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

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220• 2-cells: monad functor transformations in C.Remark 10.5. We denote by (X, 1 X ) in C the trivial monad on the object X withtrivial multiplication and unit m 1X : 1 1X · 1 1X → 1 1X and u 1X : 1 X → 1 1X .Definition 10.6. Let C, C ′ be two 2-categories and let G : C → C ′ be a pseudofunctor.Then the pseudofunctor Mnd (G) : Mnd (C) → Mnd (C ′ ) is defined asfollows:• Mnd (G) (X, A) = (G (X) , G (A))• Mnd (G) (Q, φ) = (G (Q) , G (φ))• Mnd (G) (σ) = G (σ) .Remark 10.7. Note that Cmd (C) = Mnd (C ∗ ) where C ∗ is the dual reversing2-cells of C.1<strong>1.</strong> Construction of BIM (C)The idea of defining this bicategory goes back to the strict monoidal category ofbalanced bimodule functors that we defined in Subsection 3.<strong>2.</strong> We observed that,considering bimodule functors with respect to the same monad on both sides, wehave a unit and a composition, so that they form a strict monoidal category. In thecase we consider a bimodule with respect to two different monads, the unit objectfails and the composition between them is no longer inside the class of objects of thecategory. The way to solve this problem is to look at balanced bimodule functorsas 0-cells of a bicategory, changing the definition of their product.Definition 1<strong>1.</strong><strong>1.</strong> Let X be a 0-cell and let (Y, B) be a monad in C. A left B-modulein C (or simply a left B-module) is a pair (Q, λ Q ) where Q : X → Y is a 1-cell andλ Q : B · Q → Q is a 2-cell in C, satisfying the associativity and unitality propertieswith respect to the monad B, i.e.λ Q (m B · 1 Q ) = λ Q (1 B · λ Q ) and λ Q (u B · 1 Q ) = 1 Q .Definition 1<strong>1.</strong><strong>2.</strong> Let (X, A) and (Y, 1 Y ) be monads in C. A right A-module in C(or simply a right A-module) is a monad functor in C ∗ , i.e. a 1-cell Q : X → Yand a 2-cell ρ Q : Q · A → 1 Y · Q = Q in C, satisfying the associativity and unitalityproperties with respect to the monad A, i.e.ρ Q (1 Q · m A ) = ρ Q (ρ Q · 1 A ) and ρ Q (1 Q · u A ) = 1 Q .Definition 1<strong>1.</strong>3. Let (X, A) and (Y, B) be monads in C. A B-A-bimodule in C (orsimply a B-A-bimodule) is a triple (Q, λ Q , ρ Q ) where• (Q, λ Q ) is a left B-module in C• (Q, ρ Q ) is a right A-module in C• the compatibility condition holdsλ Q (1 B · ρ Q ) = ρ Q (λ Q · 1 A ) .Lemma 1<strong>1.</strong>4. Let (X, A) , (Y, B) be monads in C and let (Q, λ Q ) be a left A-moduleand (Q, ρ Q ) be a right B-module. Then (Q, λ Q ) = Coequ C (m A · 1 Q , 1 A · λ Q ) and(Q, ρ Q ) = Coequ C (1 Q · m B , ρ Q · 1 B ).
Proof. We will only prove the statement for the left module. Similarly can be provedthe other one. Since λ Q is associative, we deduce thatNow, assume that (S, σ) is such thatThen we haveλ Q (m A · 1 Q ) = λ Q (1 A · λ Q ) .σ (m A · 1 Q ) = σ (1 A · λ Q ) .σ (u A · 1 Q ) λ Qu A= σ (1 A · λ Q ) (u A · 1 A · 1 Q )propσ= σ (m A · 1 Q ) (u A · 1 A · 1 Q ) Amonad= σ.Moreover, since λ Q is epi, we conclude that the 2-cell σ (u A · 1 Q ) is unique withrespect to the propertyσ (u A · 1 Q ) λ Q = σso that(Q, λ Q ) = Coequ C (m A · 1 Q , 1 A · λ Q ) .□Proposition 1<strong>1.</strong>5. Let (X, A) , (Y, B) and (W, C) be monads in C and let Q : Y →X and Q ′ : W → Y be respectively a A-B-bimodule with (Q, λ Q , ρ Q ) and a B-Cbimodulein C with (Q ′ , λ Q ′, ρ Q ′). Then (Q • B Q ′ , p Q,Q ′) = Coequ C (ρ Q · 1 Q ′, 1 Q · λ Q ′)is a A-C-bimodule in C via the actions λ Q•B Q ′ and ρ Q• B Q ′ uniquely determined by(236) λ Q•B Q ′ (1 A · p Q,Q ′) = p Q,Q ′ (λ Q · 1 Q ′)and(237) ρ Q•B Q ′ (p Q,Q ′ · 1 C) = p Q,Q ′ (1 Q · ρ Q ′) .Proof. Let us define the bimodule structures on Q• B Q ′ . Let us consider the followingdiagramQ · B · Q ′ · C1 Q·1 B·ρ Q ′ρ Q·1 Q ′·1 C1 Q·λ Q ′·1 CQ · B · Q ′ ρ Q·1 Q ′1 Q·λ Q ′Q · Q ′ · C1 Q·ρ Q ′p Q,Q ′·1 C Q · Q ′ p Q,Q ′Q • B Q ′ · Cρ Q•B Q ′ Q • B Q ′Note that the left square serially commutes. In fact we have(1 Q · ρ Q ′) (ρ Q · 1 Q ′ · 1 C ) ρ Q= (ρ Q · 1 Q ′) (1 Q · 1 B · ρ Q ′)and(1 Q · ρ Q ′) (1 Q · λ Q ′ · 1 C ) Qbim= (1 Q · λ Q ′) (1 Q · 1 B · ρ Q ′) .Therefore, we getp Q,Q ′ (1 Q · ρ Q ′) (ρ Q · 1 Q ′ · 1 C ) = p Q,Q ′ (1 Q · ρ Q ′) (1 Q · λ Q ′ · 1 C )and by the universal property of the coequalizer(Q • B Q ′ · C, p Q,Q ′ · 1 C ) = Coequ C (ρ Q · 1 Q ′ · 1 C , 1 Q · λ Q ′ · 1 C ), there exists a unique2-cell ρ Q•B Q ′ : Q • B Q ′ · C → Q • B Q ′ such thatρ Q•B Q ′ (p Q,Q ′ · 1 C) = p Q,Q ′ (1 Q · ρ Q ′) .221
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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
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 218 and 219: 218Let us compute, for every d ∈
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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