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 ...
116= ( Aε C Q ) ◦ ( cocan1 −1 Q ) ◦ (Qδ D ) ◦ (A µ Q D ) ◦ ( )Aρ D Q= ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ (A µ Q P Q ) ◦ (AQδ D ) ◦ ( )Aρ D Q(127)= ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ (A µ Q P Q ) ◦ (Aδ C Q) ◦ ( A C ρ Q)= ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ (cocan 1 Q) ◦ ( A C ρ Q)= ( Aε C Q ) ◦ ( A C ρ Q) C ρ Q counital= AQso we obtaincocan −12 = ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ (Qδ D ) .(b) ⇒ (a) Follows from Theorem 6.36. (a) ⇔ (c) follows by similar computations.□6.10. Coherds and distributive laws. The following result is a reformulation ofTheorem
117(112)= (Cx) ◦ (C ρ Q P ) ◦ (χP ) ◦ ( QP ε C QP ) ◦ (QP Cδ C )ε C = (Cx) ◦ (C ρ Q P ) ◦ (χP ) ◦ (QP δ C ) ◦ ( QP ε C C )Since (AC, xC) = Coequ Fun(w l C, w r C ) ,by the universal property of coequalizers,there exists a unique functorial morphism Λ : AC → CA such thatΛ ◦ (xC) = λ = (Cx) ◦ (C ρ Q P ) ◦ (χP ) ◦ (QP δ C ) .We want to prove that Λ is a mixed distributive law. We compute(Cm A ) ◦ (ΛA) ◦ (AΛ) ◦ (xxC) x = (Cm A ) ◦ (ΛA) ◦ (AΛ) ◦ (xAC) ◦ (QP xC)x= (Cm A ) ◦ (ΛA) ◦ (xCA) ◦ (QP Λ) ◦ (QP xC)defλ= (Cm A ) ◦ (CxA) ◦ (C ρ Q P A ) ◦ (χP A) ◦ (QP δ C A) ◦ (QP Cx)◦ ( QP C ρ Q P ) ◦ (QP χP ) ◦ (QP QP δ C )δ C= (Cm A ) ◦ (CxA) ◦ (C ρ Q P A ) ◦ (χP A) ◦ (QP QP x) ◦ (QP δ C QP )◦ ( QP C ρ Q P ) ◦ (QP χP ) ◦ (QP QP δ C )χ= (Cm A ) ◦ (CxA) ◦ (C ρ Q P A ) ◦ (QP x) ◦ (χP QP ) ◦ (QP δ C QP )◦ ( QP C ρ Q P ) ◦ (QP χP ) ◦ (QP QP δ C )C ρ Q= (CmA ) ◦ (CxA) ◦ (CQP x) ◦ (C ρ Q P QP ) ◦ (χP QP ) ◦ (QP δ C QP )◦ ( QP C ρ Q P ) ◦ (QP χP ) ◦ (QP QP δ C )x,(127)= (Cm A ) ◦ (Cxx) ◦ (C ρ Q P QP ) ◦ (χP QP ) ◦ (QP Qδ D P )◦ ( QP ρ D QP ) ◦ (QP χP ) ◦ (QP QP δ C )(102)= (Cx) ◦ (CχP ) ◦ (C ρ Q P QP ) ◦ (χP QP ) ◦ (QP Qδ D P )◦ ( QP ρ D QP ) ◦ (QP χP ) ◦ (QP QP δ C )χ= (Cx) ◦ (CχP ) ◦ (C ρ Q P QP ) ◦ (Qδ D P ) ◦ (χDP ) ◦ ( QP ρ D QP ) ◦ (QP χP )◦ (QP QP δ C )C ρ Q= (Cx) ◦ (CχP ) ◦ (CQδD P ) ◦ (C ρ Q DP ) ◦ (χDP ) ◦ ( QP ρ D QP ) ◦ (QP χP )◦ (QP QP δ C )(113)= (Cx) ◦ ( CQε D P ) ◦ (C ρ Q DP ) ◦ (χDP ) ◦ ( QP ρ D QP ) ◦ (QP χP )◦ (QP QP δ C )C ρ Q= (Cx) ◦( Cρ Q P ) ◦ ( Qε D P ) ◦ (χDP ) ◦ ( QP ρ D QP ) ◦ (QP χP ) ◦ (QP QP δ C )χ= (Cx) ◦ (C ρ Q P ) ◦ (χP ) ◦ ( QP Qε D P ) ◦ ( QP ρ D QP ) ◦ (QP χP ) ◦ (QP QP δ C )Qcomfun= (Cx) ◦ (C ρ Q P ) ◦ (χP ) ◦ (QP χP ) ◦ (QP QP δ C )(111)= (Cx) ◦ (C ρ Q P ) ◦ (χP ) ◦ (χP QP ) ◦ (QP QP δ C )
- Page 66 and 67: 66for every ( X, C ρ X)∈ C A, th
- Page 68 and 69: 68i.e.(44) (d ϕ K ϕ Y ) ◦ (̂η
- Page 70 and 71: 70In particular(49) d ϕ(CX, ∆ C
- Page 72 and 73: 72We have to prove that (LD ϕ , Ld
- Page 74 and 75: 74we have that Ld ϕ K ϕ Y is mono
- Page 76 and 77: and since d is mono we get that(ε
- Page 78 and 79: 78Corollary 4.63 (Beck’s Precise
- 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 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 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◦
117(112)= (Cx) ◦ (C ρ Q P ) ◦ (χP ) ◦ ( QP ε C QP ) ◦ (QP Cδ C )ε C = (Cx) ◦ (C ρ Q P ) ◦ (χP ) ◦ (QP δ C ) ