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

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240the unique A-bimodule morphism such thatm Q·A = m (Q·A,(1 Q·m A)(φ·1 A ),1 Q·m A) p Q·A,Q·A.Note that also u (Q,φ) · 1 A is an A-bimodule morphism, in factλ Q·A(1A · u (Q,φ) · 1 A)= (1Q · m A ) (φ · 1 A ) ( 1 A · u (Q,φ) · 1 A)andTherefore,(252)= (1 Q · m A ) ( u (Q,φ) · 1 A · 1 A) u (Q,φ)= ( u (Q,φ) · 1 A)mA( )ρ Q·A u(Q,φ) · 1 A · 1 A = (1Q · m A ) ( ) u (Q,φ)u (Q,φ) · 1 A · 1 A = ( )u (Q,φ) · 1 A mA .()Q · A, m (Q·A,(1Q·m A)(φ·1 A ),1 Q·m A) , u (Q·A,(1 Q·m A)(φ·1 A ),1 Q·m A)= ( Q · A, ( ) )m (Q,φ) · 1 A α, u(Q,φ) · 1 Ais an A-ring, so that the functor Mnd (F (C)) : Mnd (Mnd (C)) → Mnd (BIM (C))associates distributive laws to A-rings.Let us now consider Cmd (Mnd (C)) :• 0-cells: ((X, A) , (C, γ)) where (X, A) is a monad, C : X → X, γ : A · C →C·A together with ∆ C : C → C·C and ε C : C → 1 X satisfying coassociativityand counitality and satisfying(1 C · γ) (γ · 1 C ) ( 1 A · ∆ C) = ( ∆ C · 1 A)γ(257)1 A · ε C = ( ε C · 1 A)γ.(258)Note that, if we consider ( C · A, ∆ C·A , ε C·A) where∆ C·A = (γ · 1 C · 1 A ) ( 1 A · ∆ C · 1 A)(uA · 1 C · 1 A ) and ε C·A : C · A → A coassociativityand counitality properties are not satisfied. But, by applying thefunctor Cmd (F (C)) to the comonad ( C, ∆ C , ε C) we getCmd (F (C)) (( C, ∆ C , ε C)) = ( C · A, ∆ C · 1 A , ε C · 1 A)where∆ C · 1 A ≃ ∆ C·A : C · A → C · A • A C · Aand ∆ C · 1 A and ε C · 1 A are A-bimodule morphisms, clearly satisfying coassociativityand counitality conditions. Hence, ( )C · A, ∆ C · 1 A , ε C · 1 A is anA-coring.SinceMnd (C) ((X, 1 X ) , (Y, B)) = C(X,B) C (X, Y )dually we getCmd (C) ((X, 1 X ) , (Y, C)) = C(X,C) C (X, Y ) .Consider the objects ( (X, 1 X ) , ( ))1 (X,1X ), 1 X , ((X, A) , (C, γ)) ∈ Cmd (Mnd (C))[(C, γ) : (X, A) → (X, A) , γ : A · C → C · A] and let((Q, φ) , σ) ∈ Cmd ( Mnd (C) ( (X, 1 X ) , ( )) )1 (X,1X ), 1 X , ((X, A) , (C, γ)) be a comonadfunctor, where (Q, φ) : (X, 1 X ) → (X, A) is a 1-cell in Mnd (C), i.e. φ : A · Q → Qsatisfies φ (u A · 1 Q ) = 1 Q and φ (1 A · φ) = φ (m A · 1 Q ) and σ : (Q, φ) ( )1 (X,1X ), 1 X =(Q, φ) → (C, γ) (Q, φ) = (C · Q, (1 C · φ) (γ · 1 Q )) is a 2-cell in Mnd (C), i.e.
(1 C · φ) (γ · 1 Q ) (1 A · σ) = σφ, i.e. σ is an A-linear map. Since ((Q, φ) , σ) is acomonad functor, the 2-cell σ : Q → C · Q satisfies ( )ε C · 1 Q σ = 1Q and (1 C · φ) φ =( )C(X,C)∆C · 1 Q φ. This means that ((Q, φ) , σ) ∈C(X,A)C (X, X) (γ) is an entwined module.By applying the functor Cmd (F (C)) : Cmd (Mnd (C)) −→ Cmd (BIM (C))to the element((Q, φ) , σ) ∈ Cmd ( Mnd (C) ( (X, 1 X ) , ( )) )1 (X,1X ), 1 X , ((X, A) , (C, γ)) we getCmd (F (C)) (((Q, φ) , σ)) ∈ Cmd ( BIM (C) ( (X, 1 X ) , ( )) )1 (X,1X ), 1 X , ((X, A) , (C, γ))which is an element of C(X,C·A) BIM (X, X), i.e. it is a left C · A-comodule with respectto • A .241Appendix A. Gabriel Popescu TheoremNotation A.<strong>1.</strong> Let A be a Grothendieck category, let U be an object of A andlet B = Hom A (U, U). Assume that U is a generator of A i.e. that the functorHom A (U, −) : A → Mod-B is faithful.Lemma A.<strong>2.</strong> In the assumptions and notations of A.1, let X ∈ A and let λ :U (Hom A(U,X)) → X be the codiagonal morphism of the family (f) f∈(HomA (U,X)) . ThenIm (λ) = X.(HomProof. Let J : Ker (λ) → U A(U,X))be the canonical monomorphism and letλ : U (HomA(U,X)) → X be the codiagonal morphism of the family (f) f∈(HomA (U,X))and, for every f ∈ Hom A (U, X) let i f : U → U (HomA(U,X)) the f-th canonicalinjection. Then we have λ ◦ i f = f. Let χ : X → Coker (λ) be the canonicalprojection and let us assume that χ ≠ 0. Then there exists h : U → X such thatχ ◦ h ≠ 0. Then we have0 ≠ χ ◦ h = χ ◦ λ ◦ i h = 0 ◦ i h = 0.Contradiction. Thus Coker (λ) = 0 and hence X = KerCoker (λ) = Im (λ).□Proposition A.3. In the assumptions and notations of A.1, the functor Hom A (U, −) :A → Mod-B is full.Proof. Let ϕ ∈ Hom B (Hom A (U, X) , Hom A (U, Z)). We have to prove that thereexists a morphism g : X → Z such that ϕ = Hom A (U, g). For any subset F ofHom A (U, X) we denote byi F : U (F ) → U (Hom A(U,X))the canonical injection. If F = {f} we will write i f instead of i {f} . Let λ :U (HomA(U,X)) → X be the codiagonal morphism of the family (f) f∈(HomA (U,X))and letµ : U (HomA(U,X)) → Z be the codiagonal morphism of the family (ϕ (f)) f∈(HomA (U,X)) .Then, for every f ∈ Hom A (U, X) we haveλ ◦ i f = f and µ ◦ i f = ϕ (f) .
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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
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 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 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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