Theory and Applications of Categories, Vol. 5, No. 5, 1999, pp. 91–147.

DISTRIBUTIVE LAWS FOR PSEUDOMONADS

F. MARMOLEJO

Transmitted by Ross Street

ABSTRACT. We define distributive laws between pseudomonads in a Gray-category A, as the classical two triangles and the two pentagons but commuting only up to isomorphism. These isomorphisms must satisfy nine coherence conditions. We also define the Gray-category PSM(A) of pseudomonads in A, and define a lifting to be a pseudomonad in PSM(A). We define what is a pseudomonad with compatible structure with respect to two given pseudomonads. We show how to obtain a pseudomonad with compatible structure from a distributive law, how to get a lifting from a pseudomonad with compatible structure, and how to obtain a distributive law from a lifting. We show that one triangle suffices to define a distributive law in case that one of the pseudomonads is a (co-)KZ-doctrine and the other a KZ-doctrine.

1. Introduction

Distributive laws for monads were introduced by J. Beck in [2]. As pointed out by G. M. Kelly in [7], strict distributive laws for higher dimensional monads are rare. We need then a study of pseudo-distributive laws. The first step in this direction is quite easy: just replace the two commutative triangles, and the two commutative pentagons of [2] by appropriate invertible cells. The problem is to determine what coherence conditions to impose on these invertible cells. We should point out that, in [7], there is a step in this direction, keeping commutativity on the nose on the triangles and one of the pentagons, and asking for commutativity up to isomorphism in the remaining pentagon, plus five coherence conditions. The structure obtained from such a distributive law between two strict 2-monads is not, in general, a strict 2-monad, and since that article deals exclusively with strict 2-monads, what is obtained is a reflection result.

In this paper, instead of working with 2-monads we work with the more general pseudomonads. We will see that the structure obtained from a distributive law between pseudomonads is a pseudomonad. We define a distributive law between pseudomonads as we said above, that is to say, asking for commutativity up to isomorphism of the two triangles and the two pentagons. We propose nine coherence conditions for these isomorphisms. See section 4 below. We observe that the coherence conditions of [7] and the ones proposed in this paper coincide if in our setting we ask for commutativity on the nose of the two triangles and one of the pentagons. Thus, the examples of distributive laws given there are examples here as well.

But why exactly these nine coherence conditions?

Received by the editors 1998 June 24 and, in revised form, 1998 November 22.

Published on 1999 March 18.

1991 Mathematics Subject Classification: 18C15, 18D05, 18D20.

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

\begin{matrix}c \\ ◯ \end{matrix} F. Marmolejo 1999. Permission to copy for private use granted.

A more conceptual approach to distributive laws for monads is given by R. Street in [11]. It is shown that for a 2-category C, a distributive law is the same thing as a monad in the 2-category MND(C), whose objects are monads in C. In this paper we introduce the corresponding structure PSM(A), of pseudomonads for a corresponding three dimensional structure A, see section 7.

In M. Barr and C. Wells’ book [1], exercise (Dl) asks to prove that, for monads, a distributive law, a lifting of one monad structure to the algebras of the other, and a monad with compatible structure with the two given monads, are essentially the same thing. For pseudomonads we have already mentioned distributive laws. We define a lifting as a pseudomonad in PSM(A) in section 8. In section 6, we define what a pseudomonad whose structure is compatible with two given pseudomonads is.

We show how to obtain a composite pseudomonad with compatible structure from a distributive law between pseudomonads, how to obtain a lifting from a pseudomonad with compatible structure, and, closing the cycle, how to obtain a distributive law from a lifting.

We see then, that the nine coherence conditions can be shown to hold if we define a distributive law from a lifting. In turn, these coherence conditions allow us to define a lifting from a distributive law between pseudomonads.

The situation for distributive laws between KZ-doctrines and (co-)KZ-doctrines is a lot simpler. We show that either one of the triangles commuting up to isomorphism (satisfying coherence conditions) is enough to obtain a distributive law. One such example is the following. It is well known that adding free (finite) coproducts to categories is a KZdoctrine over Cat, and adding free (finite) products is a co-KZ-doctrine. There is a more or less obvious distributive law of the co-KZ-doctrine over the KZ-doctrine. Observe however that even if we arrange for these KZ-doctrine and co-KZ-doctrine to produce strict pseudomonads, the distributive law obtained is not strict.

This article is possible thanks to the definition of tricategories given in [6]. It is simplified by the fact that a tricategory is triequivalent to a Gray-category, a fact proved in the same paper. We thus work in the framework of Gray-categories, as in [6], continuing the development of the formal theory of pseudomonads started in [9].

This paper is organized as follows:

In section 2 we provide a brief description of the framework that we use, namely that of Gray-categories. For more details we refer the reader to [6, 5].

In section 3 we recall the definition and some properties of pseudomonads given in [9], the definition uses the definition of pseudomonoid given in [3]. We also define the change of base 2-functors, change of base strong transformations and the change of base modifications that we will need in later sections. Change of base turns out to be a Graynatural transformation.

In section 4 we define distributive laws for pseudomonads by replacing commutativity on the nose by commutativity up to isomorphism. We give here the nine coherence conditions that these isomorphisms should satisfy.

The first step to obtain compatible structures is to define a composite pseudomonad from a distributive law. This is what we do in section 5.

In section 6 we define what a pseudomonad with compatible structure is with respect to given pseudomonads. Furthermore, we exhibit the structure that makes compatible the composite pseudomonad defined in the previous section.

We introduce the Gray-category PSM(A) in section 7, to define, in section 8, a lifting as a pseudomonad in the Gray-category PSM(A).

In section 9 we show how to construct a pseudomonad in PSM(A), from given pseudomonads with compatible structure. In the following section, we go from a pseudomonad in PSM(A) to a distributive law.

Section 11 deals with distributive laws of (co-)KZ-doctrines over KZ-doctrines. We refer the reader to [9] for the definition and properties we use of KZ-doctrines, but see [8] as well. We show that one triangle, plus coherence, is enough to produce a distributive law. Compare with [10], where it is shown that one of the triangles suffices for a distributive law between idempotent monads. The case of KZ-doctrines over KZ-doctrines is formally very similar. In this latter case, we show that the composite pseudomonad is again a KZ-doctrine.

I would like to thank the referee for helping improve the readability of this paper, and for suggesting condition (12), after which all the conditions of section 6 were modeled.

2. Gray-categories

As in [9] we will work with a Gray-category A, where Gray is the symmetric monoidal closed category whose underlying category is 2-Cat with tensor product as in [6]. A Graycategory is a category enriched in the category Gray as in [4]. We will briefly spell out what this means, and we refer the reader to [6] and [4] for more details.

A Gray-category A has objects A, B, C, …. For every pair of objects A, B of A, A has a 2-category A(A, B). Given another object C in A, A has a 2-functor A(C, B) ⊗ A(A, B) → A(A, C). This 2-functor corresponds to a cubical functor M : A(B, C) × A(A, B) → A(A, C). We will denote M by juxtaposition, M(G, F) = GF for F ∈ A(A, B) and G ∈ A(B, C). Given f : F → F′ in A(A, B) and g : G → G′ in A(B, C) we will denote the invertible 2-cell M_{g,f} by

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

What the definition of being cubical means for M is the following: Given φ : f → f′ : F → F′, and f′′ : F′ → F′′ in A(A, B), and γ : g → g′ : G → G′, and g′′ : G′ → G′′ in A(B, C), we have that ( )F : A(B, C) → A(A, C) and G( ) : A(A, B) → A(A, C) are 2-functors, ( )f : ( )F → ( )F′ and g( ) : G( ) → G′( ) are strong transformations, ( )φ : ( )f → ( )f′ and γ( ) : g( ) → g′( ) are modifications, and the following three equations are satisfied

[figure: GF; gF; g′F; γF; Gf; Gf′; Gφ; g′_{f}; G′F; G′f; GF′; g′F′; G′F′] = [figure: GF; gF; Gf′; g_{f′}; G′F; G′f; G′f′; G′φ; GF′; gF′; g′F′; γF′; G′F′,]

[figure: GF; gF; Gf; g_{f}; G′F; G′f; GF′; gF′; Gf′′; g_{f′′}; G′F′; G′f′′; =; GF′′; gF′′; G′F′′; GF; gF; G(f′′∘f); g_{f′′∘f}; G′F; G′(f′′∘f); GF′′; gF′′; G′F′′,]

and

[figure: GF; gF; Gf; g_{f}; G′F; g′′F; G′f; g′′_{f}; G′′F; G′′f; =; GF′; gF′; G′F′; g′′F′; G′′F′; GF; (g′′∘g)F; Gf; (g′′∘g)_{f}; G′′F; G′′f; GF′; (g′′∘g)F′; G′′F′,]

and if either f or g is an identity, then g_{f} is an identity 2-cell. Now, for every object A of A, there is a distinguished object 1_{A}. The triangle in the definition of enriched categories means that the action of multiplying by 1_{A} is trivial. Now, the pentagon means that for another object D in A, and κ : k → k′ : K → K′ in A(C, D) the following equations hold:

(KG)F = K(GF),
(KG)f = K(Gf), (Kg)F = K(gF), (kG)F = k(GF),
(KG)φ = K(Gφ), (Kγ)F = K(γF), (κG)F = κ(GF),
(Kg)_{f} = K(g_{f}), (kG)_{f} = k_{Gf}, and (k_{g})F = k_{gF} .

We will use these properties freely, without further mention.

3. Pseudomonads

For the convenience of the reader we will recall here the definition of a pseudomonad in a Gray-category A, for more details we refer the reader to [9]. We adopt the definition of pseudomonoid given in [3].

3.1. Definition. A pseudomonad 𝔻 on an object K of a Gray-category A is a pseudomonoid in the Gray monoid A(K, K).

We give now, in elementary terms, what this means. A pseudomonad 𝔻 as above consists of an object D in A(K, K), and 1-cells d : 1_{K} → D, and m : DD → D and invertible 2-cells

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

such that the following two equations are satisfied:

[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]  (1)

[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]  (2)

It is shown in [9] that the following three equations hold for any pseudomonad 𝔻:

[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]  (3)

[figure: DD; DDD; DD; DD; D; ⇐; βD; ⟸; μ; dDD; Id_{DD}; Dm; mD; m; m] = [figure: DD; DDD; D; DD; D,; ⟸; d_{m}; ⇐; β; dDD; m; Dm; dD; Id_{D}; m]  (4)

[figure: DD; DDD; DD; DD; D; ⇐; Dη; ⟸; μ; DDd; Id_{DD}; Dm; mD; m; m] = [figure: D.; DD; D; DDD; DD; ⇐; η; ⟸; m_{d}; Id_{D}; m; m; DDd; mD; Dd]  (5)

We recall as well the 2-categories of algebras for a pseudomonad 𝔻. Let X be another object in A. An object in the 2-category 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; Id_{X}; ψ; DX; x; X; DDX; Dx; mX; χ; DX; x; DX; x; X,]

such that the following two equations are satisfied

[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]  (6)

[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]  (7)

It is shown in [9] that for every object (ψ, χ) in 𝔻-Alg_{X}, the following equality holds:

[figure: DX; dDX; Id; βX; DDX; Dx; mX; χ; DX; x; DX; x; X] = [figure: DX; dDX; x; d_{x}; DDX; Dx; X; dX; Id; ψ; DX; x; X.]  (8)

A 1-cell (h, ρ) : (ψ, χ) → (ψ′, χ′) in 𝔻-Alg_{X} consists of a 1-cell h : X → X′ in A(X , K), together with an invertible 2-cell

[figure: DX; Dh; x; ρ; DX′; x′; X; h; X′,]

that satisfies the following two equations:

[figure: X; dX; Id; ψ; DX; Dh; x; ρ; DX′; x′; X; h; X] = [figure: X; dX; h; d_{h}; DX; Dh; X′; dX′; Id; ψ′; DX′; x′; X′,]  (9)

[figure: DDX; DDX′; DX; DX′; DX; X; X′; Dρ; ⟸; χ; ⟸; ρ; ⟸; DDh; Dx; mX; Dx′; Dh; x; x′; x; h] =  [figure: DDX; DDX′; DX′; DX′; DX; X; X′.; m_{h}^{−1}; ⟸; χ′; ⟸; ρ; ⟸; DDh; mX; Dx′; mX′; x′; x′; Dh; x; h]  (10)

A 2-cell ξ : (h, ρ) → (h′, ρ′) : (ψ, χ) → (ψ′, χ′) is a 2-cell ξ : h → h′ such that the following condition is satisfied:

[figure: DX; Dh; Dh′; Dξ; x; ρ′; DX′; x′; X; h′; X′] = [figure: DX; Dh; x; ρ; DX′; x; X; h; h′; ξ; X′.]  (11)

Given another object Z of A, and K ∈ A(Z, X), we can define a change of base 2-functor K̂ : 𝔻-Alg_{X} → 𝔻-Alg_{Z}. If ξ : (h, ρ) → (h′, ρ′) : (ψ, χ) → (ψ′, χ′) is in 𝔻-Alg_{X}, then its image under K̂ is ξK : (hK, ρK) → (h′K, ρ′K) : (ψK, χK) → (ψ′K, χ′K). If k : K → K′ then we define the strong transformation k̂ : K̂ → K̂′ such that k̂_{(ψ,χ)} = (Xk, x_{k}^{−1}) and k̂_{(h,ρ)} = h_{k}^{−1}. If κ : k → k′ : K → K′ in A(Z, X), then κ̂_{(ψ,χ)} = Xκ defines a modification κ̂ : k̂ → k̂′. We have actually defined a Gray-functor 𝔻-Alg : A^{op} → Gray.

