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

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146given byWe computeσ B = m B ◦ ( σ B σ B) ◦ (P iQ) ◦ (qQ)(144)= m B ◦ ( σ B σ B) ◦ (jP Q) ◦ (κ ′ 0Q) .(σ B P Q ) ◦ (P iQ) ◦ (qQ) ◦ ( Qχ ) ◦ ( δ D QQ )q= ( σ B P Q ) ◦ (P iQ) ◦ (P Cχ) ◦ ( qQQQ ) ◦ ( δ D QQ )i= ( σ B P Q ) ◦ (P QP χ) ◦ ( P iQQQ ) ◦ ( qQQQ ) ◦ ( δ D QQ )(144)= ( σ B P Q ) ◦ (P QP χ) ◦ ( jP QQQ ) ◦ ( κ ′ 0QQQ ) ◦ ( δ D QQ )(148)= ( σ B P Q ) ◦ (P QP χ) ◦ ( jP QQQ ) ◦ ( DjQQ ) ◦ ( ∆ D QQ )(67)= ( σ B P Q ) ◦ (P QP χ) ◦ ( P τQQ ) ◦ ( jQQ )= ( σ B P Q ) ◦ ( ) (P QP µ B Q ◦ P QP A µ Q B ) ◦ ( P QP AQσ B) ◦ ( P QP σ A QP Q )◦ (P QP QP iQ) ◦ (P QP QqQ) ◦ ( P τQQ ) ◦ ( jQQ )σ= ( A σ B P Q ) ◦ ( (P QP µ Q) B ◦ P QP A µ Q B ) ◦ ( P QP σ A QB ) ◦ ( P QP QP Qσ B)◦ (P QP QP iQ) ◦ (P QP QqQ) ◦ ( P τQQ ) ◦ ( jQQ )(82)= ( σ B P Q ) ◦ ( ) (P QP µ B Q ◦ P QP µBQ B ) ◦ ( P QP Qσ B B )◦ ( P QP QP Qσ B) ◦ (P QP QP iQ) ◦ (P QP QqQ) ◦ ( P τQQ ) ◦ ( jQQ )σ= ( ) ( BBP µ B Q ◦ BP µBQ B ) ◦ ( BP Qσ B B ) ◦ ( BP QP Qσ B) ◦ (BP QP iQ)◦ (BP QqQ) ◦ ( σ B P QQQ ) ◦ ( P τQQ ) ◦ ( jQQ )defD= ( BP µ B Q)◦(BP µBQ B ) ◦ ( BP Qσ B B ) ◦ ( BP QP Qσ B) ◦ (BP QP iQ)◦ (BP QqQ) ◦ ( u B P QQQ ) ◦ ( jQQ )and thus we obtain (σ B P Q ) ◦ (P iQ) ◦ (qQ) ◦ ( Qχ ) ◦ ( δ D QQ )(164)= ( BP µ B Q)◦(BP µBQ B ) ◦ ( BP Qσ B B ) ◦ ( BP QP Qσ B) ◦ (BP QP iQ)Let us compute◦ (BP QqQ) ◦ ( u B P QQQ ) ◦ ( jQQ ) .σ B ◦ ( Qχ ) ◦ ( δ D QQ )= m B ◦ ( σ B σ B) ◦ (P iQ) ◦ (qQ) ◦ ( Qχ ) ◦ ( δ D QQ )= m B ◦ ( Bσ B) ◦ ( σ B P Q ) ◦ (P iQ) ◦ (qQ) ◦ ( Qχ ) ◦ ( δ D QQ )(164)= m B ◦ ( Bσ B) ◦ ( ) (BP µ B Q ◦ BP µBQ B ) ◦ ( BP Qσ B B )◦ ( BP QP Qσ B) ◦ (BP QP iQ) ◦ (BP QqQ) ◦ ( u B P QQQ ) ◦ ( jQQ )u B= m B ◦ (u B B) ◦ σ B ◦ ( P µ B Q)◦(P µBQ B ) ◦ ( P Qσ B B ) ◦ ( P QP Qσ B)
◦ (P QP iQ) ◦ ( jQQ )Bmonad= σ B ◦ ( P µ B Q)◦(P µBQ B ) ◦ ( P Qσ B B ) ◦ ( P QP Qσ B) ◦ (P QP iQ)◦ ( jQQ )(82)= σ B ◦ ( (P µ Q) B ◦ P A µ Q B ) ◦ ( P σ A QB ) ◦ ( P QP Qσ B) ◦ (P QP iQ)◦ (P QqQ) ◦ ( jQQ )147σ= A σ B ◦ ( ) (P µ B Q ◦ P A µ Q B ) ◦ ( P AQσ B) ◦ ( P σ A QP Q ) ◦ (P QP iQ)◦ (P QqQ) ◦ ( jQQ )(81)= m B ◦ ( σ B B ) ◦ ( P A µ Q B ) ◦ ( P AQσ B) ◦ ( P σ A QP Q ) ◦ (P QP iQ)◦ (P QqQ) ◦ ( jQQ )A µ Q= mB ◦ ( σ B B ) ◦ ( P Qσ B) ◦ ( P A µ Q P Q ) ◦ ( P σ A QP Q ) ◦ (P QP iQ)◦ (P QqQ) ◦ ( jQQ )(82)= m B ◦ ( σ B B ) ◦ ( P Qσ B) ◦ ( P µ B QP Q ) ◦ ( P Qσ B P Q ) ◦ (P QP iQ)◦ (P QqQ) ◦ ( jQQ )σ B = m B ◦ ( Bσ B) ◦ ( σ B P Q ) ◦ ( P µ B QP Q ) ◦ ( P Qσ B P Q ) ◦ (P QP iQ)◦ (P QqQ) ◦ ( jQQ )(81)= m B ◦ ( Bσ B) ◦ (m B P Q) ◦ ( σ B BP Q ) ◦ ( P Qσ B P Q ) ◦ (P QP iQ)◦ (P QqQ) ◦ ( jQQ )σ B = m B ◦ ( Bσ B) ◦ (m B P Q) ◦ ( Bσ B P Q ) ◦ (BP iQ) ◦ (BqQ) ◦ ( σ B QQ ) ◦ ( jQQ )(67)= m B ◦ ( Bσ B) ◦ (m B P Q) ◦ ( Bσ B P Q ) ◦ (BP iQ) ◦ (BqQ) ◦ ( u B QQ )◦ ( ε D QQ )u B= m B ◦ ( Bσ B) ◦ (m B P Q) ◦ (u B BP Q) ◦ ( σ B P Q ) ◦ (P iQ) ◦ (qQ) ◦ ( ε D QQ )so that we getBmonad= m B ◦ ( Bσ B) ◦ ( σ B P Q ) ◦ (P iQ) ◦ (qQ) ◦ ( ε D QQ )= σ B ◦ ( ε D QQ )(165) σ B ◦ ( Qχ ) ◦ ( δ D QQ ) = σ B ◦ ( ε D QQ )and since (B ′ , y ′ ) = Coequ Fun((Qχ)◦(δD QQ ) , ε D QQ ) there exists a unique functorialmorphism ν B : B ′ → B such that(166) ν B ◦ y ′ = σ B = m B ◦ ( σ B σ B) ◦ (P iQ) ◦ (qQ) .Now we want to prove that ν B is a morphism of monads. Let us computem B ◦ (ν B ν B ) ◦ (y ′ y ′ ) = m B ◦ (ν B B) ◦ (B ′ ν B ) ◦ (y ′ B ′ ) ◦ ( QQy ′)
