk^{∗})^{∗} and μ^{∗}l = (l^{∗}μ)^{∗}. Now μ^{∗} ↦ kμ^{∗} has, as required, a right adjoint ν^{∗} ↦ k^{∗}ν^{∗}, where k^{∗}ν^{∗} = (νk)^{∗}; for to give kμ^{∗} \overline{}ν^{∗}, that is (μk^{∗})^{∗} \overline{}ν^{∗}, is just to give μk^{∗} \overline{}ν, equivalently to give μ \overline{}νk, that is μ^{∗} \overline{}k^{∗}ν^{∗}. Similarly, μ^{∗} ↦ μ^{∗}l has a left adjoint ν^{∗} ↦ ν^{∗}l^{∗} where ν^{∗}l^{∗} = (lν)^{∗}. It is immediate that the Beck-Chevaley conditions of 2.7 are satisfied, so that M^{op} is indeed a starred L, K-module. There is an evident isomorphism gr(M^{op}) ^{\overline{}^{≃}} grM; so that each T : M \overline{}N can be seen as an arrow T^{op} : M^{op} \overline{}N^{op}, and similarly for transformations u : T \overline{}S : M \overline{}N, establishing the first of the two isomorphisms displayed above.

We can define the isomorphism (−)^{co} : ^{∗}MOD ^{\overline{}^{≃}} (^{∗}MOD)^{co} by composing (−)^{op} : ∗MOD ^{\overline{}^{≃}} ^{∗}MOD with (−)^{coop} : ^{∗}MOD ^{\overline{}^{≃}} (^{∗}MOD)^{co}. Accordingly, we have M^{co} : L^{op} \overline{}K^{op} with M^{co}(K, L) = M(K, L)^{op}, and the actions of L^{op} and K^{op} are l^{∗}, μ ↦ l^{∗}μ and μ, k^{∗} ↦ μk^{∗}. Now this action of L^{op} has ν ↦ lν as a right adjoint, and so on. Note that gr(M^{co}) ^{\overline{}^{≃}} (grM)^{op}. Of course the isomorphisms above restrict to dualities (−)^{op} : ^{∗}EQT ^{\overline{}^{≃}} ^{∗}EQT and (−)^{co} : ^{∗}EQT ^{\overline{}^{≃}} (^{∗}EQT)^{co}.

2.11. For a starred module M : L \overline{}K, besides the three duals above given by the starred modules M^{op} : K \overline{}L, M^{co} : L^{op} \overline{}K^{op}, and M^{coop} : K^{op} \overline{}L^{op}, we have from 2.7 the three modules M′ : L \overline{}K^{op}, M′′ : L^{op} \overline{}K, and M^{∗} : L^{op} \overline{}K^{op}. In general these seven modules are distinct: the only pair agreeing in domain and codomain is that given by M^{∗} and M^{co}, and these differ because M^{∗}(K, L) = M(K, L) while M^{co}(K, L) = M(K, L)^{op}.

Although M′, M′′, and M^{∗} are not in general starred, we can use the actions they involve to construct some useful starred modules. Thus M′ : L \overline{}K^{op} is given by a bi-homomorphism M′ : K, L \overline{}CAT, from which we derive a homomorphism M^{r} : K×L \overline{}CAT and so a module M^{r} : K×L \overline{}1, where 1 is the unit category; we merely set M^{r}(0, (K, L)) = M′(K, L) = M(K, L), and define the actions by taking (k, l)μ to be — we make here an arbitrary choice of bracketing — the (lμ)k^{∗} of M′ (and hence of M); then the module M^{r} is indeed starred, with (k, l)^{∗}μ = l^{∗}(μk). Similarly the actions of M′′ lead to a starred module M^{l} : 1 \overline{}K × L, with M((K, L), 0) = M(K, L) and with μ(k, l) = l^{∗}(μk). Finally M^{∗} : L^{op} \overline{}K^{op}, since it is given by a bi-homomorphism M^{∗} : K, L^{op} \overline{}CAT, which is equally a bi-homomorphism M^{∗} : L^{op}, K \overline{}CAT, gives rise to a module K \overline{}L involving the same actions as M^{∗}; however this module needs no new name, since it is clearly nothing but the starred module M^{op} : K \overline{}L.

Consider now the four categories grM, grM^{r}, grM^{op}, and grM^{l}. An object in any of these is in effect a triple (K, μ, L) where μ : K \overline{}L in M. An arrow from (K, μ, L) to (K′, μ′, L′) has in grM the form (k, Φ, l) where Φ : lμ \overline{}μ′k; it has in grM^{r} the form (k, Φ^{r}, l) where Φ^{r} : (lμ)k^{∗} \overline{}μ′; it has in grM^{op} the form (k, Φ^{∗}, l) where Φ^{∗} : μk^{∗} \overline{}l^{∗}μ′, and it has in grM^{l} the form (k, Φ^{l}, l) where Φ^{l} : μ \overline{}l^{∗}(μ′k). We may represent such arrows by their respective “squares”

[figure: L; L′; l; K; K′; k; μ; μ′; Φ; ,; L; L′; l; K; K′; k^{∗}; μ; μ′; Φ^{r}; ,; L; L′; l^{∗}; K; K′; k^{∗}; μ; μ′; Φ^{∗}; ,; L; L′; l^{∗}; K; K′; k; μ; μ′; Φ^{l}; ,]

which are unambiguous once we fix on the bracketings (lμ)k^{∗} for the domain of Φ^{r} and l^{∗}(μ′k) for the codomain of Φ^{l}.

However the adjunctions l(−) ⊣ l^{∗}(−) and (−)k^{∗} ⊣ (−)k provide a bijection between such Φ and such Φ^{r}; and equally between such Φ and such Φ^{l}, or between such Φ and such Φ^{∗}; for we are just replacing Φ by one of its mates in the sense of [K&S], under one or both of these adjunctions. It follows moreover, from the results in [K&S] on the naturality of mates, that these bijections respect composition in the various categories, thus providing isomorphisms of categories grM ≅ grM^{r} ≅ grM^{op} ≅ grM^{l}. These isomorphisms, moreover, are clearly compatible with the projections in the spans K \overline{}grM \overline{}L, 1 \overline{}grM^{r} \overline{}K × L, L \overline{}grM^{op} \overline{}K, K × L \overline{}grM^{l} \overline{}1, which constitute ĝrM, ĝrM^{r}, ĝrM^{op}, and ĝrM^{l}. It follows that an arrow T : M \overline{}N in ^{∗}MOD can be seen not only as a functor ĝrM \overline{}ĝrN, but equally as a functor ĝrM^{r} \overline{}ĝrN^{r}, and so on; similarly for transformations u : T \overline{}S in ^{∗}MOD.

Invertibility of any one of Φ, Φ^{r}, Φ^{∗}, and Φ^{l} above provides in general four distinct conditions on Φ; and such invertibilities will be important to us:

2.12. Definition. The square Φ : lμ \overline{}μ′k in M is said to be

(i) a commutative square if Φ is invertible;

(ii) a right square if Φ^{r} is invertible;

(iii) an exact square if Φ^{∗} is invertible;

(iv) a left square if Φ^{l} is invertible.

2.13. It will be necessary to have descriptions of the arrows and transformations of MOD in more primitive terms. For an arrow T : M \overline{}N in MOD it follows from the considerations of 2.5 that T is determined by its values on the objects of grM and by its values on the the three basic kinds of generating arrows of grM. For its value on the object (K, μ, L), the definitions in 2.9 oblige us to write T (K, μ, L) = (T_{0}K, (grT)μ, T_{1}L); but in cases where there is no danger of confusion we may safely write this as (T_{0}K, T μ, T_{1}L), or even as (T K, T μ, T L). Similarly, 2.9 prescribes that application of T to the canonical factorization in 2.5 of a square in M produce a diagram of the form:

[figure: T_{1}L; T_{1}L′; T_{1}l; T_{0}K; T_{0}K; 1_{T_{0}K}; T μ; T_{1}L′; 1_{T_{1}L′}; T_{0}K; 1_{T_{0}K}; T (lμ); T (μ′k); T_{1}L′; 1_{T_{1}L′}; T_{0}K′; T_{0}k; T μ′; T_{l,μ}; T Φ; T^{μ′,k}; .]

From 2.9 it follows that the restriction of grT to the fibre M(K, L) determines, for each K in K and L in L, a functor M(K, L) \overline{}N (T_{0}K, T_{1}L) that we call T_{K,L}. Thus in the middle square we have T Φ = (grT)Φ = T_{K,L′}Φ; moreover in this notation the effect of T on objects is given by T (K, μ, L) = (T_{0}K, T_{K,L}μ, T_{1}L). For the effect of T on (1_{K}, 1_{lμ}, l) and on (k, 1_{μ′k}, 1_{L′}) we have written (1_{T_{0}K}, T_{l,μ}, T_{1}l) and (T_{0}k, T^{μ′,k}, 1_{T_{1}L′}) respectively. It is convenient to refer to the T_{l,μ} and T^{μ′,k} as action comparisons. To determine the conditions to be satisfied by these derived data, one has only to determine

Theory and Applications of Categories, Vol. 4, No. 5 96

the relations satisfied by the generating arrows of grM, by examining the canonical factorization of 2.5 for each binary composite of generating arrows. We leave this task to the reader; completing it leads to the following: 2.14. Proposition. Giving an arrow T : M \overline{}N in MOD is equivalent to giving

(i) functors T_{0} : M_{0} \overline{}N_{0} and T_{1} : M_{1} \overline{}N_{1} (where we frequently write just T for both T_{0} and T_{1});

(ii) for each K in M_{0} and L in M_{1}, a functor T_{K,L} : M(K, L) \overline{}N (T_{0}K, T_{1}L) (whose

value at Φ : μ \overline{}μ′ we write for short as T Φ : T μ \overline{}T μ′); (iii) arrows T_{l,μ} : T l.T μ \overline{}T (lμ), natural in μ ∈ M(K, L); (iv) arrows T^{μ,k} : T (μk) \overline{}T μ.T k, natural in μ ∈ M(K, L); where these data are to satisfy the conditions

(T_{1_{L},μ} : T 1_{L}.T μ \overline{}T (1_{L}μ)) = (1_{T μ} : T μ \overline{}T μ),

(T^{μ,1_{K}} : T (μ1_{K}) \overline{}T μ.T 1_{K}) = (1_{T μ} : T μ \overline{}T μ),

and to make commutative the diagrams

[figure: T (l′l).T μ; T ((l′l)μ); T_{l′l,μ}; (T l′.T l).T μ; T l′.(T l.T μ); ξ; T (l′(lμ)); T ξ; T l′.T (lμ); T l′.T_{l,μ}; T_{l′,lμ}; ,]

[figure: T (l(μk)); T ((lμ)k); T η; T l.T (μk); T l.(T μ.T k); T l.T^{μ,k}; T_{l,μk}; T (lμ).T k; T^{lμ,k}; (T l.T μ).T k; η; T_{l,μ}.T k; ,] [figure: T (μk).T k′; (T μ.T k).T k′; T^{μ,k}.T k′; T ((μk)k′); T (μ(kk′)); T ζ; T^{μk,k′}; T μ.(T k.T k′); ζ; T μ.T (kk′); T^{μ,kk′}; .]

2.15. The right fibrations from L to K are described in [ST2] as the algebras for a KZdoctrine R_{L,K} on spnCAT(L, K). It takes but a simple extension of that account to show that those objects in [Λ, CAT] which are right fibrations are the algebras for a single KZdoctrine R on [Λ, CAT] — with R restricting to the R_{L,K}. Left fibrations are algebras for a dual KZ-doctrine L on [Λ, CAT]; while fibrations are algebras for a doctrine M obtained as the composite of L and R via a canonical isomorphic distributive law RL ≅ LR. Thus the objects of MOD are those M-algebras of the form ĝrM for a module M, but the arrows (and transformations) of MOD are just those of [Λ, CAT]. Again, a straightforward extension of [ST1] shows that a [Λ, CAT] arrow T : ĝrM \overline{}ĝrN is an R-homomorphism precisely when it takes right cartesian arrows to right cartesian arrows and it follows easily that this is the case precisely when all the T_{l,μ} are isomorphisms. Dually, T is an L-homomorphism precisely when all the T^{μ,k} are isomorphisms. From the general theory of KZ-doctrines (see [KCK]) we have:

2.16. Proposition. For any adjunction S ⊣ T : M \overline{}N in MOD, S is an Rhomomorphism and T is an L-homomorphism; that is, the S_{l,ν} and the T^{μ,k} are all invertible.

2.17. We now expand on the discussion begun in 2.13 for the case where T : M \overline{}N is in ^{∗}MOD. Evidently the discussion then applies equally to T^{op} : M^{op} \overline{}N^{op}. Consider the following squares:

[figure: L; L; 1^{∗}; K; K′; k^{∗}; μ; μk^{∗}; 1_{μk^{∗}}; L; L; 1; K; K′; k; μ; μk^{∗}; μk̃; T L; T L; 1; T K; T K′; T k; T μ; T (μk^{∗}); T L; T L; 1; T K; T K′; (T k)^{∗}; T μ; T (μk^{∗}); T_{k,μ}^{op}; .]

The first displays a basic right cartesian arrow in grM^{op} and the second is its mate in grM, where μk̃ is the μ-component of the unit of the adjunction (−)k^{∗} ⊣ (−)k for M. Application of T to the second square yields the third square in N, wherein the unlabelled arrow is necessarily T^{μk^{∗},k}.T (μk̃), and finally the fourth square is the mate in grN^{op} of the third square — which describes the effect of T^{op} on the first. Thus, explicitly,

[figure: T ((μk^{∗})k).(T k)^{∗}; (T (μk^{∗}).T k).(T k)^{∗}; T^{μk^{∗},k}.(T k)^{∗}; T μ.(T k)^{∗}; T (μk^{∗}); T_{k,μ}^{op}; T (μk̃).(T k)^{∗}; T (μk^{∗}).(T k); ˆ ,]

where (T k)ˆ is the counit of the adjunction (−)(T k)^{∗} ⊣ (−)T k. Dually, if the exercise above is carried out for a basic left cartesian arrow in grM^{op}, we arrive at a description of T_{μ,l}^{op}. Expressed in terms of actions in M and N it is convenient to label T_{k,μ}^{op} and (T^{op})^{μ,l} as

T_{μ,k^{∗}} : T μ.T k^{∗} \overline{}T (μk^{∗}) and T^{l^{∗},μ} : T (l^{∗}μ) \overline{}T l^{∗}.T μ

and refer to these also as action comparisons; we are now for convenience writing T k^{∗} for (T k)^{∗} and T l^{∗} for (T l)^{∗}, which are unambiguous in that T (k^{∗}) and T (l^{∗}) make no sense. Comparing the explicit formula above for T_{k,μ}^{op} = T_{μ,k^{∗}} with [K&S, section 2] reveals that

T_{−,k^{∗}} : N (k^{∗}, L)T_{K,L} \overline{}T_{K′,L}M(k^{∗}, L) : M(K, L) \overline{}M(K′, L)

is the mate under the adjunctions N (k^{∗}, L) ⊣ N (k, L) and M(k^{∗}, L) ⊣ M(k, L) in CAT of

T^{−.k} : T_{K,L}M(k, L) \overline{}N (k, L)T_{K′,L} : M(K′, L) \overline{}N (K, L);

with a dual result for T^{l^{∗},−}.

