Theory and Applications of Categories, Vol. 28, No. 14, 2013, pp. 371–402.

NO-ITERATION PSEUDOMONADS

F. MARMOLEJO AND R. J. WOOD

Abstract. We present the no-iteration version of the coherence conditions necessary to define a pseudomonad, and a description of the algebras for it in a similar fashion. We show that every no-iteration pseudomonad induces a pseudomonad, and that the corresponding algebras are equivalent. We also show that every pseudomonad induces a no-iteration pseudomonad, and again, that the corresponding algebras are equivalent. We conclude with an analysis of the algebras for the 2-monad (−)^{2} on Cat in the light of the no-iteration description of the algebras.

1. Introduction

In this paper we extend the results from [Marmolejo & Wood, 2010] to higher dimensional monads. We recall that that paper built on the idea of [Manes, 1976] (Exercise 1.3, page 32) that a monad (T, η, μ) can be presented in a way that avoids iteration of the endofunctor T. This leads in many particular cases to a significant simplification of the calculations necessary to verify that something is indeed a monad. This kind of presentation was extended to the algebras in [Marmolejo & Wood, 2010], and applied to distributive laws to achieve an alternative presentation of these that avoids iteration of the functors in question. After the latter paper had been published, we learned from R.F.C. Walters that he had a similar no iteration presentation of the monad, and also of the algebras, in his doctoral dissertation [Walters, 1970].

With these results in mind, we moved on to higher dimensions, where the no iteration idea is even more helpful; this is clear from just a glance at the coherence conditions necessary for a distributive law of one pseudomonad over another (see [Marmolejo, 1999] or [Marmolejo & Wood, 2008]). Our first stop in the direction of higher dimensional monads was [Marmolejo & Wood, 2012], where we dealt with the simplest non-trivial examples of pseudomonads, namely, (co-)lax idempotent or (co-)KZ, whose definition was originally given by Kock and Zöberlein in [Kock, 1973] and [Zöberlein, 1976]. However, [Marmolejo & Wood, 2012] did not follow strictly the logic of [Marmolejo & Wood, 2010]. This was due to the fact that, at some point, we realized that it was much more efficient to codify all the structure involved in terms of Kan extensions.

Both authors gratefully acknowledge financial support from PAPIIT UNAM, project IN110-111.

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

Received by the editors 2012-05-14 and, in revised form, 2013-06-26.

Transmitted by Ross Street. Published on 2013-07-01.

2010 Mathematics Subject Classification: 18B35, 06D10, 06B23.

Key words and phrases: Pseudomonad, algebras.

\begin{matrix}c \\ ◯ \end{matrix} F. Marmolejo and R. J. Wood, 2013. Permission to copy for private use granted.

This paper follows its lower dimensional counterpart [Marmolejo & Wood, 2010] more closely.

We begin in Section 2 with the definition of what we will call, only for the purposes of this paper, a no-iteration pseudomonad. This consists of six items of data and eight coherence conditions. Observe on the one hand that this suffices to define the endo pseudofunctor D, and also the pseudomonad over it, and on the other that there is no instance of D^{n}, for n > 1 anywhere in the coherence conditions.

In Section 3 we show first how to extend the function on objects D from a no-iteration pseudomonad to an endo pseudofunctor on a 2-category, followed by the definition of the rest of the structure that makes up a pseudomonad, the pseudomonad induced by a no-iteration pseudomonad.

In Section 4 we define the 2-category of algebras for a no-iteration pseudomonad in a similar no-iteration fashion, and in Section 5 we give the proof that this latter is equivalent to the usual algebras for the pseudomonad induced by a no-iteration pseudomonad.

Section 6 is devoted to the opposite direction, that is, every pseudomonad induces a no-iteration pseudomonad and the corresponding algebras are equivalent.

As an application we use the new description of the algebras for a pseudomonad to analyze the proof, given in [Korostenski & Tholen, 93] and [Rosebrugh & Wood, 02], of the fact that the algebras for the 2-monad (−)^{2} are factorizations systems.

This leaves the task of producing the corresponding no-iteration version of distributive laws between pseudomonads, the theme of a forthcoming paper.

2. No-iteration pseudomonads

We assume the reader is familiar with the different aspects of the theory of pseudomonads as given in, for example, [Marmolejo, 1997], with distributive laws between pseudomonads as given in [Marmolejo, 1999] and [Marmolejo, 2004], with the corresponding revision given in [Marmolejo & Wood, 2008]. We begin with the alternative no-iteration presentation of a pseudomonad in the 2-category A.

2.1. Definition. A no-iteration pseudomonad 𝔻 on A consists of

1. A function D: Ob(A) → Ob(A).

2. For every A ∈ A, a 1-cell dA: A → DA.

3. For every A, B ∈ A, a functor ( )^{𝔻}: A(A, DB) → A(DA, DB).

4. For every A ∈ A, an invertible 2-cell 𝔻_{A}: Id_{DA} → dA^{𝔻}.

5. For every f : A → DB, an invertible 2-cell [figure: A; DA; DB.; dA; f; f^{𝔻}; 𝔻_{f}]

NO-ITERATION PSEUDOMONADS  373

6. For every f : A → DB and h: B → DC, an invertible 2-cell [figure: DA; DB; DC.; f^{𝔻}; (h^{𝔻}f)^{𝔻}; h^{𝔻}; 𝔻_{f,h}] Subject to the coherence conditions

1. For every A ∈ A,

[figure: A; DA; DA; dA; Id_{DA}; dA^{𝔻}; dA; 𝔻_{dA}; 𝔻_{A}] = id_{dA}  (1)

2. For every f : A → DB,

[figure: DA; DA; DB; ^{Id^{DA}}; dA^{𝔻}; (f^{𝔻}dA)^{𝔻}; f^{𝔻}; f^{𝔻}; 𝔻_{dA,f}; 𝔻_{A}; (𝔻_{f})^{𝔻}] = id_{f^{𝔻}}  (2)

3. For every f : A → DB,

[figure: DA; DB; DB; f^{𝔻}; dB^{𝔻}; (dB^{𝔻}f)^{𝔻}; ^{Id^{DB}}; 𝔻_{f,dB}; 𝔻_{B}] = (𝔻_{B}f)^{𝔻}.  (3)

4. For every 2-cell φ: f → g: A → DB,

[figure: A; DA; DB; dA; f^{𝔻}; g^{𝔻}; g; 𝔻_{g}; φ^{𝔻}] = [figure: A; DA; DB.; dA; f^{𝔻}; f; g; 𝔻_{f}; φ]  (4)

5. For every f : A → DB, h: B → DC

[figure: A; DA; DB; DC; dA; f; f^{𝔻}; h^{𝔻}; 𝔻_{f}] = [figure: A; DA; DB; DC.; dA; h^{𝔻}f; (h^{𝔻}f)^{𝔻}; f^{𝔻}; h^{𝔻}; 𝔻_{(h^{𝔻}f)}; 𝔻_{f,h}]  (5)

6. For every φ: f → g: A → DB and every h: B → DC,

[figure: DA; DB; DC; f^{𝔻}; g^{𝔻}; (h^{𝔻}g)^{𝔻}; h^{𝔻}; 𝔻_{g,h}; φ^{𝔻}] = [figure: DA; DB; DC.; f^{𝔻}; (h^{𝔻}f)^{𝔻}; (h^{𝔻}g)^{𝔻}; h^{𝔻}; 𝔻_{f,h}; (h^{𝔻}φ)^{𝔻}]  (6)

7. For every f : A → DB and every ψ: h → k: B → DC,

[figure: DA; DB; DC; f^{𝔻}; k^{𝔻}; (h^{𝔻}g)^{𝔻}; h^{𝔻}; 𝔻_{f,k}; ψ^{𝔻}] = [figure: DA; DB; DC.; f^{𝔻}; (h^{𝔻}f)^{𝔻}; (k^{𝔻}f)^{𝔻}; h^{𝔻}; 𝔻_{f,h}; (ψ^{𝔻}f)^{𝔻}]  (7)

8. For every f : A → DB, h: B → DC and ℓ: C → DE,

[figure: DA; DB; DC; DE; f^{𝔻}; h^{𝔻}; ℓ^{𝔻}; (h^{𝔻}f)^{𝔻}; (ℓ^{𝔻}h^{𝔻}f)^{𝔻}; ((ℓ^{𝔻}h)^{𝔻}f)^{𝔻}; 𝔻_{f,h}; 𝔻_{h^{𝔻}f,ℓ}; (𝔻_{h,ℓ}f)^{𝔻}] = [figure: DA; DB; DC; DE; f^{𝔻}; h^{𝔻}; ℓ^{𝔻}; (ℓ^{𝔻}h)^{𝔻}; 𝔻_{h,ℓ}; ((ℓ^{𝔻}h)^{𝔻}f)^{𝔻}; 𝔻_{f,ℓ^{𝔻}h}] (8)

3. The pseudomonad induced by a no-iteration pseudomonad

Let 𝔻 be a no-iteration pseudomonad. We extend the function on objects D to a pseudofunctor as follows.

3.1. Proposition. Given a no-iteration pseudomonad 𝔻 on a 2-category A, we induce a pseudofunctor D: A → A as follows: D as given by 𝔻 on objects. For objects A, B we define D_{A,B} as the composite

\xymatrix{A(A, B) \ar[r]^{−∘dB} & A(A, DB) \ar[r]^{( )^{𝔻}} & A(DA, DB).}

For every A, D_{A} = 𝔻_{A}. For f : A → B and h: B → C, define D^{f,h}: Dh Df → D(hf) as (𝔻_{dC h} f)^{𝔻} · 𝔻_{dB f,dC h}.

Proof. Assume we have

[figure: A; B; C; E.; f; g; h; k; ℓ; φ; ψ]

Then D(ψ f) · D^{f,h} is the composition of the top arrow and the rightmost one in the diagram

\xymatrix{Dh Df \ar[r]^{𝔻_{dB f,dC h}} \ar[d]_{(dC ψ)^{𝔻}Df} & (Dh dB f)^{𝔻} \ar[r]^{(𝔻_{dC h}f)^{𝔻}} \ar[d]_{((dC ψ)^{𝔻}dB f)^{𝔻}} & Dhf \ar[d]^{(dC ψf)^{𝔻}} \\ Dk Df \ar[r]_{𝔻_{dB f,dC k}} & (Dk dB f)^{𝔻} \ar[r]_{(𝔻_{dC k}f)^{𝔻}} & Dkf.}

The right rectangle commutes by the functoriality of ( )^{𝔻} and (4), the left one by (7). The other composition in the diagram is D^{f,k} · Dψ Df. Similarly we have that D(h φ) · D^{f,h} = D^{g,h} · (Dh Dφ) is shown by the commutativity of

\xymatrix{Dh Df \ar[r]^{𝔻_{dB f,dC h}} \ar[d]_{Dh Dφ} & (Dh dB f)^{𝔻} \ar[r]^{(𝔻_{dC h}f)^{𝔻}} \ar[d]_{(Dh dB φ)^{𝔻}} & Dhf \ar[d]^{(dC hφ)^{𝔻}} \\ Dh Dg \ar[r]_{𝔻_{dB g,dC h}} & (Dh dB g)^{𝔻} \ar[r]_{(𝔻_{dC h}g)^{𝔻}} & Dhg,}

using naturality and (6).

