Theory and Applications of Categories, Vol. 32, No. 31, 2017, pp. 1050–1097.

CLASSIFIERS FOR MONAD MORPHISMS AND ADJUNCTION MORPHISMS

DIMITRI ZAGANIDIS

Abstract. We provide an explicit model for the free 2-category containing n composable adjunction morphisms, comparable to the Schanuel and Street model for the free adjunction. We can extract from it an explicit model for the free 2-category containing n composable lax monad morphisms. A careful proof is given, which goes through presentations of the hom-categories of our model. We use one of these hom-categories as an indexing category to construct an extended Artin-Mazur codiagonal, whose underlying bisimplicial set has the classical Artin-Mazur codiagonal as its first column.

1. Introduction

Riehl and Verity initiated a new approach to (∞, 1)-category theory, where the definitions and proofs are 2-categorical in nature. Following these authors, a good context for (∞, 1)- category theory is an ∞-cosmos, which is essentially a category enriched over quasicategories with some additional properties (see [17]).

Most of the usual models for (∞, 1)-categories do form an ∞-cosmos, such as the simplicially enriched categories of quasi-categories, Segal categories, complete Segal spaces, marked simplicial sets and iterated complete Segal spaces, as Riehl and Verity prove in their article [17]. The 2-category of small categories is also an ∞-cosmos, and many models for (∞, n)-categories are as well. In a series of articles, [14, 16, 15, 17, 18], Riehl and Verity showed that much of category theory can be done in a general ∞-cosmos, and in particular in its 2-category.

One of the key ideas is to encode homotopy coherent diagrams in a simplicial category K as simplicial functors 𝒞 → K, where 𝒞 is a well chosen simplicial category. This idea goes back at least to Cordier and Porter [5] and originated in earlier work of Vogt [22] on homotopy coherent diagrams. For instance, the homotopy coherent nerve is constructed in this way. In Riehl and Verity’s paper [16], 𝒞 is the universal 2-category containing the object of study, either a monad or an adjunction, and K is the underlying ∞-cosmos. Constructions of such universal 2-categories can generally be given by a presentation by a computad [21]. Nevertheless, some of Riehl and Verity’s arguments are combinatorial in nature and thus rely heavily on the use of a very concrete description of the universal monad and the universal adjunction. The universal adjunction was described explicitly partially by Auderset in [2] and by Schanuel and Street in [19].

Received by the editors 2016-04-19 and, in final form, 2017-08-14.

Transmitted by Stephen Lack. Published on 2017-08-23.

2010 Mathematics Subject Classification: 18C15, 18D05,18G30, 55U10, 55U35.

Key words and phrases: monad, adjunction, 2-category, Artin-Mazur codiagonal.

\begin{matrix}c \\ ◯ \end{matrix} Dimitri Zaganidis, 2016. Permission to copy for private use granted.

CLASSIFIERS FOR MONAD MORPHISMS AND ADJUNCTION MORPHISMS 1051

Lax monad morphisms are the general notion of morphisms between monads in a 2category. They were introduced by Street in his paper [20]. However, the readily available notion of morphisms between the homotopy coherent monads of Riehl and Verity is that of simplicial natural transformations, which correspond to strict monad morphisms in the 2-categorical case.

The main goal of this paper is to provide an explicit description of the universal 2category containing n composable lax monad morphisms, which we call Mnd[n]. This is a prerequisite to studying lax monad morphisms in the context of an ∞-cosmos. Based on the construction of Mnd[n], the author has been able to provide explicit descriptions of the universal 2-category Mnd_{hc}[n] containing a homotopy coherent diagram of shape [n] constituted of monads, lax monad morphisms and monad transformations. This is a part of the author’s thesis and will be explained in a forthcoming paper.

In the first part of this paper, we review the definitions of adjunction and monad and their morphisms in a 2-category. We show that if Adj[n] is the free 2-category containing n composable adjunction morphisms, there is an embedding Mnd[n] → Adj[n].

The second part of the paper consists of the construction of an explicit model for Adj[n] and a detailed proof of its 2-universal property. The proof of the result is carried out in three steps. First, we give a presentation of the hom-categories of Adj[n] (Theorem 3.3.2). In the second step, we show that the generators and relations of the hom-categories are actually 2-categorically generated by the adjunction data and adjunction morphism data (Propositions 3.3.4, 3.3.6, 3.3.7 and 3.3.8). In the last step, we prove that these relations are verified by any sequence of adjunction morphisms.