In M^{r} and M^{l} the only non-trivial cartesian arrows are respectively the right ones and the left ones, the typical such being displayed below:

[figure: L; L′; l; K; K′; k^{∗}; μ; (lμ)k^{∗}; 1_{(lμ)k^{∗}}; ,; L; L′; l^{∗}; K; K′; k; l^{∗}(μk); μ; 1_{l^{∗}(μk)}; .]

Proceeding as we did for M^{op} we find the further action comparisons

T_{l,μ,k^{∗}} : T l.(T μ.T k^{∗}) \overline{}T ((lμ)k^{∗}) and T^{l^{∗},μ,k} : T (l^{∗}(μk)) \overline{}(T l^{∗}.T μ).T k.

However, the factorization (k, 1)(1, l) = (k, l) = (1, l)(k, 1) in K × L and the observations T_{1,μ,k^{∗}} = T_{μ,k^{∗}} et cetera together with Proposition 2.14 provide:

2.18. Lemma. The following diagrams commute:

[figure: T l.T (μk^{∗}); T (lμ).T k^{∗}; T l.(T μ.k^{∗}); (T l.T μ).T k^{∗}; η; T l.T_{μ,k^{∗}}; T_{l,μ}.T k^{∗}; T (l(μk^{∗})); T ((lμ)k^{∗}); T η; T_{l,μk^{∗}}; T_{lμ,k^{∗}}; T_{l,μ,k^{∗}}; ,; T l^{∗}.T (μk); T (l^{∗}μ).T k; T (l^{∗}(μk)); T ((l^{∗}μ)k); T η; T^{l^{∗},μk}; T^{l^{∗}μ,k}; T l^{∗}.(T μ.T k); (T l^{∗}.T μ)T k; η; T l^{∗}.T^{μ,k}; T^{l^{∗},μ}.T k; T^{l^{∗},μ,k}; ;]

whence we easily conclude that

(i) each T_{l,μ,k^{∗}} is invertible if and only if each T_{l,μ} and each T_{μ,k^{∗}} are invertible,

(ii) each T^{l^{∗},μ,k} is invertible if and only if each T^{μ,k} and each T^{l^{∗},μ} are invertible.

2.19. Definition. An arrow T : M \overline{}N in ^{∗}MOD is said to be

(i) strong if each T_{l,μ} and each T^{μ,k} are invertible;

(ii) ^{∗}strong if each T_{μ,k^{∗}} and each T^{l^{∗},μ} are invertible;

(iii) a right homomorphism if each T_{l,μ,k^{∗}} is invertible; that is, if each T_{l,μ} and each T_{μ,k^{∗}} are invertible;

(iv) a left homomorphism if each T^{l^{∗},μ,k} is invertible; that is, if each T^{l^{∗},μ} and each T^{μ,k} are invertible;

(v) a homomorphism if it is both a right and a left homomorphism; that is, if it is both strong and ^{∗}strong.

Of course the term “strong” is also applicable to arrows in MOD. Preservation of the properties named in Definition 2.12 for a square in M being important properties for an arrow T : M \overline{}N in ^{∗}MOD, it is convenient to record: 2.20. Proposition. An arrow T : M \overline{}N in ^{∗}MOD

(i) preserves commutative squares if and only if it is strong;

(ii) preserves right squares if and only if it is a right homomorphism;

(iii) preserves exact squares if and only if it is ^{∗}strong;

(iv) preserves left squares if and only if it is a left homomorphism;

(v) preserves commutative squares and exact squares if and only if it preserves right squares and left squares if and only it is a homomorphism.

Because an adjunction S ⊣ T in ^{∗}MOD gives rise to further adjunctions S^{r} ⊣ T^{r} and S^{l} ⊣ T^{l}, Proposition 2.16 for MOD admits an improved form for starred modules: 2.21. Proposition. For any adjunction S ⊣ T : M \overline{}N in ^{∗}MOD, S is a right homomorphism and T is a left homomorphism. 2.22. We now complete the task set in 2.13 of also describing the transformations of MOD in more primitive terms. Certainly a transformation u : T \overline{}S : M \overline{}N in MOD involves natural transformations u_{i} : T_{i} \overline{}S_{i} : M_{i} \overline{}N_{i}, for i = 0, 1, and for each object (K, μ, L) of grM, an arrow in grN of the particular form

[figure: T_{1}L; S_{1}L; u_{1}L; T_{0}K; S_{0}K; u_{0}K; T μ; Sμ; u_{μ}; .]

Since it suffices to require naturality only with respect to the basic generating arrows of 2.5 we have:

2.23. Proposition. Giving a transformation u : T \overline{}S : M \overline{}N in MOD is equivalent to giving natural transformations u_{0} : T_{0} \overline{}S_{0} and u_{1} : T_{1} \overline{}S_{1} and the squares u_{μ} above, natural with respect to transformations Φ : μ \overline{}μ′ in M(K, L), and satisfying the following two conditions:

[figure: T_{1}L; T_{1}L′; T_{1}l; T_{0}K; T_{0}K; 1; T μ; T (lμ); T_{l,μ}; S_{1}L′; u_{1}L′; S_{0}K; u_{0}K; S(lμ); u_{lμ}; =; T_{1}L; S_{1}L; u_{1}L; T_{0}K; S_{0}K; u_{0}K; T μ; u_{μ}; S_{1}L′; S_{1}l; S_{0}K; 1; Sμ; S(lμ); S_{l,μ}; ,]

[figure: T_{1}L; S_{1}L; u_{1}L; T_{0}K′; S_{0}K′; u_{0}K′; T (μk); u_{μk}; S_{1}L; 1; S_{0}K; S_{0}k; S(μk); Sμ; S^{μ,k}; =; T_{1}L; T_{1}L; 1; T_{0}K′; T_{0}K; T_{0}k; T (μk); T μ; T^{μ,k}; S_{1}L; u_{1}L; S_{0}K; u_{0}K; u_{μ}; Sμ; .]

2.24. We will find it convenient to use “fibrational” properties of the 2-functors

((−)_{0}, (−)_{1}) : MOD \overline{}CAT × CAT and ((−)_{0}, (−)_{1}) : ^{∗}MOD \overline{}CAT × CAT.

We note that 2-fibrations have been defined and studied in [HRM], along with a characterization of adjunctions in the domain 2-category of a particular 2-fibration. Here we do no not need the full general theory of 2-fibrations, but on the other hand we require a very detailed analysis of adjunctions in MOD and ^{∗}MOD. Accordingly, we provide an independent account of the case at hand. We will see ((−)_{0}, (−)_{1}) : MOD \overline{}CAT × CAT as a fibrational structure from 1 to CAT × CAT (in the terminology of [ST1]).

The first of the 2-categories defined by the following pullbacks in 2-CAT

[figure: 1; CAT × CAT; (K, L); K, L-MOD; MOD; ((−)_{0}, (−)_{1}); and; 1; CAT × CAT; (K, L); K, L-^{∗}MOD; ^{∗}MOD; ((−)_{0}, (−)_{1})]

provides the “fibres” for ((−)_{0}, (−)_{1}). Of course, K, L-^{∗}MOD is then the full sub-2-category of K, L-MOD determined by the starred modules, while K, L-MOD itself is the (non-full) sub-2-category of MOD obtained by fixing K and L, and by requiring the T_{0}, T_{1}, u_{0}, and u_{1} of 2.9 to be identities. Thus to give an arrow T : M \overline{}N in K, L-MOD (or in K, L-^{∗}MOD) is to give functors T_{K,L} : M(K, L) \overline{}N (K, L) and to give arrows T_{l,μ} : l.T μ \overline{}T (lμ) and T^{μ,k} : T (μk) \overline{}T μ.k natural in μ, with these data satisfying what the conditions of Proposition 2.14 become when when we take T_{0} and T_{1} to be the identities of K and of L; and to give a transformation u : T \overline{}S : M \overline{}N in K, L-MOD (or in K, L-^{∗}MOD) is then to give natural transformations

u_{K,L} : T_{K,L} \overline{}S_{K,L} : M(K, L) \overline{}N (K, L)

satisfying what the conditions of Proposition 2.23 become when not only T_{0} and T_{1} but also u_{0} and u_{1} are identities: namely the equations u_{lμ}.T_{l,μ} = S_{l,μ}.lu_{μ} and S^{μ,k}u_{μk} = u_{μ}k.T^{μ,k}.

2.25. From a module N : S \overline{}R together with functors G : K \overline{}R and H : L \overline{}S, we may construct by substitution a new module G^{+}N H : L \overline{}K, for which it is often more convenient to use the alternative notation N (G, H) : L \overline{}K. As a 2-variable homomorphism, N (G, H) : K^{op}, L \overline{}CAT is defined to be the result of composing N : R^{op}, S \overline{}CAT with G^{op}, H : K^{op}, L \overline{}R^{op}, S. Thus N (G, H)(K, L) = N (GK, HL), and so on, from which it is clear that

[figure: K × L; R × S; G × H; grN (G, H); grN]

is a pullback. Moreover, N (G, H) is starred if N is so. It follows that to give an arrow (G, T, H) : (K, M, L) \overline{}(R, N , S) in MOD (respectively ^{∗}MOD) is to give an arrow T̃ : M \overline{}N (G, H) in K, L-MOD (respectively K, L-^{∗}MOD). Of course T and T̃ are related as follows: if (G, T, H) sends the arrow (k, Ψ, l) : (K, μ, L) \overline{}(K′, μ′, L′) of grM to the arrow (Gk, Φ, Hl) : (GK, T μ, HL) \overline{}(GK′, T μ′, HL′) of grN, then T̃ sends it to the arrow (k, Φ, l) : (K, T μ, L) \overline{}(K′, T μ′, L′) of grN (G, H). An evident notation allows us to regard T̃ as a “square’

[figure: K; R; G^{+}; L; S; H; M; N; T̃; ;]

compare this with the Φ^{l} of 2.11. The description of N (G, H) as a pullback also facilitates the observation that an arrow S : N̄ \overline{}N in R, S-MOD (or in R, S-^{∗}MOD) provides an arrow

S_{G,H} : N̄ (G, H) \overline{}N (G, H)

in K, L-MOD (or in K, L-^{∗}MOD) with S_{G,H}(K, ν, L) = (K, Sν, L), and so on. Similarly, a transformation u : S̄ \overline{}S : N̄ \overline{}N in R, S-MOD (or in R, S-^{∗}MOD) provides a transformation u_{G,H} : S̄_{G,H} \overline{}S_{G,H} : N̄ (G, H) \overline{}N (G, H) in K, L-MOD (or in K, L- ∗MOD) with (u_{G,H})_{K,L} = u_{GK,HL}. Observe first that these assignments define a 2functor G^{+}(−)H : R, S-MOD \overline{}K, L-MOD and similarly in the starred case. Next observe that the assignments (K, L) ↦ K, L-MOD, together with the G^{+}(−)H, are strictly functorial in (G, H) : (K, L) \overline{}(R, S) in (CAT × CAT)_{0}^{op} because on objects the G^{+}(−)H are given by composition (rather than pullback).. Finally in this context, observe that the case of (K, L) : (1, 1) \overline{}(K, L) shows that all notations of this subsection are consistent with our usage above for components, and that we have “(K, L)-component 2functors” K, L-MOD \overline{}1, 1-MOD = CAT and K, L-^{∗}MOD \overline{}1, 1-^{∗}MOD = CAT.

2.26. Continuing the considerations above, suppose next that, besides the module N : S \overline{}R and the functor G : K \overline{}R, we have a natural transformation s : J \overline{}H : L \overline{}S. We show there is then an induced arrow N (G, s) : N (G, J) \overline{}N (G, H) of K, L-modules which sends the object (K, ν : GK \overline{}JL, L) of grN (G, J) to the object (K, sL.ν : GK \overline{}HL, L) of grN (G, H). Once again, we use the description of grN (G, H) as a pullback, at the beginning of 2.25, to exhibit the requisite functor ĝrN (G, J) \overline{}ĝrN (G, H). For this we need a functor grN (G, J) \overline{}grN whose effect on (K, ν : GK \overline{}JL, L) is (GK, sL.ν, HL). Let (k, Φ, l) : (K, ν, L) \overline{}(K′, ν′, L′) be an arrow in grN (G, J) and consider the following diagram in the bicategory glN:

[figure: HL; HL′; Hl; JL; JL′; Jl; sL; sL′; HL′′; Hl′; JL′′; Jl′; sL′′; GK; GK′; Gk; ν; ν′; Φ; GK′′; Gk′; ν′′; Φ′; .]

The value at (k, Φ, l) of the required functor is (Gk, Ψ, Hl) where Ψ is the vertical pasting composite of the two left-most squares, seen as a square Hl.(sL.ν) \overline{}(sL′.ν′).Gk. Given a further arrow (k′, Φ′, l′) in grN (G, J) as above, we see that the assignment is functorial by appealing to Verity’s pasting theorem; for in the diagram above, it makes no difference whether we first paste horizontally and then vertically, or first vertically and then horizontally. So we do have the desired N (g, s), and its value at the arrow (k, Φ, l) of grN (G, J) is the arrow (k, Ψ, l) of grN (G, H) with Ψ as above. The diagram above also shows that N (G, s) : N (G, J) \overline{}N (G, H) preserves commutative squares and is thus a strong arrow in MOD. Moreover if N, and hence N (G, J) and N (G, H), are starred, it follows from the naturality of mates in [K&S, Section 2] that the arrow Ψ^{r} in grN^{r} corresponding to the vertical pasting of the left-most squares above is

[figure: HL; HL′; Hl; JL; JL′; Jl; sL; sL′; GK; GK′; Gk^{∗}; ν; ν′; Φ^{r}; .]

Thus, when N is starred, N (G, s) also preserves right squares, and accordingly is a strong right homomorphism. Dually, given t : G \overline{}F we define N (t, H) : N (F, H) \overline{}N (G, H) by pasting naturality squares on top of squares in N; and N (t, H) is a strong arrow in MOD, and a strong left homomorphism in ^{∗}MOD when N is starred.

Of course the N (G, s) are not functorial in s — there are rather isomorphisms

N (G, s′s) ^{\overline{}^{≃}} N (G, s′)N (G, s),

constructed from the ξ for N. A similar remark holds for the N (t, H) and the ζ, while the η provide isomorphisms

N (G, s)N (t, H) ^{\overline{}^{≃}} N (t, H)N (G, s).