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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
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
Page 112 and 113: 112Then ν : Y → D is the unique
Page 114 and 115: 114= A µ Q ◦ ( Aε C Q ) ◦ (AC
Page 116 and 117: 116= ( Aε C Q ) ◦ ( cocan1 −1
Page 118 and 119: 118so that we getχ= (Cx) ◦ (C ρ
Page 120 and 121: 120We want to prove that Γ is an o
Page 122 and 123: 122and since Dε D is an epimorphis
Page 124 and 125: 124χ= (Cχ) ◦ (C ρ Q P Q ) ◦
Page 126 and 127: 126Now, since cocan 1 : AC → QP i
Page 128 and 129: 1287. Herds and Coherds7.1.
Page 130 and 131: 130◦ ( σ A QQQ ) ◦ (A µ Q P Q
Page 132 and 133: 132= µ B Q ◦ (A µ Q B ) ◦ ( A
Page 134 and 135: 134Assume now that there is another
Page 136 and 137: 136and hence we get(160) x ◦ (χP
Page 138 and 139: 138Proposition 7.7. In the setting
Page 140 and 141: 140We calculateA µ Q ◦ ( σ A Q
Page 142 and 143: 142x=and=δ C=(l= QlQ ̂QQ)◦ (QP
Page 144 and 145: 144(◦ ρ D ̂QQ)Q◦ (QlQ) ◦ (Q
Page 148 and 149: 148andy= ′m B ◦ (ν B B) ◦ (y
Page 150 and 151: 150Now we compute(hQ) ◦ ( Qχ )
Page 152 and 153: 152Thus we obtainσ B ◦ ( ) (P µ
Page 154 and 155: 154Thus hQ is an isomorphism with i
Page 156 and 157: 156) ( )l=(pb Q AQ B ◦ ̂QA µ QB
Page 158 and 159: 158In fact we haveTherefore we dedu
Page 160 and 161: 160χ= h 1 ◦ (P xQ B ) ◦ (P QP
Page 162 and 163: 162so that we obtain:(190)We comput
Page 164 and 165: 164(194)=) )(p QB ̂QA ◦(Qpb Q◦
Page 166 and 167: 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
Page 176 and 177: and[µBQ ◦ ( Qσ B)] (− ⊗ T x
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
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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
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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
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218Let us compute, for every d ∈
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220• 2-cells: monad functor trans
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222We now want to prove that ρ Q·
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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,
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230On the other hand, we can first
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232so that we define the map φ F (
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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
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244Lemma A.4. Let A be an abelian c
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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
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252Proposition A.12 ([ELGO2, Propos
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254(⇒) Let {A i } i∈Ibe a famil
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256We will prove that h : ∐ B i
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258Proposition A.19. Let (T, H) be
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260Since P is finite Hom A (P, P )
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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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