Furthermore, we have

D^{Id_{A},f} · Df D_{A} = (𝔻_{dB f})^{𝔻} · 𝔻_{dA,dB f} · (dB f)^{𝔻}D_{A} = id_{Df}

by (2) applied to dB f, and

D^{f,Id_{B}} · D_{B}Df = (𝔻_{dB}f)^{𝔻} · 𝔻_{dB f,dB} · D_{B}(dB f)^{𝔻}
= (𝔻_{dB}f)^{𝔻} · (D_{B}dB f)^{𝔻}
= (𝔻_{dB}f · D_{B}dB f)^{𝔻} = (id_{dB f})^{𝔻} = id_{Df}

by (3) and (1). Finally, we show that D^{hf,ℓ} · DℓD^{f,h} = D^{f,ℓh} · D^{h,ℓ}Df by means of the following commutative diagram, where the label inside each subdiagram is the corresponding axiom needed:

[figure: Dℓ Dh Df; Dℓ(Dh dB f)^{𝔻}; Dℓ Dhf; (Dℓ Dh dB f)^{𝔻}; (Dℓ dC hf)^{𝔻}; ((Dℓ dC h)^{𝔻}dB f)^{𝔻}; Dℓhf; (Dℓ dC h)^{𝔻}Df; Dhℓ Df; (Dℓh dB f)^{𝔻}; (8); (6); (5); (7); (4); Dℓ 𝔻_{dB f,dC h}; 𝔻_{dC h,dE ℓ}Df; Dℓ(𝔻_{dC h}f)^{𝔻}; 𝔻_{Dh dB f,dE ℓ}; 𝔻_{dC hf,dE ℓ}; (Dℓ 𝔻_{dC h}f)^{𝔻}; (𝔻_{dC h,dE ℓ}dB f)^{𝔻}; (𝔻_{dE ℓ}hf)^{𝔻}; (𝔻_{Dℓ dC h}f)^{𝔻}; ((𝔻_{dE ℓ}h)^{𝔻}dB f)^{𝔻}; 𝔻_{dB f,Dℓ dC h}; (𝔻_{dE ℓ}h)^{𝔻}Df; 𝔻_{dB f,dE ℓh}; (𝔻_{dE ℓh}f)^{𝔻}]

We now extend the family of 1-cells ⟨dA: A → DA⟩_{A∈A} to a strong transformation. 3.2. Proposition. If for every f : A → B we define d_{f} := 𝔻_{dB f}:

[figure: A; DA; B; DB.; dA; f; (dB f)^{𝔻}=Df; dB; d_{f}]

then we obtain a strong transformation d: Id_{A} → D. Proof. The only non-trivial condition is

d_{h}f · Dh d_{f} = 𝔻_{dC h}f · Dh 𝔻_{dB f} = 𝔻_{dC h}f · 𝔻_{Dh dB f} · 𝔻_{dB f,dC h}dA
= 𝔻_{dC h f} · (𝔻_{dC h}f)^{𝔻}dA · 𝔻_{dB f,dC h}dA = d_{hf} · D^{f,h}

by (5) and (4).

We now define a strong transformation m: D^{2} → D. 3.3. Proposition. If for every A we define mA = (Id_{DA})^{𝔻} and for every f : A → B we define m_{f} as

[figure: D^{2}A; DA; D^{2}B; DB; (Id_{DA})^{𝔻}; D^{2}f; (Id_{DB})^{𝔻}; Df; (Df)^{𝔻}; ((Id_{DB})^{𝔻}dDB Df)^{𝔻}; 𝔻_{Id_{DA},dB f}; (𝔻_{Id_{DB}}^{−1} Df)^{𝔻}; 𝔻_{dDB Df,Id_{DB}}^{−1}]

then we obtain a strong transformation m: D^{2} → D. Proof. For φ: f → g: A → B the diagram

\xymatrix{Df mA \ar[r]^{m_{f}} \ar[d]_{Dφ mA} & mB D^{2}f \ar[d]^{mB D^{2}φ} \\ Dg mA \ar[r]_{m_{g}} & mB D^{2}g}

is the exterior of the diagram

\xymatrix{Df mA \ar[r]^{𝔻_{Id_{DA},dB f}} \ar[d]^{Dφ mA} & Df^{𝔻} \ar[r]^{(𝔻Id_{DB}^{−1} Df)^{𝔻}} \ar[d]_{Dφ^{𝔻}} & (mB dDB Df)^{𝔻} \ar[r]^{𝔻_{dDB Df,Id_{DB}}^{−1}} \ar[d]_{(mB dDB Dφ)^{𝔻}} & mB D^{2}f \ar[d]_{mB D^{2}φ} \\ Dg mA \ar[r]_{𝔻_{Id_{DA},dB g}} & Dg^{𝔻} \ar[r]_{(𝔻Id_{DB}^{−1} Dg)^{𝔻}} & (mB dDB Dg)^{𝔻} \ar[r]_{𝔻_{dDB Dg,Id_{DB}}^{−1}} & mB D^{2}g.}

The first rectangle commutes because of (7), the second by naturality, and the third by

Now take f : A → B, h: B → C. The equality (D^{2})^{f,g}·m_{h} D^{2}f ·Dh m_{f} = m_{hf} ·mA D^{f,h} is proved by the following commutative diagram:

[figure: (8); (7); (8); (6); (8); (7); (6); (∗); 𝔻_{dB f,dC h}mA; Dh𝔻_{Id_{DA},dB f}; (𝔻_{dC h}f)^{𝔻}mA; 𝔻_{Id_{DA},Dh dB f}; 𝔻_{Id_{DA},dC hf}; (𝔻_{Id_{DC}}^{−1} Dh f)^{𝔻}; 𝔻_{dDC Dhf,Id_{DC}}^{−1}; (𝔻_{dC h}f)^{𝔻𝔻}; (𝔻_{dB f,dC h}^{−1})^{𝔻}; 𝔻_{Df,dC h}; Dh(𝔻_{Id_{DB}}^{−1} Df)^{𝔻}; (Dh𝔻_{Id_{DB}}^{−1} Df)^{𝔻}; 𝔻_{mB dDB Df,dC h}; Dh𝔻_{dDB Df,Id_{DB}}; (𝔻_{Id_{DB} ,dC h}dDB Df)^{𝔻}; 𝔻_{Id_{DB} ,dC h}D^{2}f; 𝔻_{dDB Df,Dh}^{−1}; ((𝔻_{Id_{DC}}^{−1} Dh)^{𝔻}dDB Df)^{𝔻}; (𝔻_{Id_{DC}}^{−1} Dh)^{𝔻}D^{2}f; 𝔻_{dDB Df,mC dDC Dh}; 𝔻_{dDC Dh,Id_{DC}}^{−1} D^{2}f; (𝔻_{dDC Dh,Id_{DC}}^{−1} dDB Df)^{𝔻}; 𝔻_{D^{2}h dDB Df,Id_{DC}}^{−1}; (mC𝔻_{dDC Dh}Df)^{𝔻}; mC𝔻_{dDB Df,dDC Dh}; mC(𝔻_{dDC Dh}Df)^{𝔻}; (mC dDC𝔻_{dB f,dC h})^{𝔻}; (mC dDC(𝔻_{dC h}f)^{𝔻})^{𝔻}; mC(dDC𝔻_{dB f,dC h})^{𝔻}; mC D((𝔻_{dC h}f)^{𝔻})]

where the labels inside the subdiagrams indicate the equation needed for its commutativity, and (∗) is ( )^{𝔻} of the following commutative diagram:

[figure: (5); (4); (5); 𝔻_{dB f,dC h}^{−1}; id_{(Dh dB f)^{𝔻}}; Dh𝔻_{Id_{DB}}^{−1} Df; 𝔻_{Id_{DB} ,dC h}dDB Df; (𝔻_{Id_{DC}}^{−1} Dh)^{𝔻}dDB Df; 𝔻_{Dh}Df; 𝔻_{dDC Dh,Id_{DC}}^{−1} dDB Df; 𝔻_{mC dDC Dh}Df; mC𝔻_{dDC Dh}Df; mC dDC𝔻_{dB f,dC h}; 𝔻_{Id_{DC}}^{−1} Dh Df; 𝔻_{dB f,dC h}; (𝔻_{dC h}f)^{𝔻}; mC dDC(𝔻_{dC h}f)^{𝔻}; 𝔻_{Id_{DC}}^{−1} Dhf]

where the unlabeled subdiagram commutes by naturality.

Finally,

m_{Id_{DA}} · D_{A} mA = 𝔻_{dDA DId_{DA},Id_{DA}}^{−1} · (𝔻−1_{IdDA}DId_{A})^{𝔻} · 𝔻_{Id_{DA},dA} · D_{A} mA
= 𝔻_{dDA DId_{DA},Id_{DA}}^{−1} · (𝔻−1_{IdDA}DId_{A})^{𝔻} · (D_{A})^{𝔻}
= 𝔻_{dDA DId_{DA},Id_{DA}}^{−1} · (mA dDA D_{A})^{𝔻} · (𝔻−1_{IdDA})^{𝔻}
= mA D(D_{A}) · 𝔻_{dDA,Id_{DA}}^{−1} · (𝔻−1_{IdDA})^{𝔻}
= mA D(D_{A}) · mA D_{_{Id}_{DA}} = mA (D^{2})_{A}

by (3), naturality, (6) and (2).

The following two propositions give the 2-dimensional data for the construction of a pseudomonad.

378  F. MARMOLEJO AND R. J. WOOD

3.4. Proposition. If for every A in A we define βA = 𝔻_{_{Id}_{DA}} and we define ηA as

[figure: DA; D^{2}A; DA; DdA; (mA dDA dA)^{𝔻}; dA^{𝔻}; ^{Id^{DA}}; mA; 𝔻_{dDA dA,IdDA}^{−1}; (𝔻_{IdDA}^{−1}dA)^{𝔻}; D_{A}]

then we obtain modifications β: mA dD → Id_{D} and η : Id_{D} → mA DdA. Proof. The statement for β follows from the following commutative diagram for every f : A → B:

[figure: Df mA dDA; Df^{𝔻}dDA; (mB dDB Df)^{𝔻}dDA; mB D^{2}f dDA; mB dDB Df; mB dDB Df; Df; (5); (4); (5); 𝔻_{Id_{DA},dB f} dDA; Df𝔻_{Id_{DA}}; (𝔻_{Id_{DB}}^{−1} Df)^{𝔻}dDA; 𝔻_{Df}; 𝔻_{dDB Df,Id_{DB}}^{−1} dDA; 𝔻_{mB dDB Df}; mB𝔻_{dDA Df}; 𝔻_{Id_{DB}} Df]

The one for η follows from the commutative diagram

[figure: (2); (6); (8); (7); (8); (6); (3); (5); (8); (∗); Df D_{A}; id_{Df}; (𝔻_{dB f}^{−1} )^{𝔻}; D_{B} Df; Df(𝔻_{Id_{DA}}^{−1}dA)^{𝔻}; 𝔻_{dA,dB f}; 𝔻_{mA dDA dA,dB f}; Df𝔻_{dDA dA,Id_{DA}}^{−1}; (Df𝔻_{Id_{DA}}^{−1}dA)^{𝔻}; (𝔻_{Id_{DA},dB f} dDA dA)^{𝔻}; 𝔻_{Id_{DA},dB f} DdA; 𝔻_{dDA dA,Df}^{−1}; ((𝔻_{Id_{DA}}^{−1}Df)^{𝔻}dDA dA)^{𝔻}; (𝔻_{Id_{DB}}^{−1} Df)^{𝔻}DdA; (𝔻_{dDB Df,Id_{DB}}^{−1} dDA dA)^{𝔻}; 𝔻_{dDA dA,mB dDB Df}; 𝔻_{dDB Df,Id_{DB}} dDA; mB𝔻_{dDA dA,dDB Df}; 𝔻_{D^{2}f dDA dA,Id_{DB}}^{−1}; (mB𝔻_{dDB Df} dA)^{𝔻}; mB(𝔻_{dDB Df} dA)^{𝔻}; (mB dDB𝔻_{dB f})^{𝔻}; mB(dDB𝔻_{dB f})^{𝔻}; (mB𝔻_{dDB dB}^{−1}f)^{𝔻}; mB(𝔻_{dDB dB}^{−1}f)^{𝔻}; 𝔻_{DdB dB f,Id_{DB}}^{−1}; (𝔻_{dDB dB,Id_{DB}} dB f)^{𝔻}; mB𝔻_{dB f,dDB dB}^{−1}; 𝔻_{dB f,mB dDB dB}^{−1}; 𝔻_{dDB dB,Id_{DB}}^{−1} Df; (𝔻_{dB f})^{𝔻}; (𝔻_{B} dB f)^{𝔻}; (𝔻_{Id_{DB}}^{−1} dB)^{𝔻}Df; 𝔻_{dB f,dB}; ((𝔻_{Id_{DB}} dB)^{𝔻}dB f)^{𝔻}]