At the end of 2.25 we could have summarized by saying that we have a functor

(−, −)-MOD : (CAT × CAT)_{0}^{op} \overline{}(2-CAT)_{0}.

We leave it as an exercise for the interested reader to formulate the nature of the arrow with domain (CAT^{co} × CAT)^{op}, enriching the functor displayed above, which arises from the considerations of 2.26.

2.27. Remark. For the 2-functor ((−)_{0}, (−)_{1}) : MOD \overline{}CAT×CAT and fixed modules M and N, consider the effect-on-homs functor

MOD(M, N) \overline{}CAT × CAT((M_{0}, M_{1}), (N_{0}, N_{1})).

This can be seen as a span CAT(M_{0}, N_{0}) \overline{}MOD(M, N) \overline{}CAT(M_{1}, N_{1}), and the considerations of 2.26 can be adapted to show that it is a fibration from CAT(M_{1}, N_{1}) to CAT(M_{0}, N_{0}). This is a “two variable” version of the idea that appears in the “local characterization” theorem of 2-fibrations given in [HRM].

2.28. Note that N (G, s)_{K,L} : N (GK, JL) \overline{}N (GK, HL) is N (GK, sL), which for a starred N has in CAT the right adjoint N (GK, (sL)^{∗}), while we have already noted that N (G, s) is a right homomorphism. It will follow by Theorem 2.39 that for starred N the N (G, s) have right adjoints in ^{∗}MOD — in fact in K, L-^{∗}MOD. However, even now, we can define an arrow N (G, s^{∗}) : N (G, H) \overline{}N (G, J) of K, L-modules by prescribing its effect on an arrow (k, Φ, l) in N (G, H) to be the arrow in N (G, J) suggested by the symbolic diagram:

[figure: JL; JL′; Jl; HL; HL′; Hl; (sL)^{∗}; (sL′)^{∗}; GK; GK′; Gk; ν; ν′; Φ; .]

We say that this diagram is “symbolic” because it is not a diagram in any of the “gl” bicategories that have been introduced. (It involves composites of the form x.ν and y^{∗}.ν for x and y arrows of S. For this to make sense in our framework we would require a category containing both S and S^{op} which acts on the N (R, S) — but what is one to make then of the un-named transformation in the lower square?) The point here is that the equality natural transformation sL′.Jl(−) ^{\overline{}^{−}} Hl.sL(−) between the action-functors has the mate Jl.((sL)^{∗}(−) \overline{}(sL′)^{∗}.Hl(−) whose ν-component enables us to give a welldefined meaning to the displayed pasting composite. Using the mate calculus of [K&S] it is straightforward to establish a four-square interchange law, similar to that in 2.26, showing the assignment to be functorial. Moreover, contemplation of the squares in 2.11 and their relationship via mating shows that the N (G, s^{∗}) preserve exact squares and left squares and are thus by 2.20 ^{∗}strong left homomorphisms in K, L-^{∗}MOD. Dually, given t : G \overline{}F, we obtain a ^{∗}strong right homomorphism N (t^{∗}, H) : N (G, H) \overline{}N (F, H) in K, L-^{∗}MOD.

2.29. We turn now to the study of adjunctions in MOD and in the 2-categories derived from it. An adjunction x, y : S ⊣ T : M \overline{}N in MOD, as in any 2-category, consists of arrows T : M \overline{}N and S : N \overline{}M, along with transformations x : 1 \overline{}T S and y : ST \overline{}1 satisfying the triangular equations T y · xT = 1 and yS · Sx = 1. A direct expression of these triangular equations in terms of the primitive data for T and for S as given in Proposition 2.14 and the primitive data for x and for y given in Proposition 2.23 would clearly be rather complicated; yet in suitable cases we can often study adjunctions without explicit reference to the triangular equations. It is classical that adjunctions in the 2-category CAT are particularly simple to study: to find T given S, one has only to find for each M a representation N (−, T M) of the presheaf M(S−, M), whereupon the funtoriality of T, along with the existence and naturality of the unit and the counit and their satisfaction of the triangular equations, is automatic. Next, a similar simplification is available for adjunctions in [Λ, CAT], which reduce pointwise to adjunctions in CAT; so that, finally, we have a corresponding simplification for adjunctions in the full sub-2-categories MOD and ^{∗}MOD of [Λ, CAT]. It is our aim in the rest of this section to use these and other simplifications to provide, in terms of the primitive data for an arrow S in MOD or in ^{∗}MOD, simple necessary and sufficient conditions for it to admit a right adjoint.

Observe first though that adjunctions in MOD which lie in some K, L-MOD are considerably simpler than general ones, because as we noted in 2.24 the data for a transformation in K, L-MOD consist merely of a family of natural transformations. In particular, for starred N and with the definitions given in 2.26 and 2.28, the reader will have no difficulty in supplying the unit and the counit for an adjunction N (G, s) ⊣ N (G, s^{∗}) in K, L-^{∗}MOD and in verifying the triangular equations in this case. Duality then provides a further adjunction N (t^{∗}, H) ⊣ N (t, H) in K, L-^{∗}MOD.

Our goal in the next five numbered items will be to reduce an adjunction in MOD to a pair of adjunctions in CAT and an adjunction in a suitable fibre of ((−)_{0}, (−)_{1}) : MOD \overline{}CAT × CAT. (This should be compared with Lemma 2.6.)

2.30. Because ((−)_{0}, (−)_{1}) : MOD \overline{}CAT × CAT is a 2-functor it follows that an adjunction x, y : S ⊣ T : (K, M, L) \overline{}(R, N , S) in MOD gives rise to adjunctions x_{0}, y_{0} : S_{0} ⊣ T_{0} : K \overline{}R and x_{1}, y_{1} : S_{1} ⊣ T_{1} : L \overline{}S in CAT. We keep fixed here these adjunctions in CAT and consider various functors T : grM \overline{}grN for which (T_{0}, T, T_{1}) is an arrow (K, M, L) \overline{}(R, N , S) in MOD, and functors S : grN \overline{}grM for which (S_{0}, S, S_{1}) is an arrow (R, N , S) \overline{}(K, M, L) in MOD. To give such a T is by 2.25 to give an arrow T̃ : M \overline{}N (T_{0}, T_{1}) in K, L-MOD; and this gives rise, also as in 2.25, to an arrow T̃_{S_{0},1} : M(S_{0}, 1) \overline{}N (T_{0}S_{0}, T_{1}) in R, L-MOD, and hence using 2.26 to an arrow τ(T) : M(T_{0}, 1) \overline{}N (1, T_{1}) in R, L-MOD, defined as the first of the composites below. Similarly, such an S gives rise to the second composite σ(S) below in R, L-MOD.

[figure: M(S_{0}, 1); N (1, T_{1}); τ(T); N (T_{0}S_{0}, T_{1}); T̃_{S_{0},1}; N (x_{0}, T_{1}); N (1, T_{1}); M(S_{0}, 1); σ(S); M(S_{0}, S_{1}T_{1}); S̃_{1,T_{1}}; M(S_{0}, y_{1})]

On the other hand, starting with arbitrary arrows τ : M(S_{0}, 1) \overline{}N (1, T_{1}) and σ : N (1, T_{1}) \overline{}M(S_{0}, 1) in R, L-MOD, we obtain arrows

(T_{0}, T (τ), T_{1}) : (K, M, L) \overline{}(R, N , S) and (S_{0}, S(σ), S_{1}) : (R, N , S) \overline{}(K, M, L) in MOD by taking (T (τ))˜ and (S(σ))˜ to be the composites

[figure: M; N (T_{0}, T_{1}); (T (τ)); ˜ M(S_{0}T_{0}, 1); M(y_{0}, 1); τ_{T_{0},1}; N; M(S_{0}, S_{1}); (S(σ)); ˜ N (1, T_{1}S_{1}); N (1, x_{1}); σ_{1,S_{1}}]

in K, L-MOD and R, S-MOD respectively. Recall from 2.15 the meanings of R-homo-morphism and L-homomorphism.

2.31. Proposition. (i) For any such S, there is in MOD a canonical transformation of the form

(1, m, 1) : (S_{0}, S, S_{1}) \overline{}(S_{0}, S(σ(S)), S_{1}) : (R, N , S) \overline{}(K, M, L),

and it is invertible when S is an R-homomorphism; (ii) For any such T, there is in MOD a canonical transformation of the form

(1, n, 1) : (T_{0}, T (τ(T)), T_{1}) \overline{}(T_{0}, T, T_{1}) : (K, M, L) \overline{}(R, N , S),

and it is invertible when T is an L-homomorphism; (iii) For any such σ, there is in R, L-MOD a canonical transformation

σ(S(σ)) \overline{}σ : N (1, T_{1}) \overline{}M(S_{0}, 1),

and it is invertible when σis an R-homomorphism; (iv) For any such τ, there is in R, L-MOD a canonical transformation

τ \overline{}τ(T (τ)) : M(S_{0}, 1) \overline{}N (1, T_{1}),

and it is invertible when τ is an L-homomorphism.

Proof. The proofs being all of the same kind, it suffices to give that for (i). However S(σ(S)) sends ν : X \overline{}A to

(σ(S))(x_{1}A.ν) = y_{1}S_{1}A.S(x_{1}A.ν) : S_{0}X \overline{}S_{1}A,

and we have in M(S_{0}X, S_{1}A) the arrow m_{ν} given by the composite

[figure: (S(σ(S)))ν; y_{1}S_{1}A.S(x_{1}A.ν); Sν; 1.Sν; y_{1}S_{1}A.(S_{1}x_{1}A.Sν); y_{1}S_{1}A.S_{x_{1}A,ν}; (y_{1}S_{1}A.S_{1}x_{1}A).Sν; ξ; ,]

which is invertible when all the S_{s,ν} are so. That m_{ν} is natural with respect to arrows (r, Φ, s) : (X, ν, A) \overline{}(X′, ν′, A′) in grN follows immediately from the naturality of ξ and part (iii) of Proposition 2.14.

From 2.26 we have that N (x_{0}, T_{1}), M(S_{0}, y_{1}), M(y_{0}, 1), and N (1, x_{1}) all preserve commutative squares while substitution as defined in 2.25 evidently takes R-homomorphisms to R-homomorphisms and L-homomorphisms to L-homomorphisms. It follows that we may add to the above:

2.32. Lemma.

(i) If S is an R-homomorphism then so is σ(S).

(ii) If T is an L-homomorphism then so is τ(T).

(iii) If σ is an R-homomorphism then so is S(σ).

(iv) If τ is an L-homomorphism then so is T (τ).

2.33. Proposition. (i) Given S and T and an adjunction

(S_{0}, S, S_{1}) ⊣ (T_{0}, T, T_{1}) : (K, M, L) \overline{}(R, N , S)

in MOD, we may construct an adjunction

σ ⊣ τ : M(S_{0}, 1) \overline{}N (1, T_{1})

in R, L-MOD, where σ = σ(S) and τ = τ(T). (ii) Given σ and τ and an adjunction

σ ⊣ τ : M(S_{0}, 1) \overline{}N (1, T_{1})

in R, L-MOD, we may construct an adjunction

(S_{0}, S, S_{1}) ⊣ (T_{0}, T, T_{1}) : (K, M, L) \overline{}(R, N , S)

in MOD, where S = S(σ) and T = T (τ).

Proof. First suppose given an adjunction (S_{0}, S, S_{1}) ⊣ (T_{0}, T, T_{1}) as in (i). This means that, given ν : X \overline{}A, μ : K \overline{}L, r : X \overline{}T_{0}K, and l : S_{1}A \overline{}L, there is a natural bijection between transformations Φ in the left diagram, and transformations Ψ in the right diagram, of the pair

[figure: S_{1}A; L; l; S_{0}X; K; α_{0}^{−1}(r); Sν; μ; Φ; ,; A; T_{1}L; α_{1}(l); X; T_{0}K; r; ν; T μ; Ψ; ,]

where we have used α_{0} and α_{1} to denote the hom-set bijections for the adjunctions S_{0} ⊣ T_{0} and S_{1} ⊣ T_{1} respectively. Now take A here to be of the form T_{1}L′ and K to be of the form S_{0}X′ and take l and r to be the respective composites

S_{1}T_{1}L′ ^{\overline{}^{y_{1}L′}} L′ ^{\overline{}^{l′}} L , X ^{\overline{}^{r′}} X′ ^{\overline{}^{x_{0}X′}} T_{0}S_{0}X′.

Since y_{1}L′.Sν = σ(S)ν and T μ.x_{0}X′ = τ(T)μ, while

α_{1}(l′.y_{1}L′) = T_{1}l′ and α_{0}^{−1}(x_{0}X′.r′) = S_{0}r′,

absorbing isomorphisms of the forms ξ and ζ produces a bijection between transformations Φ and Ψ in the diagrams

[figure: L′; L; l′; S_{0}X; S_{0}X′; S_{0}r′; σ(S)ν; μ; Φ; ,; T_{1}L′; T_{1}L′; T_{1}l′; X; X′; r′; ν; τ(T)μ; Ψ; .]

Since it is straightforward to verify that this latter bijection is natural for arrows μ \overline{}μ′ in grM(S_{0}, 1) and ν \overline{}ν′ in grN (1, T_{1}), given that the former one is natural for arrows μ \overline{}μ′ in grM and ν \overline{}ν′ in grN, we have indeed the desired adjunction σ(S) ⊣ τ(T). Next suppose instead that we have an adjunction σ ⊣ τ as in (ii); so that given ν : X \overline{}T_{1}L′, μ : S_{0}X′ \overline{}L, r′ : X \overline{}X′, and l′ : L′ \overline{}L, there is a natural bijection between transformations Φ and Ψ as in the diagrams above, but with σ(S) replaced by σ and τ(T) replaced by τ. Now take ν and μ here to be the respective composites

X ^{\overline{}^{ν̄}} A ^{\overline{}^{x_{1}A}} T_{1}S_{1}A , S_{0}T_{0}K ^{\overline{}^{y_{0}K}} K ^{\overline{}^{μ̄}} L.

Since σ(x_{1}A.ν̄) = S(σ)ν̄ and τ(μ̄.y_{0}K) = T (τ)μ̄, while y_{0}K.S_{0}r′ = α_{0}^{−1}(r′) and T_{1}l′.x_{1}A = α_{1}(l′), we have a bijection between transformations Φ and Ψ in the diagrams

[figure: S_{1}A; L; l′; S_{0}X; K; α_{0}^{−1}(r′); S(σ)ν̄; μ̄; Φ; ,; A; T_{1}L; α_{1}(l′); X; T_{0}K; r′; ν̄; T (τ)μ̄; Ψ; ;]

and since once again the bijection is easily verified to be natural in the appropriate sense, we have the desired adjunction S(σ) ⊣ T (τ).