In the last part, we provide evidence that these hom-categories can be of interest to the homotopy theorist. First, all the hom-categories of Adj[n] are in fact Reedy categories. As an easy application of the 2-universal property of Adj[n], we provide an extended bar construction in the presence of a lax morphism of monads. We also study the extended Artin-Mazur codiagonal and show that the realization of its underlying bisimplicial set is weakly equivalent to that of the Artin-Mazur codiagonal. This provides new homotopical models for the realization of a bisimplicial set, for instance by iterating the extended Artin-Mazur codiagonal.

1.1. Notations. For a category 𝒞, |𝒞 | denotes its class of objects. If A, B ∈ |𝒞 |, 𝒞 (A, B) denotes the set of morphisms from A to B in 𝒞.

For a natural number n, we will write n instead when we consider it as an ordinal. That is, we set 0 = ∅ and n = {0, …, n − 1}.

The category Δ_{+} is the category of finite ordinals and order preserving maps. Among those maps, it is well known that the cofaces d^{i} : n − 1 → n and codegeneracies s^{j} : n + 1 → n for 0 ≤ i, j < n generate this category. The coface d^{i} is the only orderpreserving injective map not containing i in its image, while s^{j} : n + 1 → n is the only order-preserving surjective map such that j is the image of two elements of n + 1. More precisely, for k ∈ n − 1 and l ∈ n + 1,

d^{i}(k) = \{ \begin{matrix}k & : & k < i \\ k + 1 & : & k ≥ i \end{matrix} , s^{j}(l) = \{ \begin{matrix}l & : & l ≤ i \\ l − 1 & : & l > i \end{matrix} .

Remark that the more common notation [n] = {0, …, n} for the objects of Δ is such that n = [n−1]. This change of notation is more convenient when considering the ordinal sum, because [n]+[m] = [n+m−1], whereas the ordinal sum n+m is exactly the ordinal with n + m elements and is also denoted by n + m.

1.2. Acknowledgments. I would like to thank Emily Riehl for helpful discussions, my advisor Kathryn Hess for her continuous support, careful reading and suggestions, and an anonymous referee for his valuable input.

2. The relation between monads and adjunctions

In this section, we review the classical correspondence between monads and adjunctions (see Mac Lane [12]), but in the general context of monads in a 2-category that is sufficiently complete. We investigate a parallel correspondence between monad morphisms and adjunction morphisms. The theorems are phrased in term of an enriched right Kan extension (see [9]).

2.1. Monads and their Eilenberg-Moore objects of algebras. In this part, we recall briefly the definition of monad and monad morphisms in a 2-category, together with the Eilenberg-Moore object of algebras.

2.1.1. Definition. A monad over an object B in a 2-category 𝒞, is a quadruple T = (B, T, μ, η), where

• B ∈ |𝒞 |;

• T : B ⟶ B is a 1-cell;

• μ : T^{2} ⟶ T is a 2-cell;

• η : id_{B} ⟶ T is a 2-cell;

satisfying the following properties:

• μ ∘ T μ = μ ∘ μT,

• μ ∘ T η = 1_{T} = μ ∘ ηT.

Street introduced morphisms between monads under the name monad functors in [20] and later used the name monad morphisms with Lack in [11]. Some authors call them lax monad morphisms. In this paper, all monad morphisms will be lax unless otherwise mentioned, thus we will drop the adjective lax, and add the adjective strict instead when necessary.

2.1.2. Definition. Let T = (B, T, μ, η) and S = (C, S, ρ, ι) be two monads in a 2category 𝒞. A monad morphism T ⟶ S is a couple (f, γ), where

• f : B ⟶ C is a 1-cell of 𝒞;

• γ : Sf ⟶ fT is a 2-cell of 𝒞;

such that the following diagrams are commutative.

•

\xymatrix{S^{2}f \ar[d]^{ρf} \ar[r]^{Sγ} & SfT \ar[r]^{γT} & fT^{2} \ar[d]^{fμ} \\ Sf \ar[rr]^{γ} &  & fT}

•

\xymatrix{f \ar[r]^{ιf} \ar[dr]_{fη} & Sf \ar[d]^{γ} \\  & fT}

A monad morphism is said to be strict when its 2-cell is an identity.

Monad morphisms deserve special attention for several reasons. First, they are the only sensible morphisms to use when the base category is fixed. Second, they are the 1-cells of a 2-category Mnd(𝒞) of monads in 𝒞, introduced by Street in [20]. Given two monads T and S on the same object of a 2-category 𝒞, Street also explains that a 2-cell λ : T S → ST is a distributive law if and only if (S, λ) is a lax monad morphism T → T and the multiplication and unit of S are respectively monad transformations (S, λ)^{2} → (S, λ) and 1 → (S, λ) in Mnd(𝒞).

