Theory and Applications of Categories, Vol. 26, No. 1, 2012, pp. 1–29.

KAN EXTENSIONS AND LAX IDEMPOTENT PSEUDOMONADS

F. MARMOLEJO AND R.J. WOOD

Abstract. We show that colax idempotent pseudomonads and their algebras can be presented in terms of right Kan extensions. Dually, lax idempotent pseudomonads and their algebras can be presented in terms of left Kan extensions. We also show that a distributive law of a colax idempotent pseudomonad over a lax idempotent pseudomonad has a presentation in terms of Kan extensions.

1. Introduction

This paper follows [Marmolejo and Wood, 2010] and builds on the idea in [Manes, 1976], which was actually preceded by [Walters, 1970], that a monad can be presented without iterating the underlying endofunctor. [Marmolejo and Wood, 2010] extended Manes’ notion of an extension operator to handle algebras but we note now that algebras were treated in a somewhat similar manner in [Walters, 1970] too. Our treatment of algebras also enabled “no iteration” descriptions of distributive laws and wreaths. Because the values of the endofunctor of a monad are term objects, the no iteration description in effect removes the need to mention terms of terms and (terms of terms of terms). This is particularly helpful in the descriptions of distributive laws and wreaths where the intent is to rewrite M-terms of A-terms as A-terms of M-terms.

When we turn to higher dimensional monads the no iteration idea is even more helpful. For then the terms tend to be n-sorted, with n ≥ 2. For example, in completion monads with respect to classes of limits, the terms are categorical diagrams comprised of both objects and arrows. It is in fact completion monads, precisely colax idempotent pseudomonads, about which we have most to say. Such a pseudomonad (D, d, m, ⋯) is what is also called a “coKZ doctrine”, and characterized by adjunctions dD ⊣ m ⊣ Dd. We caution the reader that in [Marmolejo, 1997], our main reference for these pseudomonads, the subject matter is presented in terms of lax idempotent pseudomonads “KZ doctrines”, for which the adjunctions are reversed to give Dd ⊣ m ⊣ dD.

The extension operator in [Manes, 1976] and those in [Marmolejo and Wood, 2010] satisfy equations. It will come as no surprise that if pseudomonads (on 2-categories say) are described in similar terms then the equalities of those papers must be replaced with invertible 2-cells — which must themselves satisfy equations. However, colax idempotent pseudomonads have all but one of their 2-cell equations given by adjunction equations. Thus it might be hoped that if colax (or lax) idempotent pseudomonads are described by extension operators then their 2-cell equations might also mediate universal properties. This is the case. The extensions which appear in describing colax [lax] idempotent pseudomonads are right [left] Kan extensions! The precise definition (Definition 3.1) in terms of Kan extensions is somewhat similar to the conditions given in [Bunge, 1974] in what is called a coherently closed family of U-extensions (U is a 2-functor), furthermore, the way we extend the function of objects to a pseudofunctor from the data given in Definition 3.1 is similar to the construction of a lax adjoint to U given in [Bunge, 1974].

The first author gratefully acknowledges financial support from PAPIIT UNAM, project IN110111-2.

The second author gratefully acknowledges financial support from the Canadian NSERC.

Received by the editors 2011-09-30 and, in revised form, 2012-01-09.

Transmitted by R. Street. Published on 2011-12-31.

2000 Mathematics Subject Classification: 18B35, 06D10, 06B23.

Key words and phrases: (co-) lax idempotent pseudomonads, KZ-doctrines, pseudo-distributive laws.

\begin{matrix}c \\ ◯ \end{matrix} F. Marmolejo and R.J. Wood, 2012. Permission to copy for private use granted.

The algebras for a colax (or lax) idempotent pseudomonad are also defined in terms of Kan extensions and proven to be essentially the same as the usual algebras.

In Section 2 we begin by recalling the characterization of a colax idempotent pseudomonad 𝔻 = (D, d, ⋯) and its algebras, in terms of adjunctions, as given in [Marmolejo,1997]. Important equations involving the derived modification δ: dD → Dd are also recalled. In Section 3 we define right Kan pseudomonads and algebras for these. Section 4 provides a construction of a right Kan pseudomonad 𝔻′ from a colax idempotent pseudomonad 𝔻 and a construction of a colax idempotent pseudomonad 𝔻′ from a right Kan pseudomonad 𝔻. In Section 5 we show that starting with either notion as 𝔻, the 2-category of algebras for 𝔻 is 2-equivalent to the 2-category of algebras for 𝔻′.

We recall in Section 6 that morphisms between pseudomonads on 2-categories can be described in terms of 2-functors between their underlying 2-categories, together with liftings to their 2-categories of algebras. Moreover, these can also be described, see [Marmolejo and Wood, 2008] in terms of transitions which are a pseudo version of Street’s morphisms of monads [Street, 1972]. In Section 6 we use the work of the previous sections and these observations to give a description of transitions between colax idempotent pseudomonads in terms of extensions. Since distributive laws can be elegantly described in several ways in terms of extensions and one of their duals we are able in Section 7 to give a description of distributive laws between certain pseudomonads in terms of extensions. We note that the distributive law described in [Marmolejo, Rosebrugh, Wood, 2002], whose algebras are constructively completely distributive lattices, was produced this way, as a Kan extension. Another example is the distributive law of the small limit completion pseudomonad over the small colimit completion, whose algebras are the completely distributive categories [Marmolejo, Rosebrugh, Wood, to appear]; we also have the lextensive categories as algebras for the pseudomonad obtained from a distributive law of the finite completion pseudomonad over the finite sum completion pseudomonad; or regular categories as algebras for the finite limit completion pseudomonad over the regular factorizations pseudomonad with base cat_{ker} as defined in [Centazzo and Wood, 2002], and many more. To illustrate how these distributive laws work in the setting of Kan extensions we examine, in Section 8, the distributive law of coFam over Fam.

2. Preliminaries

For the convenience of the reader, we recall in this section the definition of co-lax idempotent pseudomonad (also known as co-KZ pseudomonad). They first appeared in the papers of Kock [Kock, 1973] and Zöberlein [Zöberlein,1976]. In this section we largely follow (the dual of) the development given in [Marmolejo, 1997].

Let K be a 2-category. A co-lax idempotent pseudomonad 𝔻 = (D, d, m, α, β, η, ε) on K consists of a pseudofunctor D: K → K, together with strong transformations d: 1_{K} → D and m: D^{2} → D, and modifications

[figure: D; D^{2}; D; dD; 1_{D}; m; α; ≃; D^{2}; D; D^{2}; m; 1_{D^{2}}; dD; β; D^{2}; D; D^{2}; m; 1_{D^{2}}; Dd; η; D; D^{2}; D,; Dd; 1_{D}; m; ε; ≃]  (1)

with α and ε invertible, that render dD ⊣ m ⊣ Dd, and such that the coherence condition

[figure: 1_{K}; D; D^{2}; D^{2}; D; d; Dd; dD; m; m; 1_{D}; ε; α] = [figure: 1_{K}; D; D; D^{2}; D; d; d; Dd; dD; m; d_{d}]  (2)

is satisfied. It is shown in [Marmolejo, 1997] that any such structure induces a pseudomonad, whose structure is given by (D, d, m, α^{−1}, ε^{−1}, μ), where μ is the pasting

[figure: D^{3}; D^{2}; D^{3}; D; D^{2}; D,; mD; 1_{D^{3}}; dD^{2}; m; Dm; dD; m; 1_{D}; ηD; d_{m}; α^{−1}]  (3)

and furthermore, that for a pseudomonad (D, d, m, α^{−1}, ε^{−1}, μ) to be co-lax idempotent it suffices that there exists a modification β such that α, β: dD ⊣ m is an adjunction; equivalently, that there exists a modification η such that η, ε: m ⊣ Dd is an adjunction.

Recall as well that we can then produce a 2-cell δ: dD → Dd as the pasting

[figure: D; D^{2}; D; D^{2},; dD; 1_{D}; m; 1_{D^{2}}; Dd; η; α^{−1}]

that this pasting is equally the pasting of ε^{−1} and β at m, that δ · d = d_{d}^{−1}, that m · δ = ε^{−1}α^{−1}, and that δ · m is the pasting of β and η at 1_{D^{2}}.

The 2-category 𝔻-Alg of 𝔻-algebras is defined as follows. Its objects are adjunctions ζ, ζ̂: dB ⊣ B,

[figure: B; B; DB; 1_{B}; dB; B; ζ; ≃; DB; DB; B; 1_{DB}; B; dB; ζ̂]  (4)

with invertible unit. The invertibility of ζ is automatic if d is fully faithful. Recall as well that ζ̂ is completely determined by ζ as the pasting

[figure: DB; D^{2}B; DB,; B; dDB; DdB; B; 1_{DB}; DB; dB; δB; Dζ^{−1}; d_{B}^{−1}]

and that all we have to do to verify that a ζ as above determines an object in 𝔻-Alg is to show that the equation

[figure: DB; D^{2}B; DB,; B; B; dDB; DdB; B; 1_{DB}; DB; B; 1_{B}; dB; δB; Dζ^{−1}; d_{B}^{−1}; ζ] = 1_{B}  (5)

is satisfied. (Note that replacing B by D, B by m, and ζ by α in the definition of ζ̂ gives us β = α̂.)

