Contents 1. Introduction 2 2. Preliminaries 4 2.1. Some results on ...
Contents 1. Introduction 2 2. Preliminaries 4 2.1. Some results on ... Contents 1. Introduction 2 2. Preliminaries 4 2.1. Some results on ...
130◦ ( σ A QQQ ) ◦ (A µ Q P QQQ )m=A A µ Q ◦ ( )Aµ B Q ◦ (mA QB) ◦ [( Aσ A QB ) ◦ ( AQP Qσ B) ◦ (AQP iQ) ◦ (AQqQ) ]◦ ( σ A QQQ ) ◦ (A µ Q P QQQ )Qis a bim= µ B Q ◦ (A µ Q B ) ◦ (m A QB) ◦ [( Aσ A QB ) ◦ ( AQP Qσ B) ◦ (AQP iQ) ◦ (AQqQ) ]◦ ( σ A QQQ ) ◦ (A µ Q P QQQ )A µ Q ass= µ B Q ◦ (A µ Q B ) ◦ [( A A µ Q B ) ◦ ( Aσ A QB ) ◦ ( AQP Qσ B) ◦ (AQP iQ) ◦ (AQqQ) ]◦ ( σ A QQQ ) ◦ (A µ Q P QQQ )(80)= µ B Q ◦ (A µ Q B ) ◦ [( A A µ Q B ) ◦ ( Aσ A QB ) ◦ ( AQP Qσ B) ◦ (AQP iQ) ◦ (AQqQ) ]◦ ( m A QQQ ) ◦ ( Aσ A QQQ )m A= µ B Q ◦ (A µ Q B ) ◦ (m A QB) ◦ [ ( AA A µ Q B ) ◦ ( AAσ A QB ) ◦ ( AAQP Qσ B) ◦ (AAQP iQ)◦ (AAQqQ)] ◦ ( Aσ A QQQ )A µ Q ass= µ B Q ◦ (A µ Q B ) ◦ ( A A µ Q B ) ◦ [ ( AA A µ Q B ) ◦ ( AAσ A QB ) ◦ ( AAQP Qσ B)◦ (AAQP iQ) ◦ (AAQqQ)] ◦ ( Aσ A QQQ )Qis a bim= A µ Q ◦ ( Aµ B Q)◦(A A µ Q B ) ◦ ( AA A µ Q B ) ◦ ( AAσ A QB ) ◦ ( AAQP Qσ B)◦ (AAQP iQ) ◦ (AAQqQ) ◦ ( Aσ A QQQ )Qis a bim= A µ Q ◦ ( A A µ Q)◦(AAµBQ)◦(AA A µ Q B ) ◦ ( AAσ A QB ) ◦ ( AAQP Qσ B)◦ (AAQP iQ) ◦ (AAQqQ) ◦ ( Aσ A QQQ )(82)= A µ Q ◦ ( A A µ Q)◦(AAµBQ)◦(AAµBQ B ) ◦ ( AAQσ B B ) ◦ ( AAQP Qσ B)◦ (AAQP iQ) ◦ (AAQqQ) ◦ ( Aσ A QQQ )σ A = A µ Q ◦ ( A A µ Q)◦(Aσ A Q ) ◦ ( AQP µ B Q)◦(AQP µBQ B ) ◦ ( AQP Qσ B B )◦ ( AQP QP Qσ B) ◦ (AQP QP iQ) ◦ (AQP QqQ)(82)= A µ Q ◦ ( (Aµ Q) ) B ◦ AQσB◦ ( ) (AQP µ B Q ◦ AQP µBQ B ) ◦ ( AQP Qσ B B )◦ ( AQP QP Qσ B) ◦ (AQP QP iQ) ◦ (AQP QqQ)Qis a bim= µ B Q◦ (A µ Q B ) ◦ ( AQσ B) ◦ ( ) (AQP µ B Q ◦ AQP µBQ B ) ◦ ( AQP Qσ B B )◦ ( AQP QP Qσ B) ◦ (AQP QP iQ) ◦ (AQP QqQ)µ B Q= assµ B Q◦ (A µ Q B ) ◦ ( AQσ B) ◦ ( ) [AQP µ B Q ◦ (AQP QmB ) ◦ ( AQP Qσ B B )]◦ ( AQP QP Qσ B) ◦ (AQP QP iQ) ◦ (AQP QqQ)(81)= µ B Q◦ (A µ Q B ) ◦ ( AQσ B) ◦ ( (AQP µ Q) ) B ◦ AQP QσB◦ ( )AQP QP µ B Q◦ ( AQP QP Qσ B) ◦ (AQP QP iQ) ◦ (AQP QqQ)