We will now define the Eilenberg-Moore object of algebras associated to a monad. Let T = (B, T, μ, η) be a monad in a 2-category 𝒞. For any X ∈ |𝒞 |, since 𝒞 (X, −) is a 2-functor,

T_{∗}(X) = (𝒞 (X, B), 𝒞 (X, T), 𝒞 (X, μ), 𝒞 (X, η))

is a monad in Cat. Moreover, any 1-cell f : X′ ⟶ X induces a strict monad morphism 𝒞 (f, B) : T_{∗}(X) ⟶ T_{∗}(X′) and thus a functor 𝒞 (X, B)^{T_{∗}(X)} ⟶ 𝒞 (X′, B)^{T_{∗}(X′)} between the categories of algebras. Also, one can check that any 2-cell induces a natural transformation between these functors. This defines a 2-functor

𝒞 (−, B)^{T_{∗}(−)} : 𝒞^{op} ⟶ Cat.  (1)

2.1.3. Definition. Let T = (B, T, μ, η) be a monad in a 2-category 𝒞. The Eilenberg-Moore object of algebras of T , if it exists, is a representing object B^{T} of the 2-functor (1).

The universal 2-category containing a monad, also called the free monad by Schanuel and Street in [19], is a 2-category Mnd such that there is a natural bijection between the monads in a 2-category 𝒞 and the 2-functors Mnd → 𝒞. Remark that by the Yoneda lemma, this property determines Mnd up to isomorphism. Its existence can be derived by constructing it by generators and relations, using presentations by computads as in [21]. A short introduction to computads is provided in Appendix A and coequalizer (3) below will provide an example of such a construction.

As in category theory, an adjunction in a 2-category induces a monad. Thus, there is a 2-functor i : Mnd → Adj, where Adj is the universal 2-category containing an adjunction, also called the free adjunction by Schanuel and Street in [19]. Let us denote by B the domain of the left adjoint in Adj and A its codomain. These two objects are distinct and the only objects of Adj. Auderset in [2, Proposition 2.2] showed that i is actually fully faithful. The 2-category Mnd can thus be described as the full sub 2-category of Adj generated by B. Moreover, in [2, Theorem 4.4], Auderset shows the following.

2.1.4. Proposition. Let T be a monad over B in a 2-category 𝒞 and 𝕋 : Mnd ⟶ 𝒞 the corresponding 2-functor. When 𝒞 is complete, the enriched right Kan extension of the 2-functor 𝕋 : Mnd ⟶ 𝒞 along i : Mnd ⟶ Adj exists and is the free forgetful adjunction in 𝒞:

\xymatrix{B \ar[r]^{f} & B^{T} . \ar[l]^{u}}  (2)

It follows that B^{T} can be expressed as the weighted limit {Adj(A, i(−)), 𝕋}.

Remark that the assumption of 𝒞 being complete is made only to ensure the existence of any enriched right Kan extension with target 𝒞, and is stronger than needed. It can be weakened to the existence of only those limits needed to construct that particular right Kan extension. We will not explore this aspect, but finitely complete is a sufficient hypothesis (see [10, page 92] for instance).

2.1.5. Example. Let 𝒞 = Cat and let \xymatrix{ℬ \ar[r]^{L} & 𝒜 \ar[l]^{R}} be an adjunction with L ⊣ R, unit

η and counit ϵ. This determines a 2-functor 𝔸 : Adj → Cat, and thus also a monad 𝔸 ∘ i : Mnd → Cat, which is T = (ℬ, RL, RϵL, η).

The object of algebras ℬ^{T} is the usual category of algebras over T , and the adjunction (2) is the usual free-forgetful adjunction. Let us describe the 2-natural transformation

[figure: Adj; 𝔸; Ran_{i}(𝔸∘i); α; Cat] given by the universal property of the enriched right Kan extension. It

is given by α_{B} = id_{ℬ} and α_{A} = Can_{𝔸} : 𝒜 → ℬ^{T} on the other object, where Can_{𝔸} is defined by • Can_{𝔸}(A) = (RA, Rϵ) for all A ∈ |𝒜 |;

• Can_{𝔸}(f) = Rf for all f ∈ Mor 𝒜.

2.2. Relations between monad morphisms and adjunction morphisms. Our goal is to generalize the results of the previous section so as to include monad morphisms in the picture. We introduce the following definitions.

2.2.1. Definition. The 2-category Mnd[n] is the universal 2-category containing n composable monad morphisms (with Mnd[0] = Mnd).

2.2.2. Definition. An adjunction morphism from an adjunction [figure: B; l; ⊥; A; r] to an ad-

junction [figure: B′; l′; ⊥; A′; r′] in a 2-category 𝒞 is a pair of 1-cells b : B ⟶ B′ and a : A ⟶ A′

such that br = r′a.

2.2.3. Definition. The 2-category Adj[n] is the universal 2-category containing n composable adjunction morphisms (with Adj[0] = Adj).