NO-ITERATION PSEUDOMONADS  379

where (∗) is ( )^{𝔻} applied to the following commutative diagram

[figure: Df dA; Df mA dDA dA; Df^{𝔻} dDA dA; Df dA; dB f; (mB dDB Df)^{𝔻}dDA dA; mB dDB Df dA; mB D^{2}f dDA dA; mB dDB dB f; mB DdB dB f; (mB dDB dB)^{𝔻}dB f; dB^{𝔻}dB f; (5); (4); (5); (4); (5); (1); 𝔻_{dB}^{−1}f; 𝔻_{B} dB f; Df𝔻_{Id_{DA}}^{−1}dA; 𝔻_{dB f}; id_{Df dA}; 𝔻_{Id_{DA},dB f} dDA dA; 𝔻_{Df} dA; (𝔻_{Id_{DB}}^{−1} Df)^{𝔻}dDA dA; 𝔻_{Id_{DB}}^{−1} Df dA; 𝔻_{Id_{DB}}^{−1} dB f; 𝔻_{mB dDB Df} dA; 𝔻_{dDB Df,Id_{DB}}^{−1} dDA dA; mB dDB 𝔻_{dB f}; mB 𝔻_{dDB Df} dA; mB𝔻_{dDB dB}^{−1}f; 𝔻_{mB dDB dB}^{−1}f; 𝔻_{dDB dB,Id_{DB}} dB f; (𝔻_{Id_{DB}}^{−1} dB)^{𝔻}dB f]

3.5. Proposition. If for every A we define μA as

[figure: D^{3}A; D^{2}A; D^{2}A; DA; DmA; mDA; mA; mA; (mA dDA mA)^{𝔻}; mA^{𝔻}; 𝔻_{dDA mA,Id_{DA}}; (𝔻_{Id_{DA}}mA)^{𝔻}; 𝔻_{Id_{D^{2}A},Id_{DA}}^{−1}]

380  F. MARMOLEJO AND R. J. WOOD

then we obtain a modification μ : mA DmA → mA mDA. Proof. The proof is obtained by glueing the following three commutative diagrams, first the last two along the dashed arrow, and then the resulting diagram with the first along the dotted arrows:

[figure: (†); Df𝔻_{dDA mA,Id_{DA}}; 𝔻_{Id_{DA},dB f} DmA; 𝔻_{mA dDA mA,dB f}; Df(𝔻_{Id_{DA}}mA)^{𝔻}; 𝔻_{dDA mA,Df}; (𝔻_{Id_{DB}}^{−1} Df)^{𝔻}DmA; (𝔻_{Id_{DA},dB f}^{−1} dDA mA)^{𝔻}; ((𝔻_{Id_{DB}} Df)^{𝔻}dDA mA)^{𝔻}; (Df𝔻_{Id_{DA}}mA)^{𝔻}; 𝔻_{mA,dB f}; Df𝔻_{Id_{D^{2}A,Id_{DA}}}^{−1}; (𝔻_{Id_{DA},dB f})^{𝔻}; (m_{f})^{𝔻}; 𝔻_{Id_{DA,dB f}} mDA; 𝔻_{Id_{D^{2}A,Df}}^{−1}; (𝔻_{Id_{DB}}^{−1} Df)^{𝔻𝔻}; (𝔻_{Id_{DB}}^{−1} Df)^{𝔻}mDA; (𝔻_{dDB Df,Id_{DB}}^{−1} )^{𝔻}; 𝔻_{Id_{D^{2}A},mB dDB Df}; (mB𝔻_{Id_{D^{2}B}}^{−1} D^{2}f)^{𝔻}; (𝔻_{Id_{D^{2}B} ,Id_{DB}} dD^{2}B D^{2}f)^{𝔻}; 𝔻_{dDA mA,mB dDB Df}; 𝔻_{dDB Df,Id_{DB}}^{−1} DmA; mB𝔻_{dDA mA,dDB Df}; (𝔻_{dDB Df,Id_{DB}}^{−1} dDA mA)^{𝔻}; mB(𝔻_{dDB Df} mA)^{𝔻}; 𝔻_{D^{2}f dDA mA,Id_{DB}}; (mB𝔻_{dDB Df} mA)^{𝔻}; 𝔻_{dDB Df mA,Id_{DB}}; mB Dm_{f}; (mB dDB m_{f})^{𝔻}; 𝔻_{dDB mB D^{2}f,Id_{DB}}^{−1}; (mB𝔻_{dDB mB}^{−1}D^{2}f)^{𝔻}; (𝔻_{dDB mB,Id_{DB}} dD^{2}B D^{2}f)^{𝔻}; ((𝔻_{Id_{DB}} mB)^{𝔻}dD^{2}B D^{2}f)^{𝔻}]

[figure: (𝔻_{dDB Df,Id_{DB}}^{−1} )^{𝔻}; 𝔻_{Id_{D^{2}A},mB dDB Df}; 𝔻_{dDB Df,Id_{DB}}^{−1} mDA; 𝔻_{D^{2}f,mB}^{−1}; (mB𝔻_{Id_{D^{2}B}}^{−1} D^{2}f)^{𝔻}; mB𝔻_{Id_{D^{2}A},dDB Df}; mB(𝔻_{Id_{D^{2}B}}^{−1} D^{2}f)^{𝔻}; 𝔻_{mDB dD^{2}B D^{2}f,Id_{DB}}^{−1}; (𝔻_{Id_{D^{2}B} ,Id_{DB}} dD^{2}B D^{2}f)^{𝔻}; mB𝔻_{dD^{2}B D^{2}f,Id_{D^{2}B}}; 𝔻_{dD^{2}B D^{2}f,mB}^{−1}; 𝔻_{Id_{D^{2}B} ,Id_{DB}} D^{3}f; 𝔻_{dD^{2}B D^{2}f,mB}^{−1}; 𝔻_{dDB mB D^{2}f,Id_{DB}}^{−1}; (mB𝔻_{dDB mB}^{−1}D^{2}f)^{𝔻}; mB(𝔻_{dDB mB}^{−1}D^{2}f)^{𝔻}; 𝔻{−1 \atop DmB dD^{2}B D^{2}f,Id_{DB}}; (𝔻_{dDB mB,Id_{DB}} dD^{2}B D^{2}f)^{𝔻}; mB𝔻_{dD^{2}B D^{2}f,dDB mB}^{−1}; 𝔻_{dD^{2}B D^{2}f,mB dDB mB}^{−1}; ((𝔻_{Id_{DB}} mB)^{𝔻}dD^{2}B D^{2}f)^{𝔻}; 𝔻_{dDB mB,Id_{DB}}^{−1} D^{3}f; (𝔻_{Id_{DB}} mB)^{𝔻}D^{3}f]

NO-ITERATION PSEUDOMONADS  381

where (†) is ( )^{𝔻} applied to the commutative diagram

[figure: 𝔻_{Id_{DA},dB f}; (𝔻_{Id_{DB}}^{−1} Df)^{𝔻}dDA mA; 𝔻_{Df} mA; Df𝔻_{Id_{DA}}mA; 𝔻_{Id_{DB}}^{−1} Df mA; m_{f}; 𝔻_{mB dDB Df} mA; 𝔻{−1 \atop dDB Df,Id_{DB}} dDA mA; mB dDB m_{f}; mB𝔻_{dDB Df} mA; 𝔻_{Id_{DB}}^{−1} dDB D^{2}f; mB𝔻_{Id_{D^{2}B}}^{−1} D^{2}f; 𝔻_{mB}^{−1}D^{2}f; 𝔻_{mB dDB mB}^{−1}D^{2}f; mB𝔻_{dDB mB}^{−1}D^{2}f; 𝔻_{Id_{D^{2}B} ,Id_{DB}} dD^{2}B D^{2}f; (𝔻_{Id_{DB}}^{−1} mB)^{𝔻}dD^{2}B D^{2}f; id; (𝔻_{Id_{DB}} mB)^{𝔻}dD^{2}B D^{2}f; 𝔻_{dDB mB,Id_{DB}} dD^{2}B D^{2}f]

We now show that we have indeed produced a pseudomonad.

3.6. Theorem. Let 𝔻 be a no-iteration monad. The above constructions of the pseudofunctor D, the strong transformations d and m, and the modifications β, η, μ constitute a pseudomonad (D, d, m, β, η, μ).

Proof. We must show the usual two coherence conditions. The equation μA DdDA = (mA DβA)(mA ηDA) is shown by the commutative diagram

[figure: (†); 𝔻_{dDA mA,Id_{DA}}DdDA; mA𝔻_{dD^{2}A dDA,dDA mA}; (𝔻_{Id_{DA}}mA)^{𝔻}DdDA; 𝔻_{dD^{2}A dDA,mA dDA mA}; 𝔻_{Id_{D^{2}A},Id_{DA}}^{−1}DdDA; ((𝔻_{Id_{DA}}mA)^{𝔻}dD^{2}A dDA)^{𝔻}; (𝔻_{dDA mA,Id_{DA}}^{−1}dD^{2}A dDA)^{𝔻}; 𝔻_{dD^{2}A,mA}^{−1}; 𝔻_{DmA dD^{2}A dDA,Id_{DA}}; mA(𝔻_{dDA mA}dDA)^{𝔻}; (mA𝔻_{dDA mA}dDA)^{𝔻}; (𝔻_{Id_{D^{2}A},Id_{DA}}dD^{2}A dDA)^{𝔻}; 𝔻_{mDA dD^{2}A dDA,Id_{DA}}^{−1}; mA𝔻_{dD^{2}A dDA,Id_{D^{2}A}}; (mA dDA𝔻_{Id_{DA}})^{𝔻}; (mA𝔻_{Id_{D^{2}A}}^{−1}dDA)^{𝔻}; 𝔻_{dDA,Id_{DA}}^{−1}; 𝔻_{dDA mA dDA,Id_{DA}}; mA(dDA𝔻_{Id_{DA}})^{𝔻}; mA(𝔻_{Id_{D^{2}A}}^{−1}dDA)^{𝔻}]

where (†) is ( )^{𝔻} applied to the commutative diagram

[figure: (𝔻_{Id_{DA}}mA)^{𝔻}dD^{2}A dDA; id; id; (𝔻_{Id_{DA}}^{−1}mA)^{𝔻}dD^{2}A dDA; 𝔻_{mA}dDA; 𝔻_{dDA mA,Id_{DA}}^{−1}dD^{2}A dDA; 𝔻_{mA dDA mA}dDA; 𝔻_{Id_{DA}}mA dDA; id; 𝔻_{Id_{D^{2}A},Id_{DA}}dD^{2}A dDA; mA𝔻_{dDA mA}dDA; mA dDA𝔻_{Id_{DA}}; 𝔻_{mA}^{−1}dDA; mA𝔻_{Id_{D^{2}A}}dDA]

The proof of the other coherence condition is given by the commutative diagram after the following one, where (†) is ( )^{𝔻} applied to the commutative diagram