For every object Z, we have an obvious forgetful 2-functor 𝔻-Alg_{Z} → A(Z, K). These 2-functors define a forgetful Gray-natural transformation Φ : 𝔻-Alg → A( , K).

4. Distributive laws

Let 𝔻 = (D, d, m, β_{𝔻}, η_{𝔻}, μ_{𝔻}) and 𝕌 = (U, u, n, β_{𝕌}, η_{𝕌}, μ_{𝕌}) be pseudomonads on the same object K of the Gray-category A. A distributive law of 𝕌 over 𝔻 consists of a 1-cell r : UD → DU in A(K, K), together with invertible 2-cells

[figure: UD; r; ω_{1}; D; uD; Du; DU; UD; r; ω_{2}; U; Ud; dU; DU]

[figure: UUD; Ur; nD; UDU; rU; ω_{3}; DUU; Dn; UD; r; DU; UD; r; DU; UDD; Um; rD; DUD; Dr; ω_{4}; DDU; mU]

subject to the following coherence conditions: (coh 1)

[figure: 1; u; d; d_{u}^{−1}; U; Ud; dU; ω_{2}; UD; r; D; Du; DU] = [figure: U; Ud; 1; u; d; u_{d}; UD; r; ω_{1}; D; uD; Du; DU.]

(coh 2)

[figure: UD; UuD; Id; η_{𝕌}D^{−1}; UUD; Ur; nD; UDU; rU; ω_{3}; DUU; Dn; UD; r; DU] = [figure: UD; r; UuD; UDu; Uω_{1}; UUD; Ur; r_{u}^{−1}; UDU; rU; DU; DUu; Id; Dη_{𝕌}^{−1}; DUU; Dn; DU.]

(coh 3)

[figure: UU; UUd; UdU; n; dUU; Uω_{2}; UUD; Ur; ω_{2}U; UDU; rU; d_{n}; DUU; Dn; U; dU; DU] = [figure: UU; UUd; n; n_{d}^{−1}; UUD; Ur; nD; U; Ud; dU; ω_{2}; UD; r; ω_{3}; UDU; rU; DUU; Dn; DU.]

(coh 4)

[figure: UUUD; UUr; UnD; nUD; UUDU; UrU; Uω_{3}; UDUU; UDn; UUD; nD; μ_{𝕌}D; UUD; Ur; nD; UDU; rU; ω_{3}; DUU; Dn; UD; r; DU] = [figure: UUUD; UUr; nUD; UUDU; UrU; nDU; n_{r}^{−1}; ω_{3}U; UDUU; UDn; rUU; UUD; Ur; nD; UDU; rU; DUUU; DUn; DnU; r_{n}^{−1}; UDU; rU; ω_{3}; DUU; Dn; Dμ_{𝕌}; DUU; Dn; UD; r; DU.]

(coh 5)

[figure: DD; uDD; DuD; DDu; ω_{1}D; Dω_{1}; UDD; Um; rD; ω_{4}; UD; r; DUD; Dr; DDU; mU; DU] = [figure: DD; uDD; m; DDu; u_{m}; UDD; Um; m_{u}; D; uD; Du; ω_{1}; UD; r; DDU; mU; DU.]

(coh 6)

[figure: UUDD; UUm; nDD; UUD; nD; Ur; n_{m}^{−1}; UDU; rU; UDD; Um; rD; UD; r; ω_{3}; DUU; Dn; DUD; Dr; ω_{4}; DDU; mU; DU] = [figure: UUDD; UUm; nDD; UrD; UUD; Ur; UDD; rD; ω_{3}D; UDUD; rUD; UDr; _{r_{r}_{−1}}; Uω_{4}; DUUD; DnD; DUr; Dω_{3}; UDDU; rDU; UmU; UDU; rU; DUD; Dr; DUDU; DrU; ω_{4}U; DDUU; DDn; mUU; m_{n}; DUU; Dn; DDU; mU; DU.] (coh 7)

[figure: UD; UdD; Id; dUD; r; Uβ_{𝔻}^{−1}; ω_{2}D; UDD; Um; rD; UDU; r; d_{r}; DUD; Dr; ω_{4}; DU; dDU; Id; β_{𝔻}U; DDU; mU; DU] = id_{r}.

(coh 8)

[figure: UD; UDd; Id; r; Uη_{𝔻}; r_{d}^{−1}; UDD; Um; rD; UD; r; DU; DUd; DdU; Id; Dω_{2}; DUD; Dr; ω_{4}; η_{𝔻}U^{−1}; DDU; mU; DU] = id_{r}.

(coh 9)

[figure: UDDD; UDm; rDD; UmD; UDD; Uμ_{𝔻}; Um; DUDD; DrD; ω_{4}D; UDD; Um; rD; UD; r; DDUD; DDr; mUD; DDDU; mDU; m_{r}; DUD; Dr; ω_{4}; DDU; mU; DU] = [figure: UDDD; UDm; rDD; r_{m}^{−1}; UDD; rD; Um; DUDD; DUm; DrD; DUD; Dr; UD; r; DDUD; DDr; Dω_{4}; ω_{4}; μ_{𝔻}U; DDDU; DmU; mDU; DDU; mU; DDU; mU; DU.]

Observe that if the pseudomonads are strict (β, η and μ are identities), and ω_{1}, ω_{2} and ω_{3} are identities, then we obtain the coherence conditions of the “mild” extensions of the classical distributive laws given in [7].

5. The composite pseudomonad given by a distributive law

Assume we have a distributive law of 𝕌 over 𝔻 as in section 4. The first question is how to produce a composite pseudomonad from 𝕌, 𝔻, and the distributive law. This is what we do in this section.

Define 𝕍 = (V, v, p, β_{𝕍}, η_{𝕍}, μ_{𝕍}) as follows: V = DU ∈ A(K, K); v is defined as the composite

\xymatrix{1_{K} \ar[r]^{u} & U \ar[r]^{dU} & DU;}

p is the composite

\xymatrix{DUDU \ar[r]^{DrU} & DDUU \ar[r]^{DDn} & DDU \ar[r]^{mU} & DU;}

β_{𝕍} is defined to be the pasting

[figure: DUDU; DrU; UDU; dUDU; DDUU; DDn; DDU; DuDU; Dω_{1}U; DDuU; DDβ_{𝕌}; d_{uDU}^{−1}; DDU; mU; DU; uDU; dDU; Id; β_{𝔻}U; DU;]

η_{𝕍} as the pasting and μ_{𝕍} as the pasting

[figure: DU; Id; DUu; Dη_{𝕌}; DU; Id; DdU; η_{𝔻}U; DU;; DUU; Dn; DdUU; DUdU; Dd_{n}^{−1}; DDU; mU; Dω_{2}U^{−1}; DDUU; DDn; DUDU; DrU]

[figure: DUDUDU; DUDrU; DrUDU; Dr_{rU}^{−1}; DUDDUU; DUDDn; DrDUU; Dr_{Dn}^{−1}; DUDDU; DUmU; DrDU; DUDU; DrU; DDUUDU; DDUrU; DDnDU; DDUDUU; DDUDn; DDrUU; DDr_{n}^{−1}; DDUDU; DDrU; Dω_{4}U; DDω_{3}U; DDDUUU; DDDUn; DDDnU; DDDμ_{𝕌}; DDDUU; DmUU; DDDn; Dm_{n}; DDUU; DDn; DDUDU; DDrU; mUDU; m_{rU}^{−1}; DDDUU; DDDn; mDUU; m_{Dn}^{−1}; DDDU; DmU; mDU; μ_{𝔻}U; DDU; mU; DUDU; DrU; DDUU; DDn; DDU; mU; DU.]

5.1. Theorem. 𝕍 = (V, v, p, β_{𝕍}, η_{𝕍}, μ_{𝕍}), as defined above, is a pseudomonad on the object K.

Proof. Observe that the pasting of μ_{𝕍} and η_{𝕍}V^{−1} is

[figure: Dη_{𝕌}DU^{−1}; Dω_{2}UDU; Dr_{rU}^{−1}; Dr_{Dn}^{−1}; Dd_{nDU}; DDr_{n}^{−1}; Dω_{4}U; DDω_{3}U; DDDμ_{𝕌}; Dm_{n}; η_{𝔻}UDU^{−1}; m_{rU}^{−1}; m_{Dn}^{−1}; μ_{𝔻}U; .]

We must show that this pasting equals p ∘ V β_{𝕍}. Substitute the pasting of Dd_{nDU} and Dη_{𝕌}DU^{−1} by the pasting of Dd_{UuDU}^{−1} and DDη_{𝕌}DU. Then use (coh 2). With the help of (2) show that

[figure: DDr_{uU}^{−1}; DDr_{n}^{−1}; DDDμ_{𝕌}; DDDη_{𝕌}U^{−1}] = [figure: DDUDβ_{𝕌}; DDrU; DDDn; ,] and make the substitution. Make the substitution

[figure: Dω_{2}UDU; Dr_{rU}^{−1}; Dr_{Dn}^{−1}] = [figure: DUd_{rU}; Dd_{UrU}^{−1}; DUd_{Dn}; Dd_{UDn}^{−1}; Dω_{2}DU; ,]

followed by the substitutions [figure: Dd_{UuDU}^{−1}; Dd_{UrU}^{−1}; DDUω_{1}U; Dd_{UDn}^{−1}; DDUDβ_{𝕌}] = [figure: DUω_{1}U; DUDβ_{𝕌}; DdUDU; ,]

and

[figure: η_{𝔻}UDU^{−1}; m_{rU}^{−1}; m_{Dn}^{−1}] = [figure: Dd_{rU}; Dd_{Dn}; η_{𝔻}DU^{−1}; ,]

and

[figure: Dd_{Dn}; Dm_{n}; μ_{𝔻}U; η_{𝔻}DU^{−1}] = [figure: Dβ_{𝔻}DUU; DDn; mU; .]

Use (coh 7), and finish with the substitution [figure: DUDβ_{𝕌}; DUω_{1}U; DUd_{rU}; DUd_{Dn}] = [figure: DUDω_{1}U; DUd_{uDU}^{−1}; DUDDβ_{𝕌}] In regards to the other condition, observe that the pasting of V μ_{𝕍}, μ_{𝕍}V and μ_{𝕍} is:

[figure: DUDr_{rU}^{−1}; DUDr_{Dn}^{−1}; DUDω_{4}U; Dr_{rUDU}^{−1}; Dr_{DnDU}^{−1}; DUDDr_{n}^{−1}; DDr_{nDU}^{−1}; DUDDω_{3}U; DUDDDμ_{𝕌}; DUDm_{n}; DUm_{rU}^{−1}; DUm_{Dn}^{−1}; DUμ_{𝔻}U; Dω_{4}UDU; Dr_{rU}^{−1}; Dr_{Dn}^{−1}; Dω_{4}U; DDω_{3}U; DDDμ_{𝕌}DU; Dm_{nDU}; DDr_{n}^{−1}; DDω_{3}U; DDDμ_{𝕌}; Dm_{n}; m_{rUDU}^{−1}; m_{DnDU}^{−1}; μ_{𝔻}UDU; m_{rU}^{−1}; m_{Dn}^{−1}; μ_{𝔻}U]

Where we have only put the name of the corresponding 2-cell in each parallelogram. To show that this pasting is equal to the pasting of p_{p}, μ_{𝕍} and μ_{𝕍} we do the following. First make the substitution

[figure: DUm_{rU}^{−1}; DUm_{Dn}^{−1}; Dω_{4}UDU; Dr_{rU}^{−1}; Dr_{Dn}^{−1}]= [figure: Dr_{DrU}^{−1}; Dr_{DDn}^{−1}; DDr_{rU}^{−1}; DDr_{Dn}^{−1}; Dω_{4}DU; Dm_{UrU}^{−1}; Dm_{UDn}^{−1}; .]

Then make the substitutions

[figure: Dm_{UrU}^{−1}; Dm_{UDn}^{−1}; Dm_{nDU}; DDr_{n}^{−1}; DDω_{3}U; DDDμ_{𝕌}]= [figure: DDDr_{n}^{−1}; Dm_{rU}; DDDω_{3}U; DDDDμ_{𝕌}; Dm_{Dn}; Dm_{rU}^{−1}; Dm_{Dn}^{−1}]

and

[figure: Dr_{DnDU}^{−1}; DUDDr_{n}^{−1}; DDr_{nDU}^{−1}; DUDDω_{3}U; DUDDDμ_{𝕌}; Dr_{DrU}^{−1}; Dr_{DDn}^{−1}; DDr_{rU}^{−1}; DDr_{Dn}^{−1}]

=

[figure: Dr_{DUrU}^{−1}; Dr_{DrU}^{−1}; Dr_{DUDn}^{−1}; DDr_{UrU}^{−1}; DDr_{rU}^{−1}; DDr_{UDn}^{−1}; Dr_{DDn}^{−1}; DDDUr_{n}^{−1}; DDr_{Dn}^{−1}; DDDUω_{3}U; DDDUDμ_{𝕌}; .]