As before, by Yoneda’s lemma, Mnd[n] and Adj[n] are determined up to isomorphism by their 2-universal property. Their existence can be proven using presentation by computads. For instance, the coequalizer ADJ[n] displayed in (3) below satisfies the universal property of Adj[n]. We notationally distinguish this coequalizer from the explicit model that we build in Section 3.2, which we denote by Adj[n]. We prove in Theorem 3.3.9 that ADJ[n] ≅ Adj[n]. Hence, this is just a notational convenience made for the proof of Theorem 3.3.9. Observe that sequences of n + 1 monads in 𝒞 together with n composable monad morphisms between them are in bijective correspondence with functors [n] → Lax(Mnd, 𝒞), where Lax(Mnd, 𝒞) denotes the category whose objects are 2-functors Mnd → 𝒞 and whose morphisms are lax natural transformations. A short introduction to lax natural transformations is provided in Appendix B.

This implies that

Mnd[n] ≅ Mnd ⊗_{Lax} [n],

where ⊗_{Lax} denotes the lax Gray tensor product on 2-Cat, as defined in [7]. This follows from the Yoneda lemma and the bijection

2-Cat(Mnd ⊗_{Lax} [n], 𝒞) ≅ 2-Cat([n], Lax(Mnd, 𝒞)).

On the other hand, Adj[n] cannot be expressed as Adj ⊗_{Lax} [n], because in Adj[n] the squares involving the right adjoints commute on the nose rather than up to a 2-cell isomorphism.

We construct now the 2-category ADJ[n] using a presentation by a computad. A similar construction can be performed to obtain Mnd[n]. A reader not familiar with presentations by computads can consult Appendix A.

We define a computad Adj[n] as follows. Its graph GrAdj[n] is given by

\xymatrix{(B, n) \ar[r]^{L_{n}} \ar[d]^{B_{n}} & (A, n) \ar[l]^{R_{n}} \ar[d]^{A_{n}} \\ (B, n − 1) \ar[d]^{B_{n−1}} \ar[r]^{L_{n−1}} & (A, n − 1) \ar[l]^{R_{n−1}} \ar[d]^{A_{n−1}} \\ ⋮ \ar[d]^{B_{1}} & ⋮ \ar[d]^{A_{1}} \\ (B, 0) \ar[r]^{L_{0}} & (A, 0) \ar[l]^{R_{0}}}

and non-trivial graphs Adj[n](X, Y) are given by Adj[n]((B, i), (B, i)) = [figure: (B, i); ∅; R_{i}L_{i}; η_{i}; (B, i)]

and Adj[n]((A, i), (A, i)) = [figure: (A, i); L_{i}R_{i}; ∅; ϵ_{i}; (A, i)] . The computad Rel[n] has underlying graph

\xymatrix{(B, n) \ar[r]^{L_{n}} \ar[d]^{B_{n}} & (A, n) \ar[l]^{R_{n}} \ar[d]^{A_{n}} \ar[dl]^{C_{n}} \\ (B, n − 1) \ar[d]^{B_{n−1}} \ar[r]^{L_{n−1}} & (A, n − 1) \ar[l]^{R_{n−1}} \ar[d]^{A_{n−1}} \ar[dl]^{C_{n−1}} \\ ⋮ \ar[d]^{B_{1}} & ⋮ \ar[d]^{A_{1}} \ar[dl]^{C_{1}} \\ (B, 0) \ar[r]^{L_{0}} & (A, 0) \ar[l]^{R_{0}}}

and non-trivial graphs Rel[n](X, Y) are given by Rel[n]((B, i), (A, i)) = [figure: (B, i); L_{i}; L_{i}; λ_{i}; (A, i)]

and Rel[n]((A, i), (B, i)) = [figure: (A, i); R_{i}; R_{i}; ρ_{i}; (B, i)] .

We define two 2-functors M, N : F(Rel[n]) → F(Adj[n]) that are identities on objects, by M(S) = N(S) = S for S ∈ {L_{i}, R_{i}, A_{i}, B_{i}}, M(C_{i}) = B_{i} · R_{i}, N(C_{i}) = R_{i−1} · A_{i}, M(ρ_{i}) = (ϵ_{i}R_{i}) ∘ (R_{i}η_{i}), N(ρ_{i}) = 1_{R_{i}}, M(λ_{i}) = (L_{i}ϵ_{i}) ∘ (η_{i}L_{i}), N(λ_{i}) = 1_{L_{i}}.

The 2-category ADJ[n] is the coequalizer (in 2-Cat) in the diagram

\xymatrix{F(Rel[n]) \ar[r]^{M} \ar[r]_{N} & F(Adj[n]) \ar[r] & ADJ[n].}  (3)

The universal properties of the free 2-category on a computad and of the coequalizer imply that ADJ[n] satisfies the 2-universal property we were looking for.