A 1-cell from (B, B, ζ) to (A, A, ξ) is a 1-cell H: B → A such that the pasting

[figure: DB; B; DB; A; DA; A; B; 1_{DB}; dB; H; DH; dA; 1_{A}; A; ζ̂; d_{H}^{−1}; ξ]  (6)

is invertible. Given H, K: (B, B, ζ) → (A, A, ξ), a 2-cell in 𝔻-Alg is simply a 2-cell τ : H → K in K. Provisionally write 𝔻′ for the pseudomonad (D, d, m, α^{−1}, ε^{−1}, μ) described above. It is shown in [Marmolejo, 1997] that 𝔻-Alg is 2-isomorphic to 𝔻′-Alg, the usual category of algebras for a pseudomonad, since the associativity constraint needed to complete a 𝔻-algebra (B, B, ζ) to a 𝔻′-algebra is given uniquely by the pasting

[figure: D^{2}B; DB; D^{2}B; B; DB; B,; mB; 1_{D^{2}B}; dDB; B; DB; dB; B; 1_{B}; ηB; d_{B}; ζ^{−1}]  (7)

while for a 1-cell H: (B, B, ζ) → (A, A, ξ), the pasting (6) uniquely completes H to a 1-cell of 𝔻′-algebras.

3. Right Kan pseudomonads and their algebras

We define co-lax pseudomonads in terms of right Kan extensions. Later on we shall show that they are the usual co-lax pseudomonads as in the previous section, but for the moment (and just to be able to distinguish one from the other in this paper) we will call them right Kan pseudomonads.

3.1. Definition. A right Kan pseudomonad 𝔻 on K is given as follows:

i) A function D: Ob(K) → Ob(K). ii) For every A ∈ K, a 1-cell dA : A → DA. iii) For every 1-cell F : B → DA, a right Kan extension of F along dB

[figure: B; DB; DA; dB; F; F^{𝔻}; 𝔻_{F}]  (8)

with 𝔻_{F} invertible (the latter being automatic if the 1-cell dB is fully faithful). Subject to the axioms

a) For every A in K,

[figure: A; DA; DA; dA; dA; 1_{DA}]

exhibits 1_{DA} as a right Kan extension of dA along dA.

b) For every G: C → DB and F : B → DA the 2-cell

[figure: C; DC; DB; DA; dC; G; G^{𝔻}; G^{𝔻}; F^{𝔻}; 𝔻_{G}]  (9)

exhibits F^{𝔻}G^{𝔻} as a right Kan extension of F^{𝔻}G along dC.

3.2. Remark. Observe that we can also define an effect ( )^{𝔻} on 2-cells: given φ : F → G: B → DA in K, we define φ^{𝔻}: F^{𝔻} → G^{𝔻} as the unique 2-cell such that

[figure: B; DB; DA; G; dB; F^{𝔻}; G^{𝔻}; φ^{𝔻}; 𝔻_{G}] = [figure: B; DB; DA.; dB; F; G; F^{𝔻}; 𝔻_{F}; φ]

We clearly obtain a functor ( )^{𝔻}: K(B, DA) → K(DB, DA).

We now define the 2-category of algebras for a a right Kan pseudomonad 𝔻 in terms of right Kan extensions. We denote it by 𝔻-\overline{Alg} and we define it as follows. An object 𝔹 in 𝔻-\overline{Alg} consists of an object B in K together with an assignment, to every F : C → B, of a right Kan extension F^{𝔹}: DC → B of F along dC

[figure: C; DC; B; dC; F; F^{𝔹}; 𝔹_{F}]  (10)

with 𝔹_{F} invertible (automatic if dC fully faithful), in such a way that for every G: X → DC in K, the diagram

[figure: X; DX; DC; B; dX; G; G^{𝔻}; F^{𝔹}; 𝔻_{G}]  (11)

exhibits F^{𝔹} · G^{𝔻} as a right Kan extension of F^{𝔹} · G along dX.

A 1-cell H: 𝔹 → 𝔸 in 𝔻-\overline{Alg} is a 1-cell H: B → A in K such that for every F : C → B, the diagram

[figure: C; DC; B; A; dC; F; F^{𝔹}; H; 𝔹_{F}]  (12)

exhibits F^{𝔹} · H as a right Kan extension of F · H along dC. A 2-cell τ : H → K: 𝔹 → 𝔸 is simply a 2-cell τ : H → K in K. Composition is as in K. It is not hard to show that composition of 1-cells in 𝔻-\overline{Alg} results in a 1-cell in 𝔻-\overline{Alg}.

3.3. Remark. As in Remark 3.2 we can, for any 𝔹 in 𝔻-\overline{Alg}, induce an effect ( )^{𝔹} on 2-cells: given φ : F → G: C → B, we define φ^{𝔹}: F^{𝔹} → G^{𝔹} as the unique 2-cell such that

[figure: C; DC; B; G; dC; F^{𝔹}; G^{𝔹}; φ^{𝔹}; 𝔹_{G}] = [figure: C; DC; B,; dC; F; G; F^{𝔹}; 𝔹_{F}; φ]

thus inducing a functor ( )^{𝔹}: K(C, B) → K(DC, B).

4. Right Kan pseudomonads versus co-lax idempotent pseudomonads 1

In this section we construct a colax idempotent pseudomonad from a right Kan pseudomonad, and vice versa. The constructions are given in the following two theorems.

4.1. Theorem. Every right Kan pseudomonad on K induces a co-lax idempotent pseudomonad on K.

Proof. Assume we have a right Kan pseudomonad 𝔻 on K. We first extend D to a pseudofunctor D: K → K. Given φ: F → F′: B → A in K, define DF = (dA · F)^{𝔻}, and define Dφ: DF → DF′ as (dA · φ)^{𝔻}, that is, Dφ is the unique 2-cell such that

[figure: B; DB; A; DA; dB; F′; dA; DF; DF′; Dφ; 𝔻_{dA·F′}] = [figure: B; DB; A; DA; dB; DF; dA; F; F′; φ; 𝔻_{dA·F}]

(using the fact that the left most square exhibits DF′ as a right Kan extension). It is then immediate that D(1_{F}) = 1_{DF} and that for ψ: F′ → F′′ we have D(ψφ) = (Dψ)(Dφ). If G: C → B, define D^{G,F} : DF · DG → D(F · G) as the unique (invertible) 2-cell such that

[figure: C; DC; B; DB; A; DA; dC; G; DG; D(F ·G); F; DF; dA; D^{G,F}; 𝔻_{dA·F ·G}] = [figure: C; DC; B; DB; A; DA; dC; G; DG; dB; F; DF; dA; 𝔻_{dB·G}; 𝔻_{dA·F}]

Observe that the inverse of the 2-cell D^{G,F} is the unique 2-cell ρ: D(F · G) → DF · DG such that

[figure: C; DC; B; DB; DA; dC; G; DG; D(F ·G); dB; DF; 𝔻_{dB·G}; ρ] = [figure: C; DC; B; DB; DB; DA.; dC; G; D(F ·G); F; dB; dA; DF; 𝔻_{dA·F ·G}; 𝔻_{dA·F}^{−1}]

(using (9)). It is not hard to see that for any γ: G → G′ and φ: F → F′

D(F · γ)D^{G,F} = D^{G′,F} (DF · Dγ) and D(φ · G)D^{G,F} = D^{G,F′}(Dφ · DG).

Since both 1_{DA} and D(1_{A}) are right Kan extensions of dA along dA, there is a unique isomorphism D_{A} : 1_{DA} → D(dA) such that

[figure: A; DA; A; DA; dA; 1_{A}; dA; 1_{DA}; D(1_{A}); D_{A}; 𝔻_{dA·1_{A}}] = 1_{dA}.

It is not hard to see that

D^{F,1_{A}}(D_{A} · DF) = 1_{DF} = D^{1_{B},F} (DF · D_{B}),

as well as

D^{G·H,F} (DF · D^{H,G}) = D^{H,F ·G}(D^{G,F} · DH),

therefore D: K → K is a pseudofunctor.

Then we extend d to a strong transformation d: 1_{K} → D by defining d_{F} = 𝔻_{dA·F} for F : B → A (all the relevant equations necessary to show that d is indeed a strong transformation appear above).

