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On the coherence conditions for pseudo-distributive laws

We survey the development of the formal theory of pseudo-monads, the analogue for pseudo-monads of the formal theory of monads. One of the main achievements of the theory is a satisfactory axiomatisation of the notion of a pseudo-distributive law between pseudo-monads.

A byproduct of the formal theory of pseudo-monads that is of particular interest for applications is the definition of a satisfactory notion of pseudodistributive law between pseudo-monads. The notion of a pseudo-distributive law between pseudo-monads should be understood as an analogue of the classical notion of a distributive law between monads introduced by Beck [2]. For this notion, the four diagrams that are required to commute in the definition of a distributive law are replaced by diagrams commuting up to invertible 3-cells, which are then required to satisfy appropriate coherence conditions. Extending the set of coherence conditions for semistrict pseudo-distributive laws between 2-monads introduced by Kelly [12], Marmolejo identified a set of nine coherence conditions for pseudo-distributive laws [20]. Later, Marmolejo and Wood [21] showed not only that an additional tenth coherence condition, introduced by Tanaka [23], can be derived from Marmolejo's conditions, but also that one of the nine conditions originally introduced by Marmolejo is derivable from the others, thereby reducing the axiomatization of the coherence conditions for a pseudo-distributive law to eight axioms.
The first main contribution of the survey is to state precisely all the coherence conditions for pseudo-monads, pseudo-monad morphisms, pseudo-monad transformations, and pseudo-monad modification, and to prove that they give rise to a tricategory that is equivalent to the Gray-category Psm K . The second main contribution is to give clearly all the coherence conditions for pseudodistributive laws: the eight core conditions that are part of the definition, the ninth, derivable, condition originally introduced in [20] and the tenth, derivable, condition stated in [23]. We also provide an interpretation of these conditions in terms of the notions of pseudo-monad morphisms, pseudo-monad transformation, and pseudo-monad modification. As direct consequence of this interpretation, we obtain an analogue of Beck's fundamental theorem relating distributive laws and liftings to categories of Eilenberg-Moore algebras [2].
We conclude these introductory remarks by recalling from [4] that the development of the formal theory of pseudo-monads in the Gray-categorical setting does not immediately account for the Kleisli construction. Indeed, the Kleisli construction for a pseudo-monad in the Gray-category 2-Cat ⊗ of 2-categories, 2-functors, pseudo-natural transformations, and modifications, produces a bicategory, not a 2-category, thus leading outside 2-Cat ⊗ . It seems therefore that a tricategorical version of the formal theory of monads, based on the existing Gray-categorical theory, will eventually be needed.

Gray-categories
We begin by reviewing the notion of a Gray-category. Let us start with some preliminaries. We write 2-Cat for the category of 2-categories and 2-functors. For 2-categories X and Y , let [X, Y ] be the 2-category of 2-functors from X to Y , pseudo-natural transformations, and modifications [14]. This definition equips the category 2-Cat with the structure of a closed category [6]. The closed structure of 2-Cat is part of symmetric monoidal structure, the tensor product of which is known as the Gray tensor product [10, Section 4.8]. We will write X ⊗ Y for the Gray tensor product of 2-categories X and Y . We write 2-Cat ⊗ to emphasise that we consider 2-Cat as being equipped with the closed symmetric monoidal structure given by the Gray tensor product.
By definition, a Gray-category is a 2-Cat ⊗ -enriched category [10, Section 5.1]. For a Gray-category K, we write K also for its set of objects. Given X, Y ∈ K, we write K(X, Y ) for the hom-2-category of maps from X to Y . We refer to the objects of K also as 0-cells, and to the n-cells of the hom-2-categories of K as the n + 1-cells of K. Following this idea, every Gray-category K can be viewed as a tricategory [10,Proposition 3.1]. Gray-categories are rather special tricategories, in that their only non-strict operation is horizontal composition of 2-cells [10,Section 5.2]. For an example of a Gray-category, recall that 2-Cat ⊗ is a monoidal closed category and so it is enriched over itself. Therefore, it can be viewed as a Gray-category, as we will do from now on. More explicitly, 2-Cat ⊗ is the Gray-category having 2-categories as 0-cells, 2-functors as 1-cells, pseudo-natural transformations as 2-cells, and modifications as 3-cells.