131A µ Q= µBQ ◦ [( Qσ B) ◦ ( (QP µ Q)] ) B ◦ QP QσB◦ ( ) (QP QP µ )B Q ◦ QP QP QσB◦ (QP QP iQ) ◦ (QP QqQ) ◦ (A µ Q P QQQ )(81)= µ B Q ◦ (Qm B ) ◦ ( Qσ B B ) ◦ ( QP Qσ B) ◦ ( QP QP µ B Q◦ (QP QP iQ) ◦ (QP QqQ) ◦ (A µ Q P QQQ )σ B = µ B Q ◦ (Qm B ) ◦ ( QBσ B) ◦ ( QBP µ B Q)◦(QP QP QσB ))◦(QBP QσB ) ◦ (QBP iQ) ◦ (QBqQ)◦ ( Qσ B QQ ) ◦ (A µ Q P QQQ )µ B Q ass= µ B Q ◦ ( µ B QB ) ◦ ( QBσ B) ◦ ( QBP µ B Q)◦(QBP QσB ) ◦ (QBP iQ) ◦ (QBqQ)◦ ( Qσ B QQ ) ◦ (A µ Q P QQQ )µ B Q= µ B Q ◦ ( Qσ B) ◦ ( ) (QP µ ) B Q ◦ QP QσB◦ (QP iQ) ◦ (QqQ) ◦ ( µ B QQQ )◦ ( Qσ B QQ ) ◦ (A µ Q P QQQ )A µ Q= µBQ ◦ ( Qσ B) ◦ ( ) (QP µ ) B Q ◦ QP QσB◦ (QP iQ) ◦ (QqQ) ◦ ( µ B QQQ )◦ (A µ Q BQQ ) ◦ ( AQσ B QQ ))◦(QP QσB ) ◦ (QP iQ) ◦ (QqQ) ◦ ( µ B QQQ )(82)= A µ Q ◦ ( σ A Q ) ◦ ( QP µ B Q◦ (A µ Q BQQ ) ◦ ( AQσ B QQ )σ A = A µ Q ◦ ( Aµ B Q)◦(AQσB ) ◦ ( σ A QP Q ) ◦ (QP iQ) ◦ (QqQ) ◦ ( µ B QQQ )◦ (A µ Q BQQ ) ◦ ( AQσ B QQ )Qis a bim= µ B Q ◦ (A µ Q B ) ◦ ( AQσ B) ◦ ( σ A QP Q ) ◦ (QP iQ) ◦ (QqQ) ◦ ( µ B QQQ )◦ (A µ Q BQQ ) ◦ ( AQσ B QQ )and thus we getχ ◦ ( QQχ ) == A µ Q ◦ (Aχ) ◦ ( σ A QQQ ) ◦ (A µ Q P QQQ ) ◦ ( σ A QP QQQ ) ◦ ( QP iQQQ )◦ ( QqQQQ )= µ B Q ◦ (A µ Q B ) ◦ ( AQσ B) ◦ ( σ A QP Q ) ◦ (QP iQ) ◦ (QqQ) ◦ ( µ B QQQ )◦ (A µ Q BQQ ) ◦ ( AQσ B QQ ) ◦ ( σ A QP QQQ ) ◦ ( QP iQQQ ) ◦ ( QqQQQ )= χ ◦ ( χQQ )2) Counitality, in the sense that χ ◦ (Qδ D ) = Qε D and χ ◦ (δ C Q) = ε C Q. We haveχ ◦ (Qδ D ) == µ B Q ◦ (A µ Q B ) ◦ ( AQσ B) ◦ ( σ A QP Q ) ◦ (QP iQ) ◦ (QqQ) ◦ (Qδ D )(144)= µ B Q ◦ (A µ Q B ) ◦ ( AQσ B) ◦ ( σ A QP Q ) ◦ (QjP Q) ◦ (Qκ ′ 0Q) ◦ (Qδ D )(148)= µ B Q ◦ (A µ Q B ) ◦ ( AQσ B) ◦ ( σ A QP Q ) ◦ (QjP Q) ◦ (QDj) ◦ ( Q∆ D)
- Page 80 and 81: 80We compute(LRɛLY ′ ) ◦ ( LR
- Page 82 and 83: 82Proof. First of all we prove that
- Page 84 and 85: 84i.e. Aα is a functorial morphism
- Page 86 and 87: 86Then we haveA µ CCX ◦ ( A∆ C
- Page 88 and 89: 884.23) is a functor à : C A → C
- Page 90 and 91: 90Let θ l = ( σ B P Q ) ◦ (P τ
