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

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122and since Dε D is an epimorphism we deduce thatFinally we compute(BεD ) ◦ Γ ◦ (Dy) defγΓ ◦ (Du B ) = u B D.= ( Bε D) ◦ (yD) ◦ ( P ρ D Q)◦ (P χ) ◦ (δD P Q)y= y ◦ ( P Qε D) ◦ ( )P ρ D Q ◦ (P χ) ◦ (δD P Q) Qcomfun= y ◦ (P χ) ◦ (δ D P Q)(106)= y ◦ ( ε D P Q ) ε D = ( ε D B ) ◦ (Dy)and since Dy is an epimorphism we get that(BεD ) ◦ Γ = ε D B.6.1<strong>1.</strong> Coherds and coGalois functors. We keep the details of this subsectionbecause the coGalois case is not as common as the Galois notion in the literature.Lemma 6.39. Let X = (C, D, P, Q, δ C , δ D ) be a formal codual structure where Q :B → A, P : A → B and C = ( C, ∆ C , ε C) is a comonad on the category A, D =(D, ∆ D , ε D) is a comonad on B. Assume that both A and B have equalizers and thatC, QD preserve them. Then, δ C : C → QP induces a morphism δ C : C U → QP C inC A so that there exists a morphism C δ C C : C A → C QP C such that(134)C U C δ C C = δ C C .Moreover δCCCF= δ C : C U C F = C → QP C C F = QP.Proof. Let us consider the following diagram with notations of Proposition 4.29C Uδ C CC Uγ CC C Uδ C C UQP C Qι P QP C U∆ C C UC C Uγ CQρ C P C UQP C Uγ CCC C Uδ C C C U QP C C USince QD preserves equalizers, by Lemma 4.18, also the functor Q preserves equalizers.Since ( δC CU) ◦ (C Uγ C) (equalizes the pair Qρ C P C U, QP C Uγ C) and ( QP C , Qι ) P =Equ Fun(Qρ C P C U, QP C Uγ C) , by the universal property of the equalizer, there existsa unique morphism δC C : C U → QP C such that((135)) QιP◦ δC C = ( δ C C U ) ◦ (C Uγ C) .We now want to prove that δ C C : C U → QP C = C U C QP C is a morphism betweenleft C-comodule functors which satisfies(CδCC)◦( CUγ C) = ( C ρ Q P C ) ◦ δ C C .□We have(CQιP ) ◦ ( Cδ C C)◦( CUγ C) (135)= ( Cδ C C U ) ◦ ( C C Uγ C) ◦ (C Uγ C)C Uγ C equ= ( Cδ C C U ) ◦ ( ∆ C C U ) ◦ (C Uγ C) (125)= ( C ρ Q P C U) ◦ ( δ C C U ) ◦ (C Uγ C)
123(135)= ( C ρ Q P C U) ◦ ( Qι P ) ◦ δ C CC ρ Q=(CQιP ) ◦ ( C ρ Q P C ) ◦ δ C Cand since C, Q preserve equalizers, CQι P is a monomorphism, so that we get(CδCC)◦( CUγ C) = ( C ρ Q P C ) ◦ δ C C .Hence, by Lemma 4.28, there exists a unique morphism there exists a unique morphismC δ C C : C A → C QP C such thatC U C δ C C = δ C C .Moreover, note that by definition of δC C we have( ) QιP◦ δC C = ( δ C C U ) ◦ (C Uγ C)so that by applying it to C F we get(QιP C F ) ◦ ( δ C C C F ) = ( δ C C U C F ) ◦ (C Uγ C C F ) .Hence, by Proposition 4.32, we obtain that(QρDQ)◦(δCC C F ) = (δ C C) ◦ ∆ C (125)= ( Qρ C P)◦ δC .Since Qρ D Q is a monomorphism, we deduce that δC C C F = δ C .Proposition 6.40. Let A and B be categories with coequalizers and let χ : QP Q →Q be a regular coherd for a formal codual structure X = (C, D, P, Q, δ C , δ D ) where theunderlying functors P : A → B, Q : B → A and C : A → A preserve coequalizers.Let• A = (A, m A , u A ) be the monad on the category A constructed in Proposition6.25;• ( Q, A µ Q)be the left A-module functor constructed in Proposition 6.25;• C Q : B → C A be the functor defined in Lemma 4.28;• Λ : AC → CA be the mixed distributive law between the comonad C and themonad A constructed in Proposition 6.38;• Ã be the lifting of A on the category C A constructed in Theorem 5.7.Then there exists a functorial morphism eA µC Q : ÃC Q → C Q such that□Moreover,C U eA µC Q = A µ Q .( )C Q, eA µC Q is a left Ã-module functor.Proof. Since χ : QP Q → Q is a regular coherd for X = (C, D, P, Q, δ C , δ D ), byProposition 6.38 the mixed distributive law Λ : AC → CA is uniquely defined byΛ ◦ (xC) = (Cx) ◦ (C ρ Q P ) ◦ (χP ) ◦ (QP δ C ) .Now we prove that A µ Q yields a functorial morphism eA µC Q. In fact we have(C A µ Q)◦ (ΛQ) ◦(A C ρ Q)◦ (xQ) x = ( C A µ Q)◦ (ΛQ) ◦ (xCQ) ◦(QP C ρ Q)defΛ= ( C A µ Q)◦ (CxQ) ◦( Cρ Q P Q ) ◦ (χP Q) ◦ (QP δ C Q) ◦ ( QP C ρ Q)(101),(127)= (Cχ) ◦ (C ρ Q P Q ) ◦ (χP Q) ◦ (QP Qδ D ) ◦ ( QP ρ D Q)
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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
