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

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260Since P is finite Hom A (P, P ) (X) ≃ Hom A(P, P(X) ) andHom A (P, P ) (M) ≃ Hom A(P, P(M) ) and then we have an exact sequence in Mod-B(273) Hom A(P, P(X) ) f−→ Hom A(P, P(M) ) −→ Q → 0where Q = Coker (f). Then Q ≃ M. Since Hom A (P, −) is full (and faithful) wehaveHom A (A, A ′ ) ≃ Hom Mod-B (Hom A (P, A) , Hom A (P, A ′ )) ,hence there exists a unique morphism g : P (X) → P (M) in A such that f =Hom A (P, −) (g). Let us consider in A(274) P (X) g−→ P (M) −→ X → 0where X = Coker (g). Since P is projective, Hom A (P, −) is exact, and applying itto (274) we get the exact sequenceHom A(P, P(X) ) f=Hom A (P,g)−→ Hom A(P, P(M) ) → Hom A (P, X) → 0.From this sequence and (273) we deduce that Q ≃ Hom A (P, X) where X =Coker (g) ∈ A.Conversely, let us assume that F : A → Mod-B is an equivalence of categories. LetG : Mod-B → A be its inverse equivalence. Since B is a progenerator and G is anequivalence of categories, by Proposition A.19 1) and 6), we deduce that G (B) is aprogenerator in A. Moreover we haveB ≃ Hom Mod-B (B, B) ≃ Hom A (G (B) , G (B)) .Observe that G is a left adjoint to F and thus we haveHom A (G (B) , −) ≃ Hom Mod-B (B, F−) .Since Hom Mod-B (B, F−) ≃ F as functors, we deduce thatF ≃ Hom A (G (B) , −)where G (B) is a progenerator in A.□Theorem A.2<strong>2.</strong> Let A be an abelian category. There exists an equivalence F : A →Mod-B, where B is a ring, if and only if A contains a progenerator P and arbitrarycoproducts of copies of P . If F is an equivalence, there exists a progenerator P inA such that Hom A (P, P ) ≃ B and F ≃ Hom A (P, −).Proof. Assume first that A contains a progenerator P and arbitrary coproducts ofcopies of P . Let B = Hom A (P, P ) and consider the functor Hom A (P, −) : A → Ab.Let us endow any abelian group Hom A (P, A) , for every A ∈ A, with a right B-module structure given by the composition with morphisms of Hom A (P, P ) = B.This means we have the following mapHom A (P, A) × Hom A (P, P ) → Hom A (P, A)(h, ξ) ↦→ h ◦ ξ.For every morphism f : A → B in A we define a morphism in Mod-B as followsHom A (P, f) : Hom A (P, A) → Hom A (P, B)
ξ ↦→ f ◦ ξ.Then it is well-defined a functor Hom A (P, −) : A → Mod-B. We want to prove thatHom A (P, −) is an equivalence of category. To be full and faithful for Hom A (P, −)means that the mapφ A,A ′ : Hom A (A, A ′ ) −→ Hom Mod-B (Hom A (P, A) , Hom A (P, A ′ ))( )HomA (P, f) : Homf ↦→A (P, A) → Hom A (P, A ′ )ξ ↦→ f ◦ ξis bijective for every A, A ′ ∈ A. Note that Hom A (P, −) induces an isomorphismφ P,P : Hom A (P, P ) −→ Hom Mod-B (Hom A (P, P ) , Hom A (P, P )) .In fact, for every ζ ∈ Hom A (P, P ) such that Hom A (P, ζ) = 0 we have that, forevery ξ ∈ Hom A (P, P ) , 0 = Hom A (P, ζ) (ξ) = ζ ◦ ξ. Since P is a generator, wededuce that ζ = 0. Now, let f : Hom A (P, P ) → Hom A (P, P ) be a morphism inMod-B and set f (Id P ) = χ. Then, for every ξ ∈ Hom A (P, P ), we haveand thusf (ξ) = f (Id P ◦ ξ) f∈Mod-B= f (Id P ) ◦ ξ = χ ◦ ξ = Hom A (P, χ) (ξ)f = Hom A (P, χ) = Hom A (P, −) (χ)so that φ P,P is an epimorphism. Let us consider families (P i ) i∈Iand (P j ) j∈JwhereP i ≃ P ≃ P j for every i ∈ I and j ∈ J. Set B i = Hom A (P, P i ) and B j =Hom A (P, P j ). Then B i = Hom A (P, P i ) ≃ Hom A (P, P ) = B and similarly B j ≃ B.Let us compute( ∐Hom A P j , ∐ )coprodP i ≃∏ (Hom A P j , ∐ )P finiteP i ≃∏ ∐Hom A (P j , P i )j∈J i∈Ij∈Ji∈Ij∈J i∈Iφ P,P≃ ∏ ∐Hom Mod-B (Hom A (P, P j ) , Hom A (P, P i )) = ∏ ∐Hom Mod-B (B j , B i )j∈J i∈I j∈J i∈I(B i ≃Bfinite∏≃ Hom Mod-B B j , ∐ )(coprod∐B i ≃ Hom Mod-B B j , ∐ )B ij∈Ji∈Ij∈J i∈Ihence(275) Hom A( ∐j∈JP j , ∐ i∈IP i)≃ Hom Mod-B( ∐j∈JB j , ∐ i∈IB i)which says that Hom A (P, −) induces a bijection between the full subcategory of thecoproducts of copies of P in A and the full subcategory of coproducts of copies ofB in Mod-B, i.e. Hom A (P, −) is full and faithful on the full subcategory of thecoproducts of copies of P in A. Let us denote by ε P i : P → P (I) , p P i : P (I) → P andε j : B = Hom A (P, P ) → B (I) = Hom A (P, P ) (I) , p j : B (I) = Hom A (P, P ) (I) → B =Hom A (P, P ) the canonical maps. Now, let A, A ′ ∈ A. Since P is a generator, wehaveP (J)e f−→ P (I)h−→ A = Coker (h) → 0261
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Contents1.
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linearity and compatibility conditi
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5and since g ◦ f is an epimorphis
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Proof. Clearly (qP )◦(αP ) = (qP
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Lemma 2.13 ([BM, L
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i.e. Hom B (Y, iX) equalizes Hom B
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13such thatd 0 ◦ v = Id Yd 1 ◦
