Theory and Applications of Categories, Vol. 3, No. 2, 1997, pp. 24–44.

DOCTRINES WHOSE STRUCTURE FORMS A FULLY FAITHFUL ADJOINT STRING

F. MARMOLEJO

Transmitted by Ross Street

ABSTRACT. We pursue the definition of a KZ-doctrine in terms of a fully faithful adjoint string Dd ⊣ m ⊣ dD. We give the definition in any Gray-category. The concept of algebra is given as an adjunction with invertible counit. We show that these doctrines are instances of more general pseudomonads. The algebras for a pseudomonad are defined in more familiar terms and shown to be the same as the ones defined as adjunctions when we start with a KZ-doctrine.

1. Introduction

Free co-completions of categories under suitable classes of colimits were the motivating examples for the definition of KZ-doctrines. We introduce in this paper a not-strict version of such doctrines defined via a fully faithful adjoint string. Thus, a non-strict KZ-doctrine on a 2-category K consists of a normal endo homomorphism D : K ⟶ K, and strong transformations d : 1_{K} ⟶ D, and m : DD ⟶ D in such a way that Dd ⊣ m ⊣ dD forms a fully faithful adjoint string, satisfying one equation involving the unit of Dd ⊣ m and the counit of m ⊣ dD. Given an object C in K, we think of DC as the co-completion of C, consisting of suitable diagrams over C, dC : C ⟶ DC as the functor that assigns to every object of C the diagram on that object with identities for every arrow in the diagram, and mC : DDC ⟶ DC as a colimit functor. The idea of pursuing the adjoint string as definition follows in the steps of [3] and was suggested by R. J. Wood.

Now, Dd ⊣ m ⊣ dD being a fully faithful adjoint string means that the counit β : m ∘ dD ⟶ Id of m ⊣ dD is invertible (equivalently, the unit η : Id ⟶ m ∘ Dd is invertible [7]).

Recall that A. Kock’s algebraic presentation of KZ-doctrines [9] asks for equalities m ∘ dD = Id and Id = m ∘ Dd, and for a 2-cell δ : Dd ⟶ dD satisfying four equations.

We can produce from the adjoint string a 2-cell δ : Dd ⟶ dD, namely, the pasting of β^{−1} and the unit for the adjunction Dd ⊣ m. This δ satisfies similar (‘non-strict’ versions of) the conditions required for a KZ-doctrine in [9]. Thus, the KZ-doctrines of [9] are particular instances of our KZ-doctrines.

Since the algebras for a KZ-doctrine are given in terms of adjunctions it seems reasonable to define the doctrine in terms of adjunctions. Instead of having equality as in [9] we have the invertible 2-cells β and η. This laxification is justified if only because associativity in [9] is deduced up to isomorphism, but that paper also mentions some shortcomings of insisting on normalized algebras. We believe also that the approach via the adjoint string gives us a better insight into the nature of δ : Dd ⟶ dD.

Received by the editors 18 October 1996 and, in revised form, 21 January 1997.

Published on 12 February 1997

1991 Mathematics Subject Classification : 18A35, 18C15, 18C20, 18D05, 18D15, 18D20.

Key words and phrases: KZ-doctrines, Pseudomonads, Algebras, Gray-categories.

\begin{matrix}c \\ ◯ \end{matrix} F. Marmolejo 1997. Permission to copy for private use granted.

We work in the framework of enriched category theory [2], where the category V is equal to the category Gray with strict tensor product [5] (see [4] as well). By working in the context of Gray-categories we are developing the ‘formal theroy of KZ-doctrines’ in the way that, by working in a 2-category, [13] develops the ‘formal theory of monads’. Notice that this is a very general setting since every tricategory is equivalent to a Gray-category [5]. The idea of defining KZ-doctrines in an enriched setting is also suggested in [9].

We adopt the definition of a pseudomonoid given in [1]. We show that every KZdoctrine is a pseudomonad (pseudomonoid in the Gray monoid determined by an object of the Gray-category), and that the 2-categories of algebras defined as adjunctions coincide with the classical algebras for a pseudomonad (Theorem 10.7). We follow [13] in defining the algebras for a pseudomonad and the algebras for a KZ-doctrine with arbitrary objects of the Gray-category as domains.

R. Street [13] gives a conceptual global account of KZ-doctrines in terms of the simplicial category Δ. Recall that in that context a doctrine on a bicategory K is a homomorphism of bicategories Δ ⟶ Hom(K, K) that preserves the monoid structure (ordinal addition on the domain and composition on the codomain), with Δ considered as a locally discrete 2-category. A KZ-doctrine is a doctrine that agrees in the common domain with a homomorphism of bicategories Δ^{+} ⟶ Hom(K, K) where Δ^{+} is the 2-category of nonempty finite ordinals, order and last element preserving functions and inequalities. As pointed out in [9] this definition explicitly excludes the left most adjoint Dd ⊣ m, without any indication as to whether it can be put back on. We show that, for a pseudomonad to be a KZ-doctrine either one of the adjunctions Dd ⊣ m or m ⊣ dD is enough.

For examples of free cocompletions of categories under different kinds of colimits we refer the reader to the bibliography of [9].

I would like express my thanks to R. J. Wood who not only provided ideas for this paper but also agreed to discuss them with me. I would like to thank Dalhousie University for its hospitality. I would like thank the referee as well, whose revisions of the different versions of this paper were very helpful on the one hand, and very fast on the other.

2. Background

We work in the context of Gray-categories, where Gray is the symmetric monoidal closed category whose underlying category is 2-Cat with the tensor product as in [5]. A Graycategory is then a category enriched in the category Gray as in [2]. If A is a Gray-category and A, B and C are objects of A, then the multiplication

A(A, B) ⊗ A(B, C) ⟶ A(A, C)

corresponds to a cubical functor of two variables

M : A(A, B) × A(B, C) ⟶ A(A, C).

We denote the action of M by juxtaposition M(F, G) = GF. Given f : F ⟶ F′ in A(A, B) and g : G ⟶ G′ in A(B, C) we denote the invertible 2-cell M_{f,g} by

[figure: GF; gF; Gf; G′F; G′f; g_{f}; GF′; gF′; G′F′.]