Let us briefly describe what the non-strictness of the horizontal composition of 2-cells in a Gray-category amounts to. Let K be a Gray-category. Let us consider 0-cells I, X, Y , 1-cells A, B : I → X and H, K : X → Y . The nonstrictness of K means that for every pair of 2-cells f : A → B and p : H → K, we have invertible 3-cells

HA HB
KA KB If, as usual in category theory [18, Section V.5], we think of 1-cells A, B : I → X as generalised elements of X, then these 3-cells are analogous to the 2-cells that are part of the structure of a pseudo-natural transformation, and indeed they satisfy very similar coherence conditions [20, Section 2]. In the following, these coherence conditions will often be used implicitly. The notions of a Gray-functor and a Gray-natural transformation are instances of the general notions of enriched functor and enriched natural transformation [13, Section 1.2]. We will use the terminology of Gray-modification and Gray-perturbation to denote the strict counterparts of the corresponding tricategorical notions [10,Section 3.3]. When working with the Yoneda embedding for Gray-categories, which is just an instance of the Yoneda embedding for enriched categories [13, Section 2.4], we often identify an object of K with the representable Gray-functor associated to it. Analogous conventions will be used also for the n-cells of K, where n = 1, 2, 3.
For further information on Gray-categories and tricategories, we invite the reader to refer to [9,10,11,16].

The Gray-category of pseudo-monads
From now on, unless otherwise specified, we work with a fixed Gray-category K.
Definition 3.1. A pseudo-monad (X, S) in K consists of a 0-cell X, a 1-cell S : X → X, 2-cells u : 1 X → S, m : S 2 → S, and invertible 3-cells satisfying the coherence axioms in (1) and (2) below: (1) Note that the notion of a pseudo-monad is self-dual, in the sense that a pseudo-monad in K is the same thing as a pseudo-monad in K op , where K op is the Gray-category obtained from K by reversing the direction of the 1-cells, but not that of the 2-cells and 3-cells.   Let (X, S) be a pseudo-monad in K. Let us recall the definition of the 2-category Ps-S-Alg(I) of I-indexed pseudo-S-algebras, pseudo-algebra morphisms, and pseudo-algebra 2-cells, where I ∈ K. An I-indexed pseudo-Salgebra consists of a 1-cell A : I → X, called the underlying 1-cell of the pseudo-algebra, a 2-cell a : SA → A, called the structure map of the pseudoalgebra, and invertible 3-cells called the associativity and unit of the pseudo-algebra, satisfying the coherence axioms (6) and (7) stated below. ? ? ? ?
. Let A be a pseudo-algebra for a pseudo-monad (X, S). The coherence condition As usual, we refer to a pseudo-algebra by the name of its underlying 1-cell, leaving the rest of its data implicit. Given pseudo-algebras A and B, a pseudoalgebra morphism f : A → B consists of a 2-cell f : A → B and an invertible 3-cell satisfying the coherence conditions (9) and (10) stated below. ? ? ? ? ? ? : : ? ? ? ? ?
Given pseudo-algebra morphisms f : A → B and g : A → B, a pseudo-algebra 2-cell consists of a 3-cell α : f → g satisfying the coherence condition (11).
There is a forgetful 2-functor U I : Ps-S-Alg(I) → K(I, X), defined by mapping a pseudo-S-algebra to its underlying 1-cell, which has a left pseudo-adjoint, defined by mapping a 1-cell A : I → X to the free pseudo-algebra on it, given by the composite 1-cell SA : I → X. The function mapping an object I ∈ K to the 2-category Ps-S-Alg(I) extends to a Gray-functor Ps-S-Alg : K op → 2-Cat ⊗ . We also have a Graytransformation U : Ps-S-Alg → X, with components given by the forgetful 2-functors U I : Ps-S-Alg(I) → K(I, X), for I ∈ K. Note that here we are using the notational conventions regarding representable Gray-functors introduced in Section 2. These conventions will be exploited repeatedly below.