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f ↦→ Rfis bijective for every X
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Remark 3.10. Let A = (A, m A , u A
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19and fromµ A P ◦ ( µ A P A ) =
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and thusk ◦ (u A QZ) ◦ (Qz) = h
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and since A preserves equalizers, A
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Conversely, let Φ be a functorial
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Proof. Apply Proposition 3.24 to th
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Since Q is a left A-module functor,
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(Q BB F, p QB F ) = Coequ Fun(µBQ
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Theorem 3.37. Let B = (B, m B , u B
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where A UG B F : B → A is such th
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Proposition 3.44. Let A = (A, m A ,
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Note that, since f and g are A-bili
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Proposition 3.54. Let (L, R) be an
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Corollary 3.58. Let (L, R) be an ad
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Definition 4.2. A
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Proposition 4.13. Let C = ( C, ∆
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Then we have(P Cx) ◦ ( ρ C P X )
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and since C preserves coequalizers,
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Proof. Apply Corollary 4.24 to the
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Let( (CQ ) ()D, ι Q) C = Equ Fun
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58F D right D-comodule functors Q :
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60prove that C ν D : C F D → (C
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624.2. The compari
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64and[(Ω ◦ Γ) (ϕ)] (Y ) = (LY,
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66for every ( X, C ρ X)∈ C A, th
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68i.e.(44) (d ϕ K ϕ Y ) ◦ (̂η
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70In particular(49) d ϕ(CX, ∆ C
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72We have to prove that (LD ϕ , Ld
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74we have that Ld ϕ K ϕ Y is mono
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and since d is mono we get that(ε
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78Corollary 4.63 (Beck’s Precise
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80We compute(LRɛLY ′ ) ◦ ( LR
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82Proof. First of all we prove that
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84i.e. Aα is a functorial morphism
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86Then we haveA µ CCX ◦ ( A∆ C
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884.23) is a functor Ã : C A → C
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90Let θ l = ( σ B P Q ) ◦ (P τ
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925)σ A = ( ε C A ) ◦ ( Cσ A)
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94(ii) the functorial morphism can
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96defΦ= ( QP A µ Q)◦(QP σ A Q
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98AU A can AA F = can AA F = ( CσA
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100Similarly, one can prove the sta
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102(b) A comonad C = ( C, ∆ C ,
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104We calculateso that we getx ◦
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106There exist functorial morphisms
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108andsatisfying(B, y) = Coequ Fun(
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1104) With notations of Theorem 6.2
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112Then ν : Y → D is the unique
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114= A µ Q ◦ ( Aε C Q ) ◦ (AC
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116= ( Aε C Q ) ◦ ( cocan1 −1
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118so that we getχ= (Cx) ◦ (C ρ
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120We want to prove that Γ is an o
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122and since Dε D is an epimorphis
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124χ= (Cχ) ◦ (C ρ Q P Q ) ◦
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126Now, since cocan 1 : AC → QP i
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1287. Herds and Coherds7.1.