M being a cubical functor implies that ( )F : A(B, C) ⟶ A(A, C) and

G( ) : A(A, B) ⟶ A(A, C)

are 2-functors. It also implies that ( )f : ( )F ⟶ ( )F′ and g( ) : G( ) ⟶ G′( ) are strong transformations. Furthermore, if φ : f ⟶ f′ and γ : g ⟶ g′ then ( )φ : ( )f ⟶ ( )f′ and γ( ) : g( ) ⟶ g′( ) are modifications. Given f′′ : F′ ⟶ F and g′′ : G′ ⟶ G′′ we also have that g_{(f′′∘f)} = (G′f′′ ∘ g_{f}) · (g_{f′′} ∘ Gf) and (g′ ∘ g)_{f} = (g′′_{f}∘gF)·(g′′F′∘g_{f}). If h : H ⟶ H′ is a 1-cell in A(C, D), then properties like h_{gF} = h_{g}F follow from the pentagon, and properties like 1_{A}F = F follow from the triangle that define a Gray-category. We will use these properties in the sequel without explicit mention.

3. KZ-Doctrines

Let A be a Gray-category and K be an object in A.

3.1. Definition. A KZ-doctrine D on K consists of an object D, 1-cells d : 1_{K} ⟶ D, and m : DD ⟶ D in A(K, K) and a fully-faithful adjoint string η, ϵ : Dd ⊣ m; and α, β : m ⊣ dD : D ⟶ DD such that

[figure: 1_{K}; D; D; DD; DD; β; ⇓; η; ⇓; d; dD; Dd; Id_{D}; m; m]  = [figure: 1_{K}; D; D; DD; D.; (d_{d})^{−1}; ⇓; d; d; dD; Dd; m]  (1)

The adjoint string being fully-faithful means that the counit β is invertible. It follows from a folklore result, whose statement and proof can be found in [7], that this is the case if and only if the unit η is also invertible.

Compare this condition with condition T0 of [9], in which strict equality m ∘ Dd = m ∘ dD = Id is asked for. As a matter of fact that paper points out some limitations that arise by requiring commutativity on the nose. Furthermore, associativity of m is deduced there only up to isomorphism.

The other piece of information given in [9] is a 2-cell from Dd to dD. In our case, this 2-cell comes from the adjoint string.

Define δ : Dd ⟶ dD to be the pasting

[figure: D; D; DD; DD.; β^{−1}; ⇓; ϵ; ⇓; Id_{D}; Id_{DD}; dD; m; Dd]  (2)

We know from [12] that δ is equal to the pasting (dD ∘ η^{−1}) · (α ∘ Dd) and that it is unique with the property m ∘ δ = β^{−1} · η^{−1}.

Condition T1 from [9] now takes the form:

3.2. Proposition.

[figure: 1_{K}; D; DD; δ; ⇓; d; Dd; dD]  =  [figure: 1_{K}; D; D; DD.; d_{d}; ⇓; d; d; dD; Dd]

Proof. Observe that as a consequence of (1), d_{d} is equal to the pasting

[figure: 1_{K}; D; D; DD; D; D; DD.; (d_{d})^{−1}; ⇓; β^{−1}; ⇓; η^{−1}; ⇓; d_{d}; ⇓; d; d; d; dD; Dd; m; Id_{D}; Id_{D}; Dd; dD]  (3)

Notice that ϵ ∘ Dd = Dd ∘ η^{−1} (consequence of one of the triangular identities). Cancel d_{d} with its inverse. Finally observe that δ ∘ d = (ϵ ∘ Dd ∘ d) · (Dd ∘ β^{−1} ∘ d).

The condition T2 of [9] takes the form of the uniqueness property for δ mentioned above. We write it as a lemma.

3.3. Lemma. m ∘ δ = β^{−1} · η^{−1}

We define the algebras for a KZ-doctrine with an arbitrary object of the Gray-category A as domain. This is in agreement with [8], where the algebras for a monad on a 2-category are defined over arbitrary objects of the 2-category.

3.4. Definition. Let X be an object of A. A D-algebra with domain X is an adjunction

φ, ψ : x ⊣ dX : X ⟶ DX

in A(X , K), with the counit ψ invertible.

A D-algebra as above, produces a co-fully-faithful adjoint string Dx ⊣ DdX ⊣ mX. As in the definition of δ, we obtain

[figure: DDX; DDX; DX; DX; ϵX; ⇓; Dψ^{−1}; ⇓; =; Id_{DDX}; Id_{DX}; mX; DdX; Dx; DDX; DDX; DX; DX.; Dφ; ⇓; ηX^{−1}; ⇓; Id_{DDX}; Id_{DX}; Dx; DdX; mX]

(4)

The following proposition tells us that for a D-algebra, the unit is uniquely determined by the counit

3.5. Proposition. If φ, ψ : x ⊣ dX : X ⟶ DX is a D-algebra, then φ is equal to the pasting

[figure: DX; DdX; dDX; δX⇓; Id_{DX}; Dψ^{−1}⇓; x; DDX; Dx; DX.; X; dX; d_{x} ⇓]  (5)

Proof. Start with the above pasting. Replace δX by (ϵX ∘ dDX) · (DdX ∘ βX^{−1}). Use (4). Since the pasting of d_{x} and d_{dX} is equal to d_{(dX∘x)}, we have that

[figure: DDX; DX; DDX; DX; X; DX; Dφ; ⇓; d_{x}; ⟹; d_{dX}; ⟹; Dx; DdX; dDX; x; dX; dX; dDX] = [figure: DX; DX; DDX.; X; φ; ⇓; x; dDX; dX]

Therefore we have that (Dφ ∘ dDX) · (DdX ∘ d_{x}) = (dDX ∘ φ) · (d_{dX}^{−1} ∘ x). Make this last substitution. As a consequence of (1) we have that the pasting of d_{dX}^{−1}, βX^{−1} and ηX^{−1} is the identity.