Next we define m: D^{2} → D such that for every A,

mA = 1_{DA}^{𝔻},

and, using (9), define for F : B → A, m_{F} : DF · mB → mA · D^{2}F as the unique 2-cell such that

[figure: DB; D^{2}B; DA; D^{2}A; DB; DA; dDB; DF; 1_{DA}; ^{1_{DB}^{𝔻}}; D^{2}F; dDA; _{1_{DA}_{𝔻}}; DF; 𝔻_{dDA·DF}; m_{F}; 𝔻_{1_{DA}}] = [figure: DB; D^{2}B; DB; DA.; dDB; 1_{DB}; ^{1_{DB}^{𝔻}}; DF; 𝔻_{1_{DB}}]  (13)

The inverse of m_{F} is the unique 2-cell θ such that

[figure: DB; D^{2}B; DB; DA; D^{2}A; =; dDB; 1_{DB}; mB; D^{2}F; DF; mA; 𝔻_{1_{DB}}; θ] [figure: DB; D^{2}B; DA; D^{2}A; DA; dDB; DF; D^{2}F; dDA; 1_{DA}; mA; 𝔻_{dDA·DF}; 𝔻_{1_{DA}}] .

It is not hard to see that m : D^{2} → D is a strong transformation.

Now define αA = 𝔻_{1_{DA}}^{−1}, then (13) tells us that α: 1_{DA} → m · dD is a modification. Define εA : mA · DdA → 1_{DA} as the unique 2-cell such that

[figure: A; DA; D^{2}A; DA; dA; DdA; 1_{DA}; mA; εA] = [figure: A; DA; DA; D^{2}A; DA; dA; dA; DdA; dDA; 1_{DA}; mA; 𝔻_{dDA·dA}; 𝔻_{1_{DA}}] .

The inverse of εA is the unique 2-cell ρ such that

[figure: A; DA; DA; D^{2}A; DA; dA; dA; DdA; 1_{DA}; dDA; 1_{DA}; mA; 𝔻_{dDA·dA}; ρ; 𝔻_{1_{DA}}] = 1_{dA}.  (14)

It is not hard to show that ε: m · Dd → 1_{D} is a modification by pasting the relevant equation with d_{F}. Define βA : dDA · mA → 1_{D^{2}A} as the unique 2-cell such that

[figure: DA; D^{2}A; DA; D^{2}A; dDA; mA; 1_{DA}; dDA; βA] = [figure: DA; D^{2}A; DA; D^{2}A.; dDA; mA; 1_{DA}; dDA; 𝔻_{1_{DA}}]

Finally define ηA : 1_{D^{2}A} → DdA · mA as the unique 2-cell such that

[figure: DA; D^{2}A; DA; D^{2}A; dDA; 1_{DA}; mA; 1_{D^{2}A}; DdA; αA^{−1}; ηA] = [figure: DA; D^{2}A; DA; D^{2}A.; DdA; 1_{DA}; mA; dDA; 1_{D^{2}A}; εA^{−1}; βA]

By Section 2, the 2-cell above is δA : dDA → DdA. It is not hard to see that β and η are modifications and that they determine, together with α and ε, adjunctions dD ⊣ m ⊣ Dd. Furthermore, the coherence condition (2) is given by (14).

4.2. Theorem. Every co-lax idempotent pseudomonad 𝔻 on K induces a right Kan pseudomonad on K.

Proof. Let 𝔻 be a co-lax idempotent pseudomonads with structure (1). We then take D and d on objects for items i) and ii) of Definition 3.1. For item iii) we define F^{𝔻} = mA·DF and show that

[figure: B; DB; DA; D^{2}A; DA; dB; F; DF; dDA; 1_{DA}; mA; d_{F}; αA^{−1}]  (15)

exhibits F^{𝔻} as a right Kan extension of F along dB. So take H: DB → DA and ψ: H · dB → F. We show that the 2-cell

[figure: DB; D^{2}B; D^{2}A; DA; DA; dDB; DdB; DF; H; DH; mA; dDA; 1_{DA}; δB; d_{H}^{−1}; Dψ; αA]  (16)

is the unique 2-cell H → F^{𝔻} that produces ψ when pasted with (15). So paste the above 2-cell with (15), substitute δB · dB by d_{dB}^{−1}, then substitute the pasting of d_{H}^{−1}, d_{dB}^{−1}, Dψ and d_{F} by dDA · ψ, and cancel αA with its inverse, thus obtaining ψ. Assume now that we have a 2-cell θ: H → F^{𝔻} such that pasting it with (15) equals ψ. Substitute Dψ in (16) by D of the pasting of θ with (15). We show that the resulting 2-cell equals θ. For this replace the pasting of δB and Dd_{F} by the pasting of d_{DF} and δDA. Now replace the pasting of d_{H}^{−1}, Dθ and d_{DF} by the pasting of θ and d_{mA}^{−1}. Paste μA and its inverse at the composite mA · DmA (where μ: m · Dm → m · mD is the pasting (3)). Replace the pasting of αA, d_{mA}^{−1} and μA by mDA · αDA, and the pasting of μ^{−1} and DαA^{−1} by mA · εDA. The pasting of αDA, δDA and εDA is the identity, leaving just θ.

The proof of a) is similar, given κ: K · dA → dA, the relevant 2-cell to consider is

[figure: DA; D^{2}A; D^{2}A; DA.; DA; dDA; DdA; DdA; 1_{DA}; K; DK; mA; dDA; 1_{DA}; δA; d_{K}^{−1}; Dκ; αA; εA]

And the proof of b) is also similar, for a 2-cell ψ: L · dC → mA · DF · G, the relevant 2-cell is

[figure: DC; D^{2}C; D^{2}A; DA.; DA; D^{2}B; D^{3}A; D^{2}A; DB; dDC; DdC; DG; L; DL; mA; dDA; 1_{DA}; D^{2}F; mB; DmA; mDA; mA; DF; δC; d_{L}^{−1}; Dψ; αA; μA; m_{F}^{−1}]

We compare these constructions in Section 6 below.

5. 𝔻-\overline{Alg} versus 𝔻-Alg

5.1. Theorem. Let 𝔻 be a right Kan pseudomonad on K, and produce the colax idempotent pseudomonad (also called 𝔻) as in Theorem 4.1. There is a 2-equivalence Φ : 𝔻- \overline{Alg} → 𝔻-Alg such that the diagram

\xymatrix{𝔻-\overline{Alg} \ar[rr]^{Φ} \ar[dr] &  & 𝔻-Alg \ar[dl] \\  & K}

commutes, where the un-labeled arrows are forgetful 2-functors.

Proof. We define Φ : 𝔻-\overline{Alg} → 𝔻-Alg as follows. For τ : H → K: 𝔹 → 𝔸 in 𝔻-\overline{Alg}, define Φ of it as τ : H → K: 𝔹_{1_{B}}^{−1} → 𝔸_{1_{A}}^{−1}. Φ will be a 2-functor if we can show that 𝔹_{1_{B}}^{−1} and H are in 𝔻-Alg. To show the first of these we must show that equation (5) is satisfied, in this case the equation is

[figure: DB; D^{2}B; DB,; B; B; dDB; DdB; ^{1_{B}^{𝔹}}; 1_{DB}; D1_{B}^{𝔹}; ^{1_{B}^{𝔹}}; 1_{B}; dB; δB; D(𝔻_{1_{B}}); d_{1_{B}^{𝔹}}^{−1}; 𝔻_{1_{𝔹}}^{−1}] = 1_{1_{B}}_{𝔹},

but this follows from the fact that (10), with F = 1_{B}, is a right Kan extension. Again, since for F = 1_{B}, (12) is a right Kan extension, we obtain the inverse of

[figure: DB; B; DB; A; DA; A; ^{1_{B}^{𝔹}}; 1_{DB}; dB; H; DH; dA; 1_{A}; _{1_{A}_{𝔸}}; 𝔹̂_{1_{B}}^{−1}; d_{H}^{−1}; 𝔸_{1_{A}}^{−1}]

as the unique 2-cell γ such that

[figure: C; DC; B; A; DA; =; dC; 1_{B}; _{1_{B}_{𝔹}}; DH; H; _{1_{A}_{𝔸}}; 𝔹_{F}; γ; B; DB; A; DA; DA; dB; H; DH; dA; dC; _{1_{A}_{𝔸}}; d_{H}; 𝔸_{1_{A}}]

In the opposite direction define Ψ : 𝔻-Alg → 𝔻-\overline{Alg} as follows. For an algebra ζ as in (4), we define Ψ(ζ) such that for every H: X → B, its extension is B · DH, and the corresponding 2-cell is

[figure: X; DX; B; DB; B; dX; H; DH; dB; 1_{B}; B; d_{H}; ζ^{−1}]  (17)

To see that Ψ(ζ) is well defined, we must show that (17) exhibits B · DH as a right Kan extension of H along dX. Given K: DX → B and κ: K · dX → H, the unique 2-cell K → B · DH that pasted with (17) equals κ is

[figure: DX; D^{2}X; DB; B.; B; dDX; DdX; DH; K; DK; B; dB; 1_{B}; δX; d_{K}^{−1}; Dκ; ζ]

Furthermore, we must show that for any G: Y → DX and H: X → B, the 2-cell

[figure: Y; DY; DX; D^{2}X; DX; DB; B; dY; G; DG; dDX; 1_{DX}; mX; DH; d_{G}; αX^{−1}; B]  (18)

exhibits B · DH · mX · DG as a right Kan extension of B · DH · G along dY. Given N : DY → B and ν: N · dY → B · DH · G, the 2-cell