We now recall from [20, Section 7] and [15, Section 6] the definition of the Gray-category Psm K of pseudo-monads in K. The 0-cells are pseudo-monads (X, S) in K. As before, the vertical arrows are the identities that are part of the assumption thatp andq are liftings of p and q, respectively. Composition and identities of Psm K are defined in the evident way, using those of K and 2-Cat ⊗ .
A Gray-category K is said to admit the construction of pseudo-algebras if for every pseudo-monad (X, S) in K, the Gray-functor Ps-S-Alg : K op → 2-Cat ⊗ is representable. When this is the case, the Yoneda lemma for Gray-categories implies that the notions of liftings given above can be given an evident alternative equivalent description, expressed purely in terms of the structure of K. Let us also recall that 2-Cat ⊗ admits the construction of pseudo-algebras: the representing object for the Gray-functor Ps-S-Alg associated to a pseudomonad (X, S) in 2-Cat ⊗ is the 2-category of pseudo-algebras, pseudo-algebra morphism, and algebra 2-cells [4,19].

Coherence axioms for the Gray-category of pseudo-monads
We provide an alternative description of the Gray-category Psm K , closer in spirit to the definition of the 2-categories of monads in a 2-category given in [22] and formulated without reference to the notion of pseudo-algebra. The material in this section is essentially an account of [23,Chapter 5] and [21, Section 3], except for some changes in terminology, that we explain later. Theorem 4.5, however, does not seem to appear in the form given here in the existing literature, even if it is closely related to [23,Theorem 5.23] and [21,Theorem 3.5]. Corollary 4.6, which is inspired by [15,Section 6], seems also to be new. These data are required to satisfy the coherence axioms in (12) and (13). By a pseudo-monad op-morphism we mean a pseudo-monad morphism in K op . Let (X, S) and (Y, T ) be pseudo-monads in K op . For a 1-cell H : X → Y , the data of a 2-cell h : T H → HS and invertible 3-cells as in Definition 4.1 is referred to as a transition from (X, S) to (Y, T ) along H in [21]. In [23], a pseudo-monad morphism of the form (H, h) : (X, S) → (X, S), for a pseudomonad (X, S), is referred to as a pseudo-distributive law of S over H.  Let (H, h) : (X, S) → (Y, T ) be a pseudo-monad morphism. We show that we can define a liftingĤ : Ps-S-Alg → Ps-T -Alg of H : X → Y . Let us consider a fixed I ∈ K. First, let us observe that if A is an I-indexed pseudo-S-algebra, then HA is naturally an I-indexed pseudo-T -algebra, with structure map given by the composite The coherence condition (6) for HA follows by an application of the coherence condition (12) for H and the coherence condition (6) for A. The coherence condition (7) for HA follows by applying the coherence condition (13) for H and the coherence condition (7) (14) and (15) To prove the condition (9) for the pseudo-algebra morphism p A , we apply the axioms for a Gray-category and then condition (14) for the pseudo-monad transformation p. To establish condition (10), it is sufficient to apply the coherence condition (15) for the pseudo-monad transformation p, and then the axioms for a Gray-category. Clearly,p is a lifting of p as required.
Finally, we describe what property a 3-cell has to satisfy in order to admit a lifting. Given a pseudo-monad modification α : (p,p) → (q,q) we can define a liftinĝ α :p →q of α as the Gray-perturbation with components given by the 3cells α A : p A → q A , for a pseudo-S-algebra A. It suffices to check that, these 3-cells are a pseudo-T -algebra 2-cells. To prove this, apply the axioms for a Gray-category and the coherence axiom (16).
