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

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2362-cells. This means that a comonad (X, C) in C is a 1-cell C : X → X togetherwith 2-cells ∆ C : C → C · C and ε C : C → 1 X called comultiplication and counitsatisfying the reversed diagrams, i.e.(1C · ∆ C) ∆ C = ( )∆ C · 1 C ∆C(249)(1C · ε C) ∆ C = 1 C = ( )(250)ε C · 1 C ∆ C .A comonad functor is a pair (P, ψ) : (X, C) → (Y, D) where P : X → Y is a 1-cellin C and ψ : P · C → D · P is a 2-cell in C satisfying(εD · 1 P)ψ = 1P · ε C and (1 D · ψ) (ψ · 1 C ) ( 1 P · ∆ C) = ( ∆ D · 1 P)ψ.Finally, a comonad functorial morphism ω : (P ′ , ψ ′ ) → (P, ψ) is ω : P ′ → P is a2-cell in C satisfyingψ (ω · 1 C ) = (1 D · ω) ψ ′ .Now, we consider the category Cmd (C) ((X, 1 X ) , (Y, C)) where (X, 1 X ) and (Y, C)are 0-cells in Cmd (C) respectively with trivial comultiplication and counit theformer and ∆ C , ε C the latter. Note that C (X, C) : C (X, Y ) → C (X, Y ) is acomonad over the category C (X, Y ) with comultiplication and counit given byC ( X, ∆ C) = ∆ C () : C (X, C) → C (X, C · C) and C ( X, ε C) = ε C () : C (X, C) →C (X, 1 Y ). The objects of such category are the comonad functors (Q, ψ) : (X, 1 X ) →(Y, C) where Q : X → Y is a 1-cell and ψ : Q · 1 X → C · Q = C (X, C)Q is a 2-cellsatisfying ( ε C · 1 Q)ψ = 1Q and ( ∆ C · 1 Q)ψ = (1C · ψ) ψ so thatCmd (C) ((X, 1 X ) , (Y, C)) = C(X,C) C (X, Y ).Following the definition of the 2-category Mnd (C) for any bicategory C, we canconsider the 2-categories Mnd (Mnd (C)) and Mnd (BIM (C)) and the functorbetween themMnd (F (C)) : Mnd (Mnd (C)) → Mnd (BIM (C)) .Let us first consider Mnd (Mnd (C)):• 0-cells: pairs ((X, A) , (Q, φ)) where (X, A) is an object in Mnd (C) and(Q, φ) is a 1-cell in Mnd (C) together with a pair of 2-cells in Mnd (C) m (Q,φ)and u (Q,φ) satisfying associativity and unitality conditions. Therefore we havethat A : X → X is a 1-cell in C together with 2-cells m A = m (X,A) : A·A → Aand u A = u (X,A) : 1 X → A satisfying associativity and unitality conditionsand we have that Q : X → X is a 1-cell in C together with the 2-cell of Cφ : A · Q → Q · A satisfying the following conditions(251)(252)(253)(254)φ (m A · 1 Q ) = (1 Q · m A ) (φ · 1 A ) (1 A · φ)φ (u A · 1 Q ) = 1 Q · u AFinally, the 2-cells of Mnd (C)m (Q,φ) : (Q, φ) · (Q, φ) = (Q · Q, (1 Q · φ) (φ · 1 Q )) → (Q, φ) and u (Q,φ) :(1 X , 1 1X ) → (Q, φ) satisfying the associativity and unitality conditions, needsto satisfy also the followingφ ( 1 A · m (Q,φ))=(m(Q,φ) · 1 A)(1Q · φ) (φ · 1 Q )φ ( 1 A · u (Q,φ))= u(Q,φ) · 1 A .