[figure: 𝔻_{mA dDA mA}DmA; 𝔻_{dDA mA,Id_{DA}}^{−1}dD^{2}A DmA; (𝔻_{Id_{DA}}mA)^{𝔻}dD^{2}A DmA; mA𝔻_{dDA mA}DmA; 𝔻_{Id_{DA}}mA DmA; mA dDA𝔻_{dDA mA,Id_{DA}}; mA dDA μA; 𝔻_{dDA mA,Id_{DA}}; mA𝔻_{Id_{D^{2}A}}DmA; 𝔻_{mA}DmA; 𝔻_{Id_{D^{2}A},Id_{DA}}^{−1}dD^{2}A DmA; 𝔻_{Id_{DA}}(mA dDA mA)^{𝔻}; mA dDA(𝔻_{Id_{DA}}mA)^{𝔻}; (𝔻_{Id_{DA}}mA)^{𝔻}; 𝔻_{Id_{DA}}mA^{𝔻}; id; mA𝔻_{dDA mA}^{−1}mDA; mA dDA𝔻_{Id_{D^{2}A},Id_{DA}}; 𝔻_{Id_{DA}}mA mDA; 𝔻_{dDA mA,Id_{DA}}dD^{2}A mDA; 𝔻_{mA dDA mA}mDA; (𝔻_{Id_{DA}}mA)^{𝔻}dD^{2}A mDA; 𝔻_{Id_{D^{2}A},Id_{DA}}^{−1}dD^{2}A mDA; 𝔻_{mA}mDA; mA𝔻_{Id_{D^{2}A}}mDA; 𝔻_{Id_{D^{2}A},Id_{DA}}]

[figure: (†); 𝔻_{dDA mA,Id_{DA}}D^{2}mA; mA𝔻_{dD^{2}A DmA,dDA mA}; 𝔻_{dD^{2}A DmA,mA dDA mA}; (𝔻_{Id_{DA}}mA)^{𝔻}D^{2}mA; ((𝔻_{Id_{DA}}mA)^{𝔻}dD^{2}A DmA)^{𝔻}; (𝔻_{dDA mA,Id_{DA}}^{−1}dD^{2}A DmA)^{𝔻}; 𝔻_{dD^{2}A DmA,mA}; 𝔻_{Id_{D^{2}A},Id_{DA}}^{−1}D^{2}mA; (𝔻_{Id_{D^{2}A},Id_{DA}}dD^{2}A DmA)^{𝔻}; 𝔻_{mDA dD^{2}A DmA,Id_{DA}}^{−1}; (mA𝔻_{Id_{D^{2}A}}DmA)^{𝔻}; mA𝔻_{dD^{2}A DmA,Id_{D^{2}A}}; mA(𝔻_{Id_{D^{2}A}}DmA)^{𝔻}; (𝔻_{dDA mA,Id_{DA}})^{𝔻}; 𝔻_{DmA,Id_{DA}}; mA𝔻_{Id_{D^{3}A},dDA mA}^{−1}; 𝔻_{Id_{D^{3}A},mA dDA mA}^{−1}; (𝔻_{Id_{DA}}mA)^{𝔻𝔻}; (𝔻_{Id_{DA}}mA)^{𝔻}mD^{2}A; 𝔻_{dDA mA,Id_{DA}}mD^{2}A; 𝔻_{Id_{D^{3}A},mA}^{−1}; 𝔻_{Id_{D^{2}A},Id_{DA}}^{−1}mD^{2}A; (𝔻_{Id_{D^{2}A},Id_{DA}})^{𝔻}; 𝔻_{mDA,Id_{DA}}^{−1}; mA𝔻_{Id_{D^{3}A},Id_{D^{2}A}}; (mA𝔻_{Id_{D^{2}A}}mDA)^{𝔻}; 𝔻_{mDA dD^{2}A mDA,Id_{DA}}; mA(𝔻_{Id_{D^{2}A}}mDA)^{𝔻}; (𝔻_{Id_{D^{2}A},Id_{DA}}^{−1}dD^{2}A mDA)^{𝔻}; 𝔻_{dD^{2}A mDA,mA}; mA𝔻_{dD^{2}A mDA,Id_{D^{2}A}}; 𝔻_{Id_{D^{2}A},Id_{DA}}^{−1}DmDA; ((𝔻_{Id_{DA}}mA)^{𝔻}dD^{2}A mDA)^{𝔻}; 𝔻_{dD^{2}A mDA,mA dDA mA}^{−1}; (𝔻_{Id_{DA}}mA)^{𝔻}DmDA; 𝔻_{DmA dD^{2}A DmA,Id_{DA}}; mA(𝔻_{dDA mA}DmA)^{𝔻}; (mA𝔻_{dDA mA}DmA)^{𝔻}; 𝔻_{dDA mA DmA,Id_{DA}}; mA(dDA μA)^{𝔻}; (mA dDA μA)^{𝔻}; 𝔻_{dDA mA mDA,Id_{DA}}; mA(𝔻_{dDA mA}^{−1}mDA)^{𝔻}; (mA𝔻_{dDA mA}^{−1}mDA)^{𝔻}; 𝔻_{DmA dD^{2}A mDA,Id_{DA}}; mA𝔻_{dD^{2}A mDA,dDA mA}^{−1}; (𝔻_{dDA mA,Id_{DA}}dD^{2}A mDA)^{𝔻}; 𝔻_{dDA mA,Id_{DA}}DmDA]

4. The algebras for a no-iteration pseudomonad

Let 𝔻 be a no-iteration pseudomonad as in Definition 2.1. Following the same no-iteration framework, we define the 2-category Âlg-𝔻 of algebras for such a presentation as follows.

An object of Âlg-𝔻 consists of

1. An object A in A.

2. For every X in A a functor ( )^{𝔸} : A(X, A) → A(DX, A).

3. For every h : X → A an invertible 2-cell

[figure: X; DX; A; dX; h^{𝔸}; h; 𝔸_{h}]

4. For every h : X → A and every g : Y → DX, an invertible 2-cell

[figure: DY; DX; A; g^{𝔻}; h^{𝔸}; (h^{𝔸}g)^{𝔸}; 𝔸_{g,h}]

subject to the axioms 1. For every φ : h → k : X → A

[figure: X; DX; A; dX; k^{𝔸}; h^{𝔸}; k; 𝔸_{k}; φ^{𝔸}] = [figure: X; DX; A; dX; k; h; h^{𝔸}; 𝔸_{h}; φ]  (9)

2. For every h: X → A

[figure: DX; DX; A; ^{Id^{DX}}; dX^{𝔻}; (h^{𝔸}dX)^{𝔸}; h^{𝔸}; h^{𝔸}; 𝔸_{dX,h}; D_{X}; (𝔸_{h})^{𝔸}] = id_{h^{𝔸}}  (10)

3. For every h: X → A and every g: Y → DX

[figure: Y; DY; DX; A; dY; g; g^{𝔻}; h^{𝔸}; 𝔻_{g}] = [figure: Y; DY; DX; A.; dY; h^{𝔸}g; (h^{𝔸}g)^{𝔸}; g^{𝔻}; h^{𝔸}; 𝔸_{(h^{𝔸}g)}; 𝔸_{g,h}]  (11)

4. For every h: X → A and every ψ: f → g : Y → DX

[figure: DY; DX; A; f^{𝔻}; g^{𝔻}; (h^{𝔸}g)^{𝔸}; h^{𝔸}; 𝔸_{g,h}; ψ^{𝔻}] = [figure: DY; DX; A; f^{𝔻}; (h^{𝔸}f)^{𝔸}; (h^{𝔸}g)^{𝔸}; h^{𝔸}; 𝔸_{f,h}; (h^{𝔸}ψ)^{𝔸}]  (12)

5. For every φ: h → k: X → A and every g: Y → DX

[figure: DY; DX; A; g^{𝔻}; k^{𝔸}; (k^{𝔸}g)^{𝔸}; h^{𝔸}; 𝔸_{g,k}; φ^{𝔸}] = [figure: DY; DX; A; g^{𝔻}; (h^{𝔸}g)^{𝔸}; (k^{𝔸}g)^{𝔸}; h^{𝔸}; 𝔸_{g,h}; (φ^{𝔸}g)^{𝔸}]  (13)

6. For every h: X → A, g: Y → DX and ℓ: Z → DY

[figure: DZ; DY; DX; A; ℓ^{𝔻}; g^{𝔻}; h^{𝔸}; (g^{𝔻}ℓ)^{𝔻}; (h^{𝔸}g^{𝔻}ℓ)^{𝔸}; ((h^{𝔸}g)^{𝔸}ℓ)^{𝔸}; 𝔻_{ℓ,g}; 𝔸_{g^{𝔻}ℓ,h}; (𝔸_{h,g}ℓ)^{𝔸}] = [figure: DZ; DY; DX; A; ℓ^{𝔻}; g^{𝔻}; h^{𝔸}; (h^{𝔸}g)^{𝔸}; ((h^{𝔸}g)^{𝔸}ℓ)^{𝔸}; 𝔸_{g,h}; 𝔸_{ℓ,h^{𝔸}g}] (14)

Given objects (A, ( )^{𝔸}) and (B, ( )^{𝔹}) in Âlg-𝔻 (we omit the rest of the structure), a 1-cell in Âlg-𝔻 is a 1-cell f : A → B in A together with an invertible 2-cell

[figure: DX; A; B; h^{𝔸}; (fh)^{𝔹}; f; f[h]]

for every h : X → A in A, subject to the coherence conditions

[figure: X; DX; A; B; dX; h; h^{𝔸}; (fh)^{𝔹}; f; 𝔸_{h}; f[h]] = [figure: X; DX; A; B; dX; h; (fh)^{𝔹}; f; 𝔹_{fh}]  (15)

and for any g: Y → DX

[figure: DY; DX; A; B; g^{𝔻}; (h^{𝔸}g)^{𝔸}; h^{𝔸}; (fh)^{𝔹}; f; 𝔸_{g,h}; f[h]] = [figure: DY; DX; A; B; g^{𝔻}; (h^{𝔸}g)^{𝔸}; ((fh)^{𝔹}g)^{𝔹}; (fh^{𝔹}g)^{𝔹}; (fh)^{𝔹}; f; 𝔹_{g,fh}; (f[h] g)^{𝔹}; f[h^{𝔸}g]]  (16)

and for every κ: h → k: X → A,

[figure: DX; A; B; k^{𝔸}; h^{𝔸}; (fh)^{𝔹}; f; κ^{𝔸}; f[h]] = [figure: DX; A; B; (fk)^{𝔹}; (fh)^{𝔹}; k^{𝔸}; f; (fκ)^{𝔹}; f[k]]  (17)

For 1-cells f : A → B, f′: B → C in Âlg-𝔻, composition is given by the usual composition f′f in A and, for every h: X → A, the 2-cell

(f′f)[h] = [figure: DX; A; B; C; h^{𝔸}; (fh)^{𝔹}; (f′fh)^{ℂ}; f; f′; f[h]; f′[fh]]

It is direct to show that the composition f′f satisfies the conditions (15), (16) and (17). A 2-cell α : f → g : A → B in Âlg-𝔻 is a 2-cell α : f → g in A such that for every h: X → A we have that

[figure: DX; A; B; h^{𝔸}; (gh)^{𝔹}; (fh)^{𝔹}; g; g[h]; (αh)^{𝔹}] = [figure: DX; A; B; h^{𝔸}; (fh)^{𝔹}; g; f; f[h]; α]  (18)

Composition of 2-cells in Âlg-𝔻 is as in A.

For every object A in Âlg-𝔻, Id_{A} is the usual Id_{A} : A → A in A together with the identity two cell

[figure: DX; A; A; h^{𝔸}; h^{𝔸}; Id_{A}; Id_{A}[h]]

on h^{𝔸} for every h : X → A in A.

5. The biequivalence 𝔻-Alg ≃ Âlg-𝔻

In this section we consider a no-iteration pseudomonad, called 𝔻, as in Definition 2.1, and induce the pseudomonad, also called 𝔻, as in Section 3. We denote the usual category of algebras for 𝔻 as 𝔻-Alg.

5.1. Theorem. The categories 𝔻-Alg and Âlg-𝔻 are biequivalent.

Proof. We define pseudofunctors in both directions.