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 76 and 77: and since d is mono we get that(ε
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 Ã : 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 124 and 125: 124χ= (Cχ) ◦ (C ρ Q P Q ) ◦
Page 126 and 127: 126Now, since cocan 1 : AC → QP i
Page 128 and 129: 1287. Herds and Coherds7.1.
Page 130 and 131: 130◦ ( σ A QQQ ) ◦ (A µ Q P Q
Page 132 and 133: 132= µ B Q ◦ (A µ Q B ) ◦ ( A
Page 134 and 135: 134Assume now that there is another
Page 136 and 137: 136and hence we get(160) x ◦ (χP
Page 138 and 139: 138Proposition 7.7. In the setting
Page 140 and 141: 140We calculateA µ Q ◦ ( σ A Q
Page 142 and 143: 142x=and=δ C=(l= QlQ ̂QQ)◦ (QP
Page 144 and 145: 144(◦ ρ D ̂QQ)Q◦ (QlQ) ◦ (Q
Page 146 and 147: 146given byWe computeσ B = m B ◦
Page 148 and 149: 148andy= ′m B ◦ (ν B B) ◦ (y
Page 150 and 151: 150Now we compute(hQ) ◦ ( Qχ )
Page 152 and 153: 152Thus we obtainσ B ◦ ( ) (P µ
Page 154 and 155: 154Thus hQ is an isomorphism with i
Page 156 and 157: 156) ( )l=(pb Q AQ B ◦ ̂QA µ QB
Page 158 and 159: 158In fact we haveTherefore we dedu
Page 160 and 161: 160χ= h 1 ◦ (P xQ B ) ◦ (P QP
Page 162 and 163: 162so that we obtain:(190)We comput
Page 164 and 165: 164(194)=) )(p QB ̂QA ◦(Qpb Q◦
Page 166 and 167: 166= Ξ ◦ (A A U A λ) ◦ (xx A
Page 168 and 169: 168)(155)= k 2 ◦(Qpb Q◦ (Ql A U
Page 170 and 171: 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 ⊗
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210Now, we consider a particular ca
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212Definition 9.27. Let k be a comm
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214∆coass= a ⊗ c (1) ⊗ A 1 A
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216Definition 9.32.</strong
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218Let us compute, for every d ∈
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220• 2-cells: monad functor trans
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222We now want to prove that ρ Q·
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224Proof. Let us consider the follo
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226and since p Q•B Q ′ ,Q ′
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228(241)= (1 Q • B l Q ′) ζ Q,
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230On the other hand, we can first
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232so that we define the map φ F (
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234Since we have(B • B (Q · A) ,
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2362-cells. This means that a comon
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238defined by settingu Q·A = ( u (
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240the unique A-bimodule morphism s
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242Let F be a finite subset of Hom
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244Lemma A.4. Let A be an abelian c
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246We haveT (ζ) ◦ ξ ◦ T H (p)
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248where k : Ker (Coker (f ◦ p))
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250be the codiagonal map of the ρ
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252Proposition A.12 ([ELGO2, Propos
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254(⇒) Let {A i } i∈Ibe a famil
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256We will prove that h : ∐ B i
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258Proposition A.19. Let (T, H) be
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260Since P is finite Hom A (P, P )
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262andP (J ′ )e f ′−→ P (I
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264hence there exists a unique morp
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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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