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

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624.<strong>2.</strong> The comparison functor for comonads.Proposition 4.42 ([GT, Proposition <strong>2.</strong>1]). Let (L, R) be an adjunction where L :B → A and R : A → B and let C = ( C, ∆ C , ε C) be a comonad on a category A.There exists a bijective correspondence between the following collections of data:M comonad morphisms ϕ : LR = (LR, LηR, ɛ) → C = ( C, ∆ C , ε C)R functorial morphism α : R → RC such that (R, α) is a right comodule functorfor the comonad CL functorial morphism β : L → CL such that (L, β) is a left comodule functorfor the comonad Cgiven byProof. For a given ϕ ∈ M, we computeΘ : M → R where Θ (ϕ) = (Rϕ) ◦ (ηR)Ξ : R → M where Ξ (α) = (ɛC) ◦ (Lα)Γ : M → L where Γ (ϕ) = (ϕL) ◦ (Lη)Λ : L → M where Λ (β) = (Cɛ) ◦ (βR) .(Θ (ϕ) C) ◦ Θ (ϕ) = (RϕC) ◦ (ηRC) ◦ (Rϕ) ◦ (ηR)η= (RϕC) ◦ (RLRϕ) ◦ (ηRLR) ◦ (ηR)η,ϕϕmorphcom (= (Rϕϕ) ◦ (RLηR) ◦ (ηR) =) R∆C◦ (Rϕ) ◦ (ηR) = ( R∆ C) ◦ Θ (ϕ)and( ) RεC◦ Θ (ϕ) = ( Rε C) ◦ (Rϕ) ◦ (ηR) ϕmorphcom= (Rɛ) ◦ (ηR) = R.Therefore we deduce that Θ (ϕ) ∈ R. For a given α ∈ R, we compute(Ξ (α) Ξ (α)) ◦ (LηR) Ξ(α)= (Ξ (α) C) ◦ (LRΞ (α)) ◦ (LηR)= (ɛCC) ◦ (LαC) ◦ (LRɛC) ◦ (LRLα) ◦ (LηR)η= (ɛCC) ◦ (LαC) ◦ (Lα) (R,α)= (ɛCC) ◦ ( LR∆ C) ◦ (Lα)ɛ= ∆ C ◦ (ɛC) ◦ (Lα) = ∆ C ◦ Ξ (α)andε C ◦ Ξ (α) = ε C ◦ (ɛC) ◦ (Lα) = ɛ ɛ ◦ ( LRε C) ◦ (Lα) (R,α)= ɛ.Therefore we deduce that Ξ (α) ∈ M. For a given ϕ ∈ M, we computeϕ[CΓ (ϕ)] ◦ Γ (ϕ) = (CϕL) ◦ (CLη) ◦ (ϕL) ◦ (Lη)= (ϕCL) ◦ (LRϕL) ◦ (LRLη) ◦ (Lη) = η (ϕCL) ◦ (LRϕL) ◦ (LηRL) ◦ (Lη)= (ϕϕL) ◦ (LηRL) ◦ (Lη) ϕmorphcom (= ∆ C L ) ◦ (ϕL) ◦ (Lη) = ( ∆ C L ) ◦ Γ (ϕ)and(ε C L ) ◦ Γ (ϕ) = ( ε C L ) ◦ (ϕL) ◦ (Lη) ϕmorphcom= (ɛL) ◦ (Lη) = L.Therefore we deduce that Γ (ϕ) ∈ L. For a given β ∈ L, we compute(Λ (β) Λ (β)) ◦ (LηR) Λ(β)= (CΛ (β)) ◦ (Λ (β) LR) ◦ (LηR)
= (CCɛ) ◦ (CβR) ◦ (CɛLR) ◦ (βRLR) ◦ (LηR)β= (CCɛ) ◦ (CβR) ◦ (CɛLR) ◦ (CLηR) ◦ (βR) = (CCɛ) ◦ (CβR) ◦ (βR)(L,β)= (CCɛ) ◦ ( ∆ C LR ) ◦ (βR) =∆ C= ∆ C ◦ (Cɛ) ◦ (βR) = ∆ C ◦ Λ (β)andε C ◦ Λ (β) = ε C ◦ (Cɛ) ◦ (βR) εC = ɛ ◦ ( ε C LR ) ◦ (βR) (L,β)= ɛ.Therefore we deduce that Λ (β) ∈ M. Let now ϕ ∈ M and let us calculateΞΘ (ϕ) = (ɛC) ◦ (LRϕ) ◦ (LηR) ɛ = ϕ ◦ (ɛLR) ◦ (LηR) = ϕ.Let now α ∈ R and let us calculateΘΞ (α) = (RΞ (α)) ◦ (ηR) = (RɛC) ◦ (RLα) ◦ (ηR) (RɛC) ◦ (ηRC) ◦ α = α.Let now ϕ ∈ M and let us calculateΛΓ (ϕ) = (Cɛ) ◦ (Γ (ϕ) R) = (Cɛ) ◦ (ϕLR) ◦ (LηR) ϕ = ϕ ◦ (LRɛ) ◦ (LηR) = ϕ.Let now β ∈ L and let us calculateΓΛ (β) = (Λ (β) L) ◦ (Lη) = (CɛL) ◦ (βRL) ◦ (Lη) β = (CɛL) ◦ (CLη) ◦ β = β.Theorem 4.43 ([D, Theorem II.<strong>1.</strong>1] and [GT, Theorem <strong>1.