Contents 1. Introduction 2 2. Preliminaries 4 2.1. Some results on ...
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Contents 1. Introduction 2 2. Preliminaries 4 2.1. Some results on ...

Contents 1. Introduction 2 2. Preliminaries 4 2.1. Some results on ... Contents 1. Introduction 2 2. Preliminaries 4 2.1. Some results on ...

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228(241)= (1 Q • B l Q ′) ζ Q,B,Q ′p Q•B B,Q ′ (p Q,B · 1 Q ′)so that, since p Q•B B,Q ′ (p Q,B · 1 Q ′) is an epimorphism, we getr Q • B 1 Q ′ = (1 Q • B l Q ′) ζ Q,B,Q ′.Proposition 1ong>1.ong>1ong>1.ong> Let (X, A) , (Y, B) be monads in C, let (P, λ P , ρ P ) , (Q, λ Q , ρ Q )be A-B-bimodules in C, let (P ′ , λ P ′, ρ P ′) , (Q ′ , λ Q ′, ρ Q ′) be B-C-bimodules in C andlet f : P → Q, f ′ : P ′ → Q ′ be bimodule morphisms in C. Then there exists a uniqueA-C-bimodule morphism f • B f ′ : P • B P ′ → Q • B Q ′ .Proof. Since f is an A-B-bimodule morphism, we have that(245) λ Q (1 A · f) = fλ P and ρ Q (f · 1 B ) = fρ P .Since f ′ is a B-C-bimodule morphism, we have that(246) λ Q ′ (1 B · f ′ ) = f ′ λ P ′ and ρ Q ′ (f ′ · 1 C ) = f ′ ρ P ′.Let us consider the following diagramP · B · P ′ ρ P ·1 P ′ 1 P ·λ P ′f·1 B·f ′ P · P ′ p P,P ′f·f ′P • B P ′f• B f ′Q · B · Q ′ ρ Q·1 Q ′ Q · Q ′ p Q,Q ′ Q • B Q ′1 Q·λ Q ′Note that the left square serially commutes, in factandThus, we get that(f · f ′ ) (ρ P · 1 P ′) = (f · 1 Q ′) (1 P · f ′ ) (ρ P · 1 P ′)ρ P= (f · 1 Q ′) (ρ P · 1 Q ′) (1 P · 1 B · f ′ )(245)= (ρ Q · 1 Q ′) (f · 1 B · 1 Q ′) (1 P · 1 B · f ′ ) = (ρ Q · 1 Q ′) (f · 1 B · f ′ )(f · f ′ ) (1 P · λ P ′) = (f · 1 Q ′) (1 P · f ′ ) (1 P · λ P ′)(246)= (f · 1 Q ′) (1 P · λ Q ′) (1 P · 1 B · f ′ )f= (1 Q · λ Q ′) (f · 1 B · 1 Q ′) (1 P · 1 B · f ′ )= (1 Q · λ Q ′) (f · 1 B · f ′ ) .p Q,Q ′ (f · f ′ ) (ρ P · 1 P ′) = p Q,Q ′ (f · f ′ ) (1 P · λ P ′)and since (P • B P ′ , p P,P ′) = Coequ C (ρ P · 1 P ′, 1 P · λ P ′) we deduce that there exists aunique 2-cell f • B f ′ : P • B P ′ → Q • B Q ′ such that(247) (f • B f ′ ) p P,P ′ = p Q,Q ′ (f · f ′ ) .□

We now want to prove that f • B f ′ is a morphism of A-C-bimodules. Note that, byProposition 1ong>1.ong>5, P • B P ′ and Q • B Q ′ are A-C-bimodules. We computeλ Q•B Q ′ (1 A · f • B f ′ ) (1 A · p P,P ′) (247)= λ Q•B Q ′ (1 A · p Q,Q ′) (1 A · f · f ′ )(236)= p Q,Q ′ (λ Q · 1 Q ′) (1 A · f · f ′ )= p Q,Q ′ (λ Q · 1 Q ′) (1 A · f · 1 Q ′) (1 A · 1 P · f ′ )(245)= p Q,Q ′ (f · 1 Q ′) (λ P · 1 Q ′) (1 A · 1 P · f ′ )λ P= p Q,Q ′ (f · 1 Q ′) (1 P · f ′ ) (λ P · 1 P ′) = p Q,Q ′ (f · f ′ ) (λ P · 1 P ′)(247)= (f • B f ′ ) p P,P ′ (λ P · 1 P ′)(236)= (f • B f ′ ) λ P •B P ′ (1 A · p P,P ′)229and since 1 A · p P,P ′is an epimorphism, we get thatλ Q•B Q ′ (1 A · f • B f ′ ) = (f • B f ′ ) λ P •B P ′i.e. f • B f ′ is a morphism of left A-modules. Similarly, we also haveρ Q•B Q ′ (f • B f ′ · 1 C ) (p P,P ′ · 1 C ) (247)= ρ Q•B Q ′ (p Q,Q ′ · 1 C) (f · f ′ · 1 C )(237)= p Q,Q ′ (1 Q · ρ Q ′) (f · f ′ · 1 C )= p Q,Q ′ (1 Q · ρ Q ′) (1 Q · f ′ · 1 C ) (f · 1 P ′ · 1 C )(246)= p Q,Q ′ (1 Q · f ′ ) (1 Q · ρ P ′) (f · 1 P ′ · 1 C )f= p Q,Q ′ (1 Q · f ′ ) (f · 1 P ′) (1 P · ρ P ′) = p Q,Q ′ (f · f ′ ) (1 P · ρ P ′)and since p P,P ′ · 1 C is epi, we get that(247)= (f • B f ′ ) p P,P ′ (1 P · ρ P ′)(237)= (f • B f ′ ) ρ P •B P ′ (p P,P ′ · 1 C)ρ Q•B Q ′ (f • B f ′ · 1 C ) = (f • B f ′ ) ρ P •B P ′i.e. f • B f ′ is also a morphism of right C-modules.□Proposition 1ong>1.ong>1ong>2.ong> For any monad (Y, B) in C, the composition denoted by • B iscompatible with the vertical canonical composition.Proof. Let (X, A) , (Y, B) , (Z, C) be monads in C, let (P, λ P , ρ P ) , (Q, λ Q , ρ Q ) ,(W, λ W , ρ W ) be A-B-bimodules in C, let (P ′ , λ P ′, ρ P ′) , (Q ′ , λ Q ′, ρ Q ′) , (W ′ , λ W ′, ρ W ′)be B-C-bimodules in C and let f : P → Q, g : Q → W be A-B-bimodule morphisms,f ′ : P ′ → Q ′ , g ′ : Q ′ → W ′ be B-C-bimodule morphisms in C. By Proposition 1ong>1.ong>11we can consider the A-C-bimodule morphisms f • B f ′ : P • B P ′ → Q • B Q ′ andg • B g ′ : Q • B Q ′ → W • B W ′ and we can compose them in order to get(g • B g ′ ) (f • B f ′ ) : P • B P ′ → W • B W ′ .

