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

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218Let us compute, for every d ∈ D,[ ]ψ ◦ (D ⊗ uA ) ◦ r −1D (d) = [ψ ◦ (D ⊗ uA )] (d ⊗ 1 k ) = ψ (d ⊗ 1 A )so that we have(A,ρ H A)H-com algDisHmodcoal= (1 A ) 0⊗ d · (1 A ) 1= 1 A ⊗ d · 1 H = 1 A ⊗ d= (u A ⊗ D) (1 k ⊗ d) = [ ](u A ⊗ D) ◦ l −1D (d)ψ ◦ (D ⊗ u A ) ◦ r −1D= (u A ⊗ D) ◦ l −1D .Let us prove (223) . Let us compute, for every a ∈ A, d ∈ D[(ψ ⊗ D) ◦ (D ⊗ ψ) ◦(∆ D ⊗ A )] (d ⊗ a)= [(ψ ⊗ D) ◦ (D ⊗ ψ)] ( d (1) ⊗ d (2) ⊗ a )= (ψ ⊗ D) ( d (1) ⊗ a 0 ⊗ d (2) · a 1)= a00 ⊗ d (1) · a 01 ⊗ d (2) · a 1A isHcomod= a 0 ⊗ d (1) · a 1(1) ⊗ d (2) · a 1(2)DisHmodcoalg= a 0 ⊗ (d · a 1 ) (1)⊗ (d · a 1 ) (2)so that we getandso that we get= ( A ⊗ ∆ D) (a 0 ⊗ d · a 1 ) = [( A ⊗ ∆ D) ◦ ψ ] (d ⊗ a)(ψ ⊗ D) ◦ (D ⊗ ψ) ◦ ( ∆ D ⊗ A ) = ( A ⊗ ∆ D) ◦ ψ[rA ◦ ( A ⊗ ε D) ◦ ψ ] (d ⊗ a) = [ r A ◦ ( A ⊗ ε D)] (a 0 ⊗ d · a 1 )(= r A a0 ⊗ ε D (d · a 1 ) ) DisHmodcoal (= r A a0 ⊗ ε D (d) ε H (a 1 ) )= aε D (d) = ε D (d) a = l A(ε D (d) ⊗ a ) = [ l A ◦ ( ε D ⊗ A )] (d ⊗ a)r A ◦ ( A ⊗ ε D) ◦ ψ = l A ◦ ( ε D ⊗ A ) .Then by Proposition 9.30, ( D = A ⊗ D, ∆ D , ε D) is the A-coring associated to theentwining (A, D, ψ) and M D A = (Mod-A)D ≃ M D A (ψ). Note that M ∈ MD A (ψ) , issuch that ( ( )M, µ M) A is a right A-module, M, ρDM is a right D-comodule satisfying(µAM ⊗ D ) ◦ (M ⊗ ψ) ◦ ( ρ D M ⊗ A ) = ρ D M ◦ µ A Mi.e. for every m ∈ M and for every a ∈ A( )ρDM ◦ µ A M (m ⊗ a) = µAM (m 0 ⊗ a 0 ) ⊗ µ H D (m 1 ⊗ a 1 )which is exactly the condition (230) for M ∈ M D A (H) . Since morphisms in bothcategories M D A (ψ) and MD A (H) are right A-linear and right D-colinear morphismswe deduce thatM D A (ψ) ≃ M D A (H) .Since by Lemma 9.33 A ∈ M D A (H) ≃ MD A (ψ) , we can apply Theorem 9.31 to thecase ”C” = D and ”M C A (ψ) ” = MD A (ψ) ≃ MD A (H) .□
219Corollary 9.35 ([Schn1, Theorem I (2) ⇔ (4)]). Let H = ( )H, ∆ H , ε H , m H , u Hbe a k-bialgebra and let ( )(A, m A , u A ) , ρ H A be a right H-comodule algebra. ThenH = A⊗H is an A-coring. Assume that A H is flat (i.e. H is k-flat). Let B = A coH .Then the following statements are equivalent:(a) the functor − ⊗ B A : Mod-B → M H A is an equivalence of categories;(b) the canonical map can : A ⊗ B A → A ⊗ H is an isomorphism (i.e. B ⊆ A isan H-Galois extension) and B A is faithfully flat.Proof. We can apply Theorem 9.34 to the case ”D” = H so that M D A (H) = MH A .□10. BicategoriesIn this last part we will change some notations to be more clear and to give moreevidence to a new product we introduce here.Let C be a bicategory. For every 0-cell X in C, we denote by 1 X : X → X theidentity 1-cell over X. For every 1-cell A in C, we denote by 1 A : A → A the identity2-cell over A. We will use juxtaposition when we compose 2-cells vertically and wewill denote by · the horizontal composition of 1-cells and 2-cells.Let us assume that