- Page 92 and 93: 925)σ A = ( ε C A ) ◦ ( Cσ A)
- Page 94 and 95: 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 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 146 and 147: 146given byWe computeσ B = m B ◦
- 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 = (
131A µ Q= µBQ ◦ [( Qσ B) ◦ ( (QP µ Q)] ) B ◦ QP QσB◦ ( ) (QP QP µ )B Q ◦ QP QP QσB◦ (QP QP iQ) ◦ (QP QqQ) ◦ (A µ Q P QQQ )(81)= µ B Q ◦ (Qm B ) ◦ ( Qσ B B ) ◦ ( QP Qσ B) ◦ ( QP QP µ B Q◦ (QP QP iQ) ◦ (QP QqQ) ◦ (A µ Q P QQQ )σ B = µ B Q ◦ (Qm B ) ◦ ( QBσ B) ◦ ( QBP µ B Q)◦(QP QP QσB ))◦(QBP QσB ) ◦ (QBP iQ) ◦ (QBqQ)◦ ( Qσ B QQ ) ◦ (A µ Q P QQQ )µ B Q ass= µ B Q ◦ ( µ B QB ) ◦ ( QBσ B) ◦ ( QBP µ B Q)◦(QBP QσB ) ◦ (QBP iQ) ◦ (QBqQ)◦ ( Qσ B QQ ) ◦ (A µ Q P QQQ )µ B Q= µ B Q ◦ ( Qσ B) ◦ ( ) (QP µ ) B Q ◦ QP QσB◦ (QP iQ) ◦ (QqQ) ◦ ( µ B QQQ )◦ ( Qσ B QQ ) ◦ (A µ Q P QQQ )A µ Q= µBQ ◦ ( Qσ B) ◦ ( ) (QP µ ) B Q ◦ QP QσB◦ (QP iQ) ◦ (QqQ) ◦ ( µ B QQQ )◦ (A µ Q BQQ ) ◦ ( AQσ B QQ ))◦(QP QσB ) ◦ (QP iQ) ◦ (QqQ) ◦ ( µ B QQQ )(82)= A µ Q ◦ ( σ A Q ) ◦ ( QP µ B Q◦ (A µ Q BQQ ) ◦ ( AQσ B QQ )σ A = A µ Q ◦ ( Aµ B Q)◦(AQσB ) ◦ ( σ A QP Q ) ◦ (QP iQ) ◦ (QqQ) ◦ ( µ B QQQ )◦ (A µ Q BQQ ) ◦ ( AQσ B QQ )Qis a bim= µ B Q ◦ (A µ Q B ) ◦ ( AQσ B) ◦ ( σ A QP Q ) ◦ (QP iQ) ◦ (QqQ) ◦ ( µ B QQQ )◦ (A µ Q BQQ ) ◦ ( AQσ B QQ )and thus we getχ ◦ ( QQχ ) == A µ Q ◦ (Aχ) ◦ ( σ A QQQ ) ◦ (A µ Q P QQQ ) ◦ ( σ A QP QQQ ) ◦ ( QP iQQQ )◦ ( QqQQQ )= µ B Q ◦ (A µ Q B ) ◦ ( AQσ B) ◦ ( σ A QP Q ) ◦ (QP iQ) ◦ (QqQ) ◦ ( µ B QQQ )◦ (A µ Q BQQ ) ◦ ( AQσ B QQ ) ◦ ( σ A QP QQQ ) ◦ ( QP iQQQ ) ◦ ( QqQQQ )= χ ◦ ( χQQ )2) Counitality, in the sense that χ ◦ (Qδ D ) = Qε D and χ ◦ (δ C Q) = ε C Q. We haveχ ◦ (Qδ D ) == µ B Q ◦ (A µ Q B ) ◦ ( AQσ B) ◦ ( σ A QP Q ) ◦ (QP iQ) ◦ (QqQ) ◦ (Qδ D )(144)= µ B Q ◦ (A µ Q B ) ◦ ( AQσ B) ◦ ( σ A QP Q ) ◦ (QjP Q) ◦ (Qκ ′ 0Q) ◦ (Qδ D )(148)= µ B Q ◦ (A µ Q B ) ◦ ( AQσ B) ◦ ( σ A QP Q ) ◦ (QjP Q) ◦ (QDj) ◦ ( Q∆ D)
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