2.34. Starting from an adjunction (S_{0}, S, S_{1}) ⊣ (T_{0}, T, T_{1}), we may construct by Proposition 2.33 (i) an adjunction σ(S) ⊣ τ(T), and then by Proposition 2.33 (ii) an adjunction (S_{0}, S(σ(S)), S_{1}) ⊣ (T_{0}, T (τ(T)), T_{1}). On the other hand, we have from Propositions 2.31 and 2.16, along with Lemma 2.32 the isomorphisms m : S \overline{}S(σ(S)) and n : T (τ(T)) \overline{}T. Comparing the construction of these in the proof of Proposition 2.31 with the constructions in the proof of Proposition 2.33, we easily verify the commutativity of

[figure: grM(Sν, μ); grN (ν, T μ); α; grM(S(σ(S))ν, μ); grN (ν, T (τ(T))μ); α′′; grM(m_{ν}, μ); grN (ν, n_{μ}); ,]

where α is the hom-set bijection for the adjunction S ⊣ T and α′′ is its analogue for the adjunction S(σ(S)) ⊣ T (τ(T)). In other words, n and m are mates under the two adjunctions. There is a corresponding result when we start from an adjunction σ ⊣ τ and form in two steps the adjunction σ(S(σ)) ⊣ τ(T (τ)). What we have therefore, for fixed x_{i}, y_{i} : S_{i} ⊣ T_{i}, i = 0, 1, is an equivalence between the category of those adjunctions of the form

(x_{0}, x, x_{1}), (y_{0}, y, y_{1}) : (S_{0}, S, S_{1}) ⊣ (T_{0}, T, T_{1}) : (K, M, L) \overline{}(R, N , S)

in MOD and the category of all adjunctions

σ ⊣ τ : M(S_{0}, 1) \overline{}N (1, T_{1})

in R, L-MOD. This completes the goal we set at the end of 2.29.

2.35. For adjunctions in ^{∗}MOD the reduction to an adjunction in a fibre can be accomplished in three other ways. In ^{∗}MOD, in the constructions of 2.30, N (x_{0}, T_{1}) has by 2.29 a left adjoint N (x_{0}^{∗}, T_{1}) in R, L-^{∗}MOD, while M(S_{0}, y_{1}) has a right adjoint M(S_{0}, y_{1}^{∗}) there. Similarly, in the second set of diagrams of 2.30, we have M(y_{0}^{∗}, 1) ⊣ M(y_{0}, 1) in K, L-^{∗}MOD and N (1, x_{1}) ⊣ N (1, x_{1}^{∗}) in R, S-^{∗}MOD. Further, if we are given an adjunction σ ⊣ τ : M(S_{0}, 1) \overline{}N (1, T_{1}), we also have σ_{T_{0},1} ⊣ τ_{T_{0},1} : M(S_{0}T_{0}, 1) \overline{}N (T_{0}, T_{1}), since the substitution T_{0}^{+}(−)1 is a 2-functor as explained in 2.25. So from the last two diagrams of 2.30 we conclude that:

If σ ⊣ τ : M(S_{0}, 1) \overline{}N (1, T_{1}) in R, L-^{∗}MOD, then

(T (τ))ˆ ⊣ (T (τ))˜ : M \overline{}N (T_{0}, T_{1}) in K, L-^{∗}MOD

and

(S(σ))˜ ⊣ (S(σ))ˇ : M(S_{0}, S_{1}) \overline{}N in R, S-^{∗}MOD,

where (T (τ))ˆ and (S(σ))ˇ are the composites

[figure: M; N (T_{0}, T_{1}); (T (τ)); ˆ M(S_{0}T_{0}, 1); M(y_{0}^{∗}, 1); σ_{T_{0},1}; ,; N; M(S_{0}, S_{1}); (S(σ)); ˇ N (1, T_{1}S_{1}); N (1, x_{1}^{∗}); τ_{1,S_{1}}; .]

In the other direction, from the first two diagrams of 2.30 we conclude that:

If T̂ ⊣ T̃ in K, L-^{∗}MOD [respectively, if S̃ ⊣ Š in R, S-^{∗}MOD], then τ(T) has the left adjoint σ [respectively σ(S) has the right adjoint τ] in R, L-^{∗}MOD, where σ and τ are the composites

[figure: M(S_{0}, 1); N (1, T_{1}); σ; N (T_{0}S_{0}, T_{1}); T̂_{S_{0},1}; N (x_{0}^{∗}, T_{1}); ,; N (1, T_{1}); M(S_{0}, 1); τ; M(S_{0}, S_{1}T_{1}); Š_{1,T_{1}}; M(S_{0}, y_{1}^{∗}); .]

In the starred case we can also use the duality (−)^{op} to get from (S_{0}, S, S_{1}) ⊣ (T_{0}, T, T_{1}) the further adjunction (S_{0}, S, S_{1})^{op} ⊣ (T_{0}, T, T_{1})^{op} in ^{∗}MOD which as in 2.30 and the items following it gives rise to σ^{∗}(S) ⊣ τ^{∗}(T) : M(1, S_{1}) \overline{}N (T_{0}, 1) in K, S-^{∗}MOD, where σ^{∗}(S) and τ^{∗}(T) are given by

[figure: N (T_{0}, 1); M(1, S_{1}); σ^{∗}(S); M(S_{0}T_{0}, S_{1}); S̃_{T_{0},1}; M(y_{0}^{∗}, S_{1}); ,; M(1, S_{1}); N (T_{0}, 1); τ^{∗}(T); N (T_{0}, T_{1}S_{1}); T̃_{1,S_{1}}; N (T_{0}, x_{1}^{∗}); ,]

and from which we recover S and T to within isomorphism by

[figure: N; M(S_{0}, S_{1}); (S(σ^{∗})); ˜ N (T_{0}S_{0}, 1); N (x_{0}^{∗}, 1); σ_{S_{0},1}^{∗}; ,; M; N (T_{0}, T_{1}); (T (τ^{∗})); ˜ M(1, S_{1}T_{1}); M(1, y_{1}^{∗}); τ_{1,T_{1}}^{∗}; .]

Using the isomorphisms m and n of 2.31 that we have in the presence of adjunctions, various pastings of the triangles in 2.30 with those above provide further isomorphisms between the arrows of ^{∗}MOD under consideration. Note too that in the starred context Lemma 2.32 can be strengthened by the systematic replacement of ‘R-homomorphism’ by ‘right homomorphism’ and ‘L-homomorphism’ by ‘left homomorphism’.

We now have, in the context of 2.34 and 2.35, equivalences between the categories of adjunctions (S_{0}, S, S_{1}) ⊣ (T_{0}, T, T_{1}), σ ⊣ τ, S̃ ⊣ Š, σ^{∗} ⊣ τ^{∗}, and T̂ ⊣ T̃. It is convenenient to record that starting with (S_{0}, S, S_{1}) ⊣ (T_{0}, T, T_{1}) we can express T̂ as the composite:

[figure: M(S_{0}T_{0}, S_{1}T_{1}); M(S_{0}T_{0}, 1); M(S_{0}T_{0}, y_{1}); N (T_{0}, T_{1}); M; T̂; S̃_{T_{0},T_{1}}; M(y_{0}^{∗}, 1); .]

In particular we have the following:

2.36. Theorem. Given adjunctions x_{0}, y_{0} : S_{0} ⊣ T_{0} : K \overline{}R and x_{1}, y_{1} : S_{1} ⊣ T_{1} : L \overline{}S in CAT, an arrow (S_{0}, S, S_{1}) : (R, N , S) \overline{}(K, M, L)

(i) in MOD has a right adjoint if and only if σ(S) : N (1, T_{1}) \overline{}M(S_{0}, 1), as defined in the second triangle of 2.30, has a right adjoint τ in R, L-MOD; and then (S_{0}, S, S_{1}) ⊣ (T_{0}, T, T_{1}), where T̃ is the (T (τ))˜ defined in the third triangle of 2.30;

(ii) in ^{∗}MOD has a right adjoint if and only if S̃ : N \overline{}M(S_{0}, S_{1}) has a right adjoint Š in R, S-^{∗}MOD; and then (S_{0}, S, S_{1}) ⊣ (T_{0}, T, T_{1}), where T̃ is given by

[figure: M(S_{0}T_{0}, 1); M(S_{0}T_{0}, S_{1}T_{1}); M(S_{0}T_{0}, y_{1}^{∗}); M; N (T_{0}, T_{1}); T̃; M(y_{0}, 1); Š_{T_{0},T_{1}}; .]

2.37. We turn now to adjunctions x, y : S ⊣ T : M \overline{}N in some K, L-MOD or in some K, L-^{∗}MOD. Given such an adjunction, it follows from the concluding remarks of 2.24 that we have adjunctions S_{K,L} ⊣ T_{K,L} : M(K, L) \overline{}N (K, L) in CAT. The right adjoint T in K, L-MOD is an L-homomorphism, and a left homomorphism in the starred case. First note that, from the general theory of KZ-doctrines, we can calculate (T^{μ,k})^{−1} for k : K′ \overline{}K and μ : K \overline{}L explicitly; it turns out to be t^{μ,k} where

t^{μ,k} = (T μ.k ^{\overline{}^{x_{T μ.k}}} T S(T μ.k) ^{\overline{}^{T (S^{T μ,k})}} T (ST μ.k) ^{\overline{}^{T (y_{μ}k)}} T (μk)).

(To help the reader unfamiliar with the general theory, we recall the essential ideas below in an appendix labelled as Section 5.) Observe now that the natural transformation

t^{−,k} : N (k, L)T_{K,L} \overline{}T_{K′,L}M(k, L) : M(K, L) \overline{}N (K′, L)

with the t^{μ,k} as components is the mate, under the adjunctions S_{K′,L} ⊣ T_{K′,L} and S_{K,L} ⊣ T_{K,L}, of

S^{−,k} : S_{K′,L}N (k, L) \overline{}M(k, L)S_{K,L} : N (K′, L) \overline{}M(K, L).

Thus given S : N \overline{}M in K, L-MOD and adjunctions S_{K,L} ⊣ T_{K,L} in CAT, we can, whether or not S has a right adjoint in K, L-MOD, define the t^{−,k} to be these mates, and the t^{μ,k} to be their components.

2.38. Lemma. If S : N \overline{}M in K, L-^{∗}MOD and there are given adjunctions S_{K,L} ⊣ T_{K,L} in CAT then to say that each t^{μ,k} is invertible is equally to say that each S_{μ,k^{∗}} is invertible. With the same hypotheses, S is a right homomorphism if and only if S is an R-homomorphism and the t^{μ,k} are invertible.

Proof. The second sentence is a trivial consequence of the first. Replacing the domain and codomain of t^{−,k} by their left adjoints gives a transformation

M(k^{∗}, L)S_{K′,L} \overline{}S_{K′,L}N (k^{∗}, L),

which by general principles must be the mate under M(k^{∗}, L) ⊣ M(k, L) and N (k^{∗}, L) ⊣ N (k, L) of S^{−,k} and which therefore by 2.17 is S_{−,k^{∗}}. It follows that each t^{μ,k} is invertible if and only if each S_{μ,k^{∗}} is so.

2.39. Theorem. An arrow S : N \overline{}M in K, L-MOD has a right adjoint if and only if the following three conditions are satisfied:

(i) S is an R-homomorphism;

(ii) for each K and L, the functor S_{K,L} : N (K, L) \overline{}M(K, L) has a right adjoint T_{K,L}, with counit say y_{K,L} : S_{K,L}T_{K,L} \overline{}1 (whose components we write as y_{μ} : ST μ \overline{}μ);

(i′) the arrows t^{μ,k} : T μ.k \overline{}T (μk), corresponding under the adjunctions in (ii) to the composite S(T μ.k) ^{\overline{}^{S^{T μ,k}}} ST μ.k ^{\overline{}^{y_{μ}k}} μk, are invertible.

For S : N \overline{}M in K, L-^{∗}MOD, conditions (i) and (i′) can be replaced by:

(i^{∗}) S is a right homomorphism.

Indeed, when these conditions hold, if we define T μ and y_{μ} by (ii) and take T (K, μ, L) to be (K, T μ, L), the arrows (1_{K}, y_{μ}, 1_{L}) : S(K, T μ, L) \overline{}(K, μ, L) in grM are universal among arrows in grM of the form (k, Φ, l) : (K′, Sν, L′) \overline{}(K, μ, L) and the right adjoint T to which they lead has for its T_{K,L} that of (ii), has for its T^{μ,k} the inverse of the t^{μ,k} of (i′), and has for its T_{l,μ} the vector transformation corresponding under the adjunctions of (ii) to the composite

S(l.T μ) ^{\overline{}^{(S_{l,T μ})^{−1}}} l.ST μ ^{\overline{}^{ly_{μ}}} lμ.

Proof. We have seen that the conditions are necessary. For sufficiency, observe first that for S : N \overline{}M to have a right adjoint in K, L-MOD is precisely for ĝrS : ĝrN \overline{}ĝrM to have a right adjoint in spnCAT(K, L) (the latter being the pullback of [Λ, CAT] over CAT × CAT along (K, L) : 1 \overline{}CAT × CAT). For ĝrS to have a right adjoint in spnCAT(K, L) is simply for grS : grM \overline{}grN to have a right adjoint in CAT which respects the ‘span’ constraints, and this can be stated simply as follows: S : N \overline{}M has a right adjoint if and only if for each object (K, μ, L) in grM there is an object (K, T μ, L) in grN and an arrow (1_{K}, y_{μ}, 1_{L}) : (K, ST μ, L) \overline{}(K, μ, L) in grM such that every arrow (k, Φ, l) : (K′, Sν, L′) \overline{}(K, μ, L) in grM is (1_{K}, y_{μ}, 1_{L}).(k, SΨ, l) for a unique arrow Ψ : lν \overline{}T μ.k in N (K′, L). Taking T μ to be T_{K,L}μ and y_{μ} to be y_{K,L}μ as provided by (ii) we have only to verify the universal property.

By 2.13, however, (k, SΨ, l) is the composite

(K′, Sν, L′) ^{\overline{}^{(1_{K′},S_{l,ν},l)}} (K′, S(lν), L) ^{\overline{}^{(1_{K′},SΨ,1_{L})}} (K′, S(T μ.k), L) ^{\overline{}^{(k,S^{T μ,k},1_{L})}} (K, ST μ, L),

so that, by the meaning of composition in grM, the desired equation reduces to

y_{μ}k.S^{T μ,k}.SΨ.S_{l,ν} = Φ,

which by the invertibility of S_{l,ν} in (i) and the definition of t^{μ,k} in (i′) is

α^{−1}(t^{μ,k}).SΨ = Φ.S_{l,ν}^{−1},

where now

α : M(K, L)(S−,?) ^{\overline{}^{≃}} N (K, L)(−, T?)

denotes the adjunction of (ii). Since the left side here, by the naturality of α^{−1}, is α^{−1}(t^{μ,k}Ψ), our equation reduces further to

t^{μ,k}Ψ = α(Φ.S_{l,ν}^{−1});

and this has a unique solution for Ψ because t^{μ,k} is invertible by (i′).