We now want to prove that f • B f ′ is a morphism of A-C-bimodules. Note that, byPropositi<strong>on</strong> 1<str<strong>on</strong>g>1.</str<strong>on</strong>g>5, P • B P ′ and Q • B Q ′ are A-C-bimodules. We computeλ Q•B Q ′ (1 A · f • B f ′ ) (1 A · p P,P ′) (247)= λ Q•B Q ′ (1 A · p Q,Q ′) (1 A · f · f ′ )(236)= p Q,Q ′ (λ Q · 1 Q ′) (1 A · f · f ′ )= p Q,Q ′ (λ Q · 1 Q ′) (1 A · f · 1 Q ′) (1 A · 1 P · f ′ )(245)= p Q,Q ′ (f · 1 Q ′) (λ P · 1 Q ′) (1 A · 1 P · f ′ )λ P= p Q,Q ′ (f · 1 Q ′) (1 P · f ′ ) (λ P · 1 P ′) = p Q,Q ′ (f · f ′ ) (λ P · 1 P ′)(247)= (f • B f ′ ) p P,P ′ (λ P · 1 P ′)(236)= (f • B f ′ ) λ P •B P ′ (1 A · p P,P ′)229and since 1 A · p P,P ′is an epimorphism, we get thatλ Q•B Q ′ (1 A · f • B f ′ ) = (f • B f ′ ) λ P •B P ′i.e. f • B f ′ is a morphism of left A-modules. Similarly, we also haveρ Q•B Q ′ (f • B f ′ · 1 C ) (p P,P ′ · 1 C ) (247)= ρ Q•B Q ′ (p Q,Q ′ · 1 C) (f · f ′ · 1 C )(237)= p Q,Q ′ (1 Q · ρ Q ′) (f · f ′ · 1 C )= p Q,Q ′ (1 Q · ρ Q ′) (1 Q · f ′ · 1 C ) (f · 1 P ′ · 1 C )(246)= p Q,Q ′ (1 Q · f ′ ) (1 Q · ρ P ′) (f · 1 P ′ · 1 C )f= p Q,Q ′ (1 Q · f ′ ) (f · 1 P ′) (1 P · ρ P ′) = p Q,Q ′ (f · f ′ ) (1 P · ρ P ′)and since p P,P ′ · 1 C is epi, we get that(247)= (f • B f ′ ) p P,P ′ (1 P · ρ P ′)(237)= (f • B f ′ ) ρ P •B P ′ (p P,P ′ · 1 C)ρ Q•B Q ′ (f • B f ′ · 1 C ) = (f • B f ′ ) ρ P •B P ′i.e. f • B f ′ is also a morphism of right C-modules.□Propositi<strong>on</strong> 1<str<strong>on</strong>g>1.</str<strong>on</strong>g>1<str<strong>on</strong>g>2.</str<strong>on</strong>g> For any m<strong>on</strong>ad (Y, B) in C, the compositi<strong>on</strong> denoted by • B iscompatible with the vertical can<strong>on</strong>ical compositi<strong>on</strong>.Proof. Let (X, A) , (Y, B) , (Z, C) be m<strong>on</strong>ads in C, let (P, λ P , ρ P ) , (Q, λ Q , ρ Q ) ,(W, λ W , ρ W ) be A-B-bimodules in C, let (P ′ , λ P ′, ρ P ′) , (Q ′ , λ Q ′, ρ Q ′) , (W ′ , λ W ′, ρ W ′)be B-C-bimodules in C and let f : P → Q, g : Q → W be A-B-bimodule morphisms,f ′ : P ′ → Q ′ , g ′ : Q ′ → W ′ be B-C-bimodule morphisms in C. By Propositi<strong>on</strong> 1<str<strong>on</strong>g>1.</str<strong>on</strong>g>11we can c<strong>on</strong>sider the A-C-bimodule morphisms f • B f ′ : P • B P ′ → Q • B Q ′ andg • B g ′ : Q • B Q ′ → W • B W ′ and we can compose them in order to get(g • B g ′ ) (f • B f ′ ) : P • B P ′ → W • B W ′ .

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