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

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114= A µ Q ◦ ( Aε C Q ) ◦ (AC A µ Q )◦ ( ACAε C Q ) ◦ ( ACcocan −11 Q ) ◦ ( cocan −11 QP Q )= A µ Q ◦ (A A µ Q )◦ ( Aε C AQ ) ◦ ( ACAε C Q ) ◦ ( ACcocan −11 Q ) ◦ ( cocan −11 QP Q )= A µ Q ◦ (m A Q)◦ ( Aε C AQ ) ◦ ( ACAε C Q ) ◦ ( ACcocan −11 Q ) ◦ ( cocan −11 QP Q )= A µ Q ◦ (m A Q)◦ ( AAε C Q ) ◦ ( Aε C CAQ ) ◦ ( ACcocan −11 Q ) ◦ ( cocan −11 QP Q )= A µ Q ◦ ( Aε C Q ) ◦ (m A CQ) ◦ ( Acocan −11 Q ) ◦ ( Aε C QP Q ) ◦ ( cocan −11 QP Q )(131)= A µ Q ◦ ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ ( A µ Q P Q)◦ ( Aε C QP Q ) ◦ ( cocan −11 QP Q )Note that we haveso we have= χ ◦ (χP Q) .(cocan 1 ) ◦ (u A C) = (A µ Q P ) ◦ (Aδ C ) ◦ (u A C)= (A µ Q P ) ◦ (u A QP ) ◦ δ C A µ Q unital= δ C(132) (cocan 1 ) ◦ (u A C) = δ C .Now we computeand so we getWe haveχ ◦ (δ C Q) = A µ Q ◦ ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ (δ C Q)(132)= A µ Q ◦ ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ (cocan 1 Q) ◦ (u A CQ)= A µ Q ◦ ( Aε C Q ) ◦ (u A CQ) = A µ Q ◦ (u A Q) ◦ ( ε C Q ) ( A µ Q unital)=χ ◦ (δ C Q) = ( ε C Q ) .(Aε C Q ) ◦ ( cocan −11 Q ) ◦ (Qδ D ) ◦ cocan 2= ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ (Qδ D ) ◦ (A µ Q D ) ◦ ( Aρ D Q= ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ (A µ Q P Q ) ◦ (AQδ D ) ◦ ( Aρ D Q(127)= ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ (A µ Q P Q ) ◦ (Aδ C Q) ◦ ( A C ρ Q)= ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ (cocan 1 Q) ◦ ( A C ρ Q)= ( Aε C Q ) ◦ ( A C ρ Q) C ρ Q counital= AQSince cocan 2 is an isomorphism, we have that(133) cocan 2 ◦ ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ (Qδ D ) = QD.Finally we computeρ D Q counital=χ ◦ (Qδ D ) = A µ Q ◦ ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ (Qδ D )))(ε C Q )A µ Q ◦ ( AQε D) ◦ ( (AρQ) D ◦ Aε C Q ) ◦ ( cocan −11 Q ) ◦ (Qδ D )= ( Qε D) ◦ ( A µ Q D) ◦ ( ) (Aρ D Q ◦ Aε C Q ) ◦ ( cocan −11 Q ) ◦ (Qδ D )
so that we obtain= ( Qε D) ◦ cocan 2◦ ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ (Qδ D ) (133)= ( Qε D)χ ◦ (Qδ D ) = ( Qε D) .Theorem 6.37. Let A and B be categories in both of which the coequalizer of anypair of parallel morphisms exist. Let X = (C, D, P, Q, δ C , δ D ) be a regular formal codualstructure on A and B such that the underlying functors C, D, P and Q preservecoequalizers, then the existence of the following structures are equivalent:(a) A coherd χ : QP Q → Q in X.(b) A monad A = (A : A → A, m A : AA → A, u A : A → A) , such that the functorA preserves coequalizers, together with a left action A µ Q : AQ → Q,subject to the following conditions:(i) The natural transformation cocan 1 := (A µ Q P ) ◦ (Aδ C ) : AC → QP isan isomorphism.(ii) The natural transformation cocan 2 := (A µ Q D ) ◦ ( AρQ) D : AQ → QD isan isomorphism.(c) A monad B = (B : B → B, m B : BB → B, u B : B → B) , such that the functorB preserves coequalizers, together with a right action µ B Q : QB → Q,subject to the following conditions:(i) The natural transformation cocan 1 := ( P µ Q) B ◦ (δD B) : DB → P Q isan isomorphism.(ii) The natural transformation cocan 2 := ( (Cµ Q) B ◦ Cρ Q B ) : QB → CQ isan isomorphism.Proof. (a) ⇒ (b) Under weaker assumptions, the monad A = (A, m A , u A ) and theaction A µ Q have been constructed in Proposition 6.25. Moreover, by Theorem 6.301) we already proved that cocan 1 is an isomorphism with (xC) ◦ ( Qρ C P)its inverse.Now we prove that ( Aε C Q ) ◦ ( cocan −11 Q ) ◦(Qδ D ) is the inverse of cocan 2 . We computeρ C P counital=cocan 2 ◦ ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ (Qδ D )= (A µ Q D ) ◦ ( ) (Aρ D Q ◦ Aε C Q ) ◦ ( cocan −11 Q ) ◦ (Qδ D )= (A µ Q D ) ◦ ( Aρ D Q)◦(Aε C Q ) ◦ (xCQ) ◦ ( Qρ C P Q ) ◦ (Qδ D )= (A µ Q D ) ◦ ( (AρQ) D ◦ (xQ) ◦ QP ε C Q ) ◦ ( Qρ C P Q ) ◦ (Qδ D )( Aµ Q D ) ◦ ( )Aρ D Q ◦ (xQ) ◦ (QδD ) = (A µ Q D ) ◦ (xQD) ◦ ( )QP ρ D Q ◦ (QδD )(101)= (χD) ◦ ( )QP ρ D Q ◦ (QδD ) (126)= (χD) ◦ (Qδ D D) ◦ ( Q∆ D)(113)= ( Qε D D ) ◦ ( Q∆ D) (D comonad)= QDso we obtaincocan 2 ◦ ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ (Qδ D ) = QD.On the other hand, we have(Aε C Q ) ◦ ( cocan −11 Q ) ◦ (Qδ D ) ◦ cocan 2115□
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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
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 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 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 ) ◦
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
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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 ⊗
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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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