Define F : Âlg-𝔻 → 𝔻-Alg as follows. Take [figure: (A, ( )^{𝔸}); (B, ( )^{𝔹}); f; g; φ] in Âlg-𝔻. Let

α_{0} = Id_{A}^{𝔸}, α_{1} = 𝔸_{Id_{A}}, β_{0} = Id_{B}^{𝔹}, β_{1} = 𝔹_{Id_{B}}, and define F A as

( [figure: A; DA; A; dA; Id_{A}; α_{0}; α_{1}] , [figure: D^{2}A; DA; DA; A; D(α_{0}); mA; (α_{0}dA α_{0})^{𝔸}; ^{α_{0}^{𝔸}}; α_{0}; α_{0}; 𝔸_{dA α_{0} ,Id_{A}}; (α_{1}α_{0})^{𝔸}; _{𝔸_{Id_{DA},Id_{A}}_{−1}}] )  (19)

For convenience, we write α_{2} for the second coordinate of F A. Define F f as

[figure: DA; DB; A; B; Df; α_{0}; (β_{0}dB f)^{𝔹}; f^{𝔹}; β_{0}; f; 𝔹_{dB f,Id_{B}}; (β_{1}f)^{𝔹}; f[Id_{A}]]

For future reference, call this two cell ρ. And define F φ = φ.

We must verify first that F A is indeed an object in 𝔻-Alg, but this is essentially the same proof given in Theorem 3.6 except that some 𝔻’s should now be 𝔸’s. We must also

NO-ITERATION PSEUDOMONADS  387

show that F f : F A → F B is in 𝔻-Alg. The diagram for this has the same shape, namely

[figure: (†); 𝔹_{dB β_{0} ,Id_{B}} D^{2}f; β_{0}𝔻_{dDB Df,dB β_{0}}; 𝔹_{dDB Df,β_{0} dB β_{0}}; (β_{1}β_{0})^{𝔹}D^{2}f; ((β_{1}β_{0})^{𝔹}dDB Df)^{𝔹}; ((𝔹_{dB β_{0} ,Id_{B}}^{−1} )^{𝔹} dDB Df)^{𝔹}; 𝔹_{dDB Df,β_{0}}; 𝔹_{Id_{DB} ,Id_{B}}^{−1} D^{2}f; (𝔹_{Id_{DB} ,Id_{B}}^{−1} dDB Df)^{𝔹}; 𝔹_{mB dDB Df,Id_{B}}^{−1}; (β_{0}𝔻_{Id_{DB}} Df)^{𝔹}; β_{0}𝔻_{dDB Df,Id_{DB}}; β_{0}(𝔻_{Id_{B}} Df)^{𝔻}; (𝔹_{dB f,Id_{B}})^{𝔹}; 𝔹_{Df,Id_{B}}; β_{0}𝔻_{Id_{DA},dB f}^{−1}; 𝔹_{Id_{DA},β_{0} dB f}^{−1}; (β_{1}f)^{𝔹𝔹}; (β_{1}f)^{𝔹}mA; 𝔹_{dB f,Id_{B}} mA; 𝔹_{Id_{DA},f}^{−1}; f[Id_{A}]mA; (f[Id_{A}]^{−1})^{𝔹}; f[α_{0}]; f𝔸_{Id_{DA},Id_{A}}^{−1}; (fα_{1}α_{0})^{𝔹}; f[α_{0}dA α_{0}]; f (α_{1}α_{0})^{𝔸}; (f[Id_{A}]dA α_{0})^{𝔹}; 𝔹_{dA α_{0} ,f}^{−1}; f 𝔸_{dA α_{0} ,Id_{A}}; f[Id_{A}]Dα_{0}; ((β_{1}f)^{𝔹}dA α_{0})^{𝔹}; 𝔹_{dA α_{0} ,β_{0} dB f}^{−1}; (β_{1}f)^{𝔹}Dα_{0}; 𝔹_{Dβ_{0} dDB Df,Id_{B}}; β_{0}(𝔻_{dB β_{0}} Df)^{𝔻}; (β_{0}𝔻_{dB β_{0}} Df)^{𝔹}; 𝔹_{dB β_{0} Df,Id_{B}}; β_{0}Dρ; (β_{0}dB ρ)^{𝔹}; 𝔹_{dB f α_{0} ,Id_{B}}; β_{0}(𝔻_{dB f}^{−1} α_{0})^{𝔻}; (β_{0}𝔻_{dB f}^{−1} α_{0})^{𝔹}; 𝔹_{Df dA α_{0} ,Id_{B}}; β_{0}𝔻_{dA α_{0} ,dB f}; (𝔹_{dB f,Id_{B}} dA α_{0})^{𝔹}; 𝔹_{dB f,Id_{B}} Dα_{0}]

where (†) is ( )^{𝔹} applied to the commutative diagram

[figure: 𝔹_{β_{0} dB β_{0}} Df; 𝔹_{dB β_{0} ,Id_{B}}^{−1} dDB Df; (β_{1}β_{0})^{𝔹}dDB Df; β_{0}𝔻_{dB β_{0}} Df; β_{1}β_{0}Df; β_{0}dB𝔹_{dB f,Id_{B}}; β_{0}dB ρ; 𝔹_{dB f,Id_{B}}; β_{0}𝔻_{Id_{DB}} Df; 𝔹_{β_{0}} Df; 𝔹_{Id_{DB} ,Id_{B}} dDB Df; β_{1}(β_{0}dB f)^{𝔹}; β_{0}dB (β_{1}f)^{𝔹}; (β_{1}f)^{𝔹}; β_{1}f^{𝔹}; id; β_{0}𝔻_{dB f}^{−1} α_{0}; β_{0}dB f[Id_{A}]^{−1}; β_{1}f α_{0}; 𝔹_{dB f,Id_{B}} dA α_{0}; 𝔹_{β_{0} dB f} α_{0}; (β_{1}f)^{𝔹}dA α_{0}; f[Id_{A}]dA α_{0}; 𝔹_{f} α_{0}; f α_{1}α_{0}; f[Id_{A}]^{−1}]

It is not hard to see that F φ is a 2-cell in 𝔻-Alg. We have that F (Id_{A}) = Id_{F A} by (10). Furthermore, if f : A → B and f′: B → C are 1-cells in Âlg-𝔻, then the proof that F (f′f) = F (f′)F (f) is given by the following commutative diagram (with γ_{0} = (Id_{C})^{ℂ} and γ_{1} = ℂ_{Id_{C}})

[figure: .; γ_{0}(𝔻_{dC f′}^{−1})^{𝔻}; ℂ_{dC f′f,Id_{C}}; γ_{0}𝔻_{dB f,dC f′}^{−1}; ℂ_{D(f′)dB f,Id_{C}}; ℂ_{dC f′,Id_{C}} Df; ℂ_{dB f,γ_{0} dC f′}; (γ_{1}f′)^{ℂ}Df; (γ_{0}𝔻_{dC f′}^{−1}f)^{ℂ}; (γ_{1}f′f)^{ℂ}; (ℂ_{dC f′,Id_{C}} dB f)^{ℂ}; ((γ_{1}f′)^{ℂ}dB f)^{ℂ}; f′[f]; (f′β_{1}f)^{ℂ}; f[β_{0}dB f]; (f′[Id_{B}]dB f)^{ℂ}; ℂ_{dB f,f′}; f′[Id_{B}]Df; f′ f[Id_{A}]; f′(β_{1}f)^{𝔹}; f′𝔹_{dB f,Id_{B}}]

In the other direction take

( [figure: A; DA; A; dA; α_{0}; Id_{A}; α_{1}] , [figure: D^{2}A; DA; DA; A; Dα_{0}; mA; α_{0}; α_{0}; α_{2}] ) [figure: DA; DB; A; B; Df; α_{0}; β_{0}; f; ρ; DA; DB; A; B; Df′; α_{0}; β_{0}; f′; ρ′; φ] ( [figure: B; DB; B; dB; β_{0}; Id_{B}; β_{1}] , [figure: D^{2}B; DB; DB; B; Dβ_{0}; mB; β_{0}; β_{0}; β_{2}] )

in 𝔻-Alg. Define G(α_{1}, α_{2}) as 1. The same object A.

2. For every X in A, ( )^{𝔸} is [figure: A(X, A); A(DX, DA); A(DX, A).; D; A(X,α_{0})]

3. For every h: X → A define 𝔸_{h} as [figure: X; DX; A; DA; A; dX; h; Dh; dA; Id_{A}; α_{0}; d_{h}; α_{1}]

4. For every g: Y → DX and h: X → A define 𝔸_{g,h} as [figure: DY; D^{2}X; DX; D^{2}A; DA; DA; A; Dg; (mX dDX g)^{𝔻}; g^{𝔻}; mX; D^{2}h; Dh; mA; Dα_{0}; α_{0}; α_{0}; (𝔻_{Id_{DX}}^{−1} g)^{𝔻}; 𝔻_{dDX g,Id_{X}}^{−1}; m_{h}; α_{2}^{−1}]

Define G(f, ρ)[h] as [figure: DX; DA; DB.; A; B; Dh; D(fh); Df; α_{0}; β_{0}; f; (D^{h,f})^{−1}; ρ] And define Gφ = φ.

To see that G(α_{1}, α_{2}) is an object in Âlg-𝔻, observe first that condition (9) is direct. The show condition (10) observe that the pasting of D_{X} and A_{dX,h} is α_{2}^{−1}D^{2}h Ddx · α_{0}m_{h} DdX · α_{0}Dh ηX, and then consider the following commutative diagram:

[figure: .; α_{0} Dh ηX; id_{α_{0} Dh}; α_{0} m_{h} DdX; α_{0} Dh ηX^{−1}; α_{2}^{−1} D^{2}h DdX; α_{0} Dα_{0} Dd_{h}; α_{0} mX Dd_{h}; α_{0} D^{dA,α_{0}} Dh; α_{2}^{−1} DdA Dh; α_{0} Dα_{1} Dh; α_{0} D_{A}^{−1} Dh; α_{0} ηX^{−1} Dh]

Condition (11) is the commutativity of the diagram

[figure: α_{0}Dh(β_{X}^{−1}g)^{𝔻}dY; α_{0}Dh𝔻_{g}; α_{0} Dh 𝔻_{dDX g,Id_{DA}}^{−1}; α_{0}Dh 𝔻_{mX dDX g}; α_{0}m_{h}Dg dY; α_{0}Dh mX d_{g}; α_{2}^{−1}D^{2}h Dg dY; α_{0}mA D^{2}h d_{g}; α_{0}Dα_{0}D^{g,Dh}dY; α_{0}Dα_{0}D^{2}h d_{g}; α_{0}D^{Dh g,α_{0}} dY; α_{0}d_{α_{0} Dh g}; α_{1}α_{0}Dh g; α_{0}Dα_{0}d_{Dh}g; α_{0}d_{α_{0}} Dh g; α_{2}^{−1}D^{2}h dDX g; α_{0}mA d_{Dh}g; α_{2}^{−1}dDA Dh g; α_{0}βA Dh g; α_{0}m_{h}dDX g; α_{0}Dh βX g]

Conditions (12) and (13) are straightforward. As for (14), observe that the right hand side of (14) is

[figure: DZ; DY; DX; D^{2}Y; D^{2}X; D^{2}A; DA; DA; A; D^{2}A; DA; (mY dDY ℓ)^{𝔻}; ℓ^{𝔻}; Dℓ; D(α_{0}D(α_{0}Dh g)ℓ); (mX dDX g)^{𝔻}; g^{𝔻}; Dg; D(α_{0} Dh g); Dh; mY; D^{2}(α_{0}Dh g); mX; D^{2}h; mA; Dα_{0}; α_{0}; α_{0}; mA; Dα_{0}; α_{0}; (βY^{−1}ℓ)^{𝔻}; (βX^{−1}g)^{𝔻}; 𝔻_{dDY ℓ,Id_{DY}}^{−1}; 𝔻_{dDX g,Id_{DX}}^{−1}; m_{h}; α_{2}^{−1}; α_{2}^{−1}; m_{α_{0} Dh g}]

where the two unlabeled 2-cells are simply the coherence isomorphisms associated to the pseudofunctor D. In the middle replace the pasting of m_{α_{0}Dh g} and the coherence isomorphism to its right by the pasting of m_{g}, m_{Dh}, m_{α_{0}} and the corresponding coherence isomorphism for D^{2}. Then replace the pasting of α_{2}^{−1}, m_{α_{0}} and α_{2}^{−1} by the pasting of μ_{A}^{−1}, α_{2}^{−1} and Dα_{2}^{−1} (in this last one we omit the corresponding coherence isomorphisms associated to D). Then replace the pasting of m_{h}, m_{Dh} and μ_{A}^{−1} by the pasting of μ_{X}^{−1}, m_{h} and Dm_{h}. Thus we have arrived at