The value T (a, Θ, b) of T at a general arrow (a, Θ, b) : (K̄, μ̄, L̄) \overline{}(K, μ, L) is of course the unique arrow for which

(1_{K}, y_{μ}, 1_{L}).ST (a, Θ, b) = (a, Θ, b)(1K̄, yμ̄, 1L̄);

we leave to the reader the easy verification that T has the stated values on the basic generating arrows of grM.

Combining Theorems 2.36 and 2.39 with the help of Lemma 2.33 and 2.34 and 2.35 we achieve our major goal of this section:

2.40. Theorem. An arrow (S_{0}, S, S_{1}) : (R, N , S) \overline{}(K, M, L) in MOD has a right adjoint if and only if the following four conditions are satisfied:

(o) the S_{i} have right adjoints T_{i}, with counits say y_{i} : S_{i}T_{i} \overline{}1;

(i) S is an R-homomorphism;

(ii) for each X in R and L in L, the functor σ(S)_{X,L} : N (X, T_{1}L) \overline{}M(S_{0}X, L), defined as the composite

N (X, T_{1}L) ^{\overline{}^{S_{X,T_{1}L}}} M(S_{0}X, S_{1}T_{1}L) ^{\overline{}^{M(S_{0}X,y_{1}L)}} M(S_{0}X, L)

has a right adjoint τ_{X,L};

(i′) if σ(S)^{ν,r} : σ(S)(νr) \overline{}σ(S)ν.S_{0}r denotes the composite

y_{1}L.S(νr) ^{\overline{}^{y_{1}L.S^{ν,r}}} y_{1}L.(Sν.S_{0}r) ^{\overline{}^{≃}^{η}} (y_{1}L.Sν).S_{0}r,

and if θ^{−,r} denotes the mate of σ(S)^{−,r}, then each θ^{μ,r} is invertible.

When these conditions are satisfied, the right adjoint of (S_{0}, S, S_{1}) is (T_{0}, T (τ), T_{1}), where the right adjoint τ of σ(S) is constructed from the data above as in Theorem 2.39.

For (S_{0}, S, S_{1}) : (R, N , S) \overline{}(K, M, L) in ^{∗}MOD, conditions (i), (i′), and (ii) can be replaced by:

(i^{∗}) S is a right homomorphism;

(ii^{∗}) for each X in R and A in S, the functor S_{X,A} : N (X, A) \overline{}M(S_{0}X, S_{1}A) has a right adjoint.

Theorem 2.40 has an obvious specialization to equipments. From the definitions in 2.9 we see that for either EQT or ^{∗}EQT we simply apply Theorem 2.40 to an arrow of the form (S_{#}, S, S_{#}) : (R, N , R) \overline{}(K, M, K). As we have said before, it is equipments rather than general modules that constitute our main interest; however, inspection of 2.36 now shows that even for (S_{#}, S, S_{#}) we are led to an adjunction in the ‘off-diagonal’ R, K-MOD.

2.41. We shall give an application of Theorem 2.40: namely to the question of limits in a starred module M. We first need to observe that the 2-category MOD admits powers (these being what most authors refer to as cotensor products), as do ^{∗}MOD, EQT, and ∗EQT. That is, for each module M = (K, M, L) and each category C, there is a module M^{C} with the universal property expressed by a 2-natural isomorphism

MOD(N , M^{C}) ^{\overline{}^{≃}} CAT(C, MOD(N , M)).

We give the construction of M^{C}, and leave the reader to verify that it does indeed have the universal property. The clue to its construction is the observation that, since the 2functors ∂_{0}, ∂_{1}, gr : MOD \overline{}CAT are representable as we remarked in 2.9, they preserve any limits that exist, and in particular powers; so that if M^{C} does exist, we must have (M^{C})_{0} = K^{C}, (M^{C})_{1} = L^{C} and gr(M^{C}) = (grM)^{C}, where the right sides are the usual functor-categories.

So M^{C} is to be a K^{C}, L^{C}-module. For P : C \overline{}K and Q : C \overline{}L, an object of M^{C}(P, Q) is a functor μ : C \overline{}grM for which ∂_{0}μ = P and ∂_{1}μ = Q; while an arrow Φ : μ \overline{}ν in M^{C}(P, Q) is a natural transformation Φ : μ \overline{}ν : C \overline{}grM for which ∂_{0}Φ and ∂_{1}Φ are identities. Natural transformations q : Q \overline{}Q′ : C \overline{}L and p : P′ \overline{}P : C \overline{}K act on such a μ to give qμ and μp where, for C ∈ C, (qμ)_{C} = q_{C}μ_{C} and (μp)_{C} = μ_{C}p_{C}, with similar definitions on a vector transformation Φ. When M is starred, so is M^{C}, and we have a completely analogous description of M^{C} in EQT and in ^{∗}EQT.

In any 2-category K with powers, we have the diagonal D : M \overline{}M^{C}, corresponding to the functor C \overline{}K(M, M) which is constant at the identity; and M is said to admit C-limits if D has a right adjoint lim : M^{C} \overline{}M in K. In our case where K = ^{∗}MOD, the functors D_{i} : M_{i} \overline{}(M^{C})_{i} are the usual diagonals M_{i} \overline{}(M_{i})^{C}. Moreover D is easily seen to be a homomorphism (in the sense of Definition 2.19), and the composite functor

M(K, L) ^{\overline{}^{D_{K,L}}} M^{C}(D_{0}K, D_{1}L) ^{\overline{}^{≃}} M(K, L)^{C}

is again the diagonal in CAT. Accordingly Theorem 2.40 for starred modules gives:

A starred module (K, M, L) admits C-limits if and only if K and L and each M(K, L) admit them.

3. [Starred] Pointed Equipments

3.1. For each category K, the hom functor K(−, −) : K^{op} × K \overline{}SET provides a K, Kmodule K(−, −) : K^{op}, K \overline{}SET \overline{}CAT, where SET \overline{}CAT sends a set to the corresponding discrete category. The structural associativites ξ, η and ζ are identities and we have an equipment (K, K). (If the category K is a groupoid then the equipment (K, K) is starred. Conversely, if (K, K) is starred then the category K is a groupoid, for given k : K′ \overline{}K in K, it follows easily that k^{∗}1_{K} is its inverse.)

Any functor G : K \overline{}R gives rise to an equipment arrow (G, G) : (K, K) \overline{}(R, R), with action comparisons given by identities. Similarly, any natural transformation t : F \overline{}G : K \overline{}R gives rise to an equipment transformation (t, t) : (F, F) \overline{}(G, G) : (K, K) \overline{}(R, R). These assignments clearly define a 2-functor CAT \overline{}EQT which, for the moment, it is best to leave un-named. (Similarly, we have a 2-functor GPD \overline{}^{∗}EQT, where GPD is the 2-category of groupoids.)

3.2. Definition. A pointed equipment is an equipment (K, M) together with a strong arrow (−)_{∗} : K \overline{}M in K, K-MOD.

In other words we require an arrow in MOD of the form

(1, (−)_{∗}, 1) : (K, K, K) \overline{}(K, M, K)

which, by Proposition 2.13 and 2.24 above, admits the following primitive description: For each arrow f : X \overline{}A in K there is a vector arrow f_{∗} : X \overline{}A in M and for each X ^{\overline{}^{f}} A ^{\overline{}^{g}} Y in K there are given isomorphisms gf_{∗} ^{\overline{}^{≃}} (gf)_{∗} and (gf)_{∗} ^{\overline{}^{≃}} g_{∗}f which satisfy

1f_{∗} ^{\overline{}^{≃}} (1f)_{∗} is the identity
(f1)_{∗} ^{\overline{}^{≃}} f_{∗}1 is the identity

[figure: (fgh)_{∗}; f(gh)_{∗}; ≃; (fg)h_{∗}; f(gh_{∗}); ξ; ≃; ≃; f(g_{∗}h); f(gh)_{∗}; ≃; η; (fg)_{∗}h; ≃; (fg_{∗})h; (fgh)_{∗}; ≃; ≃; f_{∗}(gh); (f_{∗}g)h; ζ; (fgh)_{∗}; (fg)_{∗}h; ≃; ≃; ≃; ,]

where ξ, η and ζ are the associativities for M. Observe that for any category K, the equipment (K, K) is canonically pointed by setting f_{∗} = f.

For a pointed equipment ((−)_{∗}, K, M), or (∗, K, M) for short, it is convenient to write ι_{X} : X \overline{}X for (1_{X})_{∗} : X \overline{}X. For any f : X \overline{}A in K we have

fι_{X} = f1_{∗} ^{\overline{}^{≃}} (f1)_{∗} = f_{∗} = (1f)_{∗} ^{\overline{}^{≃}} 1_{∗}f = ι_{A}f.

3.3. Lemma. For an equipment (K, M), let there be given for each object X a vector arrow ι_{X} : X \overline{}X and for each scalar arrow f : X \overline{}A an isomorphism λ_{f} : fι_{X} ^{\overline{}^{≃}} ι_{A}f, subject to

(λ_{1} : 1ι ^{\overline{}^{≃}} ι1) = (1_{ι} : ι \overline{}ι)

[figure: ξ^{−1}; f(ιg); f(gι); λ; (fg)ι; λ; η; ζ^{−1}; (ιf)g; λ; (fι)g; ι(fg); ,]

the latter of which is equally the assertion that, in terms of pasting composites, we have

[figure: X; Y; g; X; Y; g; ι_{X}; ι_{Y}; λ_{g}; Z; f; Z; f; ι_{Z}; λ_{f}; =; X; Z; fg; X; Z; fg; ι_{X}; ι_{Z}; λ_{fg}; .]

Then, defining f_{•} = fι_{X} and

gf_{•} = g(fι) ^{\overline{}^{ξ^{−1}}} (gf)ι = (gf)_{•}

(gf)_{•} = (gf)ι ^{\overline{}^{ξ}} g(fι) ^{\overline{}^{gλ}} g(ιf) ^{\overline{}^{η}} (gι)f = g_{•}f

provides a pointing (−)_{•} for (K, M). Moreover, if the isomorphisms λ : fι_{X} ^{\overline{}^{≃}} ι_{A}f come from a pointing (−)_{∗} : K \overline{}M, as in the display preceding the Lemma, then f_{•} = fι = f1_{∗} ^{\overline{}^{≃}} (f1)_{∗} = f_{∗} describes an isomorphism (−)_{•} ^{\overline{}^{≃}} (−)_{∗} in K, K-MOD(K, M). Similarly, defining f_{∘} = ι_{A}f and so on provides a pointing with (−)_{•} ^{\overline{}^{≃}} (−)_{∘} in K, K-MOD(K, M); this being the composite (−)_{•} ^{\overline{}^{≃}} (−)_{∗} ^{\overline{}^{≃}} (−)_{∘} if the data come from a given pointing (−)_{∗}. 3.4. If an equipment (K, M) has the property of being starred and carries the further structure of a pointing (−)_{∗} then we have a pointed and starred equipment. But for a starred equipment we have also the equipment (K, M^{op}) described in the context of modules in 2.10, so that it makes sense to enquire whether or not the ι determined by (−)_{∗} constitute part of the data for a pointing of (K, M^{op}). The isomorphisms λ : fι ^{\overline{}^{≃}} ιf have mates ιf^{∗} \overline{}f^{∗}ι. If these are invertible then their inverses satisfy the hypotheses of Lemma 3.3 for the equipment (K, M^{op}). (The hexagonal coherence condition follows from generalities about mating. The normality condition follows from the normality constraint for starred modules that we imposed in 2.7.) With these observations in mind we express the next definition in the vocabulary of 2.12. 3.5. Definition. A starred pointed equipment is a pointed and starred equipment for which the commutative squares

[figure: X; A; f; X; A; f; ι; ι; ≃^{λ}]

are also exact.

In many specific examples of starred pointed equipments, such as spnK for a category K with pullbacks, with the ι_{X} given by (1, X, 1) : X \overline{}X, it is consistent with our normality assumptions to take both the fι ^{\overline{}^{≃}} ιf and the f^{∗}ι ^{\overline{}^{≃}} ιf^{∗} to be identities. However, it is not true in general that an identity whose mate is an isomorphism has that mate again an identity. It is for this reason that we did not “normalize” in Definition 3.2. 3.6. Definition. An arrow of [starred] pointed equipments

(p, G, T) : (∗, K, M) \overline{}(•, R, N)

is an equipment arrow (G, T) : (K, M) \overline{}(R, N), together with an isomorphism p of equipment arrows as in for which p_{#} : (T ∗)_{#} \overline{}(•G)_{#} is the identity 1_{G}.

[figure: M; N; T; K; R; G; ∗; •; p; ≃; ,]

Using Proposition 2.23, we see that this is equivalent to giving isomorphisms

p : T (f_{∗}) ^{\overline{}^{≃}} (Gf)_{•}

which satisfy

[figure: Gg(Gf)_{•}; (Gg.Gf)_{•}; ≃; Gg.T (f_{∗}); T (gf_{∗}); T_{g,f_{∗}}; ≃^{Gg.p}; (G(gf))_{•}; 1; T ((gf)_{∗}); ≃; ≃^{p}]

and

[figure: (G(fh))_{•}; (Gf.Gh)_{•}; 1; T ((fh)_{∗}); T (f_{∗}h); ≃; ≃^{p}; (Gf)_{•}Gh; ≃; T (f_{∗}).Gh; T^{f_{∗},h}; ≃^{p.Gh}; ,]

where the unlabelled isomorphisms arise from the structural isomorphisms of the pointings.

3.7. Proposition. For an arrow

(p, G, T) : (∗, K, M) \overline{}(•, R, N)

of [starred] pointed equipments, all instances of the [T_{f_{∗},k^{∗}}, T^{l^{∗},f_{∗}},] T_{g,f_{∗}} and T^{f_{∗},h} are isomorphisms.

Proof. The unstarred part follows immediately from the preceding diagrams; it is equally the assertion that T ∗ is a strong arrow of equipments, clear since it is isomorphic to the strong arrow •G. The starred part then follows from the fact that the data for (p, G, T) determine an arrow between the pointed equipments (∗, K, M^{op}) and (•, R, N^{op}).

3.8. Proposition. An arrow of pointed equipments

(∗, K, M) \overline{}(•, R, N)

is completely determined by an equipment arrow (G, T) : (K, M) \overline{}(R, N), with either all T_{f,ι} invertible or all T^{ι,f} invertible, together with isomorphisms p : T ι_{X} ^{\overline{}^{≃}} ι_{GX} such that, for all f : X \overline{}A in K,

[figure: Gf.ι_{GX}; Gf.T ι_{X}; T (fι_{X}); T_{f,ι}; ≃^{Gf.p}; T (ι_{A}f); ≃^{T λ}; ι_{GA}.Gf; T ι_{A}.Gf; T^{ι,f}; ≃^{p.Gf}; ≃^{λ}; .]