</strong>2]). Let (L, R) be an adjunctionwhere L : B → A and R : A → B and let C = ( C, ∆ C , ε C) be a comonadon a category A. There exists a bijective correspondence between the following collectionsof data:K Functors K : B → C A such that C U ◦ K = L,M comonad morphisms ϕ : LR = (LR, LηR, ɛ) → C = ( C, ∆ C , ε C)given byΨ : K → M where Ψ (K) = (Cɛ) ◦ ([C U ( γ C K )] R )Υ : M → K where Υ (ϕ) (Y ) = (LY, (ϕLY ) ◦ (LηY )) and Υ (ϕ) (f) = L (f) .Proof. By Corollary 4.25, there exists a bijective correspondence between K and thecollection L of functorial morphisms β : L → CL such that (L, β) is a left comodulefunctor for the comonad C given byΦ : K → L where Φ (K) = C U ( γ C K ) : L → CLΩ : L → K where Ω (β) (Y ) = (LY, βY ) and Ω (β) (f) = L (f) .By Proposition 4.42, there exists a bijective correspondence between L and thecollection M of comonad morphisms ϕ : LR = (LR, LηR, ɛ) → C = ( C, ∆ C , ε C)given byWe computeΛ : L → M where Λ (β) = (Cɛ) ◦ (βR)Γ : M → L where Γ (ϕ) = (ϕL) ◦ (Lη) .(Λ ◦ Φ) (K) = (Cɛ) ◦ ([C U ( γ C K )] R ) = Ψ (K)63□
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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
Page 9 and 10:
Lemma 2.13 ([BM, L
Page 11 and 12: i.e. Hom B (Y, iX) equalizes Hom B
Page 13 and 14: 13such thatd 0 ◦ v = Id Yd 1 ◦
Page 15 and 16: f ↦→ Rfis bijective for every X
Page 17 and 18: Remark 3.10. Let A = (A, m A , u A
Page 19 and 20: 19and fromµ A P ◦ ( µ A P A ) =
Page 21 and 22: and thusk ◦ (u A QZ) ◦ (Qz) = h
Page 23 and 24: and since A preserves equalizers, A
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 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
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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
Page 158 and 159:
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
Page 172 and 173:
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
Page 178 and 179:
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
Page 188 and 189:
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
Page 202 and 203:
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
Page 218 and 219:
218Let us compute, for every d ∈
Page 220 and 221:
220• 2-cells: monad functor trans
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222We now want to prove that ρ Q·
Page 224 and 225:
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,
Page 230 and 231:
230On the other hand, we can first
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232so that we define the map φ F (
Page 234 and 235:
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
Page 244 and 245:
244Lemma A.4. Let A be an abelian c
Page 246 and 247:
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
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 260 and 261:
260Since P is finite Hom A (P, P )
Page 262 and 263:
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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