An object, or 0-cell, of Mnd (Mnd (C)) is called distributive law and it givesrise to a monad structure on Q·A. In fact, the monad functor transformationm (Q,φ) induces a multiplication on Q · Adefined by settingm Q·A : Q · A · Q · A → Q · Am Q·A = ( m (Q,φ) · m A)(1Q · φ · 1 A ) = ( m (Q,φ) · 1 A)(1Q · 1 Q · m A ) (1 Q · φ · 1 A ) .Using naturality and associativity of m (Q,φ) , naturality of φ, associativity ofm A , we havem Q·A (m Q·A · 1 Q · 1 A )= ( m (Q,φ) · 1 A)(1Q · 1 Q · m A ) (1 Q · φ · 1 A ) ( m (Q,φ) · 1 A · 1 Q · 1 A)(1 Q · 1 Q · m A · 1 Q · 1 A ) (1 Q · φ · 1 A · 1 Q · 1 A )= ( m (Q,φ) · 1 A) (m(Q,φ) · 1 Q · 1 A)(1Q · 1 Q · 1 Q · m A ) (1 Q · 1 Q · φ · 1 A )(1 Q · 1 Q · m A · 1 Q · 1 A ) (1 Q · φ · 1 A · 1 Q · 1 A )= ( m (Q,φ) · 1 A) (1Q · m (Q,φ) · 1 A)(1Q · 1 Q · 1 Q · m A ) (1 Q · 1 Q · φ · 1 A )(1 Q · 1 Q · m A · 1 Q · 1 A ) (1 Q · φ · 1 A · 1 Q · 1 A )= ( m (Q,φ) · 1 A)(1Q · 1 Q · m A ) ( 1 Q · m (Q,φ) · 1 A · 1 A)(1Q · 1 Q · φ · 1 A )(1 Q · 1 Q · m A · 1 Q · 1 A ) (1 Q · φ · 1 A · 1 Q · 1 A )(251)= ( m (Q,φ) · 1 A)(1Q · 1 Q · m A ) ( 1 Q · m (Q,φ) · 1 A · 1 A)(1Q · 1 Q · 1 Q · m A · 1 A )(1 Q · 1 Q · φ · 1 A · 1 A ) (1 Q · 1 Q · 1 A · φ · 1 A ) (1 Q · φ · 1 A · 1 Q · 1 A )= ( m (Q,φ) · 1 A)(1Q · 1 Q · m A ) (1 Q · 1 Q · m A · 1 A ) ( 1 Q · m (Q,φ) · 1 A · 1 A · 1 A)(1 Q · 1 Q · φ · 1 A · 1 A ) (1 Q · φ · 1 Q · 1 A · 1 A ) (1 Q · 1 A · 1 Q · φ · 1 A )= ( m (Q,φ) · 1 A)(1Q · 1 Q · m A ) (1 Q · 1 Q · 1 A · m A ) ( 1 Q · m (Q,φ) · 1 A · 1 A · 1 A)(1 Q · 1 Q · φ · 1 A · 1 A ) (1 Q · φ · 1 Q · 1 A · 1 A ) (1 Q · 1 A · 1 Q · φ · 1 A )= ( m (Q,φ) · 1 A)(1Q · 1 Q · m A ) ( 1 Q · m (Q,φ) · 1 A · 1 A)(1Q · 1 Q · 1 Q · 1 A · m A )(1 Q · 1 Q · φ · 1 A · 1 A ) (1 Q · φ · 1 Q · 1 A · 1 A ) (1 Q · 1 A · 1 Q · φ · 1 A )= ( m (Q,φ) · 1 A)(1Q · 1 Q · m A ) ( 1 Q · m (Q,φ) · 1 A · 1 A)(1Q · 1 Q · φ · 1 A )(1 Q · 1 Q · 1 A · 1 Q · m A ) (1 Q · φ · 1 Q · 1 A · 1 A ) (1 Q · 1 A · 1 Q · φ · 1 A )= ( m (Q,φ) · 1 A)(1Q · 1 Q · m A ) ( 1 Q · m (Q,φ) · 1 A · 1 A)(1Q · 1 Q · φ · 1 A ) (1 Q · φ · 1 Q · 1 A )(1 Q · 1 A · 1 Q · 1 Q · m A ) (1 Q · 1 A · 1 Q · φ · 1 A )(253)= ( )m (Q,φ) · 1 A (1Q · 1 Q · m A ) (1 Q · φ · 1 A ) ( )1 Q · 1 A · m (Q,φ) · 1 A(1 Q · 1 A · 1 Q · 1 Q · m A ) (1 Q · 1 A · 1 Q · φ · 1 A )so that m Q·A is associative.u (Q,φ) induces a unit of Q · A= m Q·A (1 Q · 1 A · m Q·A )237Similarly, the monad functor transformationu Q·A : 1 X → Q · A
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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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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 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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