[figure: DZ; DY; DX; D^{2}Y; D^{3}X; D^{2}X; D^{2}X; DA; DA; A; D^{2}A; DA; D^{2}A; (mY dDY ℓ)^{𝔻}; ℓ^{𝔻}; Dℓ; D(α_{0}D(α_{0}Dh g)ℓ); (mX dDX g)^{𝔻}; g^{𝔻}; Dg; Dh; mY; D^{2}g; D^{2}(α_{0}Dh g); mDX; DmX; D^{3}h; mX; mX; D^{2}h; α_{0}; DmA; D^{2}α_{0}; α_{0}; mA; Dα_{0}; Dα_{0}; (βY^{−1}ℓ)^{𝔻}; (βX^{−1}g)^{𝔻}; 𝔻_{dDY ℓ,Id_{DY}}^{−1}; 𝔻_{dDY g,Id_{DX}}^{−1}; m_{g}; μ_{X}^{−1}; m_{h}; α_{2}^{−1}; Dm_{h}; Dα_{2}^{−1}]

Comparing this pasting with the left hand side of (14), we see that we have to show that

[figure: DZ; DY; DX; D^{2}X; ℓ^{𝔻}; g^{𝔻}; (g^{𝔻}ℓ)^{𝔻}; (mX dDX g^{𝔻} ℓ)^{𝔻}; D(g^{𝔻} ℓ); D((mX dDX g)^{𝔻} ℓ); D(mX Dg ℓ); mX; 𝔻_{ℓ,g}; (βX^{−1}g^{𝔻}ℓ)^{𝔻}; 𝔻_{dDX g^{𝔻} ℓ,Id_{DX}}; D((βX^{−1}g)^{𝔻}ℓ); D(𝔻_{dDX g,Id_{DX}} ℓ)]

equals

[figure: DZ; DY; DX; D^{2}Y; D^{3}X; D^{2}X; D^{2}X; (mY dDY ℓ)^{𝔻}; ℓ^{𝔻}; Dℓ; D(mX Dg ℓ); (mX dDX g)^{𝔻}; g^{𝔻}; Dg; mY; D^{2}g; mDX; DmX; mX; mX; (βY^{−1}ℓ)^{𝔻}; (βX^{−1}g)^{𝔻}; 𝔻_{dDY ℓ,Id_{DY}}^{−1}; 𝔻_{dDX g,Id_{DX}}^{−1}; m_{g}; μX^{−1}]

On this last pasting, substitute the pasting of 𝔻_{dDY ℓ,Id_{DY}}^{−1} , 𝔻_{dDX g,Id_{DX}}^{−1}, m_{g} and μ_{X}^{−1} by the other side of the commutative diagram

[figure: (mX dDX g)^{𝔻} 𝔻_{dDY ℓ,Id_{DX}}^{−1}; 𝔻_{mY dDY ℓ,mX dDX g}; 𝔻_{dDX g,Id_{DX}} mY Dℓ; 𝔻_{Id_{DY} ,mX dDX g}Dℓ; mX 𝔻_{Id_{DX} ,dDX g Dℓ}; mX m_{g} Dℓ; 𝔻_{Dg,Id_{DX}} Dℓ; mX (βDX^{−1}Dg)^{𝔻} Dℓ; mX 𝔻_{dD^{2}X Dg,Id_{D^{2}X}} Dℓ; 𝔻_{mDX dD^{2}X Dg,Id_{DX}} Dℓ; μX^{−1} D^{2}g Dℓ; (𝔻_{Id_{DY} ,mY dDX g}dDY ℓ)^{𝔻}; 𝔻_{dDY ℓ,(mX dDX g)^{𝔻}}^{−1}; ((βX^{−1}(mX dDX)^{𝔻})^{𝔻} dDY ℓ)^{𝔻}; (𝔻_{dDX g,Id_{DX}}^{−1} )^{𝔻} Dℓ; (βX^{−1}(mX dDX g)^{𝔻})^{𝔻} Dℓ; (mX βDX^{−1} Dg)^{𝔻} Dℓ; (βX^{−1}mX Dg)^{𝔻}; (μX^{−1}dD^{2}X Dg)^{𝔻} Dℓ; 𝔻_{dDY ℓ,mX dDX(mX dDX g)^{𝔻}}^{−1}; (mX dDX 𝔻_{dDX g,Id_{DX}}^{−1} )^{𝔻}Dℓ; (mX 𝔻_{dDX mX Dg}^{−1})^{𝔻} Dℓ; 𝔻_{DmX dD^{2}X Dg,Id_{DX}}^{−1} Dℓ; mX 𝔻_{dD^{2}X Dg,dDX mX}^{−1}Dℓ]

Now replace the pasting of (βX^{−1}g)^{𝔻}, (βY^{−1}ℓ) and the top row of the previous diagram by the other side of the commutative diagram

[figure: (βX^{−1})^{𝔻}ℓ^{𝔻}; 𝔻_{ℓ,g}; (mX dDX g)^{𝔻}(βY^{−1}ℓ)^{𝔻}; 𝔻_{mY dDY ℓ,mX dDX g}; (𝔻_{Id_{DY} ,mX dDX g}dDY ℓ)^{𝔻}; ((mX dDX g)^{𝔻}βY ℓ)^{𝔻}; (𝔻_{(mX dDX g)^{𝔻}}dDY ℓ)^{𝔻}; (βX^{−1}(mX dDX g)^{𝔻} ℓ)^{𝔻}; ((βX^{−1}g)^{𝔻}ℓ)^{𝔻}; (βX^{−1}g^{𝔻}ℓ)^{𝔻}; ((mX dDX g)^{𝔻}βY^{−1}ℓ)^{𝔻}; id_{((mX dDX g)^{𝔻}ℓ)^{𝔻}}; (mX dDX(βX^{−1}g)^{𝔻} ℓ)^{𝔻}; 𝔻_{dDX g^{𝔻} ℓ,Id_{DX}}; (𝔻_{mX dDX(mX dDX g)^{𝔻}}ℓ)^{𝔻}; mX D((βX^{−1}g^{𝔻})^{𝔻}ℓ); 𝔻_{dDX(mX dDX g)^{𝔻}ℓ,Id_{DX}}]

where the rightmost arrow is ((βX^{−1}(mX dDX g)^{𝔻})^{𝔻}dDY ℓ)^{𝔻}. And finish the argument with the substitution given by the commutative diagram

[figure: 𝔻_{dDX(mX dDX g)^{𝔻}ℓ},Id_{DX}; mXD^{ℓ,(mX dDX g)^{𝔻}}; mX(𝔻_{dDX(mX dDX g)^{𝔻}}^{−1}ℓ)^{𝔻}; mX D(𝔻_{dDX g,Id_{DX}}^{−1} ℓ); (mX𝔻_{dDX(mX dDX g)^{𝔻}}^{−1}ℓ)^{𝔻}; (𝔻_{mX dDX(mX dDX g)^{𝔻}}^{−1}ℓ)^{𝔻}; 𝔻_{(dDX(mX dDX g)^{𝔻})^{𝔻}dDY ℓ,Id_{DX}}; mX𝔻_{dDY ℓ,dDX(mX dDX g)^{𝔻}}^{−1}; (𝔻_{dDY (mX dDX g)^{𝔻},Id_{DX}} dDY ℓ)^{𝔻}; 𝔻_{dDY ℓ,mX dDX(mX dDX g)^{𝔻}}^{−1}; 𝔻_{dDX(mX dDX g)^{𝔻},Id_{DX}} Dℓ; mXD(𝔻_{dDX g,Id_{DX}}^{−1} )Dℓ; (mX dDX𝔻_{dDX g,Id_{DX}}^{−1} )^{𝔻}Dℓ; (mX𝔻_{dDX mX}^{−1}Dg)^{𝔻}Dℓ; 𝔻_{dDX mX Dg,Id_{DX}} Dℓ; mX(𝔻_{dDX mX}^{−1}Dg)^{𝔻}Dℓ; mX D^{Dg,mX}Dℓ; 𝔻_{DmX dD^{2}X Dg,Id_{DX}}^{−1} Dℓ; mX𝔻_{dD^{2}X Dg,dDX mX}^{−1}Dℓ; mX D^{ℓ,mX Dg}]

This completes the proof that G(α_{0}, α_{1}) is an object in Âlg-𝔻. Conditions (15), (16),

(17) and (18) are not hard to verify, as well as the fact that G: 𝔻-Alg → Âlg-𝔻 is a pseudofunctor.

Given (A, ( )^{𝔸}) ∈ Âlg-𝔻, define t(A, ( )^{𝔸}) : (A, ( )^{𝔸}) → GF (A, ( )^{𝔸}) as Id_{A} and the assignment to every h: X → A of the invertible 2-cell

[figure: DX; A; A; DA; h^{𝔸}; (α_{0}dA h)^{𝔸}; Dh; _{Id_{A}}; α_{0}; (α_{1}h)^{𝔸}; 𝔸_{dA h,Id_{A}}]

with the notation as in (19). We have that t(A, ( )^{𝔸}) is a morphism in Âlg-𝔻 (condition (16) is a bit long), and that it defines an invertible strong transformation t: 1 → GF since for any f : (A, ( )^{𝔸}) → (B, ( )^{𝔹}) the diagram

\xymatrix{(A, ( )^{𝔸}) \ar[r]^{t(A,( )^{𝔸})} \ar[d]_{f} & GF (A, ( )^{𝔸}) \ar[d]^{GF f} \\ (B, ( )^{𝔹}) \ar[r]_{(B,( )^{𝔹})} & GF (B, ( )^{𝔹})}

commutes.