[figure: DY; D^{2}Y; DB; B,; B; D^{2}X; D^{2}B; DB; DX; dDY; DdY; DG; N; DN; B; dB; 1_{B}; D^{2}H; mX; DB; mB; B; DH; δY; d_{N}^{−1}; Dν; ζ; ζ_{2}; m_{H}^{−1}]

where ζ_{2} is the 2-cell given by (7), is the unique 2-cell that pasted with (18) equals ν. We thus conclude that Ψ(ζ) is an object of 𝔻-\overline{Alg}. Given a 1-cell L: ζ → ξ (with ξ: id_{C} → C · dC) in 𝔻-Alg, we want to show that

L: Ψ(ζ) → Ψ(ξ) is a 1-cell in 𝔻-\overline{Alg}. Thus we must show that

[figure: X; DX; B; DB; B; C; dX; H; DH; dB; 1_{B}; B; L; d_{H}; ζ^{−1}]  (19)

exhibits L · B · DH as right Kan extension of L · H along dX for any H: X → B. Given N : DX → C and ν: N · dX → L · H, the unique 2-cell N → L · B · DH that pasted with (19) equals ν is

[figure: DX; D^{2}X; DC; C,; C; DB; B; dDX; DdX; DH; N; DN; C; dC; 1_{C}; DL; L; B; δX; d_{N}^{−1}; Dν; ξ; χ]

where χ is the inverse of the 2-cell induced by L that corresponds to (6), given by the fact that L is a 1-cell of algebras. Thus we define Ψ(L) = L.

For a 2-cell λ: L → L′: ζ → ξ in 𝔻-Alg, define Ψ(λ) = λ.

It is routine to verify that Φ ∘ Ψ and Ψ ∘ Φ are isomorphic to the corresponding identities.

Similar arguments produce the following

5.2. Theorem. Given colax idempotent pseudomonad 𝔻, produce its associated right Kan pseudomonad (also called 𝔻), as in Theorem 4.2. Then 𝔻-Alg and 𝔻-\overline{Alg} are equivalent.

6. Right Kan pseudomonads versus co-lax idempotent pseudomonads 2

If 𝕌 and 𝔻 are pseudomonads on the 2-categories L and K respectively then, following [Marmolejo, 1999] we can describe morphisms from (L, 𝕌) to (K, 𝔻) in terms of liftings of 2-functors F : L → K to 2-functors F̂: 𝕌-Alg → 𝔻-Alg (that commute with the forgetful 2-functors). In Theorem 3.5 of [Marmolejo and Wood, 2008] we showed that such liftings are essentially the same as transitions from 𝕌 to 𝔻 along F. The latter are a pseudo version of the morphisms of monads found in [Street, 1972] (where they are called monad functors) and consist of strong transformations r: DF → F U together with two invertible modifications (corresponding to the two equalities of [Street, 1972]) subject to two equations. We refer the reader to [Marmolejo and Wood, 2008] for the definitions of these and of coherent isomorphisms between them.

We consider the particular case of 𝔻 = (D, d, m, α_{𝔻}, β_{𝔻}, η_{𝔻}, ε_{𝔻}) and 𝕌 = (U, u, n, α_{𝕌}, β_{𝕌}, η_{𝕌}, ε_{𝕌}) colax idempotent pseudomonads (with 𝔻 as in (1), but with subindex 𝔻 on the 2-cells that conform 𝔻, and we use the same letters for the corresponding 2-cells that conform 𝕌, but with subindex 𝕌; thus the structure for 𝕌 is

[figure: U; U^{2}; U; uU; 1_{U}; n; α_{𝕌}; ≃; U^{2}; U; U^{2}; n; 1_{U^{2}}; uU; β_{𝕌}; U^{2}; U; U^{2}; n; 1_{U^{2}}; Uu; η_{𝕌}; U; U^{2}; U.; Uu; 1_{U}; n; ε_{𝕌}; ≃]  (20)

We follow the same pattern with δ, thus δ_{𝔻}: dD → Dd, and δ_{𝕌}: uU → Uu). It follows from the previous Section that morphisms of monads between them and hence also transitions, can be described in terms of algebras for the corresponding right Kan pseudomonads.

6.1. Theorem. Let 𝕌 and 𝔻 be colax idempotent pseudomonads on 2-categories L and K respectively. A transition from 𝕌 to 𝔻 along a 2-functor F : L → K can be given by the following data: for every A in L, a 𝔻-algebra (F UA, ( )^{λ}), such that for every L: B → UA in L,

F (L^{𝕌}) : (F UB, ( )^{λ}) → (F UA, ( )^{λ})

is a morphism of 𝔻-algebras. Every transition from 𝕌 to 𝔻 along F is coherently isomorphic to one that arises in this way.

Proof. For every A in L define rA = (F uA)^{λ} and ω_{1}A = λ_{F uA}:

[figure: F A; DF A; F UA; dF A; F uA; (F uA)^{λ}= rA; ω_{1}A= λ_{F uA}]

To make r a strong transformation observe that, for every G: B → A, rA · d_{F G} = (F uA)^{λ} · 𝔻_{dF A·F G} exhibits rA · DF G as a right Kan extension of rA · dF A · F G along dF B. Thus we define r_{G} as the unique 2-cell such that

[figure: F B; DF B; F A; DF A; F UA; F UB; dF B; F G; DF G; rB; dF A; F uA; rA; F UG; d_{F G}; ω_{1}A; r_{G}] = [figure: F B; DF B; F A; DF A; F UA.; dF B; F uB; rB; F G; F uA; F UG; ω_{1}B; F u_{G}]

The inverse of r_{G} is the unique 2-cell θ (given by the fact that F UG · ω_{1}B = F ((uA · G)^{𝕌}) · λ_{F uB} exhibits F UG · rB as a right Kan extension of F UG · F uB along dB) such that

[figure: F B; DF B; F UB; F UA; DF A; dF B; F uB; rB; DF G; F UG; rA; ω_{1}B; θ] = [figure: F B; DF B; F A; DF A; F UB; F UA.; dF B; F G; F uB; DF G; dF A; F uA; rA; F UG; d_{F G}; ω_{1}A; F u_{G}^{−1}]

It is routine to verify that r: DF → F U is a strong transformation and the equation defining r_{G} above tells us that ω_{1} : r · dF → F u is a modification.

To define ω_{2}A we observe that rA · αF A^{−1} = (F uA)^{λ} · 𝔻_{1_{DF A}} exhibits rA · mF A as a right Kan extension of rA · mF A along dDF A, thus we can define ω_{2}A as the unique 2-cell such that

[figure: DF A; D^{2}F A; DF UA; F U^{2}A; DF A; F UA; dDF A; 1_{DF A}; DrA; mF A; rUA; F nA; rA; α_{𝔻}F A^{−1}; ω_{2}A] = [figure: DF A; D^{2}F A; F UA; DF UA; F U^{2}A; F UA; dDF A; rA; DrA; dF UA; F uUA; 1_{F UA}; rUA; F nA; d_{rA}; ω_{1}UA; F α_{𝕌}A^{−1}]

(This is the equation in Theorem 2.3 of [Marmolejo and Wood, 2008].) To induce the inverse of ω_{2}A we observe first that rUA · DrA ≅ (F uUA · rA)^{λ}: in one direction take the unique 2-cell χ such that

[figure: DF A; D^{2}F A; F UA; DF UA; F U^{2}A; dDF A; rA; DrA; (F uUA·rA)^{λ}; dF UA; F uUA; rUA; d_{F G}; ω_{1}UA; χ] = [figure: DF A; D^{2}F A; F UA; F U^{2}A,; dDF A; rA; (F uUA·rA)^{λ}; F uUA; λ_{F uUA·rA}]

while in the other take the unique 2-cell π such that

[figure: DF A; D^{2}F A; F UA; F U^{2}A; DF UA; dDF A; rA; DrA; (F uUA·rA)^{λ}; F uUA; rUA; λ_{F uUA·rA}; π] = [figure: DF A; D^{2}F A; F UA; DF UA; F U^{2}A.; dDF A; rA; DrA; dF UA; F uUA; rUA; d_{F G}; ω_{1}UA]

The isomorphism χ just exhibited and the equality F nA = F ((1_{UA})^{𝕌}) show that the pasting of d_{rA}, ω_{1}UA and F α_{A}^{−1} exhibits F nA · rUA · DrA as a right Kan extension of rA along dDF A. Thus, the inverse of ω_{2}A is the unique 2-cell θ such that

[figure: DF A; D^{2}F A; F UA; DF UA; F U^{2}A; F UA; DF A; dDF A; rA; DrA; mF A; dF UA; F uUA; 1_{F UA}; rUA; F nA; rA; d_{rA}; ω_{1}UA; F α_{𝕌}A^{−1}; θ] = [figure: DF A; D^{2}F A; DF A; F UA.; dDF A; 1_{DF A}; mF A; rA; α_{𝔻}F A^{−1}]

We need to verify that ω_{2} : F n·rU ·Dr → r ·mF is a modification. Given any G: B → A, one shows that the pasting of d_{DF G} and α_{𝔻}F A^{−1} followed by rA exhibits rA·mF A·D^{2}F G as a right Kan extension of rA · DF G along dF B. Then one has to prove that the two pastings that need to be equal to show that ω_{2} is a modification, are equal when preceded by dDF B and pasted with d_{DF G} and α_{𝔻}F A^{−1}.