Proof. Taking f to be an identity in each of the hexagonal diagrams following Definition 3.6 gives two expressions for a general p_{f} : T (f_{∗}) ^{\overline{}^{≃}} (Gf)_{•} in terms of p_{1} : T ι_{X} ^{\overline{}^{≃}} ι_{GX}; and the condition for these to agree is the hexagon of the proposition.

3.9. Definition. A transformation of [starred] pointed equipments

(q, F, S) \overline{}(p, G, T) : (∗, K, M) \overline{}(•, R, N)

is an equipment transformation

(t, u) : (F, S) \overline{}(G, T) : (K, M) \overline{}(R, N)

such that

[figure: M; N; K; R; G; ∗; •; T; S; u; ≃^{p}; =; M; N; S; K; R; ∗; •; G; F; t; ≃^{q}; .]

In primitive terms the definition requires

[figure: tA(F f)_{•}; (tA.F f)_{•}; ≃; tA.S(f_{∗}); ≃^{tA.q}; (Gf.tX)_{•}; 1; (Gf)_{•}tX; ≃; T (f_{∗}).tX; ≃^{p.tX}; u_{f_{∗}}; ,]

from which it follows that the u_{f_{∗}} are isomorphisms. We express this observation in the vocabulary of 2.12.

3.10. Proposition. For a transformation of [starred] pointed equipments

(t, u) : (q, F, S) \overline{}(p, G, T) : (∗, K, M) \overline{}(•, R, N)

and each f : X \overline{}A in K, the square

[figure: F A; GA; tA; F X; GX; tX; S(f_{∗}); T (f_{∗}); u_{f_{∗}}]

is a commutative one.

We can simplify Definition 3.9 in terms of the ι_{X}.

3.11. Proposition. To give a transformation of pointed equipments

(q, F, S) \overline{}(p, G, T) : (∗, K, M) \overline{}(•, R, N)

is to give an equipment transformation

(t, u) : (F, S) \overline{}(G, T) : (K, M) \overline{}(R, N)

satisfying, for each X in K,

[figure: tX.ι_{F X}; ι_{GX}.tX; λ; tX.Sι_{X}; T ι_{X}.tX; u_{ι_{X}}; tX.q; p.tX; .]

3.12. From the form in which we gave Definitions 3.6 and 3.9, it follows immediately that [starred] pointed equipments, arrows of pointed equipments, and pointed transformations, with the evident compositions, form a 2-category, which we denote by [^{∗}EQT_{∗}] EQT_{∗}, that comes with a forgetful 2-functor [^{∗}EQT_{∗} \overline{}^{∗}EQT] EQT_{∗} \overline{}EQT. The 2-category ∗EQT_{∗} is a full sub-2-category of EQT_{∗}. We write (−)_{#} for the forgetful 2-functor to CAT from any of the four 2-categories of equipments. It is now clear that the 2-functor CAT \overline{}EQT, which we introduced without a name in 3.1, factorizes canonically through EQT_{∗} \overline{}EQT. For we observed in 3.2 that an equipment of the form (K, K), with K a category, is pointed by way of identities; and the same is obviously true of an equipment arrow (G, G) coming from a functor G : K \overline{}R, while an equipment transformation (t, t) coming from a natural transformation t : F \overline{}G : K \overline{}R trivially satisfies the requirement of Proposition 3.11. We now write D : CAT \overline{}EQT_{∗} for this 2-functor and observe that (−)_{#}D = 1_{CAT}. Moreover, for a pointed equipment (∗, K, M),

(1, 1, ∗) : D((∗, K, M)_{#}) = (1, K, K) \overline{}(∗, K, M)

becomes an arrow of pointed equipments — the f-component of the stucture isomorphism is the identity vector transformation 1 : f_{∗} \overline{}f_{∗} in (K, M).

3.13. Proposition. The (1, 1, ∗) : D((∗, K, M)_{#}) \overline{}(∗, K, M) are pseudo-natural in (∗, K, M) and constitute the counit of a bi-adjunction D ⊣ (−)_{#} : EQT_{∗} \overline{}CAT.

In 3.2 we observed that a pointed equipment (∗, K, M) gives rise to pointed equipments (•, K, M) and (∘, K, M), where f_{•} = fι and f_{∘} = ιf. Observe that 1_{•} = 1ι = ι = ι1 = 1_{∘}, so that all three of these pointings for (K, M) have the same ι. Write a : (−)_{•} ^{\overline{}^{≃}} (−)_{∗} and b : (−)_{∗} ^{\overline{}^{≃}} (−)_{∘} for the previously given isomorphisms.

3.14. Lemma. The isomorphisms a and b define arrows of pointed equipments

(a, 1, 1) : (•, K, M) \overline{}(∗, K, M)

and

(b, 1, 1) : (∗, K, M) \overline{}(∘, K, M)

which exhibit (•, K, M), (∗, K, M) and (∘, K, M) as isomorphic objects of the 2-category

EQT_{∗}

3.15. We turn now to adjunctions in EQT_{∗} and ^{∗}EQT_{∗}. For arrows

[figure: N; M; S; R; K; F; •; ∗; ≃^{q}; and; M; N; T; K; R; G; ∗; •; ≃^{p}]

in EQT_{∗} to be part of an adjunction therein, say u, v : (q, F, S) ⊣ (p, G, T), it is necessary and sufficient that there be an adjunction u, v : S ⊣ T in EQT for which u and v are transformations of pointed equipments. Application of (−)_{#} to u, v : S ⊣ T provides the adjunction F ⊣ G. Write x for u_{#} and y for v_{#}. The adjunction x, y : F ⊣ G : K \overline{}R can be regarded as a further adjunction in EQT. Pointedness of each of u and v is given by an equation of the kind in Definition 3.9. Explicitly, the unit u is pointed precisely when, in the 2-category EQT, we have

[figure: N; M; R; K; F; •; ∗; N; R; G; •; S; T; 1; u; ≃^{q}; ≃^{p}; =; N; R; K; •; N; R; •; F; G; 1; x; 1; .]

We need the following purely 2-categorical result, which is closely related to Theorem 1.5 of [K&S]; it could in fact, after some preliminaries, be reduced to that theorem — but it is quicker to give a direct proof.

3.16. Lemma. Suppose that in an arbitrary 2-category we are given adjunctions x, y : F ⊣ G : K \overline{}R and u, v : S ⊣ T : M \overline{}N, along with arrows ∗ : K \overline{}M and • : R \overline{}N, and transformations p : T ∗ \overline{}• G and q : S • \overline{}∗ F (as in the squares displayed at the beginning of 3.15, except that p and q need not be invertible). Write q̄: •G \overline{}T ∗ for the mate of q under the adjunctions. Say that x and u are compatible if pF.T q.u• = •x (as in the equation displayed before the lemma), and that y and v are compatible if they satisfy the analogous equation ∗y.qG.Sp = v∗. Then in fact (i) x and u are compatible if and only if pq̄ = 1;

(ii) y and v are compatible if and only if q̄p = 1.

Thus both pairs are compatible if and only if p and q̄ are mutually inverse.

Proof. By [K&S], the mate q̄ of q is the pasting composite

[figure: 1; G; F; y; ∗; •; q; 1; S; T; u; ,]

so that pq̄ is the composite

[figure: 1; G; F; y; ∗; •; q; T; 1; S; u; G; •; p; .]

Given that the diagrams

[figure: 1; F; G; x; ,; 1; G; F; y]

are pasting-inverses, to say that pq̄ = 1 is equally to say that

[figure: ∗; •; F; q; T; 1; S; u; G; •; p; =; G; 1; F; x; •; ;]

which is the assertion that x and u are compatible. This gives (i), and the proof of (ii) is dual: one uses the expression above for q̄ to write q̄p as a pasting composite, and then uses the fact that u and v are pasting-inverses.

3.17. Corollary. (i) If

(x, u), (y, v) : (q, F, S) ⊣ (p, G, T) : (∗, K, M) \overline{}(•, R, N)

is an adjunction in EQT_{∗} then p and the mate of q are inverse to each other. (ii) If (q, F, S) : (•, R, N) \overline{}(∗, K, M) is an arrow in EQT_{∗} and

(x, u), (y, v) : (F, S) ⊣ (G, T) : (K, M) \overline{}(R, N)

is an adjunction in EQT and q̄, the mate of q, is invertible then defining p = q̄^{−1} provides an arrow (p, G, T) : (∗, K, M) \overline{}(•, R, N) and an adjunction (x, u), (y, v) : (q, F, S) ⊣ (p, G, T) in EQT_{∗}.

Corollary 3.17 furnishes us with the following coarse adjunction theorem for EQT_{∗} (and ^{∗}EQT_{∗}): An arrow

(q, S) : (•, N) \overline{}(∗, M)

in EQT_{∗} has a right adjoint if and only if S : N \overline{}M has a right adjoint in EQT and the resulting mate of q is invertible. Our goal in the next section is to refine the “invertible mate” condition so as to relate it to the hypotheses of Theorem 2.40, specialized to the case of equipments.

3.18. We focus now on the hypothesis in Corollary 3.17 (ii) and apply the general theory of 2.34. With (x, y : F ⊣ G) = (u, v : S ⊣ T)_{#} fixed, we have an equivalence between the category of adjunctions (F, S) ⊣ (G, T) : (K, M) \overline{}(R, N) in EQT and the category of all adjunctions σ ⊣ τ : M(F, 1) \overline{}N (1, G) in R, K-MOD. In particular, we have an equivalence between the category of adjunctions (F, F′) ⊣ (G, G′) : (K, K) \overline{}(R, R) in EQT and the category of all adjunctions ϕ′ ⊣ γ′ : K(F, 1) \overline{}R(1, G) in R, K-MOD; under this, let the particular adjunction (F, F) ⊣ (G, G) correspond to ϕ ⊣ γ : K(F, 1) \overline{}R(1, G). This last adjunction is trivial, in the sense that γ : K(F, 1) \overline{}R(1, G) is an isomorphism in R, K-MOD, with components the bijections K(F X, K) ^{\overline{}^{≃}} R(X, GK), and with ϕ : R(1, G) \overline{}K(F, 1) for its inverse, the unit and counit for ϕ ⊣ γ being identities.

Now a square q and its mate q̄ in EQT,

[figure: N; M; S; R; K; F; •; ∗; q; M; N; T; K; R; G; ∗; •; q̄; ,]

give rise to mates

[figure: N (1, G); M(F, 1); σ; R(1, G); K(F, 1); ϕ; •(1, G); ∗(F, 1); Q; M(F, 1); N (1, G); τ; K(F, 1); R(1, G); γ; ∗(F, 1); •(1, G); Q̄]

in R, K-MOD, with q invertible if and only if Q is invertible and q̄ invertible if and only if Q̄ is invertible. We have Q̄ = (τ ∗ (F, 1)1)(τQγ)(a • (1, G)γ), where the last 1 in τ ∗(F, 1)1 is the counit for the trivial adjunction ϕ ⊣ γ and a is the unit for the adjunction σ ⊣ τ. Under the hypotheses of Corollary 3.17 (ii), q is an isomorphism. Thus, Q is an isomorphism and it follows that q̄ is invertible if and only if Q̄ is invertible if and only if a • (1, G)γ is invertible. Since the arrow γ is invertible, we conclude that q̄ is invertible if and only if a • (1, G) is an isomorphism. In other words, q̄ is invertible if and only if, for every X in R, K in K and r : X \overline{}GK, the component a_{r_{•}} is invertible.

Since r_{•} ≅ ι_{GK}r by the remarks preceding Lemma 3.3, the invertibility of a_{r_{•}} is equivalent to that of a_{ι_{GK}r}. Recall from 2.37 that the natural transformations (τ^{−,r})^{−1} are the mates of the σ^{−,r}; in terms of the unit a : 1 \overline{}τσ, this reduces to the commutativity of

[figure: τ(σ(μr)); τ(σ(μ).F r); τ(σ^{μ,r}); μr; τ(σ(μ))r; a_{μ}r; a_{μr}; (τ^{σ(μ),r})^{−1}]

for any vector arrow μ : GK \overline{}X; we are concerned with the case μ = ι_{GK}. From the description of σ in 2.30 it follows that the σ^{ι_{GK},r} are invertible if the S^{ι_{GK},r} are so — but the latter is so by Proposition 3.7. It follows that invertibility of the a_{r_{•}}, and hence the invertibility of the mate of q, is equivalent to the invertibility of the a_{ι_{GK}}. It is worth observing that the codomain of a_{ι_{GK}} is τ(σ(ι_{GK})) ^{\overline{}^{≃}} τ((ϕ(1_{GK}))_{∗}) = τ((y_{K})_{∗}), this isomorphism being the 1_{GK}-component of the natural transformation (τQ)_{GK,K} (where Q is given above).

Before summarizing this discussion we note the variant of the “invertible mate” condition that is available for starred pointed equipments. Write c for the unit of the adjunction S̃ ⊣ Š : M(F, F) \overline{}N in R, R-^{∗}MOD, as in 2.35, that further corresponds to the adjunction (F, S) ⊣ (G, T) : (K, M) \overline{}(R, N) in ^{∗}EQT, and as usual write ỹ for the unit of the adjunction y− ⊣ y^{∗}− (where y is the counit for F ⊣ G). Now the unit a for σ ⊣ τ is given by a = (ŠỹS)c. The codomain of a_{ι_{GK}} is Š(y_{K}^{∗}(y_{K}Sι_{GK})). We have a composite of isomorphisms

Š(y_{K}^{∗}(y_{K}Sι_{GK})) ^{\overline{}^{≃}} Š(y_{K}^{∗}(y_{K}ι_{F GK})) ^{\overline{}^{≃}} Š(y_{K}^{∗}(ι_{K}y_{K})),

where the first is Š(y_{K}^{∗}(y_{K}q)), with q again being the structural isomorphism making S pointed, and where the second is provided by the “commutative square” y_{K}ι_{F GK} ^{\overline{}^{≃}} ι_{K}y_{K}. Write e_{K} : ι_{GK} \overline{}Š(y_{K}^{∗}(ι_{K}y_{K})) for the composite of a_{ι_{GK}} with the displayed isomorphism. Clearly, the a_{ι_{GK}} are invertible if and only if the e_{K} are invertible.

