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 ...
and since d is mono we get that(ε C Z ′′) ◦ C ρ Z ′′ = Z ′′76Let us prove that ( Z ′′ , C ρ Z ′′)∈ C A and thus formula (54) will say that d is amorphism in C A. Since ∆ C is a functorial morphism and by definition of C ρ Z ′′, thelower left square serially commutes. We have(CCd) ◦ ( C C ρ Z ′′)◦ C ρ Z ′′(54)= ( C C ρ Z ′)◦ (Cd) ◦ C ρ Z ′′(54)= ( )C C ρ Z ′ ◦ C ρ Z ′ ◦ d C ρ Z ′coass= ( ∆ C Z ′) ◦ C ρ Z ′ ◦ d(54)= ( ∆ C Z ′) ◦ (Cd) ◦ C ∆ρ CZ ′′ = (CCd) ◦ ( ∆ C Z ′′) ◦ C ρ Z ′′and since CCd is a monomorphism we get( )C C ρ Z ′′ ◦ C ρ Z ′′ = ( ∆ C Z ′′) ◦ C ρ Z ′′that is that C ρ Z ′′is coassociative. Moreover we haved ◦ ( ε C Z ′′) ◦ C ρ Z ′′ε C = ( ε C Z ′) ◦ (Cd) ◦ C ρ Z ′′(54)= ( ε C Z ′) ◦ C ρ Z ′ ◦ d C ρ Z ′counit= dso that C ρ Z ′′ is also counital. Therefore ( Z ′′ , C ρ Z ′′)∈ C A and d is a morphism inC A. Now we want to prove that it is an equalizer in C A. Let ( E, C ρ E)∈ C A andf : ( E, C ρ E)→(Z ′ , C ρ Z ′)be a morphism in C A such that (K ϕ d 0 ) ◦ f = (K ϕ d 1 ) ◦ f.Then, by regarding f as a morphism in A we also have that(Ld 0 ) ◦ f = (Ld 1 ) ◦ f.Since (Z ′′ , d) = Equ A (Ld 0 , Ld 1 ) , there exists a unique morphism h : E → Z ′′ suchthatd ◦ h = f.Now we want to prove that h is a morphism in C A. In fact, let us consider thefollowing diagramEC ρ EhC ρ Z ′′Z ′′ d Z ′C ρ Z ′CE Ch CZ ′′ Cd CZ ′ .Since d ∈ C A, the right square commutes. Since f ∈ C A we haveso that we have(Cd) ◦ (Ch) ◦ C ρ E = (Cf) ◦ C ρ E = C ρ Z ′ ◦ f = C ρ Z ′ ◦ d ◦ h(Cd) ◦ C ρ Z ′′ ◦ h (54)= C ρ Z ′ ◦ d ◦ h = (Cd) ◦ (Ch) ◦ C ρ Eand since Cd is a monomorphism, we deduce thatC ρ Z ′′ ◦ h = (Ch) ◦ C ρ E
i.e. h ∈ C A. Therefore (Z ′′ , d) = EquC A (K ϕ d 0 , K ϕ d 1 ). Now, since K ϕ : B → C A isan equivalence of categories, there exist X ′′ , e ∈ B such thatK ϕ X ′′ = ( Z ′′ , C ρ Z ′′)and Kϕ e = dand thus (X ′′ , e) = Equ B (d 0 , d 1 ). Moreover, sinceZ ′′ d Z ′sis a contractible coequalizer and (Z ′′ , d) = (C UK ϕ X ′′ , C UK ϕ e ) (, we deduce thatCUK ϕ X ′′ , C UK ϕ e ) (is a contractible coequalizer of (Ld 0 , Ld 1 ). Then (LX ′′ , Le) =CUK ϕ X ′′ , C UK ϕ e ) is a contractible coequalizer of (Ld 0 , Ld 1 ) so that (LX ′′ , Le) =Equ A (Ld 0 , Ld 1 ).□The following is a slightly improved version of Theorem 3.14 p. 101 [BW] for thedual case.Theorem 4.62 (Generalized Beck’s Precise Cotripleability Theorem). Let L : B →A be a functor, let C = ( C, ∆ C , ε C) be a comonad on a category A. Then L isϕ-comonadic if and only if1) L has a right adjoint R : A → B,2) ϕ : LR → C is a comonads isomorphism where LR = (LR, LηR, ɛ) with ηand ɛ unit and counit of (L, R) ,3) for every ( X, C ρ X)∈ C A, there exist Equ B(αX, R C ρ X), where α = (Rϕ) ◦(ηR), and L preserves the equalizerLd 0tLd 1 ZEqu Fun(α C U, R C Uγ C) ,4) L reflects isomorphisms.In this case in B there exist equalizers of reflexive L-contractible equalizer pairsand L preserves them.Proof. Assume first that L is ϕ-comonadic. Then by definition L has a right adjointR : A → B and a comonad morphism ϕ : LR → C such that the functor K ϕ =Υ (ϕ) : B → C A is an equivalence of categories. Let K ϕ ′ be an inverse of K ϕ . Thenin particular K ϕ ′ : C A → B is a right adjoint of K ϕ so