Observe that, for any invertible 2-cell ψ : x ∘ dX ⟶ Id_{X} the pasting (5) is always defined. Denote this pasting by ψ̂. Now we show that one of the triangular identities is always satisfied.

3.6. Lemma. If ψ : x ∘ dX ⟶ Id_{X} is an invertible 2-cell in A(X , A), then the pasting

[figure: DX; X; X; DX; _{ψ}; ⇓; ψ; ⇓; Id_{X}; dX; x; Id_{DX}; dX]

is the identity on dX.

Proof. We know from Proposition 3.2 that δX ∘ dX = d_{dX}. The pasting of d_{dX} with d_{x} is d_{(x∘dX)}. The pasting of this last 2-cell with Dψ^{−1} is equal to dX ∘ ψ^{−1}.

So, in order to see if an invertible ψ : x∘dX ⟶ Id_{X} determines a (necessarily unique, in view of Proposition 3.5) D-algebra, all we have to do is to check the other triangular identity.

3.7. Proposition. An invertible 2-cell ψ : x ∘ dX ⟶ Id_{X} in A(X , A) is the counit of an adjunction x ⊣ dX if and only if the pasting

[figure: DX; DX; X; X; _{ψ}; ⇓; ψ; ⇓; Id_{DX}; Id_{X}; x; dX; x]

is the identity on x.

Since we have m ⊣ dD with invertible counit β, we have as a corollary the condition corresponding to condition T3 in [9]

3.8. Corollary. The pasting

[figure: DD; DdD; dDD; δD⇓; Id_{DD}; Dβ^{−1}⇓; m; DDD; Dm; DD; m; β ⇓; D; dD; d_{m} ⇓; Id_{D}; D]

is the identity on m.

Observe that a KZ-doctrine in A, gives with the same data a KZ-doctrine in A^{trop} but with the roles of α, ϵ and β, η interchanged. Here A^{op} is the dual in the enriched sense, whereas A^{tr} is such that for every A and B in A, we have A^{tr}(A, B) = A(A, B)^{co}. We thus obtain the condition corresponding to T3^{∗} of [9]

3.9. Corollary. The pasting

[figure: DD; DdD; DDd; Dδ⇑; Id_{DD}; ηD^{−1}⇑; m; DDD; mD; DD; m; η ⇑; D; Dd; m_{d} ⇑; Id_{D}; D]

is the identity on m.

4. Normalized KZ-doctrines vs. KZ-doctrines

In this section we make explicit the comparison between the definition of KZ-doctrines in [9] and the definition given in this paper. Notice first that our definition is given in a general Gray-category, whereas the definition in [9] is given in 2-Cat. Notice furthermore, that we have replaced invertible 2-cells where the definition in [9] asked for strict equalities.

The definition given in [9] makes sense in a general Gray-category provided that the 2-cell d_{d} is an identity. So what we do is to compare the definitions in this more general setting.

Let’s assume first then, that we have a KZ-doctrine D in our sense, such that β, η and d_{d} are identities. Define δ = ϵ∘dD (pasting (2)). In this case the conditions corresponding to T1, T2 and T3 above are identical to the conditions T1, T2 and T3 of [9].

Conversely, assume we have (D, d, m, δ) a KZ-doctrine in the sense of [9] (where we are assuming that d_{d} is an identity). It follows from the work done in [9] that Dd ⊣ m with identity unit and m ⊣ dD with identity counit. We have therefore a KZ-doctrine in our sense. All we have to show now is that δ = ϵ ∘ dD, where ϵ is the counit of Dd ⊣ m. But this is clear since δ is unique with the property m ∘ δ = β^{−1} · η^{−1}.

5. Associativity up to isomorphism for KZ-doctrines

We deduce associativity up to isomorphism for a KZ-doctrine D as a corollary to the following technical proposition. Recall that D-algebras have objects of A as domains.

5.1. Proposition. Let ψ : x ∘ dX ⟶ Id_{X} and ζ : z ∘ dZ ⟶ Id_{Z} be D-algebras with the same object X of A as domain. Let h : X ⟶ Z be a 1-cell in A(X , K). If h has a right adjoint then the pasting

[figure: DX; DX; DZ; X; Z; Z; ⇓; _{ψ}; ⇓; d_{h}; ⇓; ζ; Id_{DX}; x; Dh; z; dX; h; dZ; Id_{Z}]

is invertible.

Proof. Assume π, χ : h ⊣ k. The inverse of the above pasting is

[figure: DX; DX; DZ; DZ; X; X; Z; Z.; ⇓; Dπ; ⇓; _{ζ}; ⇓; d_{k}; ⇓; ψ; ⇓; χ]

As a corollary we have, 5.2. Proposition. The pasting

[figure: DDD; DDD; DD; DD; D; D; ⇓; αD; ⇓; d_{m}; ⇓; β; Id_{DDD}; mD; Dm; m; dDD; m; dD; Id_{D}]  (6)

is invertible. Proof. Apply 5.1 with ψ = βD, ζ = β and h = m.

As a corollary of the following lemma, we are able to write (6) in terms of Dϵ, m_{d} and η. 5.3. Lemma. Denoting pasting (6) by μ, we have

[figure: DD; DDD; DD; DD; D; ⇐; Dη; ⟸; μ; DDd; Id_{DD}; Dm; mD; m; m; =; D.; DD; D; DDD; DD; ⇐; η; ⟸; m_{d}; Id_{D}; m; m; DDd; mD; Dd]

Proof. Start on the left hand side. Substitute (6) for μ. Make the substitution

[figure: DD; DDD; DDD; DD; αD; ⇓; DDd; mD; dDD] = [figure: DD; D; DD; DDD; DD; DDD.; α; ⇓; m_{d}; ⟸; dD_{d}; ⟸; m; dD; DDd; mD; Dd; dDD; DDd]

Then the substitution

[figure: DD; DDD; DD; D; DD; D; Dη; ⇓; dD_{d}; ⟹; d_{m}; ⟹; DDd; Dm; dD; Dd; dDD; m; dD] = [figure: D; D; DD; DD; η; ⇓; Dd; dD; m]

recalling that dD_{d} = d_{Dd}. Finally, use the fact that α and β define an adjunction.