Combining these observations with the specialization to equipments of Theorem 2.40 provides:

3.19. Theorem. An arrow (q, F, S) : (•, R, N) \overline{}(∗, K, M) in EQT_{∗} has a right adjoint if and only if the following five conditions are satisfied:

(o) the functor F has a right adjoint G, with counit say y : F G \overline{}1;

(i) S is an R-homomorphism;

(ii) for each X in R and L in L, the functor σ(S)_{X,L} : N (X, GL) \overline{}M(F X, L), defined as the composite

N (X, GL) ^{\overline{}^{S_{X,GL}}} M(F X, F GL) ^{\overline{}^{M(F X,yL)}} M(F X, L),

has a right adjoint τ_{X,L};

(i′) if σ(S)^{ν,r} : σ(S)(νr) \overline{}σ(S)ν.F r denotes the composite

yL.S(νr) ^{\overline{}^{yL.S^{ν,r}}} yL.(Sν.F r) ^{\overline{}^{≃}^{η}} (yL.Sν).F r,

and if θ^{−,r} denotes the mate of σ(S)^{−,r}, then each θ^{μ,r} is invertible;

(iii) for each object K in K, the component a_{ι_{GK}} of the unit for the adjunction σ(S)_{GK,K} ⊣ τ_{GK,K} of (ii) is an isomorphism.

When these conditions are satisfied, the right adjoint of (q, F, S) is (p, G, T (τ)), where the right adjoint τ of σ(S) is constructed from the data above as in Theorem 2.39, and where p is the inverse of the mate of q.

For (q, F, S) : (•, R, N) \overline{}(∗, K, M) in ^{∗}EQT_{∗}, conditions (i), (i′), (ii) and (iii) can be replaced by:

(i^{∗}) S is a right homomorphism;

(ii^{∗}) for each X and A in R, the functor S_{X,A} : N (X, A) \overline{}M(F X, F A) has a right adjoint.

(iii^{∗}) for each K in K, the e_{K} : ι_{GK} \overline{}Š(y_{K}^{∗}(ι_{K}y_{K})), constructed as above, is invertible.

The reader should compare the codomain of e_{K} with the diagram in Theorem 2.36 (ii). The inverses of the e_{K} provide the structure isomorphism for the right adjoint (p, T) of (q, S), where T is the right adjoint of S in ^{∗}EQT constructed by Theorem 2.40.

3.20. Quite generally, a pointing for an equipment (K, M) provides a link between the “global” scalar structure K and the “local” vector structures M(K, L). In the presence of adjunctions the link can become very strong. The following typical application of Theorem 3.19, which follows through the example in 2.41 for Theorem 2.40, will illustrate this.

For a pointed equipment M = (∗, K, M) and a category C, we have M^{C} as before, with the pointing given componentwise. Explicitly, if F is an object of M^{C} then ι_{F} (C) = ι_{F C}, for every C in C, while for every c : C \overline{}C′ in C we have ι_{F} (c) given by the commutative square F c.ι_{F C} ^{\overline{}^{≃}} ι_{F C′}.F c. If M is a starred pointed equipment then so is M^{C}. The diagonal D : M \overline{}M^{C} is a pointed homomorphism.

3.21. Definition. A pointed equipment M has C-limits if the diagonal D : M \overline{}M^{C} has a right adjoint in EQT_{∗}.

3.22. From Theorem 3.19 it follows that a starred pointed equipment M has C-limits if and only if K and each M(K, L) have C-limits and moreover, for every F : C \overline{}K with limit cone p : P \overline{}F in K, the arrows p̄_{C} : ι_{P} \overline{}p_{C}^{∗}(ι_{F C}p_{C}), corresponding by adjunction to the commutative squares p_{C}.ι_{P} ^{\overline{}^{≃}} ι_{F C}.p_{C}, constitute a limit cone in M(P, P). (The diagram in M(P, P) has, for each c : C \overline{}C′ in C, the p_{C}^{∗}(ι_{F C}p_{C}) \overline{}p_{C′}^{∗}(ι_{F C′}p_{C′}) = (F c.p_{C})^{∗}(ι_{F C′}(F c.p_{C})) given by p_{C}^{∗}(−)p_{C} applied, with appropriate parenthesization, to ι_{F C} \overline{}(F c)^{∗}(ι_{F C′}.F c), which last also corresponds to an obvious commutative square.)

To give a simple specific case, take C to be the free-living cospan a \overline{}c \overline{}b. Say that a pointed equipment M has pullbacks if M has C-limits for this C. It follows that a starred pointed M has pullbacks if and only K and each M(K, L) have pullbacks and moreover, for every pullback

[figure: A; C; f; P; B; q; p; g; r]

in K,

[figure: p^{∗}(ι_{A}p); r^{∗}(ι_{C}r); ι_{P}; q^{∗}(ι_{B}q)]

is a pullback in M(P, P).

Taking C to be the discrete category with two objects defines binary products, while taking C to be the empty category defines terminal objects, within an equipment. An equipment M has both if and only if K and each M(K, L) have finite products and moreover, for each binary product p : A \overline{}A × B \overline{}B : q in K, p^{∗}(ι_{A}p) \overline{}ι_{A×B} \overline{}q^{∗}(ι_{B}q) is a binary product in M(A × B, A × B), and ι_{1} is the terminal object of M(1, 1), where 1 is the terminal object of K. These conditions (with each ι taken to be an identity) were a crucial part of the definition of cartesian bicategory in [C&W]. The treatment of these in [CKW] can now be seen as the special case of ours, in which all equipments under consideration have each M(K, L) an ordered set.

4. The 2-functor spn : PBK ⟶ ^{∗}EQT_{∗}

4.1. In [CKW] (section 4) it was shown that rel can be regarded as a colax functor from the full and locally-full sub-2-category of CAT determined by the regular categories to a 2-category that we called F. While the foundations presented here differ somewhat from those in [CKW], it is a straightforward matter to replace F in the earlier treatment by ∗EQT_{∗}. The starred pointed equipments in question have modules M : K^{op}, K \overline{}CAT which factor through the 2-category of ordered sets. A key goal of the present treatment, as promised in [CKW], is the development of that work beyond such “locally ordered” examples. Accordingly, we give here a treatment of spn as an extended example. The following simple result motivates our preoccupation with categories that admit pullbacks.

4.2. Lemma. For any category C, the module spn′C : C \overline{}C^{op} of 2.4 (i) satisfies the conditions in the penultimate paragraph of 2.8 — namely the existence of right adjoints l^{∗}(−) for l(−) and (−)k for (−)k^{∗}, with these satisfying a Beck-Chevalley condition — if and only if C has pullbacks.

Proof. The “if” part being clear, suppose that spn′C : C \overline{}C^{op} satisfies the conditions above, and consider a cospan f : X \overline{}A \overline{}Y : g in C. Write (1, A, 1) for the object 1 : A \overline{}A \overline{}A : 1 of spn′C(A, A), and p : X \overline{}S \overline{}Y : q for the object g^{∗}((1, A, 1)f) of spn′C(X, Y). Since the identity arrow of the span (p, S, q) corresponds under the adjunctions to an arrow r : (g(p, S, q))f^{∗} \overline{}(1, A, 1) of spans, and since (g(p, S, q))f^{∗} = (fp, S, gq), we have in the category C the commutative square fp = r = gq. For a general object (x, T, y) of spn′C(X, Y), there is a bijection between arrows (x, T, y) \overline{}(p, S, q) = g^{∗}((1, A, 1)f) in spn′C(X, Y) and arrows (fx, T, gy) = (g(x, T, y))f^{∗} \overline{}(1, A, 1) in spn′C(A, A) — of which there is just one if fx = gy, and none otherwise. So p : X \overline{}S \overline{}Y : q is the pullback of f : X \overline{}A \overline{}Y : g.

4.3. We denote by PBK the full sub-2-category of CAT determined by the categories which admit pullbacks. For K in PBK we write spnK : K \overline{}K for the starred equipment obtained from spn′K by the use of Lemma 4.2; observe that (spnK)_{#} = K, in the language of 2.9, and that we may sometimes denote the equipment more fully by (K, spnK). We recall that, for a span (x, S, a) : X \overline{}A and arrows f : Y \overline{}X and g : A \overline{}B of K, we have g(x, S, a) = (x, S, ga), while (x, S, a)f is obtained by pulling back along f; similarly (x, S, a)h^{∗} is given by composition and k^{∗}(x, S, a) by pulling back. Clearly a square

[figure: A; B; g; X; Y; f; (x, S, a); (y, T, b); Σ]

in spnK is nothing but a commutative diagram

[figure: S; T; X; Y; f; x; y; A; B; g; s; a; b]

in K, so that we may as well identify Σ with s; indeed gr(spnK) is clearly isomorphic to K^{Λ}, where Λ denotes (as in 2.9) the category 0 \overline{}⋆ \overline{}1. Clearly spnK becomes a starred pointed equipment when we define the module-arrow (−)_{∗} : K \overline{}spnK of Definition 3.2 by taking h_{∗} for h ∈ K(X, A) to be the span 1_{X} : X \overline{}X \overline{}A : h; so that, in particular, ι_{X} is the span (1, X, 1). Note that, in terms of the Grothendieck categories, (−)_{∗} : grK \overline{}gr(spnK) is just the evident functor K^{2} \overline{}K^{Λ} sending the object h : H \overline{}A to the span (1_{X}, X, h) as above.

Any functor G : K \overline{}L of course induces a functor G^{Λ} : K^{Λ} \overline{}L^{Λ} — that is, a functor gr(spnK) \overline{}gr(spnL) — compatible with ∂_{0} and ∂_{1}, thus giving an arrow of equipments (G, spnG) : (K, spnK) \overline{}(L, spnL), defined on vector arrows by spnG(x, S, a) = (Gx, GS, Ga). This arrow is a right homomorphism, since clearly the comparison-arrows spnG_{g,(x,S,a)} and spnG_{(x,S,a),h^{∗}} are identities. In particular spnG_{g,ι_{X}} is an identity; whence it follows from Proposition 3.8 that spnG becomes an arrow of pointed equipments when we augment it with the isomorphism (spnG)ι_{X} ^{\overline{}^{≃}} ι_{GX} given by the identity.

Similarly each natural transformation t : F \overline{}G : K \overline{}L induces a natural transformation t^{Λ} : F^{Λ} \overline{}G^{Λ} : K^{Λ} \overline{}L^{Λ}; and since K^{Λ} and L^{Λ} are gr(spnK) and gr(spnL), this t^{Λ} is precisely an equipment-transformation (t, spnt) : (F, spnF) \overline{}(G, spnG), where (spnt)_{(x,S,a)} = t_{S}. It is clear from Definition 3.9 that (t, spnt) is is fact a transformation of pointed equipments. Since (−)^{Λ} is a 2-functor PBK \overline{}PBK, it is immediate that we have a 2-functor spn : PBK \overline{}^{∗}EQT_{∗}. (Compare this with the case of rel studied in [CKW]; there we had not a 2-functor but only a colax functor rel : REG \overline{}^{∗}EQT_{∗}.) As usual, we shall denote the functor expressing the “local” aspect of the 2-functor spn for fixed K and L by

spn_{K,L} : PBK(K, L) \overline{}^{∗}EQT_{∗}(spnK, spnL).

It is convenient to observe here that if (q, S), (p, T) : spnK \overline{}spnL are general arrows in ^{∗}EQT_{∗} then any equipment transformation u : S \overline{}T satisfies the condition of Proposition 3.11 and is thus a pointed equipment transformation. This follows directly from the description of squares in spnL and is left as an exercise.

4.4. We now consider the image of the functor spn_{K,L}. If (p, T) : spnK \overline{}spnL is an arrow in ^{∗}EQT_{∗}, the 2-functor (−)_{#} : ^{∗}EQT_{∗} \overline{}CAT (as in 3.12) sends it to an arrow T_{#} : K \overline{}L in CAT which, the inclusion PBK \overline{}CAT being fully faithful, is the same thing as an arrow T_{#} : K \overline{}L in PBK. Similarly a transformation u : S \overline{}T : spnK \overline{}spnL gives rise to a transformation u_{#} : S_{#} \overline{}T_{#} : K \overline{}L in PBK. Thus (−)_{#} provides a functor

(−)_{#} : ^{∗}EQT_{∗}(spnK, spnL) \overline{}PBK(K, L),

which from the definitions in 4.3 is seen to satisfy

(−)_{#}spn_{K,L} = 1 : PBK(K, L) \overline{}PBK(K, L).

Now, for (p, T) : spnK \overline{}spnL in ^{∗}EQT_{∗}, we compare spn(T_{#}) and (p, T). Set G = T_{#}. The typical vector arrow (x, S, a) : X \overline{}A in spnK can be written as aι_{S}x^{∗} : X \overline{}A (no parentheses being necessary for these actions in this example). We have the comparison T_{a,ι_{S},x^{∗}} : Ga.T ι_{S}.Gx^{∗} \overline{}T (aι_{S}x^{∗}) and the isomorphism p : T ι_{S} ^{\overline{}^{≃}} ι_{GS}. Together these give

spn_{K,L}(T_{#})(x, S, a) = Ga.ι_{GS}.Gx^{∗} ^{\overline{}^{≃}} Ga.T ι_{S}Gx^{∗} \overline{}T (aι_{S}x^{∗}) = T (x, S, a).

This is readily seen to be the (x, S, a)-component of an equipment transformation e_{T} : spn_{K,L}(T_{#}) \overline{}T with (e_{T})_{#} = 1_{T_{#}}. It follows that e_{T} is invertible if and only if all the T_{a,ι_{S},x^{∗}} are invertible. From the coherence conditions given in Proposition 2.14 and Lemma 2.18, invertibility of the T_{a,ι_{S},x^{∗}} implies invertibility of all the T_{g,(x,S,a),h^{∗}}. So e_{T} is invertible if and only if T is a right homomorphism. Moreover, e_{T} is natural in T and provides the counit for an adjunction 1, e : spn_{K,L} ⊣ (−)_{#} : ^{∗}EQT_{∗}(spnK, spnL) \overline{}PBK(K, L). We summarize the discussion as follows:

4.5. Proposition. An arrow (p, T) : spnK \overline{}spnL in ^{∗}EQT_{∗} is isomorphic to one of the form spnG for some functor G : K \overline{}L if and only if T is a right homomorphism, and then G is necessarily isomorphic to T_{#}. The functor spn_{K,L} is fully faithful, every transformation u : spnG \overline{}spnH being spnt for a unique t : G \overline{}H : K \overline{}L, namely t = u_{#}.

From the description of spn in 4.3, one easily calculates the comparison arrows spnG^{(x,S,a),f} and spnG^{k^{∗},(x,S,a)}. For the first, we form the pullbacks

[figure: Y; X; f; P; S; q; p; x; ,; GY; GX; Gf; Q; GS; s; r; Gx; ,]

in K and L respectively; then

spnG^{(x,S,a),f} : spnG((x, S, a).f) \overline{}spnG(x, S, a).Gf

is the induced arrow of spans given by

[figure: GP; Q; GY; GY; 1; Gp; r; GA; GA; 1; G(aq); Ga.s; ,]

and spnG^{k^{∗},(x,S,a)} is formed similarly. It follows that spnG is a left homomorphism (and hence a homomorphism) when G preserves pullbacks. Moreover the converse is also true, as we see by taking a : S \overline{}A to be 1 : S \overline{}S in the calculation above. If we start with a general homomorphism of starred pointed equipments (p, T) : spnK \overline{}spnL, then e_{T} : spn_{K,L}(T_{#}) \overline{}T is an isomorphism; and since an isomorph of a homomorphism is again a homomorphism, it follows that spnT_{#} is a homomorphism and hence that T_{#} is pullback preserving. We summarize again:

4.6. Proposition. An arrow (p, T) : spnK \overline{}spnL in ^{∗}EQT_{∗} is isomorphic to one of the form spnG for some pullback-preserving functor G : K \overline{}L if and only if T is a homomorphism; and then G is necessarily isomorphic to T_{#}.