that by Proposition 4.47 forevery ( ) ( )X, C ρ X ∈ C A, there exists Equ B αX, R C ρ X where α = Θ (ϕ) = (Rϕ) ◦(ηR) and thus ( (K ϕ, ′ k ϕ) ′ = EquFun α C U, R C Uγ C) . Then we can apply Theorem 4.55to get that L preserves the equalizer ( (K ϕ, ′ k ϕ) ′ = EquFun α C U, R C Uγ C) , L reflectsisomorphisms and ϕ : LR → C is a comonads isomorphism.Conversely, by assumption 1) L has a right adjoint R :( A → B so) that (L, R)(is an adjunction)and by assumption 2) there exist Equ B αX, R C ρ X , for everyX, C ρ X ∈ C A so that we can apply one direction of Proposition 4.47. Thus thefunctor K ϕ = Υ (ϕ) : B → C A has a right adjoint D ϕ : C A → B. Now, by applyingTheorem 4.55 in the converse direction, we deduce that K ϕ = Υ (ϕ) : B → C A is anequivalence of categories, i.e. L is ϕ-comonadic.In the case L is ϕ-comonadic, by Lemma 4.61, in B there exist equalizers ofreflexive L-contractible equalizer pairs and L preserves them.□77
- Page 25 and 26: Conversely, let Φ be a functorial
- Page 27 and 28: Proof. Apply Proposition 3.24 to th
- Page 29 and 30: Since Q is a left A-module functor,
- Page 31 and 32: (Q BB F, p QB F ) = Coequ Fun(µBQ
- Page 33 and 34: Theorem 3.37. Let B = (B, m B , u B
- Page 35 and 36: where A UG B F : B → A is such th
- Page 37 and 38: Proposition 3.44. Let A = (A, m A ,
- Page 39 and 40: Note that, since f and g are A-bili
- Page 41 and 42: Proposition 3.54. Let (L, R) be an
- Page 43 and 44: Corollary 3.58. Let (L, R) be an ad
- Page 45 and 46: Definition 4.2. A
- Page 47 and 48: Proposition 4.13. Let C = ( C, ∆
- Page 49 and 50: Then we have(P Cx) ◦ ( ρ C P X )
- Page 51 and 52: and since C preserves coequalizers,
- Page 53 and 54: Proof. Apply Corollary 4.24 to the
- Page 55 and 56: Let( (CQ ) ()D, ι Q) C = Equ Fun
- Page 58 and 59: 58F D right D-comodule functors Q :
- Page 60 and 61: 60prove that C ν D : C F D → (C
- Page 62 and 63: 624.2. The compari
- Page 64 and 65: 64and[(Ω ◦ Γ) (ϕ)] (Y ) = (LY,
- Page 66 and 67: 66for every ( X, C ρ X)∈ C A, th
- Page 68 and 69: 68i.e.(44) (d ϕ K ϕ Y ) ◦ (̂η
- Page 70 and 71: 70In particular(49) d ϕ(CX, ∆ C
- Page 72 and 73: 72We have to prove that (LD ϕ , Ld
- Page 74 and 75: 74we have that Ld ϕ K ϕ Y is mono
- Page 78 and 79: 78Corollary 4.63 (Beck’s Precise
- Page 80 and 81: 80We compute(LRɛLY ′ ) ◦ ( LR
- Page 82 and 83: 82Proof. First of all we prove that
- Page 84 and 85: 84i.e. Aα is a functorial morphism
- Page 86 and 87: 86Then we haveA µ CCX ◦ ( A∆ C
- Page 88 and 89: 884.23) is a functor à : C A → C
- Page 90 and 91: 90Let θ l = ( σ B P Q ) ◦ (P τ
- Page 92 and 93: 925)σ A = ( ε C A ) ◦ ( Cσ A)
- Page 94 and 95: 94(ii) the functorial morphism can
- Page 96 and 97: 96defΦ= ( QP A µ Q)◦(QP σ A Q
- Page 98 and 99: 98AU A can AA F = can AA F = ( CσA
- Page 100 and 101: 100Similarly, one can prove the sta
- Page 102 and 103: 102(b) A comonad C = ( C, ∆ C ,
- Page 104 and 105: 104We calculateso that we getx ◦
- Page 106 and 107: 106There exist functorial morphisms
- Page 108 and 109: 108andsatisfying(B, y) = Coequ Fun(
- Page 110 and 111: 1104) With notations of Theorem 6.2
- Page 112 and 113: 112Then ν : Y → D is the unique
- Page 114 and 115: 114= A µ Q ◦ ( Aε C Q ) ◦ (AC
- Page 116 and 117: 116= ( Aε C Q ) ◦ ( cocan1 −1