5.4. Corollary. Pasting (6) equals

[figure: DDD; DDD; DD.; DD; D; D; ⇓; Dϵ; ⇓; m_{d}; ⇓; η; Id_{DDD}; Dm; mD; m; DDd; m; Dd; Id_{D}]

Another corollary to Proposition 5.1 is 5.5. Proposition. For any D-algebra (X, x, ψ), the pasting

[figure: DDX; DDX; DX; DX; X; X; ⇓; αX; ⇓; d_{x}; ⇓; ψ; Id_{DDX}; mX; Dx; x; dDX; x; dX; Id_{X}]  (7)

is invertible. Proof. Apply 5.1 with ψ = βX, ζ = ψ and h = x.

Denote pasting (7) by χ_{ψ}. 5.6. Proposition. For any D-algebra (X, x, ψ), we have that

[figure: DX; DDX; DX; DX; X; ⇐; (ηX)^{−1}; ⟸; χ_{ψ}; DdX; Id_{DX}; Dx; mX; x; x]  =  [figure: DX; DX; DDX; X.; ⇓; Dψ; Id_{DX}; DdX; Dx; x]

Proof. Replace χ_{ψ} by (7). Notice then that the pasting of ηX with αX produces δX. Now paste with Dψ and its inverse and use 3.7.

6. 2-categories of algebras for a KZ-doctrine

Fix an object X in A. Define the 2-category D-Alg_{X} of D-algebras with domain X as follows: The objects of D-Alg_{X} are D-algebras ψ : x ∘ dX ⟶ Id_{X} with domain X. Given another D-algebra ζ : z ∘ dZ ⟶ Id_{Z} with domain X, define D-Alg_{X} (ψ, ζ) to be the full subcategory of A(X , K)(X, Z) determined by those 1-cells h : X ⟶ Z with the property that

[figure: DX; DX; DZ; X; Z; Z; ⇓; _{ψ}; ⇓; d_{h}; ⇓; ζ; Id_{DX}; x; Dh; z; dX; h; dZ; Id_{Z}]  (8)

is invertible. The horizontal composite of h : ψ ⟶ ζ and k : ζ ⟶ τ is k ∘ h.

There is a forgetful 2-functor U_{X} : D-Alg_{X} ⟶ A(X , K) with U_{X} (ψ) = X. The left biadjoint F_{X} : A(X , K) ⟶ D-Alg_{X} is defined as follows: For every X in A(X , K) define F_{X} (X) = βX. If γ : h ⟶ h′ : X ⟶ Z, define F_{X} (h) = Dh and F_{X} (γ) = Dγ. It is straightforward to show that F_{X} is a 2-functor provided we know that Dh : βX ⟶ βZ is a 1-cell in D-Alg_{X}. To see this we need a lemma.

6.1. Lemma. For every 1-cell h : X ⟶ Z in A(X , K) we have that the pasting

[figure: DDX; DDX; DDZ; DX; DZ; DZ; ⇓; αX; ⇓; d_{Dh}; ⇓; βZ; Id_{DDX}; mX; DDh; mZ; dDX; Dh; dDZ; Id_{DZ}]

is equal to m_{h}^{−1}.

Proof. Since

[figure: DX; DDX; DX; DZ; DDZ; DZ; βZ; ⇓; dD_{h}; ⟸; m_{h}; ⟸; dDX; mX; Dh; dDZ; DDh; mZ; Dh] = [figure: DX; DX; DZ; DDX; βX; ⇓; dDX; Dh; mX]

we have that (βZ∘Dh)·(mZ∘dD_{h}) = (Dh∘βX)·(m_{h}^{−1}∘dDX). Make this last substitution on the pasting of the lemma, and use the fact that α and β define an adjunction.

Notice that F_{X} ∘ U_{X} = D( ) : A(X , K) ⟶ A(X , K). The unit for the biadjunction F_{X} ⊣ U_{X} is d( ) : 1_{A(X ,K)} ⟶ D( ). The counit s : F_{X} ∘ U_{X} ⟶ 1_{D-Alg_{X}} is given by the structure maps, that is to say, for ψ : x ∘ dX ⟶ Id_{X} we put s_{ψ} = x : βX ⟶ ψ. Notice that Proposition 5.5 says that x is a 1-cell in D-Alg_{X}. Given h : ψ ⟶ ζ in D-Alg_{X}, we define the transition 2-cell s_{h} as the inverse of (8).

The invertible modification Id_{F_{X}} ⟶ (sF_{X}) ∘ (F_{X} d( )) is defined to be ηX at every X in A(X , K). The invertible modification (U_{X} s) ∘ (d( )U_{X}) ⟶ Id_{U_{X}} is defined to be ψ at every ψ in D-Alg_{X}. To see that this defines a modification we have to show:

6.2. Lemma. h ∘ ψ is equal to the pasting

[figure: X; DX; X; Z; DZ; Z.; d_{h}; ⟸; s_{h}; ⟸; ζ; ⇓; dX; h; x; Dh; h; dZ; mZ]

Proof. Consider the inverse of the above pasting composite and use the definition of s_{h}. Notice that ψ̂ ∘ dX = dX ∘ ψ^{−1}.

Change of base. Assume that we have two objects X and Z of A, and H an object in A(X , Z). Then the 2-functor ( )H : A(Z, K) ⟶ A(X , K) induces a change of base 2-functor Ĥ : D-Alg_{X} ⟶ D-Alg_{Z} such that

[figure: D-Alg_{Z}; U_{Z}; _{H}; D-Alg_{X}; U_{X}; A(Z, K); ( )H; A(X , K)]

commutes.

7. The Gray-category of D-algebras

We can, by allowing the domain to change, define the Gray-category D-Alg made up of D-algebras for a KZ-doctrine D.

The objects of D-Alg are D-algebras with any object of A as domain. Given D-algebras ψ : x ∘ dX ⟶ Id_{X} with domain X and ζ : z ∘ dZ ⟶ Id_{Z} with domain Z, the 2-category D-Alg(ψ, ζ) is defined as follows:

The objects of D-Alg(ψ, ζ) are pairs (N, h), where N is an object in A(X , Z) and h : X ⟶ ZN is a 1-cell in A(X , K), such that the pasting

[figure: DX; DX; DZN; X; ZN; ZN; ⇓; _{ψ}; ⇓; d_{h}; ⇓; ζN; Id_{DX}; x; Dh; zN; dX; h; dZN; Id_{ZN}]

is invertible.