On the other hand, given (α_{1}, α_{2}) in 𝔻-Alg define s(α_{1}, α_{2}) : F G(α_{1}, α_{2}) → (α_{1}, α_{2}) as the identity 2-cell

\xymatrix{DA \ar[rr]^{D(Id_{A})} \ar[d]_{D(Id_{A})} &  & DA \ar[dd]^{α_{0}} \\ DA \ar[d]_{α_{0}} & = \\ A \ar[rr]^{_{Id_{A}}} &  & A}

One of the conditions for s(α_{1}, α_{2}) to be a 1-cell in 𝔻-Alg is trivial and the other one is given by the following commutative diagram:

[figure: α_{0} D^{D(Id_{A}),α_{0}} D(dA α_{0}D(Id_{A})); α_{0}D^{dAα_{0} D(Id_{A}),α_{0} D(Id_{A})}; α_{0}D(α_{0}𝔻_{dA}α_{0}D(Id_{A})); α_{0}D(α_{1}α_{0}D(Id_{A})); α_{0}D^{Id_{DA},α_{0} D(Id_{A})}; α_{2}^{−1}D^{2}(Id_{A})D(dAα_{0}D(Id_{A})); α_{0}Dα_{0}(D^{2})_{A}^{−1}D(dA α_{0}D(Id_{A})); α_{0}D(α_{0}𝔻_{A})D(dA α_{0}D(Id_{A})); α_{0}D^{dA α_{0} D(Id_{A}),α_{0}}; α_{0}Dα_{0}D^{α_{0} D(Id_{A}),dA}; α_{0}D(α_{0}𝔻_{A}dA α_{0}D(Id_{A})); id; α_{0}D^{D(Id_{A}),α_{0}} D(Id_{A}); α_{0}D^{α_{0} D(Id_{A}),Id_{A}}; α_{0}D^{α_{0} D(Id_{A}),α_{0} dA}; α_{0}D(α_{1})D(α_{0}D(Id_{A})); α_{0}D^{dA,α_{0}} D(α_{0}D(Id_{A})); α_{0}(βA dA)^{𝔻}D(α_{0}D(Id_{A})); α_{0}𝔻_{dDA dA,Id_{DA}}D(α_{0}D(Id_{A})); α_{2}^{−1}DdA D(α_{0}D(Id_{A})); α_{0}mA D^{α_{0} D(Id_{A}),dA}; α_{2}^{−1}D(dAα_{0}D(Id_{A})); α_{0}m_{Id_{A}}^{−1}D(dA α_{0}D(Id_{A})); α_{0}𝔻_{A}^{−1}mA D(dA α_{0}D(Id_{A})); α_{0}mA(D^{2})_{A}^{−1} D(dA α_{0}D(Id_{A})); α_{0}D(Id_{A})𝔻_{dDA dA α_{0} D(Id_{A}),Id_{DA}}^{−1}; α_{0}𝔻_{A}^{−1}(mA dDA dA α_{0}D(Id_{A}))^{𝔻}; α_{0}𝔻_{dDA dA α_{0} D(Id_{A}),Id_{A}}^{−1}; α_{0}D(Id_{A})(βA^{−1}dA α_{0}D(Id_{A}))^{𝔻}; α_{0}(βA^{−1}dA α_{0}D(Id_{A}))^{𝔻}; α_{0}D^{α_{0} D(Id_{A}),Id_{A}}; id_{α_{0} D(α_{0} D(Id_{A}))}; α_{0}D^{Id_{A},α_{0}}; id_{α_{0} Dα_{0} D^{2}(Id_{A})}; α_{0}mA D^{α_{0} ,dA}; α_{0}𝔻_{dDA dA α_{0} ,Id_{DA}}^{−1}; α_{0}(βA^{−1}dA α_{0})^{𝔻}; α_{0}D^{Id_{DA},α_{0}}; α_{0}Dα_{0}D(𝔻_{A}^{−1}); α_{2}D^{2}(Id_{A}); α_{0}m_{Id_{A}}^{−1}; α_{0}D(Id_{A})(βA)^{𝔻}; α_{0}D(Id_{A})𝔻_{dDA,Id_{DA}}; α_{0}D(Id_{A})mA D_{DA}^{−1}; α_{0}Dα_{0}D^{2}(Id_{A})D_{DA}^{−1}; α_{2}D^{2}(Id_{A})D(Id_{DA}); α_{0}mA D^{2}(Id_{A})D_{DA}^{−1}; α_{0}m_{Id_{A}}D(Id_{A}); α_{0}mA DdA D(α_{0}𝔻_{A})]

Then the diagram

\xymatrix{F G(α_{1}, α_{2}) \ar[r]^{s(α_{1},α_{2})} \ar[d]_{F G(f,ρ)} & (α_{1}, α_{2}) \ar[d]^{(f,ρ)} \\ F G(β_{1}, β_{2}) \ar[r]_{s(β_{1},β_{2})} & (β_{1}, β_{2})}

commutes, and we obtain s: F G → 1 invertible.

6. Every pseudomonad induces a no-iteration pseudomonad

6.1. Theorem. Let 𝕌 = (U, u, n, β, η, μ) be a pseudomonad on A. Define D: Ob(A) → Ob(A) as DA = UA. For every A define dA = uA. For every A, B in A define ( )^{𝔻} as the composite

\xymatrix{A(A, DB) \ar[r]^{U} & A(DA, D^{2}B) \ar[r]^{A(DA,nB)} & A(DA, DB).}

Define 𝔻_{A} = ηA: Id_{DA} → nA UdA. For f : A → DB in A, define 𝔻_{f} as the pasting

[figure: A; DA; DB; D^{2}B; DB.; uA; f; Uf; uDB; id_{DB}; nA; u_{f}; βB]

And for h : B → DC, define 𝔻_{f,h} as the pasting

[figure: DA; D^{2}B; DB; D^{3}C; D^{2}C; D^{2}C; DC.; Uf; nB; U^{2}h; Uh; nDC; UnC; nC; nC; n_{h}; μC^{−1}]

The data defined above is a no-iteration pseudomonad.

Proof. The proof is direct.

Starting with a pseudomonad 𝕌 on A, produce the no-iteration pseudomonad as in the theorem above. Now produce the pseudomonad 𝔻 induced by this presentation. We describe in detail the pseudomonad 𝔻 = (D, d, m, β, η, μ).

The action of D on the objects is U. For φ: f → g: A → B in A, the action of D on it is given by

[figure: DA; DB; D^{2}B; DB; Uf; Ug; UuB; nB; Uφ]

D_{A} = ηA. If h: B → C in A, a brief calculation tells us that D^{f,g} is

Now d: 1_{A} → D is given by dA = uA for every A, and for every f : A → B we have that d_{f} is

[figure: DA; DB; D^{2}B; DB; DC; D^{2}C; DC; Uf; UuB; nB; Id_{DB}; Uh; UuC; nC; ηB^{−1}]

[figure: A; DA; B; DB; DB; D^{2}B; DB; dA; f; Uf; dB; dB; UdB; dDB; nB; Id_{DB}; u_{f}; u_{uB}; βB]

For every A in A, mA = nA. And for f : A → B in A, m_{f} is (after one substitution)

[figure: D^{2}A; DA; D^{2}B; DB; D^{3}B; D^{2}B; D^{2}B; D^{2}B; DB; D^{3}B; nA; U^{2}f; Uf; nB; U^{2}uB; UuB; nDB; UnB; nB; Id_{D^{2}B}; nB; UuDB; nDB; n_{f}; n_{uB}; μB^{−1}; ηDB]

We also have that β_{𝔻}A = β_{𝕌}A whereas η_{𝔻}A is the pasting

[figure: DA; DA; D^{2}A; D^{2}A; D^{3}A; Id_{DA}; UuA; Id_{D^{2}A}; UuDA; nA; nDA; ηA; ηDA]

Finally μ_{𝔻}A is (after one substitution) the pasting

[figure: D^{3}A; D^{2}A; D^{3}A; D^{2}A; D^{2}A; DA; UmA; mDA; Id_{D^{2}A}; UdDA; nDA; nA; nA; ηDA^{−1}; μA]

6.2. Theorem. The bicategories 𝕌-Alg and 𝔻-Alg, for the pseudomonads 𝕌 and 𝔻 considered above, are isomorphic.

Proof. Define r: U → D such that for any A ∈ A, rA = Id_{UA}: UA → DA and for f : A → B, r_{f} = ηB^{−1} Uf. It is not hard to see that r is an invertible strong transformation, and that the diagrams

[figure: 1; U; D; u; d; r; U^{2}; UD; D^{2}; U; D; Ur; n; rD; m; r]

commute. It is not hard to see that r, together with the two commutative diagrams given above is a transition from 𝔻 to 𝕌 along 1 : A → A in the notation of [Marmolejo & Wood, 2008], and thus induces a lifting 𝔻-Alg → 𝕌-Alg of 1_{A}. The inverse is the lifting given by the transition s: D → U from 𝕌 to 𝔻 along 1_{A} given as follows. sA = Id_{UA} for every A in A, for f : A → B, s_{f} = ηB Uf. Now the diagram

[figure: 1; D; U; d; s; u]

commutes, and we define

[figure: D^{2}; DU; U^{2}; D; U; Ds; m; sU; n; s; ω_{2}]

such that ω_{2}A = nA ηUA^{−1}.

7. Factorization systems

We take from [Korostenski & Tholen, 93] and [Rosebrugh & Wood, 02] the example of the monad (−)^{2} : Cat → Cat, where 2 = 0 [figure: α]1. It is given on every category A by DA = A^{2}, ηA : A → A^{2} is given on a: A → A′ as (a, a) : 1_{A} → 1_{A′}. If we denote by d_{0}, d_{1} : A^{2} → A the domain and codomain functors, then for F : X → A^{2} we define F^{𝔻}: X^{2} → A^{2} such that, for f : Y → X in X, F^{𝔻}(f) is

\xymatrix{d_{0}F Y \ar[r]^{F Y} & d_{1}F Y \ar[r]^{d_{1}F f} & d_{1}F X,}

and for (h_{0}, h_{1}) : f → f′, with f′: Y′ → X′,

F^{𝔻}(h_{0}, h_{1}) = (d_{0}F h_{0}, d_{1}F h_{1}).

Given φ: F → G: X → A^{2}, define, for f : Y → X, φ^{𝔻}f = (d_{0}φY, d_{1}φX).

Then ηA^{𝔻} = 1_{A^{2}}, F^{𝔻} ∘ ηA = F and for G: Z → X we have that F^{𝔻} ∘ G^{𝔻} = (F^{𝔻} ∘ G)^{𝔻} and thus we take 𝔻_{A} = id, 𝔻_{F} = id, and 𝔻_{G,F} = id. Then the coherence conditions are trivial.

It is well known that every category with a factorization system is an algebra for this monad. We prove this explicitly with the new formulation for the algebras. So take A with a factorization system (E, M). Given F : X → A we define F^{𝔸}: X^{2} → A as follows. Given f : Y → X in X^{2}, we define F^{𝔸}(f) as in the diagram

[figure: F Y; F X; F^{𝔸}(f); F f; e_{F f}; m_{F f}]

where m_{F f} ∘e_{F f} is the E-M factorization of F f. Given f′: Y′ → X′ in X^{2} and a morphism (h, k) : f → f′ in X^{2}, F^{𝔸}(h, k) is the unique arrow in A such that the diagram

[figure: F Y; F X; F^{𝔸}(f); F Y′; F X′; F^{𝔸}(f′); F f; e_{F f}; m_{F f}; F f′; e_{F f′}; m_{F f′}; F h; F k; F^{𝔸}(h,k)]

commutes (given by the diagonal fill in property). It is clear that F^{𝔸} is a functor. Furthermore, given φ: F → F′: X → A, we define φ^{𝔸}: F^{𝔸} → F′^{𝔸}: X^{2} → A such that φ^{𝔸}f : F^{𝔸}(f) → F′^{𝔸}(f) is the unique arrow that makes the diagram

[figure: F Y; F X; F^{𝔸}(f); F′Y; F′X.; F′^{𝔸}(f); F f; e_{F f}; m_{F f}; F′f; e_{F′f}; m_{F′f}; φY; φX; φ^{𝔸}f]

commutes. It is direct to see that φ^{𝔸} is natural, and that in fact we have defined a functor

( )^{𝔸}: Cat(X, A) → Cat(X^{2}, A).

We also have that the diagram

\xymatrix{X \ar[r]^{dX} \ar[dr]_{F} & X^{2} \ar[d]^{F^{𝔸}} \\  & A}

commutes, provided that the E-M factorization of 1_{A} is 1_{A} ∘ 1_{A} for every A ∈ A (which we assume). We thus take 𝔸_{F} = id.