2.2.4. Proposition. An adjunction morphism between two adjunctions induces a monad morphism between the induced monads.

Proof. Let b : B ⟶ B′ and a : A ⟶ A′ form an adjunction morphism from an

adjunction [figure: B; l; ⊥; A; r] to an adjunction [figure: B′; l′; ⊥; A′; r′] .

Let β be the mate of the identity br → r′a, that is, the composite

[figure: B; l; 1; η; A; a; r; A′; r′; 1; ϵ′; B; b; B′; l′; A′.]

We claim that (b, r′β) is a monad morphism (rl, rϵl, η) → (r′l′, r′ϵ′l′, η′).

Remark first that r′β : r′l′b ⟶ r′al = brl, as desired. The compatibility with the unit is given by the triangle identity of the second adjunction and the interchange law

[figure: B; l; 1; η; A; a; r; A′; r′; 1; ϵ′; B; b; B′; l′; 1; η′; A′; r′; B′; =; B; l; 1; η; A; a; r; A′; r′; B; b; B′.]

The compatibility condition with the multiplication is given by the triangle identity of the first adjunction and the interchange law

[figure: B; l; 1; η; B; b; B′; l′; A; r; a; r; 1; ϵ; A′; r′; 1; ϵ′; A′; r′; B; 1; η; l; B; b; B′; l′; A; r; a; r; A′; r′; 1; ϵ′; A′; r′; B; b; B′; =; B; l; 1; η; B; b; B′; l′; A; a; r; A′; r′; 1; ϵ′; A′; r′; 1; ϵ′; B′; l′; A′; r′; B′.]

Proposition 2.2.4 implies that there is an induced 2-functor j : Mnd[n] → Adj[n], which satisfies the following important property.

2.2.5. Proposition. Let 𝕋 : Mnd[n] ⟶ Cat be the 2-functor induced by the n com-

posable morphisms of monads \xymatrix{T_{n} \ar[r]^{(B_{n},γ_{n})} & … \ar[r]^{(B_{1},γ_{1})} & T_{0}} where T_{i} is a monad (T_{i}, μ_{i}, η_{i}) on a category ℬ_{i}. There is a diagram

\xymatrix{ℬ_{n} \ar[r]^{F_{n}} \ar[d]^{B_{n}} & ℬ_{n}^{T_{n}} \ar[l]^{U_{n}} \ar[d]^{\overline{(B_{n},γ_{n})}} \\ ℬ_{n−1} \ar[r]^{F_{n−1}} \ar[d] & ℬ_{n−1}^{T_{n−1}} \ar[l]^{U_{n−1}} \ar[d] \\ ⋮ \ar[d]^{B_{1}} & ⋮ \ar[d]^{\overline{(B_{1},γ_{1})}} \\ ℬ_{0} \ar[r]^{F_{0}} & ℬ_{0}^{T_{0}} \ar[l]^{U_{0}}}

which determines a 2-functor 𝔸 : Adj[n] ⟶ Cat. Moreover, this 2-functor is the enriched right Kan extension of 𝕋 along j : Mnd[n] → Adj[n] and 𝔸j = 𝕋 (the natural transformation is the identity).

Proof. Let 0 < i ≤ n be an integer. Recall that if (X, m) is a T_{i}-algebra, then

\overline{(B_{i}, γ_{i})}(X, m) = (B_{i}X, B_{i}(m) ∘ (γ_{i})_{X}),

and for f a morphism of algebras, \overline{(B_{i}, γ_{i})}(f) = B_{i}(f), thus the squares involving the right adjoints commute on the nose. This determines n composable adjunction morphisms and thus a 2-functor 𝔸 : Adj[n] ⟶ Cat.

We compute now the monad morphism (B_{i}, γ̃_{i}) induced by the morphism of adjunction

\xymatrix{ℬ_{i} \ar[r]^{F_{i}} \ar[d]^{B_{i}} & ℬ_{i}^{T_{i}} \ar[l]^{U_{i}} \ar[d]^{\overline{(B_{i},γ_{i})}} \\ ℬ_{i−1} \ar[r]^{F_{i−1}} & ℬ_{i−1}^{T_{i−1}}. \ar[l]^{U_{i−1}}}

Let us recall the unit and counit of the free forgetful adjunction on level i.

• The unit is η_{i} : 1_{ℬ_{i}} → U_{i}F_{i} = T_{i}.

• The counit ϵ_{i} is given on the T_{i}-algebra (X, m) by ϵ_{(X,m)} = m : T_{i}X → X.