A 1-cell (n, n̄) : (N, h) ⟶ (N′, h′) in D-Alg(ψ, ζ) consists of a 1-cell n : N ⟶ N′ in A(X , Z) and a 2-cell n̄ : Zn ∘ h ⟶ h′ in A(X , K).

A 2-cell ν : (n, n̄) ⟶ (n′, n̄′) is a 2-cell ν : n ⟶ n′ in A(X , Z) such that n̄ = n̄′ · (Y ν ∘ h). Vertical composition is the obvious one.

Define Id_{(N,h)} = (Id_{N}, id_{h}).

Given (n, n̄) : (N, h) ⟶ (N′, h′), and (ℓ, ℓ̄) : (N′, h′) ⟶ (N′′, h′′) define (ℓ, ℓ̄) ∘ (n, n̄) = (ℓ ∘ n, ℓ̄ · (Zℓ ∘ n̄)). If λ : (ℓ, ℓ̄) ⟶ (ℓ′, ℓ̄′) and ν : (n, n̄) ⟶ (n′, n̄′) define λ ∘ (n, n̄) = λ ∘ n and (ℓ, ℓ̄) ∘ ν = ℓ ∘ ν. This completes the definition of the 2-category D-Alg(ψ, ζ).

Define 1_{ψ} = (1_{X} , Id_{X}).

For another D-algebra τ : y ∘dY ⟶ Id_{Y} with domain Y, we define the cubical functor

M : D-Alg(ψ, ζ) × D-Alg(ζ, τ) ⟶ D-Alg(ψ, τ)

denoted by juxtaposition as for A, as follows:

Given (N, h) in D-Alg(ψ, ζ) and ω : (o, ō) → (o′, ō′) : (O, g) ⟶ (O′, g′) in D-Alg(ζ, τ), define (O, g)(N, h) = (ON, gN ∘ h), and (o, ō)(N, h) = (oN, ōN ∘ h), and ω(N, h) = ωN.

On the other hand, given ν : (n, n̄) ⟶ (n′, n̄′) : (N, h) ⟶ (N′, h′) in D-Alg(ψ, ζ) and (O, g) in D-Alg(ζ, τ) we define (O, g)(N, h) = (ON, gN ∘ h), and (O, g)(n, n̄) = (On, (gN′ ∘ n̄) · (g_{n} ∘ h)), and (O, g)ν = Oν. The proof that we obtain 2-functors with these definitions is fairly straightforward.

For (n, n̄) : (N, h) ⟶ (N′, h′) and (o, ō) : (O, g) ⟶ (O′, g′) we define the invertible 2-cell (o, ō)_{(n,n̄)} = o_{n} : (O′, g′)(n, n̄) ∘ (o, ō)(N, h) ⟶ (o, ō)(N′, h′) ∘ (O, g)(n, n̄).

These definitions give us a cubical functor since we have a cubical functor A(X , Z) × A(Z, Y) ⟶ A(X , Y).

We have to show now that the diagrams required for a Gray-category are satisfied. We only do the pentagon. Given another D-algebra θ : w ∘ dW ⟶ Id_{W} with domain W, we have that the pentagon commutes if and only if the diagram of cubical functors

[figure: D-Alg(ψ, ζ) × D-Alg(ζ, τ) × D-Alg(τ, θ); D-Alg(ψ,ζ)×M; M×D-Alg(τ,θ); D-Alg(ψ, τ) × D-Alg(τ, θ); M; D-Alg(ψ, ζ) × D-Alg(ζ, θ); M; D-Alg(ψ, θ)]

commutes. This is equivalent to the following six conditions for (n, n̄) : (N, h) ⟶ (N′, h′) in D-Alg(ψ, ζ), (o, ō) : (O, g) ⟶ (O′, g′) in D-Alg(ζ, τ) and (p, p̄) : (P, k) ⟶ (P′, k′) in D-Alg(τ, θ):

1. (( )(N, h)) ∘ (( )(O, g)) = ( )((O, g)(N, h)) : D-Alg(τ, θ) ⟶ D-Alg(ψ, θ).

2. ((P, k)( )) ∘ (( )(N, h)) = (( )(N, h)) ∘ ((P, k)( )) : D-Alg(ζ, τ) ⟶ D-Alg(ψ, θ).

3. (P, k)( )) ∘ (O, g)( )) = ((P, k)(O, g))( ) : D-Alg(ψ, ζ) ⟶ D-Alg(ψ, θ).

4. (p, p̄)_{(o,ō)(N,h)} = ((p, p̄)_{(o,ō)})(N, h).

5. (p, p̄)_{(O,g)(n,n̄)} = ((p, p̄)(O, g))_{(n,n̄)}.

6. (P, k)((o, ō)_{(n,n̄)}) = ((P, k)(o, ō))_{(n,n̄)}.

All the above conditions follow from the definitions and the corresponding facts for the Gray-category A.

8. Pseudomonads

We adopt the definition of pseudomonoid given in [1]. That is, given a Gray-category A, and an object K in A, we define a pseudomonad 𝔻 on K to be a pseudomonoid in the Gray monoid A(K, K). Explicitly, 𝔻 consists of an object D in A(K, K) together with 1-cells d : 1_{K} ⟶ D and m : DD ⟶ D and invertible 2-cells

[figure: D; DD; D; D; ⇐; β; ⇐; η; dD; Id_{D}; Dd; Id_{D}; m; DDD; DD; DD; D; ⟸; μ; Dm; mD; m; m]

satisfying the following two conditions

[figure: DDDD; DDD; DDD; DD; DDD; DD; D; Dμ; ⟸; μD; ⟸; μ; ⟸; DDm; DmD; mDD; Dm; Dm; mD; m; mD; m] = [figure: DDDD; DDD; DD; DD; DDD; DD; D; m_{m}^{−1}; ⟸; μ; ⟸; μ; ⟸; DDm; mDD; Dm; mD; m; m; Dm; mD; m] (9)

[figure: DD; DDD; DD; D; DD; μ; ⇓; DdD; Dm; mD; m; m] = [figure: DD; DDD; DDD; DD; D.; Dβ; ⇓; ηD; ⇓; DdD; DdD; Dm; mD; m]

(10)

Warning: The direction of the arrows η and μ is the opposite to that given in [1]. Since they are invertible this represents no problem.

