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 ...
142x=and=δ C=(l= QlQ ̂QQ)◦ (QP Aτ) ◦ (QP xQ) ◦ (δ C QP Q) ◦ (CQδ D ) ◦ (C ρ Q D ) ◦ ρ D Q(QlQ ̂QQ) (◦ QP xQ ̂QQ)◦ (QP QP τ) ◦ (δ C QP Q) ◦ (CQδ D ) ◦ (C ρ Q D ) ◦ ρ D Q(δ=CQlQ ̂QQ) (◦ QP xQ ̂QQ) (◦ δ C QP Q ̂QQ)◦ (CQP τ) ◦ (CQδ D )δ C=(125)=◦ (C ρ Q D ) ◦ ρ D QC ρ Q=(QlQ ̂QQ) (◦ QP xQ ̂QQ) (◦ δ C QP Q ̂QQ) (◦C ρ Q P Q ̂QQ)◦ (QP τ)(127)=(QlQ ̂QQ) (◦ QP xQ ̂QQ(QlQ ̂QQ) (◦ QP xQ ̂QQ)◦(C ρ Q P Q ̂QQ)◦ (Qδ D ) ◦ ρ D Q)◦(δ C QP Q ̂QQ)◦◦ (δ C Q) ◦ C ρ Q(δ C QP Q ̂QQ)◦◦ (Cτ) ◦ C ρ Q(δ C Q ̂QQ)◦ (Cτ) ◦ C ρ Q(C ρ Q P Q ̂QQ)◦ (QP τ)(C ρ Q P Q ̂QQ) (◦ δ C Q ̂QQ)◦(C ρ Q P Q ̂QQ) (◦ δ C Q ̂QQ)◦ (CQlQ) ◦ (CQP xQ) ◦ (Cδ C QP Q) ◦ (CCQδ D )◦ ( C C ρ Q D ) ◦ ( CρQ) D ◦ C ρ Q(C ρ Q P Q ̂QQ) (◦ δ C Q ̂QQ)◦ (CQlQ) ◦ (CQP xQ) ◦ (CQP Qδ D ) ◦ (Cδ C QD)◦ ( C C ρ Q D ) ◦ ( CρQ) D ◦ C ρ Q(Cδ C Q ̂QQ) (◦ ∆ C Q ̂QQ)◦ (CQlQ) ◦ (CQP xQ) ◦ (CQP Qδ D ) ◦ (Cδ C QD)Since◦ ( C C ρ Q D ) ◦ ( Cρ D Q)◦ C ρ Q(C C ρ Q D ) ◦ ( )Cρ D Q ◦ C ρ QQis a bicom=( ) ( )CCρDQ ◦ C C ρ Q ◦ C Qis a comρ Q = ( (CCρQ) D ◦ ∆ C Q ) ◦ C ρ Q)◦ C Qis a bicom (ρ Q = ∆ C QD ) ◦ (C ρ Q D ) ◦ ρ D Q∆ C= ( ∆ C QD ) ◦ ( Cρ D QQis a com= ( C C ρ Q D ) ◦ (C ρ Q D ) ◦ ρ D Qwe obtain(Cδ C Q ̂QQ) (◦ ∆ C Q ̂QQ)◦ (CQlQ) ◦ (CQP xQ) ◦ (CQP Qδ D ) ◦ (Cδ C QD)◦ ( C C ρ Q D ) ◦ ( CρQ) D ◦ C ρ Q(= Cδ C Q ̂QQ) (◦ ∆ C Q ̂QQ)◦ (CQlQ) ◦ (CQP xQ) ◦ (CQP Qδ D ) ◦ (Cδ C QD)◦ ( C C ρ Q D ) ◦ (C ρ Q D ) ◦ ρ D Q
(127)=(Cδ C Q ̂QQ) (◦ ∆ C Q ̂QQ)◦ (CQlQ) ◦ (CQP xQ) ◦ (CQP Qδ D )143∆ C=Qis a com=Qis a bicom=◦ (CQδ D D) ◦ ( Cρ D QD ) ◦ (C ρ Q D ) ◦ ρ D Q(Cδ C Q ̂QQ)◦ (CCQlQ) ◦ (CCQP xQ) ◦ (CCQP Qδ D ) ◦ (CCQδ D D)◦ ( CCρ D QD ) ◦ ( ∆ C QD ) ◦ (C ρ Q D ) ◦ ρ D Q(Cδ C Q ̂QQ)◦ (CCQlQ) ◦ (CCQP xQ) ◦ (CCQP Qδ D ) ◦ (CCQδ D D)◦ ( CCρ D QD ) ◦ ( C C ρ Q D ) ◦ (C ρ Q D ) ◦ ρ D Q(Cδ C Q ̂QQ)◦ (CCQlQ) ◦ (CCQP xQ) ◦ (CCQP Qδ D ) ◦ (CCQδ D D)◦ ( C C ρ Q DD ) ◦ ( Cρ D QD ) ◦ (C ρ Q D ) ◦ ρ D QC ρ Q=(Cδ C Q ̂QQ))◦(C C ρ Q ̂QQ ◦ (CQlQ) ◦ (CQP xQ) ◦ (CQP Qδ D ) ◦ (CQδ D D)Qis a bicom=◦ ( Cρ D QD ) ◦ (C ρ Q D ) ◦ ρ D Q(Cδ C Q ̂QQ))◦(C C ρ Q ̂QQ ◦ (CQlQ) ◦ (CQP xQ) ◦ (CQP Qδ D )◦ (CQδ D D) ◦ (C ρ Q DD ) ◦ ( ρ D QD ) ◦ ρ D QC ρ Q=(Cδ C Q ̂QQ))◦(C C ρ Q ̂QQ (C )◦ ρ Q ̂QQ ◦ (QlQ) ◦ (QP xQ) ◦ (QP Qδ D )(127)=(127)=Qis a bicom=δ C=◦ (Qδ D D) ◦ ( ρ D QD ) ◦ ρ D Q) ((CQδ D ̂QQ ◦ Cρ D ̂QQ) (C )Q ◦ ρ Q ̂QQ ◦ (QlQ) ◦ (QP xQ) ◦ (QP Qδ D )◦ (Qδ D D) ◦ ( ρ D QD ) ◦ ρ D Q) ((CQδ D ̂QQ ◦ Cρ D ̂QQ) (C )Q ◦ ρ Q ̂QQ ◦ (QlQ) ◦ (QP xQ) ◦ (QP Qδ D )◦ (δ C QD) ◦ (C ρ Q D ) ◦ ρ D Q) ((CQδ D ̂QQ ◦C ρ Q D ̂QQ) (◦ ρ D ̂QQ)Q ◦ (QlQ) ◦ (QP xQ) ◦ (QP Qδ D )◦ (δ C QD) ◦ (C ρ Q D ) ◦ ρ D Q) ((CQδ D ̂QQ ◦C ρ Q D ̂QQ) (◦ ρ D ̂QQ)Q ◦ (QlQ) ◦ (QP xQ) ◦ (δ C QP Q)Hence we obtain◦ (CQδ D ) ◦ (C ρ Q D ) ◦ ρ D Q(Q ̂Qτ)◦ τ(= QlQ ̂QQ) (◦ QP xQ ̂QQ) (◦ δ C QP Q ̂QQ) (◦C ρ Q P Q ̂QQ) (◦ δ C Q ̂QQ)=(QlQ ̂QQ) (◦ QP xQ ̂QQ)◦◦ (Cτ) ◦ C ρ Q(δ C QP Q ̂QQ))◦(CQδ D ̂QQ ◦(C ρ Q D ̂QQ)
- 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 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 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 = (
- 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
142x=and=δ C=(l= QlQ ̂QQ)◦ (QP Aτ) ◦ (QP xQ) ◦ (δ C QP Q) ◦ (CQδ D ) ◦ (C ρ Q D ) ◦ ρ D Q(QlQ ̂QQ) (◦ QP xQ ̂QQ)◦ (QP QP τ) ◦ (δ C QP Q) ◦ (CQδ D ) ◦ (C ρ Q D ) ◦ ρ D Q(δ=CQlQ ̂QQ) (◦ QP xQ ̂QQ) (◦ δ C QP Q ̂QQ)◦ (CQP τ) ◦ (CQδ D )δ C=(125)=◦ (C ρ Q D ) ◦ ρ D QC ρ Q=(QlQ ̂QQ) (◦ QP xQ ̂QQ) (◦ δ C QP Q ̂QQ) (◦C ρ Q P Q ̂QQ)◦ (QP τ)(127)=(QlQ ̂QQ) (◦ QP xQ ̂QQ(QlQ ̂QQ) (◦ QP xQ ̂QQ)◦(C ρ Q P Q ̂QQ)◦ (Qδ D ) ◦ ρ D Q)◦(δ C QP Q ̂QQ)◦◦ (δ C Q) ◦ C ρ Q(δ C QP Q ̂QQ)◦◦ (Cτ) ◦ C ρ Q(δ C Q ̂QQ)◦ (Cτ) ◦ C ρ Q(C ρ Q P Q ̂QQ)◦ (QP τ)(C ρ Q P Q ̂QQ) (◦ δ C Q ̂QQ)◦(C ρ Q P Q ̂QQ) (◦ δ C Q ̂QQ)◦ (CQlQ) ◦ (CQP xQ) ◦ (Cδ C QP Q) ◦ (CCQδ D )◦ ( C C ρ Q D ) ◦ ( CρQ) D ◦ C ρ Q(C ρ Q P Q ̂QQ) (◦ δ C Q ̂QQ)◦ (CQlQ) ◦ (CQP xQ) ◦ (CQP Qδ D ) ◦ (Cδ C QD)◦ ( C C ρ Q D ) ◦ ( CρQ) D ◦ C ρ Q(Cδ C Q ̂QQ) (◦ ∆ C Q ̂QQ)◦ (CQlQ) ◦ (CQP xQ) ◦ (CQP Qδ D ) ◦ (Cδ C QD)Since◦ ( C C ρ Q D ) ◦ ( Cρ D Q)◦ C ρ Q(C C ρ Q D ) ◦ ( )Cρ D Q ◦ C ρ QQis a bicom=( ) ( )CCρDQ ◦ C C ρ Q ◦ C Qis a comρ Q = ( (CCρQ) D ◦ ∆ C Q ) ◦ C ρ Q)◦ C Qis a bicom (ρ Q = ∆ C QD ) ◦ (C ρ Q D ) ◦ ρ D Q∆ C= ( ∆ C QD ) ◦ ( Cρ D QQis a com= ( C C ρ Q D ) ◦ (C ρ Q D ) ◦ ρ D Qwe obtain(Cδ C Q ̂QQ) (◦ ∆ C Q ̂QQ)◦ (CQlQ) ◦ (CQP xQ) ◦ (CQP Qδ D ) ◦ (Cδ C QD)◦ ( C C ρ Q D ) ◦ ( CρQ) D ◦ C ρ Q(= Cδ C Q ̂QQ) (◦ ∆ C Q ̂QQ)◦ (CQlQ) ◦ (CQP xQ) ◦ (CQP Qδ D ) ◦ (Cδ C QD)◦ ( C C ρ Q D ) ◦ (C ρ Q D ) ◦ ρ D Q
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