Given G: Z → X^{2}, F : X → A and h: W → Z in Z^{2}, we have that F^{𝔸}G^{𝔻}(f) is given by the E-M factorization of F d_{1}Gh · F GW:

[figure: F d_{0}GW; F d_{1}GW; F d_{1}GZ,; F^{𝔸}G^{𝔻}(h); F GW; F d_{1}Gh; e_{F d_{1}Gh·F GW}; m_{F d_{1}Gh·F GW}]

whereas the calculation of (F^{𝔸}G)^{𝔸}(h) can be read from the following diagram

[figure: d_{0}GW; d_{1}GW; F^{𝔸}GW; d_{0}GZ; d_{1}GZ; F^{𝔸}GZ; (F^{𝔸}G)^{𝔸}(h); GW; e_{GW}; m_{GW}; GZ; e_{GZ}; m_{GZ}; d_{0}Gh; d_{1}Gh; F^{𝔸}Gh; e_{F^{𝔸}Gh}; m_{F^{𝔸}Gh}]

We then define 𝔸_{G,F} h as the unique isomorphism such that the diagram

[figure: F d_{0}GW; F^{𝔸}GW; (F^{𝔸}G)^{𝔸}(h); F^{𝔸}GZ; F d_{1}GZ; F^{𝔸}G^{𝔻}(h); e_{F GW}; e_{F d_{1}Gh·F GW}; e_{F^{𝔸}Gh}; m_{F^{𝔸}Gh}; m_{F GZ}; m_{F d_{1}Gh·F GW}; 𝔸_{G,F} h]  (20)

commutes. It is not hard to see that 𝔸_{G,F} : F^{𝔸}G^{𝔻} → (F^{𝔸}G)^{𝔸} is a natural transformation. As for the coherence conditions, (9) is immediate, for (10) we must show that 𝔸_{dX,F} = id, but this follows from the fact that (20) is transformed into

[figure: F W; F W; F^{𝔸}(h); F Z; F Z; F^{𝔸}(h); 1_{F W}; e_{F h}; e_{F h}; m_{F h}; 1_{F Z}; m_{F h}; 𝔸_{dX,F} h]

for the particular case G = dX. (11) is the fact that 𝔸_{G,F} (1_{Z}) = id, which is not hard to see. (12) is given by the fact that both composites involved, namely 𝔸_{H,F} h · F^{𝔸}ψ^{𝔸}h and (F^{𝔸}ψ)^{𝔸}h · 𝔸_{G,F} h make commutative the diagram

\xymatrix{F d_{0}GW \ar[rr]^{e_{F d_{1}Gh·F GW}} \ar[d]_{F d_{0}ψW} &  & F^{𝔸}G^{𝔻}h \ar[d]_{𝔸_{H,F} h·F^{𝔸}ψ^{𝔸}h}^{(F^{𝔸}ψ)^{𝔸}h·𝔸_{G,F} h} \ar[rr]^{m_{F d_{1}Gh·F GW}} &  & F d_{1}GZ \ar[d]^{F d_{1}ψZ} \\ F d_{0}HW \ar[r]_{e_{F HW}} & F^{𝔸}HW \ar[r]_{e_{F^{𝔸}Hh}} & (F^{𝔸}H)^{𝔸}h \ar[r]_{m_{F^{𝔸}Hh}} & F^{𝔸}HZ \ar[r]_{m_{F HZ}} & F d_{1}HZ.}

(13) is given by the fact that both composites involved, namely 𝔸_{G,F′}h · φ^{𝔸}G^{𝔻}h and (φ^{𝔸}G)^{𝔸}h · 𝔸_{G,F} h, make commutative the diagram

\xymatrix{F d_{0}GW \ar[rr]^{e_{F d_{1}Gh·F GW}} \ar[d]_{φd_{0}GW} &  & F^{𝔸}G^{𝔻}h \ar[d]_{𝔸_{G,F′}h·φ^{𝔸}G^{𝔻}h}^{(φ^{𝔸}G)^{𝔸}h·𝔸_{G,F} h} \ar[rr]^{m_{F d_{1}Gh·F GW}} &  & F d_{1}GZ \ar[d]^{φd_{1}GZ} \\ F′d_{0}GW \ar[r]_{e_{F′GW}} & F′^{𝔸}GW \ar[r]_{e_{F′^{𝔸}Gh}} & (F′^{𝔸}G)^{𝔸}h \ar[r]_{m_{F′^{𝔸}Gh}} & F′^{𝔸}GZ \ar[r]_{m_{F′GZ}} & F′d_{1}GZ.}

Finally, (14) is given by the fact that for a k: S → U in U and K: U → Z^{2}, both compositions involved, namely 𝔸_{K,F^{𝔸}G}k·𝔸_{F,G}K^{𝔻}k and (𝔸_{G,F} K)^{𝔸}k·𝔸_{G^{𝔻}K,F} k, make commutative the diagram

\xymatrix{F d_{0}G^{𝔻}KS \ar[rr]^{e_{F d_{1}G^{𝔻}Kk·F G^{𝔻}KS}} \ar[d]^{e_{F Gd_{0}KS}} &  & F^{𝔸}(G^{𝔻}K)^{𝔻} \ar[d]_{𝔸_{K,F^{𝔸}G}k·𝔸_{F,G}K^{𝔻}k}^{(𝔸_{G,F} K)^{𝔸}k·𝔸_{G^{𝔻}K,F} k} \ar[rr]^{m_{F d_{1}G^{𝔻}Kk·F G^{𝔻}KS}} &  & F d_{1}G^{𝔻}KU \\ F^{𝔸}Gd_{0}KS \ar[r]_{e_{F^{𝔸}GKS}} & (F^{𝔸}G)^{𝔸}KS \ar[r]_{e_{(F^{𝔸}G)^{𝔸}Kk}} & ((F^{𝔸}G)^{𝔸}K)^{𝔸}k \ar[r]_{m_{(F^{𝔸}G)^{𝔸}Kk}} & (F^{𝔸}G)^{𝔸}KU \ar[r]_{m_{F^{𝔸}KU}} & F^{𝔸}Gd_{1}KU. \ar[u]^{m_{F Gd_{1}KU}}}

In the opposite direction, assume we have an algebra (A, ( )^{𝔸}) for the monad ( )^{2}. According to [Korostenski & Tholen, 93] we have to define the following:

1. for every f in A, a factorization of f: [figure: •; f; e_{f}; •;; •; m_{f}]

2. for every (u, v) : f → g in A^{2}, an arrow t(u, v) that makes the following diagram

commute

[figure: •; u; e_{f}; •; m_{f}; t(u,v); •; v; •; e_{g}; •; m_{g}; •;]

such that, with E = {f|m_{f} is iso} and M = {f|e_{f} is iso}, the following conditions are satisfied:

a. if f is an isomorphism, then e_{f} and m_{f} are isomorphisms;

b. the assignment of t(u, v) is functorial on (u, v);

c. e_{f} ∈ E and m_{f} ∈ M.

Take an arrow f : A → B. This defines a functor ⌜f⌝: 2 → A in the obvious way. Thus we have the isomorphism

[figure: 2; 2^{2}; A.; d2; ⌜f⌝; ⌜f⌝^{A}; 𝔸_{⌜f⌝}]

Observe that 2^{2} is isomorphic to 3 = [figure: 0; 1; 2; α; γ; β] with βα = γ, and that d2 takes as

value γ. We then obtain the following commutative diagram

\xymatrix{⌜f⌝^{𝔸}0 \ar[r]^{⌜f⌝^{𝔸}α} \ar[d]_{𝔸_{⌜f⌝}0} & ⌜f⌝^{𝔸}1 \ar[r]^{⌜f⌝^{𝔸}β} \ar@{-->}[dr]_{m_{f}} & ⌜f⌝^{𝔸}2 \ar[d]^{𝔸_{⌜f⌝}2} \\ A \ar[rr]_{f} \ar@{-->}[ur]_{e_{f}} &  & B,}

where we define e_{f} = ⌜f⌝^{𝔸}α · 𝔸_{⌜f⌝}0^{−1} and m_{f} = 𝔸_{⌜f⌝}2 · ⌜f⌝^{𝔸}β. This gives the desired factorization for f.

If g: C → D is another arrow and (u, v) : f → g is an arrow in A^{2}, then (u, v) defines a natural transformation φ: ⌜f⌝ → ⌜g⌝, and then a natural transformation φ^{𝔸}: ⌜f⌝^{𝔸} → ⌜g⌝^{𝔸}. Using condition (9) it is not hard to see that the diagram

\xymatrix{A \ar[r]^{e_{f}} \ar[d]_{u} & ⌜f⌝^{𝔸}1 \ar[r]^{m_{f}} \ar[d]_{φ^{𝔸}1} & B \ar[d]_{v} \\ C \ar[r]_{e_{g}} & ⌜g⌝^{𝔸}1 \ar[r]_{m_{g}} & D}

commutes. That this assignment results functorial on (u, v) follows at once from the fact that ( )^{𝔸}: Cat(2, A) → Cat(2^{2}, A) is a functor.

We show now that m_{e_{f}} is an isomorphism. For this we observe that since the assignment f ↦ (e_{f}, m_{f}) is functorial on arrows of A^{2} and e_{f} is isomorphic in A^{2} to ⌜f⌝^{𝔸}α, then it suffices to show that m_{⌜f⌝^{𝔸}α} is an isomorphism. Define G: 2 → 2^{2} the functor whose value is the arrow α. Then the image of G^{𝔻}: 2^{2} → 2^{2} is

\xymatrix{ & 1 \ar[dr]^{id} \\ 0 \ar[ur]^{α} \ar[rr]_{α} &  & 1.}

We have that ⌜f⌝^{𝔸}G: 2 → A takes the value ⌜f⌝^{𝔸}α, and thus the natural isomorphism 𝔸_{G,⌜f⌝}: ⌜f⌝^{𝔸}G^{𝔻} → (⌜f⌝^{𝔸}G)^{𝔸} produces the following commutative diagram

\xymatrix{⌜f⌝^{𝔸}0 \ar[d]_{𝔸_{G,⌜f⌝^{𝔸}}0} \ar[r]^{⌜f⌝^{𝔸}α} & ⌜f⌝^{𝔸}1 \ar[r]^{id} \ar[d]_{𝔸_{G,⌜f⌝^{𝔸}}1} & ⌜f⌝^{𝔸}1 \ar[d]^{𝔸_{G,⌜f⌝^{𝔸}}2} \\ (⌜f⌝^{𝔸}G)^{𝔸}0 \ar[r]^{(⌜f⌝^{𝔸}G)^{𝔸}α} \ar[d]_{𝔸_{⌜f⌝^{𝔸}G}0} & (⌜f⌝^{𝔸}G)^{𝔸}1 \ar[r]^{(⌜f⌝^{𝔸}G)^{𝔸}α} \ar@{-->}[dr]_{m_{⌜f⌝^{𝔸}α}} & (⌜f⌝^{𝔸}G)^{𝔸}2 \ar[d]^{𝔸_{⌜f⌝^{𝔸}G}1} \\ ⌜f⌝^{𝔸}0 \ar[ur]_{e_{⌜f⌝^{𝔸}α}} &  & ⌜f⌝^{𝔸}1}

The upper right square tells us that (⌜f⌝^{𝔸}G)^{𝔸}α is an isomorphism, thus m_{(⌜f⌝^{𝔸}α} is also an isomorphism. Therefore m_{e_{f}} is an isomorphism.

The proof that e_{m_{f}} is similar, just take G with value β instead of α.

We are left with showing that for f an isomorphism, e_{f} and m_{f} are isomorphisms. For this we take the category I of the “generic” isomorphism, that is I is the category with two objects and two no identity arrows as in

\xymatrix{0 \ar[r]^{α} & 1. \ar[l]^{β}}

Then a functor F : I → A is the same thing as an isomorphism f in A. Now apply the same ideas as above observing that every arrow in I^{2} is an isomorphism, and taking G: 2 → I^{2} with value the arrow that goes from ⌜0⌝: 2 → I^{2} to ⌜1⌝: 2 → I^{2} in I^{2}.

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

F. Marmolejo and R.J. Wood. Kan extensions and lax idempotent pseudomonads. Theory and Applications of Categories, Vol. 26, 2012, No. 1, pp 1-29.

Robert Rosebrugh and R.J. Wood. Coherence for Factorization Algebras. Theory and Applications of Categories, Vol. 10, 2002, No. 6, pp 134-147.

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

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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

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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 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 Edinburgh: Tom.Leinster@ed.ac.uk Ieke Moerdijk, Radboud University Nijmegen: i.moerdijk@math.ru.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