As pointed out in [1], a pseudomonoid in the cartesian closed 2-category Cat of categories, functors and natural transformations is precisely a monoidal category, where condition (9) corresponds to the pentagon and condition (10) corresponds to the triangle that has the distinguished object I in the middle. It is well known that in this case the commutativity of these diagrams implies the commutativity of the two triangles that have I on one extreme or the other, and that the ‘right’ and ‘left’ arrows I ⊗ I ⟶ I coincide [6]. (This in turn implies the commutativity of all the diagrams [11]). Results like those of [6] can be shown in the present context.

8.1. Proposition. If 𝔻 = (D, d, m, β, η, μ) is a pseudomonad on an object K, then we have the following equalities:

[figure: 1.; 1_{K}; D; D; DD; DD; β; ⇓; η; ⇓; d; dD; Dd; Id_{D}; m; m; =; 1_{K}; D; D; DD; D.; (d_{d})^{−1}; ⇓; d; d; dD; Dd; m]

[figure: 2.; DD; DDD; DD; DD; D; ⇐; βD; ⟸; μ; dDD; Id_{DD}; Dm; mD; m; m; =; DD; DDD; D; DD; D.; ⟸; d_{m}; ⇐; β; dDD; m; Dm; dD; Id_{D}; m]

[figure: 3.; DD; DDD; DD; DD; D; ⇐; Dη; ⟸; μ; DDd; Id_{DD}; Dm; mD; m; m; =; D.; DD; D; DDD; DD; ⇐; η; ⟸; m_{d}; Id_{D}; m; m; DDd; mD; Dd]

Proof. To show 2 start with the following pasting Make the substitution

[figure: DD; DDD; DD; DD; D; DDD; DDD; DD; DDD; DD; D.; βD; ⇓; μ; ⇓; d_{m}^{−1}; ⟸; ηDD; ⇓; μ; ⇓; β^{−1}; ⇐; dDD; dDD; Dm; mD; m; m; dD; Dm; mD; DdDD; m; mDD; m]

[figure: DD; DDD; DD; DD; D; DDD; DDD; DD; βD; ⇓; μ; ⇓; d_{m}^{−1}; ⟸; dDD; dDD; Dm; mD; m; m; dD; Dm] = [figure: DD; DDD; DD; DDDD; D; DDD; DDD; DD; DDD; DβD; ⇓; d_{dDD}^{−1}; ⟸; d_{Dm}^{−1}; ⟸; d_{m}^{−1}; ⟸; Dμ; ⟸; dDD; dDD; Dm; dDDD; m; dDD; DmD; DDm; dD; DdDD; Dm; Dm]

(using the fact that d_{m∘Dm∘dDD}^{−1} is equal to the pasting of d_{dDD}^{−1}, d_{Dm}^{−1} and d_{m}^{−1}). Make the substitution (10) multiplied on the right by D. Now make the substitution (9). Make the substitution

[figure: DD; DDD; DD; DDD; DDDD; DDD; d_{dDD}^{−1}; ⟸; d_{Dm}^{−1}; ⟸; dDD; dDD; Dm; dDDD; dDD; DdDD; DDm] = [figure: DDD; DD; D; DD; DDD; DD; DDD; DDD; d_{m}; ⇓; Dd_{m}^{−1}; ⇓; d_{m}^{−1}; ⟸; d_{dD}^{−1}; ⟸; Dm; dDD; m; dDD; dD; dD; dDD; Dm; DdDD; DdD; DDm]

Then the substitution

[figure: D; DD; D; DD; DDD; DD; d_{dD}^{−1}; ⟸; d_{m}^{−1}; ⟸; dD; dD; m; dDD; dD; DdD; Dm] = [figure: DD; D; D; DD; DD.; DDD; β; ⇓; Dβ^{−1}; ⇓; m; dD; dD; dD; DdD; Dm]

Notice that, as consequence of (10), the pasting of Dβ^{−1} and μ is equal to m ∘ ηD. The pasting of ηD, Dd_{m}^{−1} and m_{m}^{−1} is equal to Dm ∘ ηDD. Observe that the bottom part of the resulting diagram is equal to the bottom part of the pasting we started from. Since all the 2-cells are invertible, we conclude 2.

3 can be proved similarly or by duality.

To show 1, we show first that the pasting

[figure: 1; D; DD; DD; DDD; DD; D; d_{dD}; ⇓; ηD; ⇓; βD; ⇓; d; dD; dD; DdD; mD; m; dDD]  (11)

is the identity. To do this, replace m ∘ ηD by a pasting of Dβ^{−1} and μ, using (10). Use condition 2 of the proposition proved above. The pasting of Dβ^{−1}, d_{dD} and d_{m} is dD∘β^{−1}. We thus obtain an identity.

Start again with (11). Paste d_{d} and its inverse on top of it. Now, ηD ∘ Dd is equal to the pasting of Dd_{d}, m_{d} and η. The pasting of Dd_{d}, d_{d} and d_{dD} is equal to the pasting of d_{d}, dD_{d} and d_{d}. The pasting of dD_{d}, m_{d} and βD is Dd ∘ β. Since (11) is an identity, the resulting pasting is an identity. We thus obtain another identity if we remove d_{d} and its inverse. Now paste with η and η^{−1}.

9. 2-categories of algebras for a Pseudomonad

As in the case of algebras for a KZ-doctrine we define the algebras for a pseudomonad with an object of A for domain.

Let 𝔻 be a pseudomonad on an object K of the Gray-category A. Let X be an object of A. We define the 2-category 𝔻-Alg_{X} of 𝔻-algebras with domain X as follows.