Thus, the 2-cell γ̃_{i} that makes (B_{i}, γ̃_{i}) the associated monad morphism T_{i} → T_{i−1} at the object X ∈ ℬ_{i} is the composite

\xymatrix{T_{i−1}B_{i}(X) \ar[r]^{(T_{i−1}B_{i}η_{i})_{X}} & T_{i−1}B_{i}(T_{i}X) \ar[r]^{(γ_{i}T_{i})_{X}} & B_{i}T_{i}^{2}X \ar[r]^{(B_{i}μ_{i})_{X}} & B_{i}T_{i}X.}

Since (B_{i}, γ_{i}) is a monad morphism, the unit condition implies that this is equal to the composite

\xymatrix{T_{i−1}B_{i}(X) \ar[r]^{(T_{i−1}η_{i−1}B_{i})_{X}} & T_{i−1}^{2}B_{i}(X) \ar[r]^{(T_{i−1}γ_{i})_{X}} & T_{i−1}B_{i}(T_{i}X) \ar[r]^{(γ_{i}T_{i})_{X}} & B_{i}T_{i}^{2}X \ar[r]^{(B_{i}μ_{i})_{X}} & B_{i}T_{i}X.}

Now the compatibility with the multiplications implies that this is equal to

\xymatrix{T_{i−1}B_{i}(X) \ar[r]^{(T_{i−1}η_{i−1}B_{i})_{X}} & T_{i−1}^{2}B_{i}(X) \ar[r]^{(μ_{i−1}B_{i})_{X}} & T_{i−1}B_{i}(X) \ar[r]^{(γ_{i})_{X}} & B_{i}T_{i}X}

and thus γ̃_{i} = γ_{i}. This shows that 𝔸j = 𝕋.

We are going to use the fact that the enriched right Kan extension can be recognized by its universal property. Its existence is a consequence of Cat being complete as a 2-category. Let

\xymatrix{𝒞_{i} \ar[r]^{L_{i}} \ar[d]^{C_{i}} & 𝒟_{i} \ar[l]^{R_{i}} \ar[d]^{D_{i}} \\ 𝒞_{i−1} \ar[r]^{L_{i−1}} & 𝒟_{i−1} \ar[l]^{R_{i−1}}}

be a morphism of adjunctions for all i = 1, …, n where the adjunction L_{i} ⊣ R_{i} has unit η̃_{i} and counit ϵ̃_{i}. The adjunction L_{i} ⊣ R_{i} determines a 2-functor 𝔹_{i} : Adj → Cat. The whole structure determines a 2-functor 𝔹 : Adj[n] ⟶ Cat. Let P_{i} be the monad on 𝒞_{i} induced by the adjunction L_{i} ⊣ R_{i}. By 2.2.4, the monad morphism P_{i} → P_{i−1} induced by the adjunction morphism (C_{i}, D_{i}) is (C_{i}, δ_{i}), where

δ_{i} = R_{i−1}((ϵ̃_{i−1}D_{i}L_{i}) ∘ (L_{i−1}C_{i}η̃_{i})).

Let ψ : 𝔹j ⟶ 𝕋 be a 2-natural transformation. We thus have functors ψ_{i} : 𝒞_{i} ⟶ ℬ_{i} such that

(i) ψ_{i}R_{i}L_{i} = T_{i}ψ_{i} and ψ_{i}η̃_{i} = η_{i}ψ_{i}, ψ_{i}R_{i}ϵ̃_{i}L_{i} = μ_{i}ψ_{i};

(ii) ψ_{i−1}C_{i} = B_{i}ψ_{i} and γ_{i}ψ_{i} = ψ_{i−1}R_{i−1}((ϵ̃_{i−1}D_{i}L_{i}) ∘ (L_{i−1}C_{i}η̃_{i})).

We need to show that there is a unique 2-natural transformation ω : 𝔹 ⟶ 𝔸 such that ωj = ψ. Said differently, we have to show that ψ extends uniquely to a 2-natural transformation 𝔹 ⟶ 𝔸.

The relations (i) above show that ψ_{i} is a strict monad morphism from the monad to the monad T_{i} and thus induces a functor (𝒞_{i})^{P_{i}} ⟶ (ℬ_{i})^{T_{i}}. Precomposing with Can_{𝔹_{i}} : 𝒟_{i} ⟶ (𝒞_{i})^{P_{i}} (see Example 2.1.5) yields ρ_{i} : 𝒟_{i} ⟶ (ℬ_{i})^{T_{i}}.