- Page 118 and 119: 118so that we getχ= (Cx) ◦ (C ρ
- Page 120 and 121: 120We want to prove that Γ is an o
- Page 122 and 123: 122and since Dε D is an epimorphis
- Page 124 and 125: 124χ= (Cχ) ◦ (C ρ Q P Q ) ◦
i.e. h ∈ C A. Therefore (Z ′′ , d) = EquC A (K ϕ d 0 , K ϕ d 1 ). Now, since K ϕ : B → C A isan equivalence of categories, there exist X ′′ , e ∈ B such thatK ϕ X ′′ = ( Z ′′ , C ρ Z ′′)and Kϕ e = dand thus (X ′′ , e) = Equ B (d 0 , d 1 ). Moreover, sinceZ ′′ d Z ′sis a c<strong>on</strong>tractible coequalizer and (Z ′′ , d) = (C UK ϕ X ′′ , C UK ϕ e ) (, we deduce thatCUK ϕ X ′′ , C UK ϕ e ) (is a c<strong>on</strong>tractible coequalizer of (Ld 0 , Ld 1 ). Then (LX ′′ , Le) =CUK ϕ X ′′ , C UK ϕ e ) is a c<strong>on</strong>tractible coequalizer of (Ld 0 , Ld 1 ) so that (LX ′′ , Le) =Equ A (Ld 0 , Ld 1 ).□The following is a slightly improved versi<strong>on</strong> of Theorem 3.14 p. 101 [BW] for thedual case.Theorem 4.62 (Generalized Beck’s Precise Cotripleability Theorem). Let L : B →A be a functor, let C = ( C, ∆ C , ε C) be a com<strong>on</strong>ad <strong>on</strong> a category A. Then L isϕ-com<strong>on</strong>adic if and <strong>on</strong>ly if1) L has a right adjoint R : A → B,2) ϕ : LR → C is a com<strong>on</strong>ads isomorphism where LR = (LR, LηR, ɛ) with ηand ɛ unit and counit of (L, R) ,3) for every ( X, C ρ X)∈ C A, there exist Equ B(αX, R C ρ X), where α = (Rϕ) ◦(ηR), and L preserves the equalizerLd 0tLd 1 ZEqu Fun(α C U, R C Uγ C) ,4) L reflects isomorphisms.In this case in B there exist equalizers of reflexive L-c<strong>on</strong>tractible equalizer pairsand L preserves them.Proof. Assume first that L is ϕ-com<strong>on</strong>adic. Then by definiti<strong>on</strong> L has a right adjointR : A → B and a com<strong>on</strong>ad morphism ϕ : LR → C such that the functor K ϕ =Υ (ϕ) : B → C A is an equivalence of categories. Let K ϕ ′ be an inverse of K ϕ . Thenin particular K ϕ ′ : C A → B is a right adjoint of K ϕ so that by Propositi<strong>on</strong> 4.47 forevery ( ) ( )X, C ρ X ∈ C A, there exists Equ B αX, R C ρ X where α = Θ (ϕ) = (Rϕ) ◦(ηR) and thus ( (K ϕ, ′ k ϕ) ′ = EquFun α C U, R C Uγ C) . Then we can apply Theorem 4.55to get that L preserves the equalizer ( (K ϕ, ′ k ϕ) ′ = EquFun α C U, R C Uγ C) , L reflectsisomorphisms and ϕ : LR → C is a com<strong>on</strong>ads isomorphism.C<strong>on</strong>versely, by assumpti<strong>on</strong> 1) L has a right adjoint R :( A → B so) that (L, R)(is an adjuncti<strong>on</strong>)and by assumpti<strong>on</strong> 2) there exist Equ B αX, R C ρ X , for everyX, C ρ X ∈ C A so that we can apply <strong>on</strong>e directi<strong>on</strong> of Propositi<strong>on</strong> 4.47. Thus thefunctor K ϕ = Υ (ϕ) : B → C A has a right adjoint D ϕ : C A → B. Now, by applyingTheorem 4.55 in the c<strong>on</strong>verse directi<strong>on</strong>, we deduce that K ϕ = Υ (ϕ) : B → C A is anequivalence of categories, i.e. L is ϕ-com<strong>on</strong>adic.In the case L is ϕ-com<strong>on</strong>adic, by Lemma 4.61, in B there exist equalizers ofreflexive L-c<strong>on</strong>tractible equalizer pairs and L preserves them.□77
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