The coherence conditions in Definition 2.1 of [Marmolejo and Wood, 2008] remain to be shown. The first ends in r, so it suffices to show that both pastings are equal when preceded by dF and pasted with ω_{1}. The following commutative diagram shows this:

[figure: F n·rU·Dr·DdF ·dF; r·mF ·DdF ·dF; F n·rU·Dr·dDF ·dF; r·mF ·dDF ·dF; r·dF; F n·rU·dF U·r·dF; F n·F uU·r·dF; F n·rU·DF u·dF; F n·rU·dF U·F u; F n·F uU·F u; F u.; F n·F Uu·r·dF; F n·F Uu·F u; ω_{2}·DdF ·dF; F n·rU·Dr·d_{dF}; F n·rU·Dω_{1}·dF; r·mF ·d_{dF}; r·ε_{𝔻}F ·dF; ω_{2}·dDF ·dF; F n·rU·d_{r}; r·α_{𝔻}F^{−1}·dF; ω_{1}; F n·rU·dF U·ω_{1}; F n·ω_{1}U·r·dF; α_{𝕌}^{−1}·r·dF; F n·F uU·ω_{1}; F n·rU·d_{F u}; F n·r_{u}^{−1}·dF; F n·ω_{1}U·F u; α_{𝕌}^{−1}·F u; F n·F Uu·ω_{1}; F n·F u_{u}; F ε_{𝕌}·F u]

For the other condition, observe first that, for every A, rA · mF A · α_{𝔻}DF A^{−1} exhibits rA·mF A·mDF A as a right Kan extension of rA·mF A along dD^{2}F A. It then suffices to show that both pastings are the same when preceded by dD^{2}F and pasted with α_{𝔻}DF^{−1}.

18  F. MARMOLEJO AND R.J. WOOD

The following commutative diagram shows that these are equal:

[figure: r·mF ·DmF ·dD^{2}F; r·mF ·mDF ·dD^{2}F; F n·rU·Dr·DmF ·dD^{2}F; F n·rU·Dr·dDF ·mF; r·mF ·dDF ·mF; F n·rU·DF n·DrU·D^{2}r·dD^{2}F; F n·rU·DF n·DrU·dDF U·Dr; F n·rU·DF n·dF U^{2}·rU·Dr; F n·rU·dF U·r·mF; F n·F Un·rU^{2}·DrU·D^{2}r·dD^{2}F; F n·F Un·rU^{2}·dF U^{2}·rU·Dr; F n·rU·dF U·F n·rU·Dr; F n·F uU·r·mF; F n·F nU·rU^{2}·DrU·D^{2}r·dD^{2}F; F n·F nU·rU^{2}·dF U^{2}·rU·Dr; F n·F uU·F n·rU·Dr; F n·F Un·F uU^{2}·rU·Dr; F n·rU·Dr; r·mF; F n·F nU·rU^{2}·DrU·dDF U·Dr; F n·F nU·F uU^{2}·rU·Dr; F n·rU·mF U·dDF U·Dr; r·mF ·mDF ·dD^{2}F; F n·rU·mF U·D^{2}r·dD^{2}F; F n·rU·Dr·mDF ·dD^{2}F.; r·μ_{𝔻}F ·dD^{2}F; r·mF ·d_{mF}; r·mF ·α_{𝔻}DF^{−1}; ω_{2}·DmF ·dD^{2}F; F n·rU·Dr·d_{mF}; ω_{2}·dDF ·mF; F n·rU·dr·mF; r·α_{𝔻}F^{−1}·mF; \begin{matrix}F n·rU· \\ Dω_{2}·dD^{2}F \end{matrix}; F n·rU·DF n·DrU·d_{Dr}; F n·r{−1 \atop n}·DrU·D^{2}r·dD^{2}F; F n·rU·DF n·d_{rU} ·Dr; F n·rU·d_{F n}·rU·Dr; F n·r{−1 \atop n}·dF U^{2}·rU·Dr; F n·ω_{1}U·r·mF; F μ_{𝕌}·rU^{2}·DrU·D^{2}r·dD^{2}F; F μ_{𝕌}·rU^{2}·dF U^{2}·rU·Dr; F n·F Un·ω_{1}U^{2}·rU·Dr; F n·rU·dF U·ω_{2}; F n·ω_{1}U·F n·rU·Dr; F α{−1 \atop 𝕌}·r·mF; \begin{matrix}F n·ω_{2}U· \\ Dr·dD^{2}F \end{matrix}; F n·F nU·rU^{2}·DrU·d_{Dr}; F n·F nU·ω_{1}U^{2}·rU·Dr; F n·F uU·ω_{2}; F α{−1 \atop 𝕌}·F n·rU·Dr; F n·F un·rU·Dr; F μ_{𝕌}·F uU^{2}·rU·Dr; ω_{2}; F n·F nU·rU^{2}·d_{rU} ·Dr; F n·ω_{2}U·dDF U·Dr; F n·F α_{𝕌}U^{−1}·rU·Dr; F n·rU·α_{𝔻}F U^{−1}·Dr; r·mF ·α_{𝔻}DF^{−1}; F n·rU·mF U·d_{Dr}; F n·rU·m{−1 \atop r}·dD^{2}F; F n·rU·Dr·α_{𝔻}DF^{−1}; \begin{matrix}ω_{2}·mDF · \\ dD^{2}F \end{matrix}]

Assume now that we have a transition (r, ω_{1}, ω_{2}) from 𝕌 to 𝔻 along F. Consider the composite

[figure: 𝕌-Alg; 𝔻-Alg; 𝔻-\overline{Alg,}; F̂; Ψ]

where F̂: 𝕌-Alg → 𝔻-Alg is the lifting of F determined by the transition (r, ω_{1}, ω_{2}) as in Proposition 2.2 of [Marmolejo and Wood, 2008], and Ψ : 𝔻-Alg → 𝔻-\overline{Alg} was defined in the proof of Theorem 5.1. If we apply this composite to the free 𝕌-algebra α_{𝕌}A, A in L, we obtain the following 𝕌-algebra structure on F UA: for H: X → F UA, H^{λ} is the composite

[figure: DX; DF UA; F U^{2}A; F UA; DH; rUA; DnA]

and λ_{H} is the pasting

[figure: X; DX; F UA; DF UA; F U^{2}A; F UA.; dX; H; DH; dF UA; F uUA; 1_{F UA}; rUA; F nA; d_{H}; ω_{1}UA; F α_{𝕌}A^{−1}]  (21)

Furthermore, since for every L: B → UA we have that L^{𝕌}: α_{𝕌}B → α_{𝕌}A is a morphism of 𝕌-algebras, the same composite of functors tells us that F (L^{𝕌}) : (F UB, ( )^{λ}) → (F UA, ( )^{λ}) is a 𝔻-algebra morphism. According to the first part of this proof, the ω̂_{1} of the induced transition from 𝕌 to 𝔻 along F is

[figure: F A; DF A; F UA; DF UA; F U^{2}A; F UA,; dF A; F uA; DF uA; dF UA; F uUA; 1_{F UA}; rUA; F nA; d_{F uA}; ω_{1}UA; F α_{𝕌}A^{−1}]

with its corresponding ω̂_{2}, and the invertible modification that makes this and (r, ω_{1}, ω_{2}) coherently isomorphic is given by

[figure: DF; DF U; F U^{2}; F U.; F U; DF u; r; rU; F n; F Uu; 1_{F U}; r_{u}; F ε_{𝕌}^{−1}]

This completes the proof.

7. Distributive laws

In this section we deal with distributive laws. We treat the particular case of a distributive law of a colax idempotent pseudomonad over a lax idempotent pseudomonad, but observe that the other cases are similar. We point out that the composite pseudomonad resulting from a distributive law of a colax idempotent pseudomonad over another colax idempotent pseudomonad turns out to be colax idempotent (see Theorem 11.7 in [Marmolejo, 1999]). We begin with the following 7.1. Lemma. Let 𝔻 be a pseudomonad on K and let 𝕌 be a colax idempotent monad (as in (20)) on K. If there is a distributive law of 𝕌 over 𝔻, then

(i) For every A, d_{uA}^{−1} exhibits dUA as a right Kan extension of DuA · dA along uA.

(ii) For every L: B → UA in K,

[figure: B; UB; UA; DUA; uB; L; L^{𝕌}; dUA; 𝕌_{L}]

exhibits dUA · L^{𝕌} as a right Kan extension of dUA · L along uB.