We are now going to show that the 2-transformation ω : 𝔹 → 𝔸 given on level i by ψ_{i}, ρ_{i} is 2-natural. It is enough to check naturality with respect to the generating 1-cells and 2-cells. To deal with naturality with respect to 1-cells, it is enough to show that in the diagram

[figure: ℬ_{i}; F_{i}; B_{i}; ℬ_{i}^{T_{i}}; U_{i}; \overline{(B_{i},γ_{i})}; 𝒞_{i}; ψ_{i}; L_{i}; C_{i}; 𝒟_{i}; R_{i}; D_{i}; ρ_{i}; ℬ_{i−1}; F_{i−1}; ℬ_{i−1}^{T_{i−1}},; U_{i−1}; 𝒞_{i−1}; ψ_{i−1}; L_{i−1}; 𝒟_{i−1}; R_{i−1}; ρ_{i−1}]

all squares other than the front and back squares are commutative. Example 2.1.5 implies that the top and bottom squares are commutative, while the left square is commutative by definition. For the right square, relation (ii) implies that (ψ_{i−1}, 1) ∘ (C_{i}, δ_{i}) = (B_{i}, γ_{i}) ∘ (ψ_{i}, 1), and thus the right square of the diagram

[figure: 𝒟_{i}; D_{i}; ρ_{i}; ^{Can^{𝔹_{i}}}; 𝒞_{i}^{P_{i}}; \overline{(ψ_{i},1)}; \overline{(C_{i},δ_{i})}; ℬ_{i}^{T_{i}}; \overline{(B_{i},γ_{i})}; 𝒟_{i−1}; _{Can_{𝔹_{i−1}}}; ρ_{i−1}; 𝒞_{i−1}^{P_{i−1}}; \overline{(ψ_{i−1},1)}; ℬ_{i−1}^{T_{i−1}}]

is commutative. The triangle identity of the adjunction L_{i} ⊣ R_{i} implies that the left square commutes also. Finally, we need to check naturality with respect to the counits. Recall that the counit ϵ_{i} of the free-forgetful adjunction is given by (ϵ_{i})_{(B_{i},m_{i})} = m_{i}. Thus,

ϵ_{i}ρ_{i} = ψ_{i}R_{i}ϵ̃_{i} = U_{i}ρ_{i}ϵ̃_{i} = ρ_{i}ϵ̃_{i}.

Remark that the equation U_{i}ρ_{i} = R_{i}ψ_{i} determines the underlying object of the T_{i−1}- algebra ρ_{i}(D) for all D ∈ |𝒟_{i}|, and also how ρ_{i} acts on morphisms. The equation ϵ_{i}ρ_{i} = ρ_{i}ϵ̃_{i} determines the multiplication of ρ_{i}(D), establishing uniqueness.

The proposition below is a general result of enriched category theory. One can find it in a slightly different form together with its proof in [9, Paragraph 1.10].

2.2.6. Proposition. Let 𝒱 be a symmetric monoidal category and let 𝒳 , 𝒞 be 𝒱 - categories. Let F : 𝒳 → [𝒞^{op}, 𝒱 ] be a 𝒱 -functor such that its image is contained in the essential image of the Yoneda embedding Y : 𝒞 → [𝒞^{op}, 𝒱 ]. For each X ∈ |𝒳 |, choose a representation of F (X), α_{X} : 𝒞 (−, KX) → F (X) (α_{X} is a 𝒱 -natural isomorphism).

Then, there exists a unique 𝒱 -functor K : 𝒳 → 𝒞 such that

α : Y K → F

is 𝒱 -natural.

2.2.7. Definition. Let F : 𝒳 → [𝒞^{op}, 𝒱 ] be a 𝒱 -functor such that its image is contained in the essential image of the Yoneda embedding Y : 𝒞 → [𝒞^{op}, 𝒱 ]. A representation of F is defined to be the choice of a 𝒱 -functor K : 𝒳 → 𝒞 together with a 𝒱 -isomorphism Y K → F.

2.2.8. Theorem. Let 𝒞 be a complete 2-category and 𝕋 : Mnd[n] ⟶ 𝒞 a 2-functor

corresponding to n composable morphisms of monads \xymatrix{T_{n} \ar[r]^{(b_{n},γ_{n})} & … \ar[r]^{(b_{1},γ_{1})} & T_{0}} where T_{i} is a monad (T_{i}, μ_{i}, η_{i}) on an object B_{i}.