An object of 𝔻-Alg_{X} consists of an object X in A(X , K), together with a 1-cell x : DX ⟶ X, and invertible 2-cells

[figure: X; DX; X; ⇐; ψ; dX; Id_{X}; x; DDX; DX; DX; X.; ⟸; χ; Dx; mX; x; x]

This data must satisfy the following two conditions

[figure: DDDX; DDX; DDX; DX; DDX; DX; X; Dχ; ⟸; μX; ⟸; χ; ⟸; DDx; DmX; mDX; Dx; Dx; mX; x; mX; x] = [figure: DDDX; DDX; DX; DX; DDX; DX; X; m_{x}^{−1}; ⟸; χ; ⟸; χ; ⟸; DDx; mDX; Dx; mX; x; x; Dx; mX; x] (12)

[figure: DX; DDX; DX; X; DX; χ; ⇓; DdX; Dx; mX; x; x] = [figure: DX; DDX; DDX; DX; X.; Dψ; ⇓; ηX; ⇓; DdX; DdX; Dx; mX; x]

(13)

We denote an object in 𝔻-Alg_{X} by the pair (ψ, χ).

Given another 𝔻-algebra (ζ, θ) with ζ : z ∘ dZ ⟶ Id_{Z}, a 1-cell in 𝔻-Alg_{X} is a pair (h, ρ) : (ψ, χ) ⟶ (ζ, θ), where h : X ⟶ Z is a 1-cell in A(X , K) and

[figure: DX; DZ; X; Z; ⟸; ρ; Dh; x; z; h]

is an invertible 2-cell in A(X , K), such that the following two conditions are satisfied.

[figure: X; DX; DZ; X; Z; ⇐; ψ; ⟸; ρ; dX; Id_{X}; Dh; x; h; z]  =  [figure: X; DX; Z; DZ; Z; ⇓; d_{h}; ⇓; ζ; dX; h; Dh; dZ; Id_{Z}; z]  (14)

[figure: DDX; DDZ; DX; DX; DZ; X; Z; ⟸; χ; ⟸; ρ; ⟸; Dρ; DDh; mX; Dx; Dz; x; Dh; x; z; h]  = [figure: DDX; DDZ; DX; DZ; DZ; X; Z.; ⟸; θ; ⟸; ρ; ⟸; m_{h}^{−1}; DDh; mX; mZ; Dz; x; Dh; z; z; h]  (15)

Given (h, ρ), (h′, ρ′) : (ψ, χ) ⟶ (ζ, θ), a 2-cell ξ : (h, ρ) ⟶ (h′, ρ′) is a 2-cell ξ : h ⟶ h′ in A(X , K) such that (ξ ∘x)·ρ = ρ′ ·(z ∘Dξ). Vertical composition is the obvious one.

Horizontal composition: for (h, ρ) : (ψ, χ) ⟶ (ζ, θ) and (k, π) : (ζ, θ) ⟶ (τ, σ) we define (k, π) ∘ (h, ρ) = (k ∘ h, (k ∘ ρ) · (π ∘ Dh)).

This completes the definition of 𝔻-Alg_{X}.

A proof very similar to that of condition 2 of Proposition 8.1 produces:

9.1. Lemma. For every 𝔻-algebra (ψ, χ) we have

[figure: DX; DDX; DX; DX; X; ⇐; βX; ⟸; χ; dDX; Id_{DX}; Dx; mX; x; x] = [figure: DX; DDX; X; DX; X.; ⟸; d_{x}; ⇐; ψ; dDX; x; Dx; dX; Id_{X}; x]  (16)

As a matter of fact, condition 2 of Proposition 8.1 is the above lemma applied to the 𝔻-algebra (β, μ).

The Gray-category 𝔻-Alg of algebras for a pseudomonad 𝔻 can be defined along the same lines as the Gray-category D-Alg of algebras for a KZ-doctrine.

10. Every KZ-doctrine is a pseudomonad

Assume we have a KZ-doctrine D = (D, d, m, α, β, η, ϵ) as in Section 3. Define μ as pasting (6). We already know that μ is invertible.

10.1. Proposition. 𝔻 = (D, d, m, β, η, μ) is a pseudomonad.

Proof. Condition (10) is Proposition 5.6 applied to the D-algebra β. As for the other condition, start on the left hand side of (9). Substitute (6) and (6) multiplied by D on the right for μ and μD respectively. The pasting of βD and αD is the identity. The pasting of d_{mD}, d_{m} and Dμ equals the pasting of d_{Dm}, d_{m} and μ. Paste with (dDD ∘ β) · (αD ∘ dDD) in the middle. Use Lemma 6.1.

To be able to say anything meaningful on this connection between KZ-doctrines and pseudomonads, we must show first that the categories of algebras D-Alg_{X} and 𝔻-Alg_{X} for any X are essentially the same. We devote the rest of this section to show that they are 2-isomorphic. So we fix an object X of A, and a KZ-doctrine D on K. We take 𝔻 as the pseudomonad induced by D as in the above proposition.

We start by stating the recognition lemma [13] in the form we will use it

10.2. Lemma. Given ψ : x∘dX ⟶ Id_{X} and ζ : z∘dZ ⟶ Id_{Z} in D-Alg_{X}, h : X ⟶ Z a 1-cell in A(X , K) and ρ : z ∘ Dh ⟶ h ∘ x a 2-cell, we have that

ρ = [figure: DX; DX; DZ; X; Z; Z; ⇓; _{ψ}; ⇓; d_{h}; ⇓; ζ; Id_{DX}; x; Dh; z; dX; h; dZ; Id_{Z}]

if and only if

[figure: X; DX; DZ; X; Z; ⇐; ψ; ⟸; ρ; dX; Id_{X}; Dh; x; h; z]  =  [figure: X; DX; Z; DZ; Z.; ⇓; d_{h}; ⇓; ζ; dX; h; Dh; dZ; Id_{Z}; z]

Let ψ : x ∘ dX ⟶ Id_{X} be an object in D-Alg_{X}. Let χ_{ψ} be equal to pasting (7).

10.3. Lemma. (ψ, χ_{ψ}) is a 𝔻-algebra.

Proof. Condition (12) is shown as condition (9) in Proposition 10.1. Condition (13) is Proposition 5.6.

Conversely

10.4. Lemma. If (ψ, χ) is a 𝔻-algebra with ψ : x ∘ dX ⟶ Id_{X}, then ψ is a D-algebra and χ = χ_{ψ} (pasting 7).

Proof. To show that ψ is a D-algebra it suffices to show that the pasting in Proposition 3.7 is the identity on x. Substitute pasting (5) for ψ̂. Paste with χ and its inverse. Use (13) on the pasting of Dψ^{−1} and χ. By Lemma 3.3 the pasting of ηX and δX is βX^{−1}. Now use (16). The condition for χ follows from Lemma 10.2 and (16).

10.5. Lemma. Let ψ : x ∘ dX ⟶ Id_{X} and ζ : z ∘ dZ ⟶ Id_{Z} be objects and h : ψ ⟶ ζ be a 1-cell in D-Alg_{X}. Define ρ_{h} as pasting (8). Then we have that (h, ρ_{h}) : (ψ, χ_{ψ}) ⟶ (ζ, χ_{ζ}) is a 1-cell in 𝔻-Alg_{X}.

Proof. Condition (14) follows immediately from the definition of ρ_{h}. The proof of (15) is very similar to the proof of condition (9) in Proposition 10.1.

Conversely

10.6. Lemma. If (h, ρ) : (ψ, χ) ⟶ (ζ, θ) is a 1-cell in 𝔻-Alg_{X}, then h : ψ ⟶ ζ is a 1-cell in D-Alg_{X} and ρ = ρ_{h} (pasting (8)).

The situation for 2-cells is similar. We thus have

10.7. Theorem. If we define Φ : D-Alg_{X} ⟶ 𝔻-Alg_{X} such that for every ξ : h ⟶ h′ : ψ ⟶ ζ in D-Alg_{X} we have Φ(ψ) = (ψ, χ_{ψ}), Φ(h) = (h, ρ_{h}) and Φ(ξ) = ξ, we obtain a 2-isomorphism.

It can also be shown that the Gray-categories D-Alg and 𝔻-Alg are isomorphic.

11. Pseudomonads vs. KZ-doctrines

In [13], the leftmost adjoint in the definition of KZ-doctrine is explicitly excluded. A question raised in [9] asks whether it can be put back on. The answer given here is in the affirmative.

11.1. Theorem. If 𝔻 = (D, d, m, β, η, μ) is a pseudomonad on an object K of a Graycategory A, then, the following statements are equivalent

1. m ⊣ dD with counit β.

2. Dd ⊣ m with unit η.

Proof. Assume α, β : m ⊣ dD. Notice that we can still define δ as

[figure: .; ⇓; η^{−1}; ⇓; α]

Define ϵ as the pasting

[figure: D; DD; DDD; DD.; ⇓; m_{d}; ⇓; Dδ; ⇓; ηD^{−1}; Dd; m; DDd; DdD; mD]

Then ϵ is the counit for an adjunction η, ϵ : Dd ⊣ m. The converse follows similarly or by duality.

References

[1] B. Day and Ross Street. Monoidal bicategories and Hopf algebroids. Advances in Math. (to appear).

[2] S. Eilenberg and G. M. Kelly. Closed categories. In: Proceedings of the Conference on Categorical Algebra, La Jolla 1965, ed. S. Eilenberg et. al., Springer Verlag 1966; pp. 421-562.

[3] B. Fawcett and R. J. Wood. Constructive complete distributivity I. Math. Proc. Camb. Phil. Soc. 107, 1990, pp. 81-89.

[4] J. W. Gray, Formal category theory: Adjointness for 2-categories. Lecture Notes in Mathematics 391, Springer-Verlag, 1974

[5] R. Gordon, A. J. Power and Ross Street. Coherence for tricategories. Memoirs of the American Mathematical Society 558. 1995.

[6] G. M. Kelly. On MacLane’s conditions for coherence of natural associativities, commutativities, etc. Journal of Algebra 1, 1964, pp. 397-402.

[7] G. M. Kelly and F. W. Lawvere. On the complete lattice of essential localizations. Bulletin de la Société Mathématique de Belgique (Serie A) Tome XLI. 1989, pp 179-185.

[8] G. M. Kelly and Ross Street. Review of the elements of 2-categories. In: Proceedings Sydney category theory seminar 1972/1973. Lecture Notes in Mathematics 420, 1974, pp. 75-103.

[9] A. Kock. Monads for which structures are adjoint to units. JPAA2 104(1), 1995, pp. 41-59.

[10] S. Mac Lane. Categories for the working mathematician, Graduate Texts in Mathematics, vol. 5, Springer-Verlag, 1971.

[11] S. Mac Lane. Natural commutativity and associativity. Rice University Stud. 49, 1963, pp. 28-46.

[12] R. Rosebrugh and R. J. Wood. Distributive adjoint strings. Theory and applications of categories, vol 1, 1995.

[13] Ross Street, Fibrations in bicategories, Cahiers de topologie et géométrie différentielle, vol. XXI-2, 1980, pp. 111-159.

[14] Ross Street, The formal theory of monads. JPAA 2, 1972, pp. 149-168.

Instituto de Matemáticas Universidad Nacional Autónoma de México Area de la Investigación Científica México D. F. 04510 México

Email: quico@matem.unam.mx

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John Baez, University of California, Riverside: baez@math.ucr.edu Michael Barr, McGill University: barr@triples.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, University of Genoa: carboni@vmimat.mat.unimi.it P. T. Johnstone, University of Cambridge: ptj@pmms.cam.ac.uk G. Max Kelly, University of Sydney: kelly m@maths.su.oz.au Anders Kock, University of Aarhus: kock@mi.aau.dk F. William Lawvere, State University of New York at Buffalo: mthfwl@ubvms.cc.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@gar.union.edu Robert Paré, Dalhousie University: pare@cs.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.su.oz.au R. J. Wood, Dalhousie University: rjwood@cs.da.ca