C is a bicategory with completeness requirement (all thecategories C ((X, A) , (Y, B)) have coequalizers which are preserved by compositionwith 1-cells).We keep denoting by (A, m A , u A ) a monad with its multiplication and unit.We now want to define the 2-category Mnd (C) following the definition given in[St]. For simplicity we will always assume to work with a 2-category C even if onecan prove similar <strong>results</strong> for an arbitrary bicategory.Definition 10.<strong>1.</strong> Let C be a 2-category. A monad in C is a pair (X, A) where Xis an object of C, A : X → X is a 1-cell in C together with 2-cells m A : 1 A · 1 A → 1 Aand u A : 1 X → 1 A satisfying associativity and unitality conditions, i.e.(231)(232)m A (1 A · m A ) = m A (m A · 1 A )m A (u A · 1 A ) = 1 A = m A (1 A · u A ) .Definition 10.<strong>2.</strong> Let C be a 2-category and let (X, A) , (Y, B) be monads in C. Amonad functor in C is a pair (Q, φ) : (X, A) → (Y, B) where Q : X → Y is a 1-celland φ : B · Q → Q · A satisfying the following conditions(233)(234)(1 Q · m A ) (φ · 1 A ) (1 B · φ) = φ (m B · 1 Q )φ (u B · 1 Q ) = 1 Q · u A .Definition 10.3. Let C be a 2-category, let (X, A) , (Y, B) be monads in C and let(Q, φ) , (Q ′ , φ ′ ) : (X, A) → (Y, B) be monad functors. A monad functor transformationσ : (Q, φ) → (Q ′ , φ ′ ) in C is a 2-cell σ : Q → Q ′ such that(235) φ ′ (1 B · σ) = (σ · 1 A ) φ.Definition 10.4. The 2-category Mnd (C) consists of• Objects: monads in C• 1-cells: monad functors in C
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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
Page 168 and 169: 168)(155)= k 2 ◦(Qpb Q◦ (Ql A U
Page 170 and 171: 170) ) (χ= ρ ◦(p QB ̂QA ◦(Qp
Page 172 and 173: 172Theorem 8.13. Let A and B be cat
Page 174 and 175: 174l = eC ρ L : L = − ⊗ B A
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Page 178 and 179: 178so that− ⊗ R 1 A ⊗ R c = (
Page 180 and 181: 180− ⊗ T x ⊗ R 1 A ⊗ A f
Page 182 and 183: 182(208)(209)(210)(211)(h1 ) 0 ⊗
Page 184 and 185: 184= abd 0 ⊗ d 1 1 ⊗ d 2 1b⊗d
Page 186 and 187: 186so that h 1 ⊗ h 2 ⊗ a ∈ A
Page 188 and 189: 188= 〈( h (1) y (1))εH ( h (2) y
Page 190 and 191: 190H C is faithfully coflat. Assume
Page 192 and 193: 192=(Qε C H C) ( ∑ )−□ C k i
Page 194 and 195: 194Following Theorem 6.29, we now c
Page 196 and 197: 196)û E(ε C H C (h) = û E (π (h
Page 198 and 199: 198Letandα l = (ϕ ⊗ H) ( (x ⊗
Page 200 and 201: 200This map is well-defined, in fac
Page 202 and 203: 202We now have to prove that this m
Page 204 and 205: 2041) − ⊗ B Σ A preserves the
Page 206 and 207: 206functorial isomorphism. In parti
Page 208 and 209: 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 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 260 and 261: 260Since P is finite Hom A (P, P )
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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