Proof. Let (r, ω_{1}, ω_{2}, ω_{3}, ω_{4}) be a distributive law of 𝕌 over 𝔻 as in Proposition 5.1 in [Marmolejo and Wood, 2008]. Actually, the only part of the structure for a distributive law that we need to prove this is r, w_{1}, w_{2} and the coherence condition (10) of that article. For (i) let H: UA → DUA and θ: H · uA → DuA · dA, then the unique 2-cell H → dUA that pasted with d_{uA}^{−1} is θ is given by the pasting

[figure: UA; U^{2}A; UDUA; DU^{2}A; DUA.; DUA; UDA; DUA; uUA; UuA; H; UdA; dUA; UH; rUA; DnA; uDUA; DuUA; 1_{DUA}; UDuA; rA; DUuA; 1_{DUA}; δ_{𝕌}A; u_{H}^{−1}; ω_{1}UA^{−1}; Dα_{𝕌}A; Uθ; ω_{2}A; r_{uA}^{−1}; Dε_{𝕌}A]

For (ii) let M : UB → DUA and λ: M · uB → dUA · L, then the unique 2-cell M → dUA · L^{𝕌} that pasted with dUA · 𝕌_{L} is λ is given by the pasting

[figure: UB; U^{2}B; UDUA; DU^{2}A; DUA,; DUA; U^{2}A; UA; uUB; UuB; M; UL; L^{𝕌}; UM; rUA; DnA; uDUA; DuUA; 1_{DUA}; UdUA; dU^{2}A; nA; dUA; δ_{𝕌}B; u_{M}^{−1}; ω_{1}UA^{−1}; Dα_{𝕌}A; Uλ; σ; ω_{2}UA; d_{nA}]

where σ: nA · UL → L^{𝕌} is the unique 2-cell that pasted with 𝕌_{L} equals the pasting of u_{L} and α_{𝕌}A.

For the next theorem we take 𝕌 a colax idempotent pseudomonad with structure as in (20), but 𝔻 is now a lax idempotent pseudomonad. We take the data for 𝔻 as follows:

[figure: D; D^{2}; D; Dd; 1_{D}; m; η_{𝔻}; ≃; D^{2}; D; D^{2}; m; 1_{D^{2}}; Dd; ε_{𝔻}; D^{2}; D; D^{2}; m; 1_{D^{2}}; dD; α_{𝔻}; D; D^{2}; D; dD; 1_{D}; m; β_{𝔻}; ≃]

so that Dd ⊣ m ⊣ dD.

7.2. Theorem. Assume that 𝕌 is a colax idempotent monad on K, and that 𝔻 is a lax idempotent monad on K such that the conditions (i) and (ii) of Lemma 7.1 are satisfied. Then a distributive law of 𝕌 over 𝔻 can be given by the following data:

(iii) For every A in K, a 𝕌-algebra structure (DUA, ( )^{λ}),

such that the following two conditions are satisfied:

(iv) For every L: B → UA, D(L^{𝕌}) : (DUB, ( )^{λ}) → (DUA, ( )^{λ}) is 1-cell of 𝕌-algebras.

(v) For every H: C → DUA, (H^{λ})^{𝔻}: (DUC, ( )^{λ}) → (DUA, ( )^{λ}) is an algebra morphism.

Proof. According to Theorem 6.1 we get a transition from 𝕌 to 𝕌 along D if we define, for every A in K, rA = (DuA)^{λ}, define ω_{1}A as the 2-cell λ_{DuA}:

[figure: DA; UDA; DUA,; uA; DuA; (DuA)^{λ}; λ_{DuA}]

define r_{G}, for G: B → A, as the unique 2-cell such that

[figure: DB; UDB; DA; UDA; DUA; DUB; uDB; DG; UDG; rB; uDA; DuA; rA; F UG; u_{DG}; ω_{1}A; r_{G}] = [figure: DB; UDB; DA; UDA; DUA,; uDB; DuB; rB; DG; DuA; DUG; ω_{1}B; Du_{G}]

and define ω_{3}A as the unique 2-cell such that

[figure: UDA; U^{2}DA; UDUA; DU^{2}A; UDA; DUA; uUDA; 1_{UDA}; UrA; nDA; rUA; DnA; rA; α_{𝕌}DA^{−1}; ω_{3}A] = [figure: UDA; U^{2}DA; DUA; UDUA; DU^{2}A; DUA; uUDA; rA; UrA; uDUA; DuUA; 1_{DUA}; rUA; DnA; u_{rA}; ω_{1}UA; Dα_{𝕌}A^{−1}]

We define ω_{2}A as the unique 2-cell such that such that

[figure: A; UA; UDA; DA; DUA; uA; dA; UdA; DuA; dUA; rA; d_{uA}^{−1}; ω_{2}A] = [figure: A; UA; DA; UDA; DUA.; uA; dA; UdA; uDA; DuA; rA; u_{dA}; ω_{1}A]

Then ω_{2} is an invertible modification. Now we define ω_{4}A as the unique 2-cell such that

[figure: D^{2}A; UD^{2}A; DUDA; D^{2}UA; DA; DUA; UDA; uD^{2}A; mA; rDA; UmA; DrA; DrA; mUA; DuA; uDA; rA; u_{mA}; ω_{4}A; ω_{1}A] = [figure: D^{2}A; UD^{2}A; DUDA; D^{2}UA; DA; DUA.; uD^{2}A; DuDA; D^{2}uA; mA; rDA; DrA; DrA; mUA; DuA; ω_{1}DA; Dω_{1}A; m_{uA}^{−1}]

One induces the inverse of ω_{4}A using the fact that

[figure: D^{2}A; UD^{2}A; DUDA; D^{2}UA; DUA; uD^{2}A; DuDA; rDA; DrA; mUA; ω_{1}DA]

exhibits mUA·DrA·rDA as a right Kan extension of mUA·DrA·DuDA along uD^{2}A; the proof that it is indeed a right Kan extension follows from the fact that mUA · DrA ≃ (rA)^{𝔻} = ((DuA)^{λ})^{𝔻}: (DUDA, ( )^{λ}) → (DUA, ( )^{λ}) is a 1-cell of 𝕌-algebras. To show that ω_{4} is a modification, one shows that for every G: B → A,

[figure: D^{2}B; UD^{2}B; D^{2}A; UD^{2}A; DA; UDA; DUA; uD^{2}A; D^{2}G; UD^{2}G; uD^{2}A; mA; UmA; uDA; DuA; rA; u_{D^{2}G}; u_{mA}; ω_{1}A]

exhibits rA · UmA · UD^{2}G as a right Kan extension of DuA · mA · D^{2}G, since rA · UmA · UD^{2}G ≃ (DuA · mA · D^{2}G)^{λ}.

Next we show that (r, ω_{2}, ω_{4}) is an op-transition from 𝔻 to 𝔻 along U. Coherence condition (7) of [Marmolejo and Wood, 2008] follows from the fact that ω_{1}A exhibits rA as a right Kan extension of DuA along uDA, using the defining equation of ω_{2}A. And coherence condition (8) of [Marmolejo and Wood, 2008] follows from the fact that

[figure: D^{3}A; UD^{3}A; D^{2}A; UD^{2}A; DA; UDA; DUA; uD^{3}A; DmA; UDmA; uD^{2}A; mA; UmA; uDA; DuA; rA; u_{DmA}; u_{mA}; ω_{1}A]

exhibits rA · UmA · UDmA as a right Kan extension of DuA · mA · DmA along uD^{3}A, this because rA · UmA · UDmA ≃ (DuA · mA · DmA)^{λ}.

Thus we have a transition (r, ω_{1}, ω_{3}) from 𝕌 to 𝕌 along D and an op-transition (r, ω_{2}, ω_{4}) from 𝔻 to 𝔻 along U. We are left with the verification that the coherence conditions of Proposition 5.1 of [Marmolejo and Wood, 2008] are satisfied. Condition (10) of that paper is the defining equation of ω_{2}, while (11) of the same paper follows from the fact that dUA · α_{𝕌}^{−1} exhibits dUA · nA as a right Kan extension of dUA along uUA. And (12) of that paper is the defining equation for ω_{4}, leaving us only with coherence condition (13) of the same paper. This coherence condition follows from the fact that rA · UmA · α_{𝕌}DA exhibits rA · UmA · nD^{2}A as a right Kan extension of rA · UmA along uUD^{2}A (since rA · UmA · nD^{2}A ≃ (rA · UmA)^{λ}).

We must now show that every distributive law of 𝕌 over 𝔻, with 𝕌 colax idempotent and 𝔻 lax idempotent, arises essentially in this way. Let (r, ω_{1}, ω_{2}, ω_{3}, ω_{4}) be a distributive law of 𝕌 over 𝔻. Then we have that conditions (i) and (ii) of Lema 7.1 are satisfied, and we must obtain conditions (iii), (iv) and (v) of Theorem 7.2, and show that the distributive law obtained from Theorem 7.2 is essentially the distributive law (r, ω_{1}, ω_{2}, ω_{3}, ω_{4}).