Then, the enriched right Kan extension of 𝕋 along j is given by a representation of the diagram

\xymatrix{𝒞 (−, B_{n}) \ar[r]^{𝒞(−,f_{n})} \ar[d]^{𝒞(−,b_{n})} & 𝒞 (−, B_{n})^{T_{n∗}(−)} \ar[l]^{𝒞(−,u_{n})} \ar[d]^{\overline{𝒞(−,b_{n}),𝒞(−,γ_{n})}} \\ 𝒞 (−, B_{n−1}) \ar[r]^{𝒞(−,f_{n−1})} \ar[d] & _{𝒞 (−, B_{n−1})_{T_{n−1∗}(−)}} \ar[l]^{𝒞(−,u_{n−1})} \ar[d] \\ ⋮ \ar[d] & ⋮ \ar[d] \\ 𝒞 (−, B_{0}) \ar[r]^{𝒞(−,f_{0})} & _{𝒞 (−, B_{0})_{T_{0∗}(−)}} \ar[l]^{𝒞(−,u_{0})}}  (4)

which can be chosen to be

\xymatrix{B_{n} \ar[r]^{f_{n}} \ar[d]^{b_{n}} & B_{n}^{T_{n}} \ar[l]^{u_{n}} \ar[d] \\ B_{n−1} \ar[d] \ar[r]^{f_{n−1}} & B_{n−1}^{T_{n−1}} \ar[l]^{u_{n−1}} \ar[d] \\ ⋮ \ar[d] & ⋮ \ar[d] \\ B_{0} \ar[r]^{f_{0}} & B_{0}^{T_{0}}. \ar[l]^{u_{0}}}

The lemma below is the key of the proof of Theorem 2.2.8.

2.2.9. Lemma. Let \xymatrix{𝒜 \ar[r]^{F} \ar[d]^{J} & [𝒞^{op}, 𝒱 ] \\ ℬ} be a diagram of 𝒱 -functors. The enriched right

Kan extension of F along J can be obtained as the adjunct of the composite

\xymatrix{𝒞^{op} \ar[r] & [𝒜 , 𝒱 ] \ar[r]^{^{Ran^{J}}} & [ℬ, 𝒱 ].}

Proof of lemma 2.2.9. Consider the diagram

\xymatrix{[𝒜 , [𝒞^{op}, 𝒱 ]] \ar[d]^{_{Ran_{J}}} \ar[r]^{≅} & [𝒞^{op}, [𝒜 , 𝒱 ]] \ar[l] \ar[d]^{[1,Ran_{J}]} \\ [ℬ, [𝒞^{op}, 𝒱 ]] \ar[u]^{J^{∗}} \ar[r]^{≅} & [𝒞^{op}, [ℬ, 𝒱 ]] \ar[l] \ar[u]^{[1,J^{∗}]}}

Since the diagram of left adjoints commutes, the diagram of right adjoints commutes up to isomorphism.

Proof of theorem 2.2.8. Let Ran_{j}𝕋 : Adj[n] → 𝒞 be the enriched right Kan extension. Since the Yoneda embedding Y : 𝒞 → [𝒞^{op}, Cat] preserves weighted limits, Y Ran_{j}𝕋 ≅ Ran_{j}(Y 𝕋). By the uniqueness statement of Proposition 2.2.6, we only have to show that for X ∈ |Adj[n]|, Ran_{j}(Y 𝕋)(X) : 𝒞^{op} → Cat is the corresponding representable.

By the previous lemma, Ran_{j}(Y 𝕋) is isomorphic to the adjunct of the composite

\xymatrix{𝒞^{op} \ar[r]^{𝒞 (−,𝕋(−))} & [Mnd[n], Cat] \ar[r]^{^{Ran^{j}}} & [Adj[n], Cat],}

which we denote by ad_{1}(Ran_{j} ∘ 𝒞 (−, 𝕋(−))). We also denote by ad_{2}(Ran_{j} ∘ 𝒞 (−, 𝕋(−))) its adjunct 𝒞^{op} × Adj[n] → Cat.

By Proposition 2.2.5 and its proof, we can describe

Ran_{j} : [Mnd[n], Cat] → [Adj[n], Cat].

To a 2-functor 𝕏 : Mnd[n] → Cat, it associates its enriched right Kan extension given by morphisms of adjunctions induced by the monad morphisms, between the corresponding free-forgetful adjunctions.

To a 2-natural transformation α : 𝕏 ⇒ 𝕏′, it associates the 2-natural transformation Ran_{j}α : Ran_{j}𝕏 ⇒ Ran_{j}𝕏′ determined by the universal property of Ran_{j}𝕏′ applied to the couple (Ran_{j}𝕏, α · j). From the proof, this extension of α is given by the induced (strict) morphisms of monads.

For a modification χ : α → α′, its extension Ran_{j}χ : Ran_{j}α → Ran_{j}α′ is given on the algebras object by the same natural transformation as on the base object. This is the only possible choice because of the relation it must satisfy with respect to the forgetful functor, and thus a valid choice.

As a consequence,

ad_{2}(Ran_{j} ∘ 𝒞 (−, 𝕋(−))) : 𝒞^{op} × Adj[n] → Cat

is the diagram (4).