Now use (coh 4). Proceed with the following substitutions

[figure: DDDr_{nU}^{−1}; DDDUDμ_{𝕌}; DDDDμ_{𝕌}U; DDDr_{n}^{−1}; DDDDμ_{𝕌}] = [figure: DDDr_{Un}^{−1}; DDDr_{n}^{−1}; DDDDr_{n}^{−1}; DDDDμ_{𝕌}; DDDDμ_{𝕌}; ,]

followed by

[figure: DDDω_{3}UU; DDDUr_{n}^{−1}; DDDr_{Un}^{−1}; DDDDr_{n}^{−1}] = [figure: DDDn_{Dn}^{−1}; DDDω_{3}U; DDDr_{n}^{−1}; ,]

and then

[figure: Dm_{rU}^{−1}; μ_{𝔻}UDU; Dm_{Dn}^{−1}; m_{rU}^{−1}; m_{Dn}^{−1}] = [figure: m_{DrU}^{−1}; m_{DDn}^{−1}; μ_{𝔻}DU; m_{rU}^{−1}; m_{Dn}^{−1}; ,]

followed by the substitution

[figure: m_{DnDU}^{−1}; DDDr_{n}^{−1}; DDDω_{3}U; DDDDμ_{𝕌}; m_{DrU}^{−1}; m_{DDn}^{−1}]

=

[figure: m_{DUrU}^{−1}; m_{DrU}^{−1}; m_{DUDn}^{−1}; DDr_{n}^{−1}; m_{DDn}^{−1}; DDω_{3}U; DDDμ_{𝕌}; .]

Use (coh 9) to show that

[figure: DUμ_{𝔻}U; Dω_{4}DU; Dω_{4}U; Dm_{rU}; Dm_{Dn}; μ_{𝔻}DU; Dm_{n}; μ_{𝔻}U] = [figure: Dr_{mU}^{−1}; DDω_{4}U; Dω_{4}U; DDm_{n}; Dm_{n}; m_{mU}; μ_{𝔻}U; μ_{𝔻}U] and make the substitution. Next the substitution

[figure: DUDr_{rU}^{−1}; DUDr_{Dn}^{−1}; Dr_{rUDU}^{−1}; Dr_{DUrU}^{−1}; Dr_{DUDn}^{−1}; DDr_{UrU}^{−1}; DDr_{UDn}^{−1}; DDω_{3}UDU; DDDn_{rU}^{−1}; DDDn_{Dn}^{−1}; m_{rUDU}^{−1}; m_{DUrU}^{−1}; m_{DUDn}^{−1}] = [figure: Dr_{UDrU}^{−1}; Dr_{UDDn}^{−1}; DDn_{DrU}^{−1}; DDn_{DDn}^{−1}; Dr_{rDU}^{−1}; m_{UDrU}^{−1}; m_{UDDn}^{−1}; DDω_{3}DU; Dr_{rU}^{−1}; Dr_{Dn}^{−1}; m_{rDU}^{−1}; ,]

followed by the substitution

[figure: Dr_{rDU}^{−1}; DUDω_{4}U; Dr_{DrU}^{−1}; Dr_{DDn}^{−1}; DUDm_{n}; =; Dr_{mU}^{−1}] [figure: Dr_{rU}^{−1}; Dr_{UmU}^{−1}; DDUω_{4}U; Dr_{Dn}^{−1}; DDUm_{n}; .]

Substitute the pasting of DDUm_{n}, DDω_{4}U, DDr_{Dn}^{−1} and DDDr_{n}^{−1} by the pasting of DDω_{4}UU, DDr_{n}^{−1} and DDm_{Un}. Then observe that the pasting of DDm_{Un}, DDm_{n} and DDDDμ_{𝕌} equals the pasting of DDDμ_{𝕌}, DDm_{nU} and DDm_{n}. Use (coh 6). To finish the proof, make the substitution

[figure: m_{rDU}^{−1}; DDω_{4}U; m_{DrU}^{−1}; m_{DDn}^{−1}; DDm_{n}; =; m_{mU}^{−1}] [figure: m_{rU}^{−1}; m_{UmU}^{−1}; Dω_{4}U; m_{Dn}^{−1}; Dm_{n}; .]

6. Compatible pseudomonad structures

We consider now the question of when can a pseudomonad be considered as the composite of two pseudomonads.

Let 𝔻, 𝕌 be pseudomonads on the same object K of the Gray-category A. Given another pseudomonad 𝕍 = (V, v, p, β_{𝕍}, η_{𝕍}, μ_{𝕍}) on the same object K, we say that 𝕍 is compatible with the pseudomonads 𝔻 and 𝕌 if V = DU and there are invertible 2-cells:

[figure: U; dU; θ_{1}; 1_{K}; v; u; DU,; DUDU; p; θ_{2}; DUU; Dn; DUdU; DU,; DUDU; p; θ_{3}; DDU; DuDU; mU; DU,]

subject to coherence conditions. To describe the coherence conditions introduce the following pastings:

θ_{4} = [figure: DUDUU; DUDn; DUDUdU; pU; DUθ_{2}^{−1}; DUDU; p; p_{dU}^{−1}; DUDUDU; pDU; DUp; μ_{𝕍}; DUDU; p; DUU; DUdU; Dn; θ_{2}; DU,]

and

θ_{5} = [figure: DDUDU; mUDU; DuDUDU; Dp; θ_{3}DU^{−1}; DUDU; p; Du_{p}; DUDUDU; DUp; pDU; μ_{𝕍}^{−1}; DUDU; p; DDU; DuDU; mU; θ_{3}; DU.]

The coherence conditions are:

[figure: DU; DUu; DUv; Id; η_{𝕍}; DU,; DUθ_{1}^{−1}; DUDU; p; θ_{2}; =; Dη_{𝕌},; DUU; DUdU; Dn]  (12)

[figure: DUU; DUdU; Dn; Du_{dU}^{−1}; DUDU; p; θ_{3}; θ_{2}^{−1}; DU; DuU; DdU; Id; DDU; mU; DuDU; η_{𝔻}U^{−1}; DU; =; Dβ_{𝕌},]  (13)

[figure: DDU; DuDU; mU; d_{uDU}; θ_{1}DU; UDU; dUDU; DUDU; p; θ_{3}^{−1}; β_{𝕍}; DU; Id; uDU; dDU; DU; =; β_{𝔻}U,]  (14)

[figure: DDUU; mUU; DuDUU; DDn; θ_{3}U^{−1}; DUU; Dn; Du_{Dn}; DUDUU; pU; DUDn; DUθ_{4}^{−1}; =; m_{n}; DUDU; p; DDU; DuDU; mU; θ_{3}; DU]  (15)

[figure: DUUU; DnU; DUdUU; DUn; θ_{2}U^{−1}; DUU; Dn; DUd_{n}; DUDUU; pU; DUDn; θ_{4}^{−1}; =; Dμ_{𝕌}^{−1}; DUDU; p; DUU; DUdU; Dn; θ_{2}; DU]  (16)

[figure: DDDU; DmU; DDuDU; mDU; Dθ_{3}^{−1}; DDU; mU; m_{uDU}^{−1}; DDUDU; Dp; mUDU; θ_{5}^{−1}; =; μ_{𝔻}U; DUDU; p; DDU; DuDU; mU; θ_{3}; DU]  (17)

Some other equations we will need later are contained in the following:

Theory and Applications of Categories, Vol. 5, No. 5 110

6.1. Proposition. Assume we have invertible 2-cells θ_{1}, θ_{2}, and θ_{3}, satisfying the coherence conditions. Then

(Dn, θ_{4}) : (β_{𝕍}U, μ_{𝕍}U) → (β_{𝕍}, μ_{𝕍})  (18)

is a 1-cell in 𝕍-Alg_{K}, and the following are 2-cells in 𝕍-Alg_{K}:

[figure: (β_{𝕍}, μ_{𝕍}); Id; (DUu,p_{u}^{−1}); Dη_{𝕌}; (β_{𝕍}, μ_{𝕍}); (β_{𝕍}U, μ_{𝕍}U); (Dn,θ_{4})]  (19)

[figure: (β_{𝕍}U, μ_{𝕍}U); (Dn,θ_{4}); (β_{𝕍}UU, μ_{𝕍}UU); (DUn,p_{n}^{−1}); (DnU,θ_{4}U); Dμ_{𝕌}; (β_{𝕍}, μ_{𝕍}),; (β_{𝕍}U, μ_{𝕍}U); (Dn,θ_{4})]  (20)

We also have that

(p, θ_{5}^{−1}) : (β_{𝔻}UDU, μ_{𝔻}UDU) → (β_{𝔻}U, μ_{𝔻}U)  (21)

is a 1-cell in 𝔻-Alg_{K}, and the following are 2-cells in 𝔻-Alg_{K}:

[figure: (β_{𝔻}UDU, μ_{𝔻}UDU); (p,θ_{5}^{−1}); (β_{𝔻}UDUDU, μ_{𝔻}UDUDU); (DUp,m_{Up}^{−1}); (pDU,θ_{5}DU^{−1}); μ_{𝕍}; (β_{𝔻}U, μ_{𝔻}U),; (β_{𝔻}UDU, μ_{𝔻}UDU); (p,θ_{5}^{−1})]  (22)

[figure: (β_{𝔻}UDU, μ_{𝔻}UDU); (p,θ_{5}^{−1}); θ_{2}; (β_{𝔻}UU, μ_{𝔻}UU); (DUdU,m_{UdU}^{−1} ); (Dn,m_{n}^{−1}); (β_{𝔻}U, μ_{𝔻}U),]  (23)

[figure: (β_{𝔻}UDU, μ_{𝔻}UDU); (p,θ_{5}^{−1}); (β_{𝔻}UDUU, μ_{𝔻}UDUU); (DUDn,m_{UDn}^{−1}); (pU,θ_{5}U^{−1}); θ_{4}; (β_{𝔻}U, μ_{𝔻}U),; (β_{𝔻}UU, μ_{𝔻}UU); (Dn,m_{n}^{−1})]  (24)

Furthermore, if we define σ_{1} as,

[figure: UDU; dUDU; θ_{1}DU; DUDU; p; β_{𝕍}; DU; uDU; vDU; Id; DU,]

σ_{2} as

[figure: UUDU; UdUDU; nDU; dUUDU; d_{UdUDU}^{−1}; UDUDU; Up; dUDUDU; d_{Up}^{−1}; UDU; dUDU; d_{nDU}; DUUDU; DUdUDU; DnDU; θ_{2}DU; DUDUDU; DUp; pDU; μ_{𝕍}; DUDU; p; UDU; dUDU; DUDU; p; DU,]

γ as

[figure: UU; n; UdU; dUU; d_{UdU}^{−1}; UDU; dUDU; DUU; DUdU; Dn; d_{n}; θ_{2}; DUDU; p; U; dU; DU.]

then (σ_{1}, σ_{2}) is an object in 𝕌-Alg_{K}, and (dU, γ) : (β_{𝕌}, μ_{𝕌}) → (σ_{1}, σ_{2}) is a 1-cell in 𝕌-Alg_{K}.

Proof. We show first that (Dn, θ_{4}) is a 1-cell in 𝕍-Alg_{K}. To show that (Dn, θ_{4}) satisfies (9), start on the left hand side and make the substitutions:

[figure: β_{𝕍}U; p_{dU}^{−1}] = [figure: ,; v_{DUdU}; β_{𝕍}DU]

[figure: β_{𝕍}DU; μ_{𝕍}] = [figure: ,; v_{p}; β_{𝕍}]

and

[figure: θ_{2}; DUθ_{2}^{−1}; v_{DUdU}; v_{p}] = v_{Dn}.

To show that (Dn, θ_{4}) satisfies (10), start on the left hand side, cancel DUθ_{2} and its inverse, make the substitution

[figure: DUp_{dU}^{−1}; μ_{𝕍}U; p_{dU}^{−1}; =; p_{DUdU}^{−1}; p_{dU}^{−1}; μ_{𝕍}DU; .]

Then use (1), and conclude with

[figure: DUDUθ_{2}^{−1}; p_{DUdU}^{−1}; p_{p}^{−1}] = [figure: p_{Dn}^{−1}; DUθ_{2}^{−1}]

Next we show that Dη_{𝕌} in (19) is a 2-cell in 𝕍-Alg_{K}. To show that satisfies (11), start on the left hand side of (11). Use (12) to substitute the pasting of DUDη_{𝕌} and DUθ_{2}^{−1} for the pasting of DUβ_{𝕍} and DUDUθ_{1}^{−1}. Then make the substitution

[figure: DUDUθ_{1}^{−1}; p_{u}^{−1}; p_{dU}^{−1}] = [figure: p_{v}^{−1}; .; DUθ_{1}^{−1}]

Now use (5), and then use (12) once again.