4.7. It is convenient to write PBK_{pbk} for the locally-full sub-2-category of PBK determined by the pullback-preserving functors, and ^{∗}EQT_{∗hom} [^{∗}EQT_{∗rhm}, ^{∗}EQT_{∗lhm}] for the locally-full sub-2-category of ^{∗}EQT_{∗} determined by the homomorphisms [right homomorphisms, left homomorphisms]. We may as well regard spn as a 2-functor spn : PBK \overline{}^{∗}EQT_{∗rhm}, so that the adjunction 1, e : spn_{K,L} ⊣ (−)_{#} of 4.4 provides by Proposition 4.5 an (adjoint) equivalence of categories

1, e : spn_{K,L} ⊣ (−)_{#} : ^{∗}EQT_{∗rhm}(spnK, spnL) \overline{}PBK(K, L).

By restriction we have spn : PBK_{pbk} \overline{}^{∗}EQT_{∗hom} and hence by Proposition 4.6 a further equivalence of categories

1, e : spn_{K,L} ⊣ (−)_{#} : ^{∗}EQT_{∗hom}(spnK, spnL) \overline{}PBK_{pbk}(K, L).

4.8. For any 2-category K and objects K and L therein, we write LAJ(K)(L, K) for the evident category whose objects are adjunctions x, y : F ⊣ G : K \overline{}L and whose arrows from x, y : F ⊣ G to x′, y′ : F′ ⊣ G′ are transformations F \overline{}F′. Since any 2-functor, such as spn : PBK \overline{}^{∗}EQT_{∗}, takes adjunctions to adjunctions, we have a functor

spn_{L,K} : LAJ(PBK)(L, K) \overline{}LAJ(^{∗}EQT_{∗})(spnL, spnK).

Since (−)_{#} is a 2-functor we have also a functor

(−)_{#} : LAJ(^{∗}EQT_{∗})(spnL, spnK) \overline{}LAJ(PBK)(L, K),

for which the e of 4.4 lifts to provide an adjunction 1, e : spn_{K,L} ⊣ (−)_{#}, where the (u, v : S ⊣ T)-component of this new e is the S-component of the former one.

4.9. Proposition. The adjunction

1, e : spn_{L,K} ⊣ (−)_{#} : LAJ(^{∗}EQT_{∗})(spnL, spnK) \overline{}LAJ(PBK)(L, K)

is an equivalence of categories.

Proof. For an object u, v : S ⊣ T in LAJ(^{∗}EQT_{∗})(spnL, spnK), we have only to show that e_{S⊣T} = e_{S} is invertible, which is the case if and only if S is a right homomorphism. From Proposition 2.21, however, any left adjoint in ^{∗}EQT_{∗} is a right homomorphism.

Moreover, the right-adjoint component T of the object S ⊣ T in the proof of Proposition 4.9 is clearly isomorphic to spn(T_{#}). So, it is like S a right homomorphism, and the adjunction T ⊣ U in ^{∗}EQT_{∗} is also an adjunction in ^{∗}EQT_{∗rhm}. Thus:

4.10. Corollary. The inclusion of categories

LAJ(^{∗}EQT_{∗rhm})(spnL, spnK) \overline{}LAJ(^{∗}EQT_{∗})(spnL, spnK)

is an equality.

Combining Propositions 4.6 and 4.9 brings us to the image under spn of adjunctions F ⊣ G : K \overline{}L in PBK with F pullback-preserving — which is to say adunctions in PBK_{pbk}. (Such adjunctions were studied in [R&W] from points of view that enabled them to be seen both as partial geometric morphisms and families of geometric morphisms.) Clearly, they correspond to adjunctions S ⊣ T : spnK \overline{}spnL in ^{∗}EQT_{∗} with S a homomorphism. On the face of it, such an adjunction is in ^{∗}EQT_{∗lhm}; but we have already observed that T is also a right homomorphism. We conclude that:

4.11. Corollary. The inclusion of categories

LAJ(^{∗}EQT_{∗hom})(spnL, spnK) \overline{}LAJ(^{∗}EQT_{∗rhm})(spnL, spnK)

is an equality.

4.12. Corollary. The adjunction

spn_{L,K} ⊣ (−)_{#} : LAJ(^{∗}EQT_{∗hom})(spnL, spnK) \overline{}LAJ(PBK_{pbk})(L, K)

is an equivalence of categories.

4.13. Suppose that K is a category with pullbacks and C-limits. The adjunction in CAT which provides the C-limits has for its left-adjoint component the diagonal K \overline{}K^{C} and can be seen as an adjunction in PBK_{pbk}. The 2-functors spn : PBK \overline{}^{∗}EQT_{∗} and spn : PBK_{pbk} \overline{}^{∗}EQT_{∗hom} can be shown to preserve powers. (Powers in PBK and PBK_{pbk} are as in CAT. Powers in ^{∗}EQT_{∗hom} are as in ^{∗}EQT_{∗}.) It follows from 3.22 and the above that if K has C-limits then spnK has C-limits as a pointed equipment, and that the resulting spnK^{C} \overline{}spnK is a homomorphism. For a general starred pointed equipment M which has C-limits, there is no reason to suppose that M^{C} \overline{}M is a right homomorphism.

For T : M \overline{}N in any 2-category with powers, and a category C, we have a commutative square of arrows

[figure: N; N^{C}; D; M; M^{C}; D; T; T^{C}; 1; ,]

where the D’s are diagonals. If each of the diagonals has a right adjoint L, then the identity transformation has a mate

[figure: N; N^{C}; L; M; M^{C}; L; T; T^{C}; T̃; .]

4.14. Definition. If, in a 2-category with C-powers, objects M and N have C-limits in the sense above, then an arrow T : M \overline{}N is said to preserve C-limits if T̃ is invertible.

4.15. In section 5 of [CKW] a very detailed general discussion of preservation is given for finite products. From that discussion it follows easily that (i) if G : K \overline{}L in PBK preserves C-limits then spnG in ^{∗}EQT_{∗} preserves C-limits and (ii) right adjoints in ^{∗}EQT_{∗} preserve C-limits.

In the case of pointed equipments, Definition 4.14 requires that, for each vector arrow θ : F \overline{}G in M^{C}, the square

[figure: T LG; LT^{C}F; T̃G; T LF; LT^{C}F; T̃F; T Lθ; LT^{C}θ; T̃θ]

be an isomorphism in grN. By Lemma 2.6, this is equivalent to asking for invertibility of each T̃F and commutativity of each such square. Invertibility of the T̃F means precisely that T_{#} preserves C-limits. In the presence of this condition, for pointed T between starred pointed equipments, commutativity of the squares is equivalent to preservation of C-limits by the T_{K,L} : M(K, L) \overline{}N (T K, T L) — apparently an additional condition.

However, in the case of starred pointed equipments with C the empty category, a pointed T for which T_{#} preserves C-limits does itself preserve C-limits. For here we are speaking of terminal objects and the only square under consideration is

[figure: T_{#}1; 1; T_{#}1; 1; T ι_{1}; ι_{1}; ,]

where each 1 denotes a terminal object in the appropriate category of scalars. Recall from the discussion in 3.22 that ι_{1} is the terminal object in N (1, 1). For any r : X \overline{}A in N_{#} the (ordinary) functor (−)r preserves limits because it has the left adjoint (−)r^{∗}. Thus, in the diagram above, the upper leg is the terminal object in N (T_{#}1, 1) and the vector transformation is unambiguous. Moreover, the square

[figure: T_{#}1; 1; T_{#}1; 1; ι_{T_{#}1}; ι_{1}]

is commutative, and the square above is obtained from it by pasting the isomorphism T ι_{1} ^{\overline{}^{≃}} ι_{T_{#}1} at the left edge. Thus an arrow T in ^{∗}EQT_{∗} preserves terminal objects if and only if T_{#}1 \overline{}1 is invertible.

4.16. For any 2-category K and objects K and L therein admitting terminal objects, let us write K_{1}(K, L) for the full subcategory of K(K, L) determined by the arrows K \overline{}L that preserve terminal objects. Write LEX for the full sub-2-category of PBK determined by the categories which have terminal objects. Then the familiar 2-category LEX_{lex} of finitely complete categories, left exact functors and arbitrary natural transformations has LEX_{lex}(K, L) = PBK_{pbk,1}(K, L). The following proposition then follows immediately from the last equivalence in 4.7 and the conclusion of 4.15:

4.17. Proposition. For categories K and L with finite limits, the adjunction

spn_{K,L} ⊣ (−)_{#} : ^{∗}EQT_{∗hom,1}(spnK, spnL) \overline{}LEX_{lex}(K, L)

is an equivalence of categories.

By a geometric morphism K \overline{}L between categories with finite limits we understand an adjunction x, y : F ⊣ G : K \overline{}L with F left exact. Write LEX_{geo} for the 2category of finitely-complete categories and geometric morphisms. Now LEX_{geo}(K, L) = LAJ(PBK_{pbk,1})(L, K), and the following proposition follows immediately from Corollary 4.12.

4.18. Proposition. The adjunction

spn_{L,K} ⊣ (−)_{#} : LAJ(^{∗}EQT_{∗hom,1})(spnL, spnK) \overline{}LEX_{geo}(K, L)

is an equivalence of categories.

5. Appendix

Let K be a 2-category and L a dual KZ-doctrine (in the sense of [KCK]) on K, with the 2-natural transformation λ : 1_{K} \overline{}L as its unit. An L-algebra is an object M of K together with an arrow M : LM \overline{}M which is a reflection right adjoint for λM : M \overline{}LM. If (M, M) and (N , N) are two such algebras and T : M \overline{}N is any arrow in K, the identity square 1 : λN .T \overline{}LT.λM (given by the naturality of λ) has, under the adjunctions λM ⊣ M and λN ⊣ N, a mate

[figure: M; N; T; LM; LN; LT; M; N; T^{−,−}]

called the canonical comparison for T; and T is said to be an L-homomorphism precisely when T^{−,−} is invertible.

Proposition 2.5 of [KCK] asserts that such a T is certainly an L-homomorphism if it has a left adjoint, and the proof of that proposition describes the inverse of T^{−,−} explicitly in terms of the adjunction η, ϵ : S ⊣ T. In fact the inverse t^{−,−} : N.LT \overline{}T.M of T^{−,−} is itself the mate, under the adjunctions S ⊣ T and LS ⊣ LT, of the canonical comparison S^{−,−} : S.N \overline{}M.LS for S.

Take K to be [Λ, CAT], or spnCAT(L, K) as in 2.15. In either case define

L(K ^{\overline{}^{∂_{0}}} M ^{\overline{}^{∂_{1}}} L)

to be the span from L to K given by

[figure: K; M; K/∂_{0}; K; 1; ∂_{0}; L; ∂_{1}; ,]

where the square is a comma square. Define λM : M \overline{}LM in spnCAT(L, K) to be the functor which sends μ to (∂_{0}μ, 1_{∂_{0}μ}, μ). It is well known that L extends to a 2-functor that underlies a dual KZ-doctrine, the algebras for which are the left-fibration spans. Among these are the spans of the form ĝrM, for a module M, and these now constitute our sole interest. Consistently with the notation of the paper, let us write M for ĝrM and

K′ ^{\overline{}^{k}} K ^{\overline{}^{μ}} L

for a typical object of LM.

The structure arrow M : LM \overline{}M is the action functor that sends K′ ^{\overline{}^{k}} K ^{\overline{}^{μ}} L to K′ ^{\overline{}^{μk}} L. Observe that, for k and L fixed, M restricts to the functor M(k, L). Moreover if T : M \overline{}N is an arrow in MOD, and thus an arrow T : ĝrM \overline{}ĝrN in [Λ, CAT], then the (k, μ)-component of T^{−,−}, as in the first square of this appendix, is the action comparison T^{μ,k} : T (μk) \overline{}T μ.T k. If T is in K, L-MOD, as in 2.37, then the scalar components of T are identity functors, so that we have T^{μ,k} : T (μk) \overline{}T μ.k. This being so, it follows from the general result above that (T^{μ,k})^{−1} is indeed the t^{μ,k} of 2.37.

References

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[KCK] A. Kock. Monads for which structures are adjoint to units. Journal of Pure and Applied Algebra, 104:41–59, 1995.

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[PAJ] A. J. Power. A 2-categorical pasting theorem. Journal of Algebra, 129:439–445, 1990.

[R&W] Robert Rosebrugh and R. J. Wood. Pullback preserving functors. Journal of Pure and Applied Algebra, 73:73–90, 1991.

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[S&W] R. Street and R. Walters. Yoneda structures on 2-categories Journal of Algebra, 50(2):350–379, 1978.

[VTY] Dominic Verity. Enriched Categories, Internal Categories and Change of Base. PhD thesis, Cambridge, 1992.

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II Facoltà di Scienze MM. FF. NN. Università di Milano Via Lucini 3, 22100 Como Italy and School of Mathematics and Statistics FO7 University of Sydney, N.S.W. 2006 Australia and School of Mathematics, Physics, Computing and Electronics Macquarie University, N.S.W. 2109 Australia and Department of Mathematics, Statistics and Computing Science Dalhousie University, Halifax, N.S. B3H 3J5 Canada

Email: carboni@vmimat.mat.unimi.it and maxk@maths.usyd.edu.au and
domv@macadam.mpce.mq.edu.au and rjwood@mscs.dal.ca

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Editorial board.

John Baez, University of California, Riverside: baez@math.ucr.edu Michael Barr, McGill University: barr@math.mcgill.ca Lawrence Breen, Université de 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 della Calabria: carboni@unical.it P. T. Johnstone, University of Cambridge: ptj@pmms.cam.ac.uk G. Max Kelly, University of Sydney: kelly m@maths.usyd.edu.au Anders Kock, University of Aarhus: kock@mi.aau.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.ruu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mscs.dal.ca Andrew Pitts, University of Cambridge: ap@cl.cam.ac.uk Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz James Stasheff, University of North Carolina: jds@charlie.math.unc.edu Ross Street, Macquarie University: street@macadam.mpce.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 Sydney: walters b@maths.usyd.edu.au R. J. Wood, Dalhousie University: rjwood@mscs.dal.ca