Observe that (D, ω_{1}, ω_{3}) is a transition from 𝕌 to 𝕌 along D. Then Theorem 6.1 gives us the 𝕌-algebra structure on DUA corresponding to (21), which in this case assigns to an H: X → DUA the right Kan extension

[figure: X; UX; DUA; UDUA; DU^{2}A; DUA,; uX; H; UH; uDUA; DuUA; 1_{DUA}; rUA; DnA; u_{H}; ω_{1}UA; Dα_{𝕌}A^{−1}]

and for every L: B → UA, D(L^{𝕌}) is a 1-cell of 𝕌-algebras. This gives us conditions (iii) and (iv) of Theorem 7.2.

We are left with showing that, for any H: C → DUA, (H^{λ})^{𝔻}: (DUC, (−)^{λ}) → (DUA, (−)^{λ}) is a 𝕌-algebra morphism. To do this we observe that (DUC, (−)^{λ}) and (DUA, (−)^{λ}) are the images under the 2-functor Ψ : 𝕌-Alg→ 𝕌-\overline{Alg} of the 𝕌-algebras given by

[figure: DUC; UDUC; DU^{2}C; DUC; uDUC; DuUC; 1_{DUC}; rUC; DnC; ω_{1}UC^{−1}; Dα_{𝕌}C; and; DUA; UDUA; DU^{2}A; DUA; uDUA; DuUA; 1_{DUA}; rUA; DnA; ω_{1}UA^{−1}; Dα_{𝕌}A]

respectively (these in turn are the images of the free algebras α_{𝕌}C and α_{𝕌}A under the lifting 𝕌-Alg → 𝕌-Alg induced by the transition (D, ω_{1}, ω_{3})), thus it suffices to show that

(H^{λ})^{𝔻} = [figure: DUC; DUDUA; D^{2}U^{2}A; D^{2}UA; DUA; DUH; DrUA; D^{2}nA; mUA]

is a 1-cell between these latter 𝕌-algebras. According to (6), we must show that

DnA · rUA · UmUA · UD^{2}nA · UDrUA · UDUH · UDnC · UrUC · δ_{𝕌}DUC

is invertible; and one uses the available isomorphisms to produce nDUC just after δ_{𝕌}DUC to conclude that the above 2-cell is indeed invertible. One then applies the construction given in Theorem 7.2 to produce a new distributive law (s, π_{1}, π_{2}, π_{3}, π_{4}). The claim is that the original distributive law (r, ω_{1}, ω_{2}, ω_{3}, ω_{4}) is coherently isomorphic to this new one in the following sense:

7.3. Definition. Let 𝕌 and 𝔻 be pseudomonads on K, and let (r, ω_{1}, ω_{2}, ω_{3}, ω_{4}) and (s, π_{1}, π_{2}, π_{3}, π_{4}) be distributive laws of 𝕌 over 𝔻. We say that the distributive laws are coherently isomorphic if there is an invertible α: r → s that makes the transitions (r, ω_{1}, ω_{3}) and (s, π_{1}, π_{3}) coherently isomorphic, and makes the op-transitions (r, ω_{2}, ω_{4}) and (s, π_{2}, π_{4}) coherently isomorphic.

7.4. Theorem. Let 𝕌 be a colax idempotent monad, 𝔻 a lax idempotent monad on K, and (r, ω_{1}, ω_{2}, ω_{3}, ω_{4}) a distributive law of 𝕌 over 𝔻. If (s, π_{1}, π_{2}, π_{3}, π_{4}) is the distributive law produced just before Definition 7.3, then (r, ω_{1}, ω_{2}, ω_{3}, ω_{4}) and (s, π_{1}, π_{2}, π_{3}, π_{4}) are coherently isomorphic distributive laws.

Proof. Theorem 6.1 already gives us (r, ω_{1}, ω_{3}) and (s, π_{1}, π_{3}) coherently isomorphic by the 2-cell

[figure: UD; UDU; DU^{2}; DU.; DU; UDu; r; rU; Dn; DUu; 1_{DU}; r_{u}; Dε_{𝕌}^{−1}]  (22)

We must show that it also makes (r, ω_{2}, ω_{4}) and (s, π_{2}, π_{4}) coherently isomorphic. We have that sA = DnA · rUA · UDuA and, π_{1} is the pasting

[figure: DA; UDA; DUA; UDUA; DU^{2}A; DUA.; uDA; DuA; UDuA; uDUA; DuUA; 1_{DUA}; rUA; DnA; u_{DuA}; ω_{1}UA; Dα_{𝕌}A^{−1}]

To show that (22) at A pasted with π_{2} equals ω_{2} we use the fact that d_{uA}^{−1} exhibits dUA as a right Kan extension of DuA · dA along uA and the defining equation of π_{2}, namely

[figure: A; UA; UDA; DA; DUA; uA; dA; UdA; DuA; dUA; DnA·rUA·UDuA; d_{uA}^{−1}; π_{2}A] = [figure: A; UA; DA; UDA; DUA.; uA; dA; UdA; uDA; DuA; DnA·rUA·UDuA; u_{dA}; π_{1}A]

The case for ω_{3} and π_{3} is similar to the one just shown.

7.5. Remark. Of course we still have not shown that Definition 7.3 is good, in the sense that the structures induced (liftings, composite pseudomonads, coherent structures) are essentially the same for two coherently isomorphic distributive laws. However, this would take us too far from the objectives of the present paper. We defer the treatment of this issue to a paper that will deal with the “no-iteration” version of the algebras for a general pseudomonad, and the corresponding version of a distributive law.

8. Example

Let 𝕌 be coFam on Cat. That is, U : Ob(Cat) → Ob(Cat) is given as follows. For a category A, the objects of UA are finite families ⟨A_{i}⟩_{i∈I} of objects of A. A morphism ⟨A_{i}⟩_{i∈I} → ⟨B_{j}⟩_{j∈J} in UA consists of a function φ: J → I together with a family of morphisms ⟨f_{j}: A_{φ(j)} → B_{j}⟩_{j∈J} in A. The identity on ⟨A_{i}⟩_{i∈I} is (1_{I}, ⟨1_{A_{i}}⟩_{i∈I}), whereas composition of (φ, ⟨f_{j}⟩_{j∈J}) : ⟨A_{i}⟩_{i∈I} → ⟨B_{j}⟩_{j∈J} and (ψ, ⟨g_{k}⟩_{k∈K}) : ⟨B_{j}⟩_{j∈J} → ⟨C_{k}⟩_{k∈K} is (φψ, ⟨g_{k} · f_{ψ(k)}⟩_{k∈K}).

The functor uA : A → UA sends an object A to the family with exactly one element ⟨A⟩_{{∗}}, and f : A → B to (1_{{∗}}, ⟨f⟩_{{∗}}).

We observe that UA has finite products. Given a finite set I, and for every i ∈ I an element ⟨A_{ij}⟩_{j∈J_{i}} in UA, then

∏_{i∈I}⟨A_{ij}⟩_{j∈J_{i}} = ⟨A_{ij}⟩_{(i,j)∈∐_{i∈I} J_{i}},

with the i-th projection given by

(σ_{i}: J_{i} → _{∐i∈I}J_{i}, ⟨1_{A_{ij}}⟩_{j∈J_{i}}) : ⟨A_{ij}⟩_{(i,j)∈∐_{i∈I} J_{i}} → ⟨A_{ij}⟩_{j∈J_{i}}.

Given a functor F : B → UA, F^{𝕌}: UB → UA is such that F (⟨B_{j}⟩_{j∈J}) = ∏_{j∈J} F B_{j}, and given a morphism (γ, ⟨g_{j}⟩_{j∈J}) : ⟨C_{k}⟩_{k∈K} → ⟨B_{j}⟩_{j∈J}, define F^{𝕌} on it such that the diagram

[figure: F^{𝕌}(⟨C_{k}⟩_{k∈K}); F C_{γ(j)}; F^{𝕌}(⟨B_{j}⟩_{j∈J}); F B_{j}; π_{γ(j)}; F^{𝕌}((γ,⟨g_{j}⟩_{j∈J})); g_{j}; π_{j}]

commutes for every j ∈ J. Then the diagram

[figure: B; UB; UA; uB; F; F^{𝕌}]

commutes (provided we make the convention that a unary product is simply the object involved). And it exhibits F^{𝕌} as a right Kan extension of F along uB. Indeed, given

[figure: B; UB; UA,; uB; F; H; θ]

then the unique natural transformation θ̂: H → F^{𝕌} that preceded by uA is θ, is given, at ⟨B_{j}⟩_{j∈J}, by the morphism that makes the diagram

[figure: H(⟨B_{j}⟩_{j∈J}); H(⟨B_{j}⟩_{{∗}}); F^{𝕌}(⟨B_{j}⟩_{j∈J}); F^{𝕌}(⟨B_{j}⟩_{j∈J}); H((⌜j⌝,⟨1_{B_{j}} ⟩_{{∗}})); θ̂⟨B_{j}⟩_{j∈J}; F^{𝕌}((⌜j⌝,⟨1_{B_{j}} ⟩_{{∗}})); θB_{j}]

commute for all j ∈ J.