Next we show that Dμ_{𝕌} in (20) is a 2-cell of 𝕍-algebras. Start on the right hand side of (11). Use (16) to substitute the pasting of θ_{2} and Dμ_{𝕌}, by the pasting of DUd_{n}^{−1}, DUθ_{4} and θ_{2}U. Make the substitution

[figure: p_{n}^{−1}; DUd_{n}^{−1}; p_{dU}^{−1}; ,; =; DUDUd_{n}^{−1}; p_{dUU}^{−1}; p_{Dn}^{−1}; .]

Since we already know that (Dn, θ_{4}) is a 𝕍-algebra morphism, we can substitute the pasting of μ_{𝕍}, p_{Dn}^{−1}, and θ_{4}, by the pasting of DUθ_{4}, θ_{4} and μ_{𝕍}. By the definition of θ_{4}, we can substitute the pasting of μ_{𝕍}U, p_{dUU}^{−1} and θ_{2}U by the pasting of DUθ_{2}U and θ_{4}U. Finally, use (16).

That (p, θ_{5}^{−1}) is a morphism of 𝔻-algebras, and that μ_{𝕍}, θ_{2}, and θ_{4} are 2-cells of 𝔻-algebras are similar and left to the reader.

Now we show that (σ_{1}, σ_{2}) is an object in 𝕌-Alg_{K}. Paste η_{𝕌}DU^{−1} onto the left hand side of (7), and make a substitution on it to obtain

[figure: d_{UuDU}^{−1}; d_{UdUDU}^{−1}; d_{Up}^{−1}; η_{𝕌}DU; θ_{2}DU; μ_{𝕍}; .]  (25)

Also paste η_{𝕌}DU^{−1} on what turns out to be the right hand side of (7), and make the following two substitutions:

[figure: Uθ_{1}DU; Uβ_{𝕍}; dUDU]= [figure: d_{UdUDU}^{−1}; d_{UuDU}^{−1}; d_{Up}^{−1}; DUθ_{1}DU; DUβ_{𝕍}; ,]

[figure: DUβ_{𝕍}; p] = [figure: η_{𝕍}DU; μ_{𝕍}; .]

Compare the pasting we arrive at with (25), and use equation (12).

Now, the left hand side of (6) in this case, is the following pasting:

Make the following sequence of substitutions:

[figure: Ud_{UdUDU}^{−1}; Ud_{UnDU}; Ud_{Up}^{−1}; Uθ_{2}DU; Uμ_{𝕍}; μ_{𝕌}DU; d_{UdUDU}^{−1}; d_{Up}^{−1}; d_{nDU}; θ_{2}DU; μ_{𝕍}; .]

[figure: μ_{𝕌}DU; d_{nDU}; =; d_{UnDU}^{−1}; d_{nUDU}; Dμ_{𝕌}DU; ,; d_{nDU}] [figure: Uμ_{𝕍}; d_{Up}^{−1}; =; d_{UDUp}^{−1}; d_{UpDU}; DUμ_{𝕍}; d_{Up}^{−1}; ,] [figure: DUμ_{𝕍}; μ_{𝕍}; =; p_{p}^{−1}; μ_{𝕍}DU^{−1}; μ_{𝕍}; μ_{𝕍}; ,]

[figure: Ud_{nDU}; d_{UnDU}^{−1}; d_{UdUDU}^{−1}; =; d_{UdUUDU}^{−1}; DUd_{nDU}; d_{UDnDU}^{−1}; ,]

[figure: Uθ_{2}DU; d_{UDnDU}; d_{UpDU}; =; d_{UDUdUDU}^{−1}; DUθ_{2}DU; ,]

and

[figure: Ud_{UdUDU}; Ud_{Up}^{−1}; d_{UdUUDU}^{−1}; d_{UDUdUDU}^{−1}; d_{UDUp}^{−1}; =; d_{UUdUDU}^{−1}; d_{UUp}^{−1}; DUd_{UdUDU}^{−1}; DUd_{Up}^{−1}; d_{UdUDU}^{−1}]

On the other hand, the right hand side of (6) in this case is the pasting

[figure: n_{p}^{−1}; d_{UdUDU}^{−1}; d_{Up}^{−1}; n_{dUDU}^{−1}; d_{nDU}; θ_{2}DU; μ_{𝕍}; .; d_{Up}^{−1}; μ_{𝕍}; d_{UdUDU}^{−1}; θ_{2}DU; d_{nDU}]

On this pasting make the following substitutions:

[figure: n_{dUDU}^{−1}; n_{p}^{−1}; d_{UdUDU}^{−1}; d_{Up}^{−1}; d_{nDU}; =; d_{UUdUDU}^{−1}; d_{UUp}^{−1}; d_{nUDU}; Dn_{dUDU}^{−1}; Dn_{p}^{−1}; ,]

[figure: Dn_{dUDU}^{−1}; Dn_{p}^{−1}; θ_{2}DU; =; DUd_{UdUDU}^{−1}; DUd_{Up}^{−1}; θ_{2}UDU; p_{dUDU}^{−1}; p_{p}^{−1}]

and

[figure: p_{dUDU}^{−1}; θ_{2}DU; =; DUθ_{2}DU; θ_{4}DU; μ_{𝕍}DU^{−1}; (definition of θ_{4}).]

Compare both results, and use equation (16).

The claim about (dU, γ) is left to the reader.

Next we prove that the composite of two pseudomonads via a distributive law as defined in section 5, is compatible. 6.2. Theorem. Assume we have a distributive law as in section 4 and let 𝕍 the composite pseudomonad defined in section 5. If we define θ_{1} = id_{dU∘u}, and θ_{2} as the pasting

[figure: DUDU; DrU; DUU; DUdU; DdUU; Dn; Dω_{2}U; Dd_{n}; DDUU; DDn; DU; DdU; Id; η_{𝔻}U^{−1}; DDU; mU; DU,]

and θ_{3} as the pasting

then the pseudomonad 𝕍 is compatible with 𝕌 and 𝔻.

[figure: DUDU; DrU; DDU; DuDU; DDuU; Dω_{1}U; Id; DDβ_{𝕌}; DDUU; DDn; DDU; mU; DU,]

Proof. Conditions (12), (13), and (14) are fairly easy and left to the reader. The remaining ones are also easy once we have shown that

θ_{4} = [figure: DUDUU; DUDn; DrUU; Dr_{n}^{−1}; DUDU; DrU; DDUUU; DDUn; DDnU; DDμ_{𝕌}; DDUU; DDn; DDUU; DDn; mUU; m_{n}^{−1}; DDU; mU; DUU; Dn; DU,] and θ_{5} = [figure: DDUDU; mUDU; DDrU; m_{rU}; DUDU; DrU; DDDUU; mDUU; DDDn; m_{Dn}; DDUU; DDn; DDDU; mDU; DmU; μ_{𝔻}U; DDU; mU; DDU; mU; DU.]

To show the first equality above, start with the definition of θ_{4} and make the following substitutions:

[figure: DUDd_{n}^{−1}; DUDω_{2}U^{−1}; Dr_{rU}^{−1}; Dr_{Dn}^{−1}; Dr_{UdU}^{−1}] = [figure: Dr_{n}^{−1}; Dr_{dU}^{−1}; ,; DDUω_{2}U^{−1}; DDUd_{n}^{−1}]

and

[figure: m_{UdU}^{−1}; m_{rU}^{−1}; m_{Dn}^{−1}; Dω_{2}U; Dd_{n}] = [figure: DDω_{2}U; DDd_{n}; m_{n}^{−1}; m_{dU}^{−1}; .]

Now use (coh 3) on the pasting of DDUω_{2}U^{−1}, DDω_{3}U, DDn_{dU}^{−1} and DDω_{2}U. Next make the substitutions:

[figure: DDω_{2}UU; DDUd_{n}^{−1}; DDr_{n}^{−1}] = [figure: DDd_{Un}^{−1}; DDω_{2}U; ,]

[figure: DDd_{Un}^{−1}; DDd_{nU}; DDDμ_{𝕌}; DDd_{n}] = [figure: DDμ_{𝕌}; DDd_{n}; ,]

and

[figure: DDd_{n}; Dm_{n}; m_{dU}^{−1}; μ_{𝔻}U; η_{𝔻}U^{−1}] = mU ∘ DDn ∘ Dη_{𝔻}UU^{−1}.

Finally, use (coh 8).

The proof for θ_{5} is very similar, except that we need the equation:

[figure: UD; uUD; Id; β_{𝕌}D^{−1}; UUD; Ur; nD; UDU; rU; ω_{3}; DUU; Dn; UD; r; DU] = [figure: UD; uUD; r; u_{r}; UUD; Ur; DU; uDU; DuU; Id; ω_{1}U; UDU; rU; Dβ_{𝕌}; DUU; Dn; DU.]

To prove this last equation, observe that Id_{DU} is isomorphic to Dn ∘ rU ∘ uDU, thus it suffices to show that the equation holds when followed by Dn ∘ rU ∘ uDU. And, since u_{r} and ω_{3} are invertible, we can paste them below on both sides of the equation. This last equation is not hard to prove, using (coh 4) and (coh 2). The rest of the proof is left to the reader.

7. The Gray-category of pseudomonads in a Gray-category

In this section we define the Gray-category PSM(A) of pseudomonads on a Gray-category A.

The objects of PSM(A) are pseudomonads in A.

Given pseudomonads 𝔻 on K, and 𝕌 on L, we denote the 2-category PSM(A)(𝔻, 𝕌) as [𝔻, 𝕌], and we define it as follows. The objects of [𝔻, 𝕌] are pairs (G, G_{0}), where G : 𝕌-Alg → 𝔻-Alg : A^{op} → Gray is a Gray-natural transformation, G_{0} ∈ A(L, K), such that

\xymatrix{𝕌-Alg \ar[r]^{G} \ar[d]_{Φ} & 𝔻-Alg \ar[d]^{Φ} \\ A( , L) \ar[r]_{G_{0}( )} & A( , K)}

commutes, where Φ is the forgetful Gray-natural transformation defined in section 3. We can consider 𝕌-Alg and 𝔻-Alg as trihomomorphisms and G as a tritransformation. A 1-cell (G, G_{0}) → (H, H_{0}) in [𝔻, 𝕌] is a pair (g, g_{0}), where g : G → H is a strict trimodification, and g_{0} : G_{0} → H_{0} is in A(L, K), such that

[figure: 𝕌-Alg; G; H; g; 𝔻-Alg; Φ; A( , K)] = [figure: 𝕌-Alg; Φ; A( , L); G_{0}( ); H_{0}( ); g_{0}( ); A( , K).]

What we mean by strict trimodification is, a trimodification in which all the invertible modifications required in the definition ([6],pg. 25) are identities. A 2-cell (g, g_{0}) → (h, h_{0}) in [𝔻, 𝕌] is a pair (γ, γ_{0}), where γ : g → h is a perturbation, and γ_{0} : g_{0} → h_{0} is in A(L, K), such that

[figure: ^{𝕌-Alg^{X}}; ^{𝔻-Alg^{X}}; A(X , K); g_{X} ⇓; ⇓ g′_{X}; Φ_{X}; γ_{X}] = [figure: ^{𝕌-Alg^{X}}; A(Z, L); A(X , K).; g_{0}( ) ⇓; ⇓ h_{0}( ); Φ_{Z}; γ_{0}( )]

Vertical and horizontal compositions are the obvious ones. The rest of the operations are defined as follows:

(H, H_{0})(G, G_{0}) = (HG, H_{0}G_{0})
(H, H_{0})(g, g_{0}) = (Hg, H_{0}g_{0}) (h, h_{0})(G, G_{0}) = (hG, h_{0}G_{0})
(H, H_{0})(σ, σ_{0}) = (Hσ, H_{0}σ_{0}) (τ, τ_{0})(G, G_{0}) = (τG, τ_{0}G_{0})

8. Liftings

8.1. Definition. A lifting is a pseudomonad in the Gray-category PSM(A).

In more detail, observe that a pseudomonad 𝔻̃ in PSM(A) has to have domain a pseudomonad 𝕌 = (U, u, n, β_{𝕌}, η_{𝕌}, μ_{𝕌}) in A, with domain some object K of A. Observe that 𝔻̃ is of the form

𝔻̃ = ((D̃, D), (d̃, d), (m̃, m), (β̃, β), (η̃, η), (μ̃, μ)).

Taking the second entries, we obtain a pseudomonad 𝔻 = (D, d, m, β_{𝔻}, η_{𝔻}, μ_{𝔻}) in A on the same object K. Furthermore, given an object X of A, we can define D_{X} = D̃_{X}, and d_{X} = d̃_{X} etcetera, to obtain a pseudomonad 𝔻_{X} = (D_{X} , d_{X} , m_{X} , β_{X} , η_{X} , μ_{X}) in Gray, on the 2-category 𝕌-Alg_{X}. 𝔻 and the family ⟨𝔻_{X} ⟩_{X} behave well with forgetful 2-functors and change of base.

9. From a pseudomonad with compatible structure to a pseudomonad in PSM(A)

Assume we have a pseudomonad 𝕍 that is compatible with pseudomonads 𝔻 and 𝕌 as in section 6. We want to define a pseudomonad 𝔻̃ on the object 𝕌 of PSM(A). Let X be an object of A. We begin by defining a 2-functor D_{X} : 𝕌-Alg_{X} → 𝕌-Alg_{X}. Given an object (ψ, χ) in 𝕌-Alg_{X}, with

[figure: UX; x; ψ; X; uX; Id; X,; UUX; nX; Ux; χ; UX; x; UX; x; X,]

Define the first entry of 𝔻_{X} (ψ, χ) as and the second entry of 𝔻_{X} (ψ, χ) as

[figure: UDX; UDuX; dUDX; θ_{1}DX; d_{UDuX}^{−1}; UDUX; dUDUX; DUDX; DUDuX; v_{DuX}; DUDUX; pX; DX; uDX; DuX; Id; vDX; DUX; vDUX; β_{𝕍}X; DUX; Dx; Dψ; DX,]

[figure: UUDX; UUDuX; nDX; UUDUX; UdUDUX; dUUDUX; nDUX; UDUDUX; UpX; dUDUDUX; d_{UdUDUX}^{−1}; UDUX; UDx; UDuUX; Id; UDβ_{𝕌}X; UDu{−1 \atop x}; UDX; UDuX; d_{UDnX}; UDUUX; UDnX; UDUx; dUDUUX; d{−1 \atop UDUx}; UDUX; dUDUX; d{−1 \atop UpX}; UDUX; dUDUX; θ_{4}X^{−1}; DUDUUX; DUDnX; DUDUx; pUX; p{−1 \atop x}; DUDUX; p; n{−1 \atop DuX}; d_{nDUX}; DUUDUX; DnDUX; θ_{2}DUX; DUdUDUX; DUDUDUX; pDUX; DUpX; μ_{𝕍}X; DUDUX; pX; DUUX; DnX; DUx; Dχ; DUX; Dx; UDUX; UDuX; UDUX; dUDUX; DUDUX; pX; DUX; Dx; DX.]

If (h, ρ) : (ψ, χ) → (ψ′, χ′) in 𝕌-Alg_{X}, then define the first entry of D_{X} (h, ρ) as Dh, and the second as the pasting

[figure: UDX; UDh; UDuX; UDu_{h}^{−1}; UDX′; UDuX′; UDUX; UDUh; dUDUX; d_{UDUh}^{−1}; UDUX′; dUDUX′; DUDUX; DUDUh; pX; p_{h}^{−1}; DUDUX′; pX′; DUX; DUh; Dx; Dρ; DUX′; Dx′; DX; Dh; DX′]

Given ξ : (h, ρ) → (h′, ρ′) in 𝕌-Alg_{X}, define D_{X} (ξ) = Dξ.

9.1. Proposition. The above definitions make D_{X} : 𝕌-Alg_{X} → 𝕌-Alg_{X} a 2-functor. Furthermore, if we define D̃ : 𝕌-Alg → 𝕌-Alg as D_{X} at every object X of A, then D̃ is a Gray-transformation, and (D̃, D) ∈ [𝕌, 𝕌].

Proof. The hardest part of the proof is to show that D_{X} (ψ, χ) is indeed an object in 𝕌-Alg_{X}. This is what we do, and we leave the rest to the reader. We must show that (6) and (7) are satisfied. To show (7), pass η_{𝕌}DX to the left hand side, and perform the following sequence of substitutions:

[figure: η_{𝕌}DX^{−1}; n_{DuX}^{−1}; d_{nDUX}] = [figure: Dη_{𝕌}DUX^{−1}; Uu_{DuX}; d_{UuDUX}^{−1}; ,]

[figure: Dη_{𝕌}DUX^{−1}; θ_{2}DUX] = [figure: η_{𝕍}DUX^{−1}; DUθ_{1}DUX; ,] (see (12))

[figure: d_{UuDUX}^{−1}; d_{UdUDUX}^{−1}; DUθ_{1}DUX] = [figure: Uθ_{1}DUX; ,; d_{UvDUX}^{−1}]

[figure: η_{𝕍}DUX^{−1}; μ_{𝕍}X] = [figure: pX; DUβ_{𝕍}X; ,]

[figure: d_{UvDUX}^{−1}; d_{UpX}^{−1}; DUβ_{𝕍}X] = [figure: dUDUX; Uβ_{𝕍}X; ,]

[figure: Uu_{DuX}; Uθ_{1}DUX] = [figure: Uθ_{1}DX; Ud_{UDuX}^{−1}; Uv_{DuX}]

[figure: UDuX; UDβ_{𝕌}X] = [figure: UDu_{uX}^{−1}; UDη_{𝕌}X^{−1}; ,]

[figure: UDη_{𝕌}X^{−1}; d_{UDnX}] = [figure: d_{UDUuX}^{−1}; DUDη_{𝕌}X^{−1}; ,]

[figure: DUDη_{𝕌}X^{−1}; θ_{4}X^{−1}] = [figure: p_{uX}^{−1}; Dη_{𝕌}X^{−1}; ,] (see (19))

[figure: Dη_{𝕌}X^{−1}; Dχ] = [figure: Dx; ,; DUψ] and

[figure: UDu_{uX}^{−1}; UDu_{x}^{−1}; d_{UDUuX}^{−1}; d_{UDUx}^{−1}; p_{uX}^{−1}; p_{x}^{−1}; DUψ] = [figure: UDψ; UDuX; dUDUX; pX; .]

To get from the left hand side of (6) to the right hand side, make the following sequence of substitutions:

[figure: Un_{DuX}^{−1}; μ_{𝕌}DX; n_{DuX}^{−1}; =; n_{UDuX}^{−1}; n_{DuX}^{−1}; μ_{𝕌}DUX; ,]

[figure: μ_{𝕌}DUX; d_{nDUX}] = [figure: d_{UnDUX}^{−1}; d_{nUDUX}; Dμ_{𝕌}DUX; ,; d_{nDUX}]

[figure: Dμ_{𝕌}DUX; θ_{2}DUX] = [figure: θ_{2}UDUX; DUd_{nDUX}^{−1}; θ_{4}DUX; ,] (see (16))

[figure: Ud_{nDUX}; d_{UnDUX}^{−1}; d_{UdUDUX}^{−1}; ,; DUd_{nDUX}^{−1}] = [figure: d_{UdUUDUX}^{−1}; d_{UDnDUX}^{−1}]

[figure: Uθ_{2}DUX; Uμ_{𝕍}X; d_{UDnDUX}^{−1}; d_{UpX}^{−1}] = [figure: d_{UDUdUDUX}^{−1}; d_{UDUpX}^{−1}; DUθ_{2}DUX; DUμ_{𝕍}X; d_{UpX}^{−1}]

[figure: DUθ_{2}DUX; θ_{4}DUX] = [figure: p_{dUDUX}^{−1}; μ_{𝕍}DUX; θ_{2}DUX; ,] (definition of θ_{4})

[figure: DUμ_{𝕍}X; μ_{𝕍}DUX; μ_{𝕍}X; =; p_{pX}^{−1}; μ_{𝕍}X; μ_{𝕍}X; ,]

[figure: Ud_{UdUDUX}^{−1}; Ud_{UpX}^{−1}; d_{UdUUDUX}^{−1}; d_{UDUdUDUX}^{−1}; d_{UDUpX}^{−1}] = [figure: d_{UUdUDUX}^{−1}; d_{UUpX}^{−1}; DUd_{UdUDUX}^{−1}; DUd_{UpX}^{−1}; d_{UdUDUX}^{−1}; ,]

[figure: θ_{2}UDUX; DUd_{UdUDUX}^{−1}; DUd_{UpX}^{−1}; p_{dUDUX}^{−1}; p_{pX}^{−1}] = [figure: Dn_{dUDUX}^{−1}; Dn_{pX}^{−1}; θ_{2}DUX; ,]

[figure: d_{UUdUDUX}^{−1}; d_{UUpX}^{−1}; d_{nUDUX}; Dn_{dUDUX}^{−1}; Dn_{pX}^{−1}] = [figure: n_{dUDUX}^{−1}; n_{pX}^{−1}; d_{UdUDUX}^{−1}; d_{UpX}^{−1}; d_{nDUX}; ,]

[figure: Ud_{UDnX}; UUDβ_{𝕌}X] = [figure: UDUDβ_{𝕌}X; Ud_{UDuUX}^{−1}; ,]

[figure: UDUDβ_{𝕌}X; dUDUDUX; pDUX] = [figure: d_{UDUDuUX}^{−1}; d_{UDUDnX}^{−1}; p_{DuUX}^{−1}; p_{DnX}^{−1}; DUDβ_{𝕌}X; ,]

[figure: θ_{2}DUX; DUd_{UDuUX}^{−1}; p_{DuUX}^{−1}] = [figure: Dn_{DuUX}^{−1}; θ_{2}DUUX; ,]

[figure: Ud_{UDuUX}^{−1}; d_{UdUDUX}^{−1}; d_{UDUDuUX}^{−1}; =; d_{UUDuUX}^{−1}; DUd_{UDuUX}^{−1}; d_{UdUDUUX}^{−1}; ,]

[figure: d_{UUDuUX}^{−1}; d_{nDUX}; Dn_{DuUX}^{−1}; DUDβ_{𝕌}X; =; n_{DuUX}^{−1}; d_{nDUUX}; UDβ_{𝕌}X; d_{UDnX}; ,]

[figure: Uθ_{4}X^{−1}; d_{UDUDnX}^{−1}; d_{UpX}^{−1}; =; d_{UpUX}^{−1}; DUθ_{4}X^{−1}; d_{UDnX}^{−1}; ,]

[figure: DUθ_{4}X^{−1}; p_{DnX}^{−1}; μ_{𝕍}X; =; μ_{𝕍}UX; θ_{4}X^{−1}; θ_{4}X; ,; (see (18))]

[figure: UDnX; UDβ_{𝕌}X] = [figure: UDβ_{𝕌}UX; UDu_{nX}^{−1}; UDμ_{𝕌}X; ,]

[figure: UDμ_{𝕌}X; d_{UDnX}; d_{UDnX}^{−1}] = [figure: d_{UDUnX}^{−1}; d_{UDnUX}; DUDμ_{𝕌}X]

[figure: DUDμ_{𝕌}X; θ_{4}X^{−1}; θ_{4}X] = [figure: p_{nX}^{−1}; θ_{4}UX^{−1}; Dμ_{𝕌}X] (see (20))

[figure: UDχ; UDu_{nX}^{−1}; UDu_{x}^{−1}; d_{UDUnX}^{−1}; d_{UDUx}^{−1}; p_{nX}^{−1}; p_{x}^{−1}; ,] = [figure: UDu_{Ux}^{−1}; UDu_{x}^{−1}; d_{UDUUx}^{−1}; d_{UDUx}^{−1}; p_{Ux}^{−1}; p_{x}^{−1}; ,; DUχ]

[figure: DUχ; Dμ_{𝕌}X; Dχ; =; Dn_{x}^{−1}; Dχ; Dχ; ,]

[figure: UDβ_{𝕌}UX; UDu_{Ux}^{−1}; d_{UDnUX}; d_{UDUUx}^{−1}; θ_{4}UX^{−1}; p_{Ux}^{−1}; Dn_{x}^{−1}] = [figure: UDβ_{𝕌}X; d_{UDnX}; d_{UDUx}^{−1}; θ_{4}X^{−1}; p_{x}^{−1}]

[figure: d_{UDUx}^{−1}; d_{UdUDUUX}^{−1}; Ud_{UDUx}^{−1}; d_{UpUX}^{−1}; Up_{x}^{−1}] = [figure: d_{UUDUx}^{−1}; d_{UdUDUX}^{−1}; d_{UpX}^{−1}; DUd_{UDUx}^{−1}; DUp_{x}^{−1}]

[figure: DUp_{x}^{−1}; μ_{𝕍}UX; p_{x}^{−1}; =; p_{DUx}^{−1}; p_{x}^{−1}; μ_{𝕍}X; ,]

[figure: θ_{2}DUUX; DUd_{UDUx}^{−1}; p_{DUx}^{−1}] = [figure: Dn_{DUx}^{−1}; θ_{2}DUX; ,]

[figure: d_{UUDUx}^{−1}; d_{nDUUX}; Dn_{DUx}^{−1}; =; n_{DUx}^{−1}; d_{UDUx}^{−1}; d_{nDUX}; ,]

and

[figure: UUDu_{x}^{−1}; n_{DuUX}^{−1}; n_{DUx}^{−1}; =; n_{Dx}^{−1}; UDu_{x}^{−1}; n_{DuX}^{−1}; .]

We are now ready to define d_{X} : 1 → D_{X}, and m_{X} : D_{X} D_{X} → D_{X}. The first entry of (d_{X})_{(ψ,χ)} is dX, while the second is the pasting

[figure: UX; UdX; UuX; dUX; x; Ud_{uX}; UDX; UDuX; d_{UuX}^{−1}; UUX; dUUX; UdUX; d_{UdUX}^{−1}; UDUX; dUDUX; DUX; DUuX; Id; Dη_{𝕌}X^{−1}; DUUX; DUdUX; DnX; θ_{2}X; DUDUX; pX; d_{x}; DUX; Dx; X; dX; DX,]

and (d_{X})_{(h,ρ)} = d_{h}.

Define the first entry of (m_{X})_{(ψ,χ)} as mX, and the second as the pasting

[figure: UDDX; UmX; UDDuX; UDuDX; Um_{uX}; UDU; UDuX; UDUDX; dUDUDX; UDUDuX; UDu_{DuX}^{−1}; UDDUX; UDuDUX; dUDDUX; UmUX; d_{UDuDUX}; d_{UmUX}^{−1}; UDUX; dUDUX; DUDUDX; pDX; DUDUDuX; d_{UDUDuX}^{−1}; p_{DuX}^{−1}; UDUDUX; dUDUDUX; DUDDUX; DUDuDUX; DUmUX; DUDX; DUDuX; DUDUDUX; pDUX; DUpX; DUθ_{3}X; DUDUX; pX; DUDUX; DdUDUX; Id; η_{𝔻}UDUX; μ_{𝕍}X; DDUDUX; mUDUX; DpX; θ_{5}X; DUDUX; pX; DDUX; DDx; mUX; m_{x}; DUX; Dx; DDX; mX; DX.]

Define (m_{X})_{(h,ρ)} = m_{h}.

9.2. Proposition. With the above definitions, d_{X} : 1 → D_{X} and m_{X} : D_{X} D_{X} → D_{X} are strong transformations. Furthermore, defining d̃ as d_{X}, and m̃ as m_{X} at every object X of A, we have that, (d̃, d) : 1 → (D̃, D) and (m̃, m) : (D̃, D)(D̃, D) → (D̃, D) are 1-cells in the 2-category [𝕌, 𝕌].

Proof. The size of the diagrams obtained is what makes it hard to prove that (m_{X})_{(ψ,χ)} satisfies (10). To get from the left hand side of (10) to the right hand side make the following three substitutions:

[figure: Uθ_{5}X; UDu_{pX}^{−1}; Um_{x}; d_{UDUpX}^{−1}; p_{pX}^{−1}; UDu_{Dx}^{−1}; Um_{uX}; DUDβ_{𝕌}X; d_{UDUDx}^{−1}; d_{UDuDUX}; UDu_{DuX}^{−1}; d_{UmUX}^{−1}; p_{Dx}^{−1}; p_{DuX}^{−1}; d_{UDUDuX}^{−1}; DUθ_{3}X; Dd_{UDnX}; DUDu_{x}^{−1}; μ_{𝕍}X; Dd_{UpX}^{−1}; Dθ_{4}X^{−1}; η_{𝔻}UDUX; Dd_{UDUx}^{−1}; Dμ_{𝕍}X; Dp_{x}^{−1}; θ_{5}X; DDχ; m_{x}]

=

[figure: Uθ_{3}DUX; d_{UpDUX}^{−1}; UDβ_{𝕌}X; UDu_{x}^{−1}; d_{UDnX}; d_{UDUx}^{−1}; μ_{𝕍}DUX; d_{UpX}^{−1}; θ_{4}X^{−1}; p_{x}^{−1}; η_{𝔻}UDUDUX; μ_{𝕍}X; θ_{5}DUX; Dχ; θ_{5}X; m_{x}]

[figure: Up_{DuX}^{−1}; Uμ_{𝕍}X; UDβ_{𝕌}DX; Uη_{𝔻}UDUX; UDu_{UDuX}^{−1}; d_{UDnDX}; d_{UDUUDuX}^{−1}; UDu_{dUDUX}^{−1}; Uθ_{3}DUX; d_{UpDUX}^{−1}; d_{UpX}^{−1}; d_{UpDX}^{−1}; θ_{4}DX^{−1}; p_{UDuX}^{−1}; d_{UDUdUDUX}^{−1}; μ_{𝕍}DX; p_{dUDUX}^{−1}; μ_{𝕍}DUX; μ_{𝕍}X; Dn_{DuX}^{−1}; Dd_{UdUDUX}; η_{𝔻}UDUDUX; Dd_{nDUX}; Dθ_{2}DUX; θ_{5}DUX]

=

[figure: d_{UDUDUDuX}^{−1}; d_{UDUpX}^{−1}; d_{UpX}^{−1}; p_{DUDuX}^{−1}; p_{pX}^{−1}; μ_{𝕍}X; p_{DuX}^{−1}; μ_{𝕍}X; η_{𝔻}UDUX]

and

[figure: UUm_{uX}; UUDu_{DuX}^{−1}; Ud_{UDuDUX}; UD_{UmUX}^{−1}; n_{DuDX}^{−1}; Ud_{UDUDuX}^{−1}; UDUθ_{3}X; θ_{2}DUX; d_{UdUDUDX}^{−1}; d_{UDUDUDuX}^{−1}; d_{UDUpX}^{−1}; d_{nDUDX}; p_{DUDuX}^{−1}; p_{pX}^{−1}]

=

[figure: n_{mX}^{−1}; n_{DuX}^{−1}; Um_{uX}; d_{nDUX}; θ_{2}DUX; d_{UdUDUX}^{−1}; UDu_{DuX}^{−1}; d_{UmUX}^{−1}; d_{UDUDuX}^{−1}; d_{UDuDUX}; DUθ_{3}X; .]

The fact that the three equalities above are indeed satisfied and the rest of the proof are left to the reader.

It is easily seen that, defining η̃, β̃, and μ̃ at every X as (η_{X})_{(ψ,χ)} = η_{𝔻}X, (β_{X})_{(ψ,χ)} = β_{𝔻}X, and (μ_{X})_{(ψ,χ)} = μ_{𝔻}X, we obtain 2-cells (η̃, η), (β̃, β), and (μ̃, μ) in [𝕌, 𝕌]. We thus have:

9.3. Theorem. 𝔻̃ = ((D̃, D), (d̃, d), (m̃, m), (β̃, β), (η̃, η), (μ̃, μ)) is a pseudomonad in PSM(A), on the object 𝕌.

10. From a pseudomonad in PSM(A) to a distributive law

Assume that we have a pseudomonad in PSM(A) as in section 8. We produce a distributive law as follows. Consider D_{K} : 𝕌-Alg_{K} → 𝕌-Alg_{K}. Since the diagram

\xymatrix{𝕌-Alg_{K} \ar[r]^{D_{K}} \ar[d]_{Φ_{K}} & 𝕌-Alg_{K} \ar[d]^{Φ_{K}} \\ A(K, K) \ar[r]_{D( )} & A(K, K)}

commutes, we have that D_{K}(β_{𝕌}, μ_{𝕌}) is of the form (σ_{1}, σ_{2}) with

[figure: UDU; s; σ_{1}; DU; uDU; Id; DU,; UUDU; Us; nDU; σ_{2}; UDU; s; UDU; s; DU.]

Furthermore, since the diagram

\xymatrix{𝕌-Alg_{K} \ar[r]^{D_{K}} \ar[d]^{Û} & 𝕌-Alg_{K} \ar[d]^{Û} \\ _{𝕌-Alg_{K}} \ar[r]^{D_{K}} & _{𝕌-Alg_{K}}}

commutes, where Û is the change of base 2-functor defined in section 3, then D_{K}(n, μ_{𝕌}) : D_{K}(β_{𝕌}U, μ_{𝕌}U) → D_{K}(β_{𝕌}, μ_{𝕌}) is of the form (Dn, σ_{3}) : (σ_{1}U, σ_{2}U) → (σ_{1}, σ_{2}) with σ_{3} of the form

[figure: UDUU; UDn; sU; σ_{3}; UDU; s; DUU; Dn; DU.]

Similarly, it can be shown that (d_{K})_{(β_{𝕌},μ_{𝕌})} is of the form (dU, σ_{4}), with

[figure: UU; UdU; n; σ_{4}; UDU; s; U; dU; DU.]

If we apply D_{K} to (σ_{1}, σ_{2}) we obtain another 𝕌-algebra (σ′_{1}, σ′_{2}), with

[figure: UDDU; s′; σ′_{1}; DDU; uDDU; DDU,; UUDDU; Us′; nDDU; σ′_{2}; UDDU; s′; UDDU; s′; DU.]

Applying D_{K} to (s, σ_{2}) : (β_{𝕌}DU, μ_{𝕌}DU) → (σ_{1}, σ_{2}), we obtain a morphism (Ds, τ_{1}) with

[figure: UDUDU; UDs; sDU; τ_{1}; UDDU; s′; DUDU; Ds; DDU.]

We also have that (m_{K})_{(β_{𝕌},μ_{𝕌})} is of the form (mU, γ_{1}), with

[figure: UDDU; UmU; s′; γ_{1}; UDU; s; DDU; mU; DU.]

Define σ_{5} as the pasting It is with the help of σ_{1} to σ_{5} that we define the distributive law as follows: Define r = s ∘ UDu, ω_{1} as the pasting

[figure: UDDU; UmU; UDσ_{1}^{−1}; Id; UDuDU; UDU; s; UDUDU; UDs; sDU; τ_{1}; DUDU; Ds; UDDU; s′; γ_{1}; DDU; mU; DU.]

[figure: UD; UDu; D; uD; Du; u_{Du}; UDU; s; σ_{1}; DU; Id; uDU; DU,]

ω_{2} as the pasting

[figure: UD; UDu; U; Ud; Uu; Ud_{u}; Id; UDU; s; UU; n; η_{𝕌}^{−1}; UdU; σ_{4}; DU,; U; dU]

ω_{3} as the pasting

[figure: UUD; UUDu; nD; UUDU; Us; nDU; UDU; UDuU; Id; UDUU; sU; UDn; UDβ_{𝕌}; DUU; Dn; n_{Du}^{−1}; σ_{2}; UDU; s; σ_{3}^{−1}; UD; UDu; UDU; s; DU,]

and ω_{4} as the pasting

[figure: UD; UDu; UDU; s; DU; Um_{u}; UDDU; UmU; UDuDU; UDUDU; sDU; σ_{5}; UDD; UDuD; UDDu; Um; UDu_{Du}^{−1}; UDUD; sD; UDUDu; s_{Du}^{−1}; DUD; DUDu; DUDU; Ds; DDU.; mU]

The proof that we obtain a distributive law in this way is based on the behavior of the 2-cells σ_{1}-σ_{5}. Aside from the obvious conditions that come from the facts that (σ_{1}, σ_{2}) is a 𝕌-algebra, (Dn, σ_{3}) and (dU, σ_{4}) are homomorphisms of 𝕌-algebras, we have the following conditions.

10.1. Proposition. The following are 2-cells in 𝕌-Alg_{K}:

Dη_{𝕌} : Id → (Dn, σ_{3}) ∘ (DUu, s_{u}^{−1}).  (26)

Dμ_{𝕌} : (Dn, σ_{3}) ∘ (DUn, s_{n}^{−1}) → (Dn, σ_{3}) ∘ (DnU, σ_{3}U).

d_{n} : (Dn, σ_{3}) ∘ (dUU, σ_{4}U) → (dU, σ_{4}) ∘ (n, μ_{𝕌}).  (27)

Furthermore, the following equations are satisfied:

[figure: DDU; uDDU; DuDU; Id; u_{DuDU}; UDDU; UDuDU; UmU; UDU; s; DUDU; uDUDU; σ_{1}DU; Dσ_{1}; UDUDU; sDU; σ_{5}; DUDU; Ds; DDU; mU; DU; =; DDU; uDDU; mU; u_{mU}; UDDU; UmU; DU; uDU; Id; σ_{1}; UDU; s; DU,]

[figure: Uσ_{5}; ⇓; σ_{5}; ⇐; n_{DuDU}^{−1}; ⇐; σ_{2}DU; ⇐; UDβ_{𝕌}DU; ⇐; UDu_{s}^{−1}; ⇐; σ_{3}DU^{−1}; ⇐; s_{s}^{−1}; ⇐; Dσ_{2}; ⇐]

=

[figure: UUDDU; UUmU; nDDU; n_{mU}^{−1}; UUDU; nDU; Us; UDDU; UmU; UDuDU; UDU; s; UDUDU; sDU; DUDU; Ds; σ_{5}; σ_{2}; UDU; s; DDU; mU; DU]  (28)

[figure: UDDUU; UmUU; UDDn; UDUU; UDn; Um_{n}; UDUDUU; sDUU; UDUDn; UDu_{Dn}^{−1}; UDDU; UmU; UDuDU; UDU; s; DUDUU; DsU; DUDn; s_{Dn}^{−1}; UDUDU; sDU; DDUU; DDn; Dσ_{3}; DUDU; Ds; σ_{5}; DDU; mU; DU] = [figure: UDDUU; UmUU; UDuDUU; UDUU; UDn; sU; UDUDUU; sDUU; UDU; s; DUDUU; DsU; σ_{5}U; DDUU; DDn; mUU; DUU; σ_{3}; Dn; m_{n}; DDU; mU; DU.]  (29)

[figure: UDU; UDuU; UDdU; Uη_{𝔻}U; Id; UDUU; sU; UDUdU; UDu_{dU}^{−1}; UDDU; UmU; UDuDU; UDU; s; DUU; Dn; DUdU; s_{dU}^{−1}; UDUDU; sDU; DU; DdU; Dσ_{4}; DUDU; Ds; σ_{5}; DDU; mU; DU] = [figure: UDU; UDβ_{𝕌}^{−1}; Id; UDuU; UDUU; UDn; sU; σ_{3}; UDU; s; DUU; Dn; DU; Id; DdU; η_{𝔻}U; DU,; DDU; mU]  (30)

[figure: UDU; UuDU; UdDU; Ud_{uDU}; UDDU; UDuDU; UmU; UUDU; UdUDU; nDU; σ_{4}DU; UDUDU; sDU; UDU; s; UDU; dUDU; s; d_{s}; DUDU; Ds; σ_{5}; DU; dDU; Id; DDU; mU; β_{𝔻}U; DU] = [figure: UDU; UuDU; Id; UdDU; η_{𝕌}DU; Uβ_{𝔻}U; UUDU; nDU; UDDU; UmU; UDU; s; DU,]

[figure: UDu_{mU}^{−1}; UDDU; UDuDU; UmU; UDU; s; UDDDU; UDmU; UDuDDU; s_{mU}^{−1}; UDUDU; sDU; UDUDDU; UDUmU; sDDU; DUDU; Ds; σ_{5}; DUDDU; DUmU; DUDuDU; DUDUDU; DsDU; Dσ_{5}; DDUDU; DDs; μ_{𝔻}U; DDU; mU; DU; DDDU; DmU; mDU; DDU; mU]

=

[figure: Uμ_{𝔻}U; UDDU; UmU; UDU; s; UDDDU; UDmU; UmDU; UDDuDU; UDuDDU; Um_{uDU}; UDu_{DuDU}^{−1}; UDDU; UmU; UDuDU; UDUDDU; UDUDuDU; sDDU; s_{DuDU}^{−1}; UDDUDU; UmUDU; UDuDUDU; UDUDU; sDU; DUDDU; DUDuDU; UDUDUDU; sDUDU; σ_{5}DU; DUDUDU; DsDU; σ_{5}; DDUDU; mUDU; DDs; m_{s}; DUDU; Ds; DU.; DDDU; mDU; DDU; mU]  (31)

Observation: The equations that appear in the proposition can be written as conditions in 𝕌-Alg_{K}. For instance, it can be shown that the pair of pastings:

φ = [figure: UDDU; u_{DuDU}; UDuDU; UDUDU; sDU; σ_{1}DU; DDU; DuDU; uDDU; Dσ_{1}; Id; DUDU; Id; uDUDU; DUDU; Ds; DDU,]

and

ϑ = [figure: UUDuDU; nDDU; UsDU; nDUDU; UDs; UDuUDU; Id; UDβ_{𝕌}DU; UDu_{s}^{−1}; UDuDU; UDUs; UDnDU; sUDU; sDU; sDU; σ_{3}DU^{−1}; s_{s}^{−1}; n_{DuDU}^{−1}; σ_{2}DU; DUs; DnDU; Dσ_{2}; Ds; UDuDU; sDU; Ds; ,]

define an object in 𝕌-Alg_{K}. Thus, the first two equations say that (mU, σ_{5}) : (φ, ϑ) → (σ_{1}, σ_{2}) is a 𝕌-algebra morphism. Similarly the rest of the equations. We leave the details for the interested reader.

Proof. The fact that Dη_{𝕌} and Dμ_{𝕌} are 2-cells in 𝕌-Alg_{K} follow from applying D_{K} to η_{𝕌} and μ_{𝕌}, respectively. The statement about d_{n} follows from pseudonaturality of d_{K}. As for the equations, the proof of the last equation is fairly typical, and it is the only one we do. We begin by introducing some notation and deducing some equations we will need. The notation is

\begin{aligned}D_{K}( \xymatrix{(β_{𝕌}DDU, μ_{𝕌}DDU) \ar[r]^{(s′,σ′_{2})} & (σ′_{1}, σ′_{2}))} & = & \xymatrix{(σ_{1}DDU, σ_{2}DDU) \ar[r]^{(Ds′,τ_{2})} & (σ′′_{1}, σ′′_{2}),} \\ D_{K} \xymatrix{((σ_{1}DU, σ_{2}DU) \ar[r]^{(Ds,τ_{1})} & (σ′_{1}, σ′_{2}))} & = & \xymatrix{(σ′_{1}DU, σ′_{2}DU) \ar[r]^{(DDs,τ_{3})} & (σ′′_{1}, σ′′_{2}),} \\ D_{K} \xymatrix{((σ′_{1}, σ′_{2}) \ar[r]^{(mU,γ_{1})} & (σ_{1}, σ_{2}))} & = & \xymatrix{(σ′′_{1}, σ′′_{2}) \ar[r]^{(DmU,γ_{2})} & (σ′_{1}, σ′_{2}),} \end{aligned}

and (m_{K})_{(σ_{1},σ_{2})} = (mDU, γ_{3}) : (σ′′_{1}, σ′′_{2}) → (σ′_{1}, σ′_{2}).

Now the equations. Since

[figure: 𝕌-Alg_{K}; D_{K}D_{K}; D_{K}; m_{K}; 𝕌-Alg_{K}; Φ_{K}; A(K, K)] = [figure: 𝕌-Alg_{K}; Φ_{K}; A(K, K); DD( ); D( ); m( ); A(K, K),]

we have that m_{s} = (m_{K})_{(s,σ_{2})} is a 2-cell

[figure: (σ′_{1}DU, σ′_{2}DU); (mUDU,γ_{1}DU); (DDs,τ_{3}); m_{s}; (σ_{1}DU, σ_{2}DU); (Ds,τ_{1}); (σ′′_{1}, σ′′_{2}); (mDU,γ_{3}); (σ′_{1}, σ′_{2})]

in 𝕌-Alg_{K}. From this we obtain the equation

[figure: γ_{1}DU; m_{s}; τ_{1}] = [figure: Um_{s}; τ_{3}; γ_{3}; .]  (32)

Similarly, (μ_{K})_{(β_{𝕌},μ_{𝕌})} is a 2-cell

[figure: (σ′′_{1}, σ′′_{2}); (DmU,γ_{2}); (mDU,γ_{3}); μ_{𝔻}U; (σ′_{1}, σ′_{2}); (mU,γ_{1}); (σ′_{1}, σ′_{2}); (mU,γ_{1}); (σ_{1}, σ_{2}).]

We thus obtain the equation

[figure: Uμ_{𝔻}U; γ_{3}; γ_{1}] = [figure: γ_{2}; μ_{𝔻}U; γ_{1}; ,]  (33)

Observe that

[figure: (β_{𝕌}DUDU, μ_{𝕌}DUDU); (UDs,n_{Ds}^{−1}); (sDU,σ_{2}DU); τ_{1}; (β_{𝕌}DDU, μ_{𝕌}DDU); (s′,σ′_{2}); (σ_{1}DU, σ_{2}DU); (Ds,τ_{1}); (σ′_{1}, σ′_{2})]

is a 2-cell in 𝕌-Alg_{K}. Since D_{K} behaves well with change of base, then D_{K}(UDs, n_{Ds}^{−1}) = (DUDs, s_{Ds}^{−1}). Thus, applying D_{K}, we have that

[figure: (σ_{1}DUDU, σ_{2}DUDU); (DUDs,s_{Ds}^{−1}); (DsDU,τ_{1}); Dτ_{1}; (σ_{1}DDU, σ_{2}DDU); (Ds′,τ_{2}); (σ′_{1}DU, σ′_{2}DU); (DDs,τ_{3}); (σ′′_{1}, σ′′_{2})]

is a 2-cell in 𝕌-Alg_{K}. From this we obtain the equation

[figure: τ_{1}DU; τ_{3}] = [figure: UDτ_{1}^{−1}; s_{Ds}^{−1}; τ_{2}; .; Dτ_{1}]  (34)

We also have that

[figure: (β_{𝕌}DDU, μ_{𝕌}DDU); (UmU,n_{mU}^{−1} ); (s′,σ′_{2}); γ_{1}; (β_{𝕌}DU, μ_{𝕌}DU); (s,σ_{2}); (σ′_{1}, σ′_{2}); (mU,γ_{1}); (σ_{1}, σ_{2})]

is a 2-cell in 𝕌-Alg_{K}. Applying D_{K} to it we obtain that the equation

[figure: UDγ_{1}; τ_{2}; γ_{2}] = [figure: s_{mU}^{−1}; Dγ_{1}; τ_{1}; ,]  (35)

is satisfied.

We are ready to show (31). Start with the second member of the equation, and use (32). Next make the substitution

[figure: Um_{uDU}; UDσ_{1}^{−1}; Um_{s}] = [figure: UDDσ_{1}^{−1}; UmDU; .]

Use now (33) and (34). Make the substitutions

[figure: UDσ_{1}DU^{−1}; UDτ_{1}^{−1}; ,] = [figure: UDσ′_{1}^{−1}; UDu_{Ds}^{−1}; ,]

and

[figure: UDσ′_{1}^{−1}; UDmU] = [figure: UDu_{mU}^{−1}; UDσ_{1}^{−1}; UDγ_{1}; ,]

using that (Ds, τ_{1}) : (σ_{1}DU, σ_{2}DU) → (σ′_{1}, σ′_{2}) and (mU, γ_{1}) : (σ′_{1}, σ′_{2}) → (σ_{1}, σ_{2}) are 1-cells in 𝕌-Alg_{K}. Then the substitution

[figure: UDDσ_{1}^{−1}; UDu_{DuDU}^{−1}; UDu_{Ds}^{−1}; s_{DuDU}^{−1}; s_{Ds}^{−1}] = [figure: UDuDDU; sDDU; DUDσ_{1}^{−1}; .]

Finally, use (35).

We show now that we do have a distributive law. 10.2. Theorem. With the above definitions, r : UD → DU and the invertible 2-cells ω_{1}, ω_{2}, ω_{3}, and ω_{4}, define a distributive law of 𝕌 over 𝔻. Proof. We show (coh 3, 6 and 8), leaving the rest to the reader. Start on the left hand side of (coh 3) and make the following substitutions:

[figure: η_{𝕌}U^{−1}; n] = [figure: Uβ_{𝕌}; μ_{𝕌}; ,]

[figure: σ_{4}U; μ_{𝕌}; d_{n}] = [figure: Ud_{n}; σ_{4}; σ_{3}^{−1}; using (27),]

[figure: Uβ_{𝕌}; Ud_{uU}; Ud_{n}] = [figure: UdU; UDβ_{𝕌}; ,]

[figure: Uσ_{4}; σ_{4}] = [figure: n_{dU}^{−1}; μ_{𝕌}^{−1}; σ_{4}; σ_{2}; ,]

using that (dU, σ_{4}) : (β_{𝕌}, μ_{𝕌}) → (σ_{1}, σ_{2}) is a 1-cell in 𝕌-Alg_{K}; [figure: Uη_{𝕌}^{−1}; μ_{𝕌}^{−1}] = [figure: ,; n_{u}^{−1}; η_{𝕌}^{−1}]

and

[figure: UUd_{u}; n_{u}^{−1}; n_{dU}^{−1}] = [figure: n_{d}^{−1}; Ud_{u}; n_{Du}^{−1}; .]

To prove (coh 6), start on the right hand side, and make the following substitutions: [figure: UUDu_{Du}^{−1}; n_{DuD}^{−1}; Us_{Du}^{−1}; σ_{2}D; UDβ_{𝕌}D; UDu_{UDu}^{−1}; s_{UDu}^{−1}; σ_{3}D^{−1}; Dn_{Du}^{−1}] = [figure: n_{DDu}^{−1}; UDu_{Du}^{−1}; n_{DuDU}^{−1}; s_{Du}^{−1}; σ_{2}DU; DUβ_{𝕌}DU; σ_{3}DU^{−1}; ,]

[figure: σ_{5}U; Dσ_{3}^{−1}; m_{n}] = [figure: Um_{n}; UDu_{Dn}^{−1}; σ_{5}; σ_{3}^{−1}; using (29); s_{Dn}^{−1}; ,]

[figure: UDu_{DuU}^{−1}; s_{DuU}^{−1}; Udu_{Dn}^{−1}; DUDβ_{𝕌}; s_{Dn}^{−1}] = [figure: UDDβ_{𝕌}; UDuDU; sDU; ,]

and

[figure: UDDβ_{𝕌}; Um_{uU}; Um_{n}] = [figure: UmU; UDβ_{𝕌}; .]

Now use (28). Conclude with the substitution

[figure: UUm_{u}; n_{DDu}^{−1}; n_{mU}^{−1}] = [figure: n_{m}^{−1}; Um_{u}; n_{Du}^{−1}; .]

To prove (coh 8), start on the left hand side, and make the substitutions:

[figure: UDu_{d}^{−1}; s_{d}^{−1}; UDu_{Du}^{−1}; DUd_{u}; s_{Du}^{−1}] = [figure: UDd_{u}; UDu_{u}^{−1}; UDu_{dU}^{−1}; s_{u}^{−1}; s_{dU}^{−1}; ,]

and

[figure: Uη_{𝔻}; UDd_{u}; Um_{u}] = [figure: UDu; Uη_{𝔻}U; .]

Now, use (30), then the equation obtained from (26). It is easily seen that what remains equals the identity.

11. (Co-)KZ-doctrines over KZ-doctrines

Distributive laws for KZ-doctrines over KZ-doctrines are formally very similar to distributive laws of co-KZ-doctrines over KZ-doctrines. We start this section by recalling the definition and some aspects of KZ-doctrines [9] (see [8] as well). Then we define distributive laws of co-KZ-doctrines over KZ-doctrines, and we show that such a distributive law induces a distributive law between the induced pseudomonads, as defined in section 4. Finally, we show that the composite pseudomonad given by a distributive law of a KZ-doctrine over a KZ-doctrine is again a KZ-doctrine.

11.1. We are still working in a Gray-category A, and K is an object of A. 11.2. 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]  (36)

The 2-cell δ : Dd → dD is defined as the pasting

[figure: D; D; DD; DD,; β^{−1}; ⇓; ϵ; ⇓; Id_{D}; Id_{DD}; dD; m; Dd]

and it satisfies the equations δ ∘ d = d_{d} and m ∘ δ = β^{−1} · η^{−1}.

In a co-KZ-doctrine U, with u : 1 → U and n : UU → U, the fully-faithful adjoint string is of the form uU ⊣ n ⊣ Uu.

Proposition 10.1 of [9] says that D induces a pseudomonad 𝔻 = (D, d, m, β, η, μ) if μ is defined as the pasting

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

11.3. Assume we have a KZ-doctrine

D = (D, d, m, α_{D}, β_{D}, η_{D}, ϵ_{D}),

and a co-KZ-doctrine U = (U, u, n, α_{U}, β_{U}, η_{U}, ϵ_{U}), with

[figure: DD; Id; m; α_{D}; DD,; D; dD; DD; m; D; dD; Id; β_{D}; D,; D; Id; Dd; η_{D}; D,; DD; m; D; Dd; DD; m; Id; ϵ_{D}; DD,]

and

[figure: U; Id; uU; α_{U}; U,; UU; n; U; uU; UU; n; Id; β_{U}; UU,; UU; Id; n; η_{U}; UU,; U; Uu; UU; n; U; Uu; Id; ϵ_{U}; U.]

Since we are assuming D to be a KZ-doctrine, and U to be a co-KZ-doctrine, then β_{D} and η_{D}, and α_{U}, and ϵ_{U} are invertible. Let δ : Dd → dD and υ : uU → Uu be the corresponding induced 2-cells.

Theory and Applications of Categories, Vol. 5, No. 5 142

11.4. Definition. A distributive law of U over D consists of a 1-cell r : UD → DU in A(K, K), together with an invertible 2-cell

[figure: UD; r; ω_{1}; D; uD; Du; DU,]

such that the pasting

[figure: u_{dU}^{−1}; DU; uDU; DuU; U; uU; Uu; υ; dU; Ud; Ud_{u}^{−1}; UU; UdU; ω_{1}U^{−1}; r_{u}^{−1}; UDU; rU; Dα_{U}; Dϵ_{U}; DUU; Dn; DU; UD; r; UDu; DU; DUu]

is invertible. We denote the inverse by ω_{2}. We further require the pasting ω_{3} = [figure: UUD; nD; Id; η_{U}D; Uω_{1}; UUD; Ur; r_{u}^{−1}; UDU; rU; DUU; Dn; UD; r; UuD; UDu; DU; DUu; Id; Dϵ_{U}; DU]

to be invertible, and the pasting ω_{4} = [figure: UD; UDd; r; DU; DUd; DdU; Id; η_{D}U; DU; UDD; Um; Id; Uϵ_{D}; r_{d}; UDD; rD; Dω_{2}^{−1}; DUD; Dr; DDU; mU]

to be invertible. We also require the following two coherence conditions: [figure: UD; uUD; UuD; υD; r; u_{r}^{−1}; DU; uDU; DuU; ω_{1}U^{−1}; UUD; Ur; UDU; rU; DUU; Dn; DU] = [figure: UD; UuD; UDu; Uω_{1}^{−1}; r; r_{u}; DU; DuU; DuU; Dυ; UUD; Ur; UDU; rU; DUU; Dn; DU,]

and [figure: UD; UDd; UdD; Uδ; r; r_{d}; DU; DUd; DdU; Dω_{2}^{−1}; UDD; rD; DUD; Dr; DDU; mU; DU] = [figure: UD; UdD; dUD; ω_{2}D^{−1}; r; d_{r}^{−1}; DU; DdU; dDU; δU; UDD; rD; DUD; Dr; DDU; mU; DU.]

Using the equations υ ∘ n = η_{U} · β_{U} and δ ∘ m = α_{D} · ϵ_{D}, and the coherence conditions, it is not hard to see that the inverse of ω_{3} is

[figure: UD; uUD; r; DU; uDU; DuU; Id; Dα_{U}; DU,; UUD; nD; Id; β_{U}D; u_{r}^{−1}; UUD; Ur; ω_{1}U^{−1}; UDU; rU; DUU; Dn]

and the inverse of ω_{4} is

[figure: UDD; Um; Id; Uα_{D}; ω_{2}D; UDD; rD; d_{r}; DUD; Dr; DDU; mU; UD; r; UdD; dUD; DU; dDU; Id; β_{D}U; DU.]

Looking at the definition it seems possible to define a distributive law in terms of ω_{2}, defining ω_{1} with a similar pasting as ω_{2} was defined above. It can be shown that if we start with ω_{1}, define ω_{2} as above, and then use a similar pasting to define a new ω′_{1} we have that ω_{1} = ω′_{1}. And similarly if we start with ω_{2}.

11.5. Theorem. If 𝔻 and 𝕌 are the pseudomonads determined by the KZ-doctrine D and the co-KZ-doctrine U respectively, then r together with ω_{1}, ω_{2}, ω_{3} and ω_{4} define a distributive law of 𝕌 over 𝔻.

Proof. Since U is a co-KZ-doctrine, we have that β_{𝕌} = α_{U}^{−1} and η_{𝕌} = ϵ_{U}^{−1}. With this in mind, we must show that the coherence conditions (coh 1) to (coh 9) of section 4 are satisfied. We show that (coh 1), (coh 3) and (coh 4) are satisfied and leave the others to the reader.

Consider the inverse of the left hand side of (coh 1). To get the inverse of the right hand side make the following four substitutions

υ ∘ u = u_{u}^{−1},
[figure: d_{u}; u_{u}^{−1}; u_{dU}^{−1}; Ud_{u}^{−1}] = [figure: u_{d}^{−1}; u_{Du}^{−1}; ,]

[figure: ω_{1}U^{−1}; u_{Du}^{−1}; r_{u}^{−1}] = [figure: ω_{1}^{−1}; ,; Du_{u}^{−1}]

and

[figure: Dα_{U}; Du_{u}^{−1}; Dϵ_{U}] = id_{Du}

To prove (coh 3) start with the inverse of the right hand side and make the following substitutions:

υ ∘ n = η_{U} · β_{U},
β_{U} = [figure: u_{n}^{−1}; υU; Uα_{U}^{−1}; ,]

[figure: u_{n}^{−1}; u_{dU}^{−1}] = [figure: d_{n}^{−1}; u_{dUU}^{−1}; u_{Dn}^{−1}; ,; Ud_{n}]

[figure: u_{Dn}^{−1}; ω_{1}U^{−1}] = [figure: Du_{n}^{−1}; ω_{1}UU^{−1}; ,; r_{n}]

[figure: Uα_{U}^{−1}; Ud_{n}; r_{n}] = [figure: ,; Ud_{uU}^{−1}; r_{uU}^{−1}; DUα_{U}^{−1}]

[figure: Du_{n}^{−1}; Dα_{U}] = [figure: Dα_{U}U; ,; Dμ_{𝕌}^{−1}]

and

[figure: DUα_{U}^{−1}; Dμ_{𝕌}^{−1}] = [figure: Dϵ_{U}U; Dn; .]

Comparing the pasting obtained with the left hand side of (coh 3), we see that we must show that

[figure: η_{U}; Ud_{u}^{−1}; r_{u}^{−1}; Dϵ_{U}; n_{d}; ω_{3}^{−1}] = Dn ∘ rU ∘ Uω_{2}^{−1}.

Now, pass every 2-cell of the left hand side to the right hand side, except for η_{U}. It is not hard to see, with the help of (coh 1), that the resulting equation is satisfied.

Start with the inverse of the left hand side of (coh 4), and make the following sequence of substitutions:

[figure: μ_{𝕌}D^{−1}; β_{U}D] = [figure: β_{U}UD; u_{n}^{−1}; ,]

[figure: u_{n}^{−1}; Uβ_{U}D] = [figure: β_{U}D; u_{uUD}; ,]

[figure: u_{r}^{−1}; u_{uUD}; Uu_{r}^{−1}] = [figure: u_{r}^{−1}; u_{Ur}^{−1}; u_{uDU}; ,]

[figure: β_{U}UD; u_{Ur}^{−1}] = [figure: ,; n_{r}; β_{U}DU]

[figure: u_{uDU}; Uω_{1}U^{−1}] = [figure: u_{DuU}; ω_{1}U^{−1}; ,; u_{rU}^{−1}]

[figure: u_{DuU}; UDα_{U}] = [figure: Dα_{U}; ,; u_{Dn}^{−1}]

[figure: u_{Dn}^{−1}; ω_{1}U^{−1}] = [figure: Du_{n}^{−1}; ω_{1}UU^{−1}; ,; r_{n}]

and

[figure: Du_{n}^{−1}; Dα_{U}] = [figure: Dα_{U}U; ,; Dμ_{𝕌}^{−1}]

11.6. Assume now that D is a KZ-doctrine as before, but that U is also a KZ-doctrine. The data for D is unchanged, and the data for U is now:

[figure: UU; Id; n; α_{U}; UU,; U; uU; UU; n; U; uU; Id; β_{U}; U,; U; Id; Uu; η_{U}; U,; UU; n; U; Uu; UU; n; Id; ϵ_{U}; UU.]

We have υ : Uu → uU.

Copying almost exactly the definition of a distributive law of a co-KZ-doctrine over a KZ-doctrine, we obtain the concept of a distributive law of the KZ-doctrine U over the KZ-doctrine D. If 𝕌 and 𝔻 are the pseudomonads induced by U and D respectively, we obtain, in a very similar way, a distributive law between the pseudomonads 𝕌 and 𝔻. Let 𝕍 be the composite pseudomonad obtained from this last distributive law, as in section

Theory and Applications of Categories, Vol. 5, No. 5 146

11.7. Theorem. The composite pseudomonad obtained from a distributive law between KZ-doctrines is again a KZ-doctrine. Proof. We follow the notation introduced before the statement of the theorem, and the notation for 𝕍 is the same as in section 5. According to theorem 11.1 of [9], it suffices to show that p ⊣ vV with counit β_{𝕍}. Define ϑ : V v → vV as the pasting

[figure: DUU; DUdU; Du_{dU}; DU; uDU; DUu; DuU; Dυ; DdU; dDU; δU; d_{uDU}; DDU; DuDU; DUDU.; UDU; dUDU]

It is not hard to see that ϑ ∘ v = p_{p} and that p ∘ ϑ = β_{𝕍}^{−1}· η_{𝕍}^{−1}. Using these two equations, an easy calculation shows that p ⊣ vV with unit

\xymatrix{V V \ar[r]^{V vV} \ar[r]_{vV V}|{ϑV ⇓} \ar[rr]^{Id}^{V β_{𝕍}^{−1}⇓} \ar[dr]_{p} & V V V \ar[r]_{V p} & V V \\  & V \ar[ur]_{vV}^{v_{p} ⇓}}

and counit β_{𝕍}.

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[4] 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.

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

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

[7] G. M. Kelly. Coherence theorems for lax algebras and for distributive laws. Lecture Notes in Mathematics 420, Springer-Verlag, 1974.

[8] A. Kock. Monads for which structures are adjoint to units. Journal of Pure and Applied Algebra 104(1), 1995, pp. 41-59.

[9] F. Marmolejo. Doctrines whose structure forms a fully faithful adjoint string. Theory and applications of categories, Vol. 3, No. 2, pp. 24-44. 1997.

http://tac.mta.ca/tac/volumes/1997/n3/n3.dvi

[10] R. Rosebrugh and R. J. Wood. Distributive adjoint strings. Theory and applications of categories,

Vol. 1, No. 6, 1995, pp. 119-145. http://tac.mta.ca/tac/volumes/1995/n6/v1n6.dvi

[11] Ross Street. The formal theory of monads. Journal of Pure and Applied Algebra 2, 1972, pp. 149-168.

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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 dell Insubria: carboni@fis.unico.it P. T. Johnstone, University of Cambridge: ptj@pmms.cam.ac.uk G. Max Kelly, University of Sydney: maxk@maths.usyd.edu.au Anders Kock, University of Aarhus: kock@imf.au.dk F. William Lawvere, State University of New York at Buffalo: wlawvere@acsu.buffalo.edu Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.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@math.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Sydney: walters b@maths.usyd.edu.au R. J. Wood, Dalhousie University: rjwood@mscs.dal.ca