Given two pseudo-monads (X, S) and (Y, T ) in K, we defineK (X, S), (Y, T ) to be the 2-category having pseudo-monad morphisms from (X, S) to (Y, T ) as 0-cells, pseudo-monad transformations as 1-cells, and pseudo-monad modifications as 2-cells. The development in this section shows that we have a 2-functor Theorem 4.5 below can be read as saying that the coherence axioms for pseudomonad morphisms, pseudo-monad transformations, and pseudo-monad modifications are not only sufficient, but also necessary in order to obtain liftings. Proof. Let us begin by considering a liftingĤ : Ps-S-Alg → Ps-T -Alg of a 1-cell H : X → Y . By the definition of a lifting, the following diagram of 2-categories and 2-functors commutes: Let us now observe that S : X → X can be regarded as an X-indexed pseudo-S-algebra, with structure map given by the 2-cell m : S 2 → S. By the commutativity of the diagram above, this pseudo-S-algebra is mapped by the 2-functorĤ X into a pseudo-T -algebra with underlying 1-cell HS : X → Y , with structure map a 2-cell of the form h ′ : T HS → HS, and invertible 3-cells fitting in the diagrams The in which the bottom 3-cell is part of the structure making pS : HS → KS into a pseudo-algebra morphism. Finally, if (α,α) : (p,p) → (q,q) is a lifting of a 3-cell α : p → q, then α : p → q is a pseudo-monad modification. Lengthy calculations show that these definitions determine a 2-functor which provides the required quasi-inverse to F . We omit the construction of the required invertible pseudo-natural transformations η : 1 → GF and ε : F G → 1, since this is not difficult.
Corollary 4.6. For every Gray-category K, there exist a tricategoryK having pseudo-monads in K as 0-cells, pseudo-monad morphisms as 1-cells, pseudomonad transformations as 2-cells, and pseudo-monad modifications as 3-cells, and a triequivalence F :K → Psm K .
Proof. Theorem 4.5 allows us to apply the lemma on transport of structure in [10, Section 3.6].
It would be of interest to define a tricategoryK as in Corollary 4.6 without reference to the Gray-category Psm K and to verify whether this is indeed only a tricategory, and not a Gray-category, as anticipated in [15,Section 6].
Remark. For a pseudo-monad (X, S) in K, there is a Gray-natural family of isomorphisms of 2-categories Ps-S-Alg(I) ∼ =K (I, 1 I ), (X, S) for I ∈ K, where (I, 1 I ) denotes the identity pseudo-monad on I. Hence, K admits the construction of pseudo-algebras if and only if for every pseudo-monad (X, S) in K there exists an object X S ∈ K and a pseudo-monad morphism ε S : X S , 1 X S → (X, S) such that, for every I ∈ K, the pseudo-functor K I, X S →K (I, 1 I ), (X, S) defined by composition with ε S is an isomorphism. More explicitly, to give a pseudo-monad morphism ε S as above is to give a morphism U : X S → X, a transformation u : SU → U , and invertible modifications satisfying the coherence conditions for a pseudo-monad morphism.
5. Pseudo-distributive laws  [21,Proposition 4.2], they imply (C10). Let us also note that the axioms (C3) and (C4) express that (n,n) : (T, d) 2 → (T, d) is a pseudo-monad transformation; the axioms (C5) and (C6) express that (v,v) : (X, 1 X ) → (T, d) is a pseudo-monad transformation; and finally the axioms (C7), (C8), and (C10), express that α, ρ, and λ, and are pseudo-monad modifications, respectively. It is then clear that giving a pseudo-distributive law of T over S is equivalent to giving a lifting of T to Ps-S-Alg, by which we mean a lifting of all the data that is part of the pseudo-monad T , thus obtaining an analogue of Beck's fundamental result on the equivalence between giving a distributive law of a monad T over a monad S and a lifting of the monad T to the category of Eilenberg-Moore algebras for S [2].
We limit ourselves to drawing the boundaries of these diagrams and explain in text which 3-cells should be inserted in them, except from the 3-cells coming from the structure of a Gray-category of K. In (C1), the left-hand side pasting is obtained using Sm,m, and the associativity of the pseudo-monad S; the right-hand side pasting is obtained using the associativity of the pseudo-monad S andm.  (C5) For (C5), the left-hand side pasting is obtained using Sn,nS andmT ; the right-hand side pasting is obtained usingm andmut.
To state the coherence conditions (D8), (D9) and (D10), let α, λ, and ρ be the associativity, left unit, and right unit for the pseudo-monad T . For (C7), the left-hand side pasting is obtained using Sα,n,nT ; the right-hand side pasting is obtained usingn and αS. For (C8), the left-hand side pasting is obtained using Sρ andn,vS; the righthand side pasting is ρS. For (C9), the left-hand side pasting is obtained using the right unit of the pseudo-monad S,ū; the right-hand side pasting is obtained using the right unit of the pseudo-monad S.