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130◦ ( σ A QQQ ) ◦ (A µ Q P Q
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132= µ B Q ◦ (A µ Q B ) ◦ ( A
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134Assume now that there is another
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136and hence we get(160) x ◦ (χP
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138Proposition 7.7. In the setting
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140We calculateA µ Q ◦ ( σ A Q
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142x=and=δ C=(l= QlQ ̂QQ)◦ (QP
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144(◦ ρ D ̂QQ)Q◦ (QlQ) ◦ (Q
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146given byWe computeσ B = m B ◦
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148andy= ′m B ◦ (ν B B) ◦ (y
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150Now we compute(hQ) ◦ ( Qχ )
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152Thus we obtainσ B ◦ ( ) (P µ
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154Thus hQ is an isomorphism with i
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156) ( )l=(pb Q AQ B ◦ ̂QA µ QB
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158In fact we haveTherefore we dedu
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160χ= h 1 ◦ (P xQ B ) ◦ (P QP
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162so that we obtain:(190)We comput
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164(194)=) )(p QB ̂QA ◦(Qpb Q◦
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166= Ξ ◦ (A A U A λ) ◦ (xx A
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168)(155)= k 2 ◦(Qpb Q◦ (Ql A U
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170) ) (χ= ρ ◦(p QB ̂QA ◦(Qp
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172Theorem 8.13. Let A and B be cat
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174l = eC ρ L : L = − ⊗ B A
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and[µBQ ◦ ( Qσ B)] (− ⊗ T x
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178so that− ⊗ R 1 A ⊗ R c = (
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180− ⊗ T x ⊗ R 1 A ⊗ A f
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182(208)(209)(210)(211)(h1 ) 0 ⊗
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184= abd 0 ⊗ d 1 1 ⊗ d 2 1b⊗d
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186so that h 1 ⊗ h 2 ⊗ a ∈ A
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188= 〈( h (1) y (1))εH ( h (2) y
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190H C is faithfully coflat. Assume
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192=(Qε C H C) ( ∑ )−□ C k i
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194Following Theorem 6.29, we now c
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196)û E(ε C H C (h) = û E (π (h
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198Letandα l = (ϕ ⊗ H) ( (x ⊗
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200This map is well-defined, in fac
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202We now have to prove that this m
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2041) − ⊗ B Σ A preserves the
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206functorial isomorphism. In parti
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208coaction ρ C Σ : Σ → Σ ⊗
Page 210 and 211: 210Now, we consider a particular ca
Page 212 and 213: 212Definition 9.27. Let k be a comm
Page 214 and 215: 214∆coass= a ⊗ c (1) ⊗ A 1 A
Page 216 and 217: 216Definition 9.32.</strong
Page 218 and 219: 218Let us compute, for every d ∈
Page 220 and 221: 220• 2-cells: monad functor trans
Page 222 and 223: 222We now want to prove that ρ Q·
Page 224 and 225: 224Proof. Let us consider the follo
Page 226 and 227: 226and since p Q•B Q ′ ,Q ′
Page 228 and 229: 228(241)= (1 Q • B l Q ′) ζ Q,
Page 230 and 231: 230On the other hand, we can first
Page 232 and 233: 232so that we define the map φ F (
Page 234 and 235: 234Since we have(B • B (Q · A) ,
Page 236 and 237: 2362-cells. This means that a comon
Page 238 and 239: 238defined by settingu Q·A = ( u (
Page 240 and 241: 240the unique A-bimodule morphism s
Page 242 and 243: 242Let F be a finite subset of Hom
Page 244 and 245: 244Lemma A.4. Let A be an abelian c
Page 246 and 247: 246We haveT (ζ) ◦ ξ ◦ T H (p)
Page 248 and 249: 248where k : Ker (Coker (f ◦ p))
Page 250 and 251: 250be the codiagonal map of the ρ
Page 252 and 253: 252Proposition A.12 ([ELGO2, Propos
Page 254 and 255: 254(⇒) Let {A i } i∈Ibe a famil
Page 256 and 257: 256We will prove that h : ∐ B i
Page 258 and 259: 258Proposition A.19. Let (T, H) be
Page 262 and 263: 262andP (J ′ )e f ′−→ P (I
Page 264 and 265: 264hence there exists a unique morp
Page 266: 266[RW] R. Rosebrugh, R.J. Wood, Di
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Contents 1. Introduction 2 2. Preliminaries 4 2.1. Some results on ...

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