It is a routine exercise to verify that F^{𝕌}: UB → UA preserves finite products, and that ⟨B_{j}⟩_{j∈J} = ∏_{j∈J}⟨B_{j}⟩_{{∗}} in UB. Then we can verify condition b) of Theorem 3.1. Indeed, given G: C → UB, H: UC → UA and θ: H · uC → F^{𝕌} · G, then the unique natural transformation θ̂: H → F^{𝕌} · G^{𝕌} that preceded by uC is θ is given, at ⟨C_{i}⟩_{i∈I} in UC, by the unique arrow that makes the diagram

[figure: H(⟨C_{i}⟩_{i∈I}); H(⟨C_{i}⟩_{{∗}}); F^{𝕌}G^{𝕌}(⟨C_{i}⟩_{i∈I}); F^{𝕌}G^{𝕌}(⟨C_{i}⟩_{{∗}}); H(π_{i}); θ̂⟨C_{i}⟩_{i∈I}; F^{𝕌}G^{𝕌}(π_{i}); θC_{i}]

commute for all i ∈ I.

We have shown, using the techniques of this paper, that 𝕌 is a colax idempotent monad. It is well known that the algebras for 𝕌 are categories with finite products and functors that preserve finite products.

Dually, as 𝔻 we take Fam. Thus DA = (U(A^{op}))^{op}, and the rest of the structure can be read from this from the description of 𝕌. Of course, 𝔻 is a lax idempotent monad.

It is well known that there is a distributive law of 𝕌 over 𝔻; the main ingredient being the fact that if A has finite products then FamA also has finite products. Here we verify the conditions of this paper.

We observe first that DUA has finite products. Indeed, given a finite set I, and for every i ∈ I an element ⟨⟨A_{ijk}⟩_{k∈K_{ij}}⟩_{j∈J_{i}} in DUA, then the product of the family is given by the object

⟨⟨A_{it(i)k}⟩_{k∈∐_{i∈I} K_{i,t(i)}}⟩_{t∈∏_{i∈I} J_{i}},

with the i-th projection given by the projection π_{i}: ∏_{i∈I} J_{i} → J_{i} together with, for every t ∈ ∏_{i∈I} J_{i}, the morphism

(σ_{i}: K_{it(i)} → ∐_{i∈I}K_{it(i)}, ⟨1_{A_{it(i)k}}⟩_{k∈K_{it(i)}}) : ⟨A_{it(i)k}⟩_{k∈∐_{i∈I} K_{i,t(i)}} → ⟨A_{ijk}⟩_{k∈K_{ij}}.

It is not hard to verify that the conditions of Lemma 7.1 are satisfied. Indeed, to see that d_{uA}^{−1} exhibits dUA as a right Kan extension of DuA · dA along uA, take θ: H · uA → DuA · dA, then the unique 2-cell θ̂: H → dUA that pasted with d_{uA}^{−1} produces θ is constructed as follows. Given ⟨A_{i}⟩_{i∈I} in UA, we observe that

∏_{i∈I}⟨⟨A_{i}⟩_{{∗}}⟩_{{∗}} = ⟨⟨A_{i}⟩_{i∈I}⟩_{{∗}}

in DUA. Thus θ̂⟨A_{i}⟩_{i∈I} : H(⟨A_{i}⟩_{i∈I}) → dUA(⟨A_{i}⟩_{i∈I}) = ⟨⟨A_{i}⟩_{i∈I}⟩_{{∗}} is the unique arrow such that the diagram

[figure: H(⟨A_{i}⟩_{i∈I}); H(⟨A_{i}⟩_{{∗}}); ⟨⟨A_{i}⟩_{i∈I}⟩_{{∗}}; ⟨⟨A_{i}⟩_{{∗}}⟩_{{∗}}; H((⌜i⌝,⟨A_{i}⟩_{{∗}})); θ̂⟨A_{i}⟩_{i∈I}; θA_{i}; π_{i}]

commutes.

References

Marta C. Bunge Coherent extensions and relational algebras. Trans. Amer. Math. Soc. 197 (1974), 355-390.

Marta Bunge and Jonathon Funk. Singular coverings of toposes. Lecture Notes in Mathematics 1890, Springer. 2006.

C. Centazzo and R.J. Wood. An extension of the regular completion. Journal of pure and applied algebra, Vol. 175, No. 1, 2002, 93-108

A. Kock. Monads for which structures are adjoint to units, preprint, Aarhus Univ. (1973); revised version published in J. Pure Appl. Alg. 104 (1995), 41-59.

E. G. Manes. Algebraic Theories. Springer-Verlag, 1976.

F. Marmolejo. Doctrines whose structure forms a fully faithful adjoint string. Theory and Applications of Categories, Vol. 3, 1997, No. 2, pp 23-24.

F. Marmolejo. Distributive laws for pseudomonads. Theory and Applications of Categories, Vol. 5, No. 5, 1999, pp. 91-147.

F. Marmolejo, R. D. Rosebrugh, and R.J. Wood. A basic distributive law. Journal of Pure and Applied Algebra, Vol. 168 (2-3), 2002, pp. 209-226.

F. Marmolejo and R.J. Wood. Coherence for pseudodistributive laws revisited. Theory and Applications of Categories, Vol. 20, No. 6, 2008, pp. 74-84.

F. Marmolejo and R.J. Wood. Monads as extension systems - no iteration is necessary. Theory and Applications of Categories, Vol. 24, No. 4, 2010, pp. 84-113.

F. Marmolejo, R. Rosebrugh and R.J. Wood. Completely and totally distributive categories. Journal of Pure and Applied Algebra. To appear.

R. Street. The formal theory of monads. Journal of Pure and Applied Algebra, Vol. 2, 1972. 149-168.

R.F.C. Walters. A categorical approach to universal algebra. PhD Thesis. Australian National University, 1970.

V. Zöberlein. Doctrines on 2-categories, Math. Zeitschrift 148 (1976), 267-279.

Instituto de Matemáticas, Universidad Nacional Autónoma de México Area de la Investigación Científica, Circuito Exterior, Ciudad Universitaria Coyoacán 04510, México, D.F. México

Department of Mathematics and Statistics, Dalhousie University Chase Building, Halifax, Nova Scotia, Canada B3H 3J5

Email: quico@matem.unam.mx
      rjwood@mathstat.dal.ca

This article may be accessed at http://www.tac.mta.ca/tac/ or by anonymous ftp at

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/26/1/26-01.{dvi,ps,pdf}

THEORY AND APPLICATIONS OF CATEGORIES (ISSN 1201-561X) will disseminate articles that significantly advance the study of categorical algebra or methods, or that make significant new contributions to mathematical science using categorical methods. The scope of the journal includes: all areas of pure category theory, including higher dimensional categories; applications of category theory to algebra, geometry and topology and other areas of mathematics; applications of category theory to computer science, physics and other mathematical sciences; contributions to scientific knowledge that make use of categorical methods. Articles appearing in the journal have been carefully and critically refereed under the responsibility of members of the Editorial Board. Only papers judged to be both significant and excellent are accepted for publication. Full text of the journal is freely available in .dvi, Postscript and PDF from the journal’s server at http://www.tac.mta.ca/tac/ and by ftp. It is archived electronically and in printed paper format.

Subscription information Individual subscribers receive abstracts of articles by e-mail as they are published. To subscribe, send e-mail to tac@mta.ca including a full name and postal address. For institutional subscription, send enquiries to the Managing Editor, Robert Rosebrugh, rrosebrugh@mta.ca.

Information for authors The typesetting language of the journal is TEX, and L^{A}TEX2e strongly encouraged. Articles should be submitted by e-mail directly to a Transmitting Editor. Please obtain detailed information on submission format and style files at http://www.tac.mta.ca/tac/.

Managing editor Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca

TEXnical editor Michael Barr, McGill University: barr@math.mcgill.ca

Assistant TEX editor Gavin Seal, Ecole Polytechnique Fédérale de Lausanne:

gavin seal@fastmail.fm

Transmitting editors

Clemens Berger, Université de Nice-Sophia Antipolis, cberger@math.unice.fr Richard Blute, Université d’ Ottawa: rblute@uottawa.ca Lawrence Breen, Université de Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: ronnie.profbrown(at)btinternet.com

Valeria de Paiva: valeria.depaiva@gmail.com

Ezra Getzler, Northwestern University: getzler(at)northwestern(dot)edu Kathryn Hess, Ecole Polytechnique Fédérale de Lausanne : kathryn.hess@epfl.ch Martin Hyland, University of Cambridge: M.Hyland@dpmms.cam.ac.uk P. T. Johnstone, University of Cambridge: ptj@dpmms.cam.ac.uk Anders Kock, University of Aarhus: kock@imf.au.dk Stephen Lack, Macquarie University: steve.lack@mq.edu.au F. William Lawvere, State University of New York at Buffalo: wlawvere@buffalo.edu Tom Leinster, University of Glasgow, Tom.Leinster@glasgow.ac.uk Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.uu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz Giuseppe Rosolini, Università di Genova: rosolini@disi.unige.it Alex Simpson, University of Edinburgh: Alex.Simpson@ed.ac.uk James Stasheff, University of North Carolina: jds@math.upenn.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 Insubria: robert.walters@uninsubria.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca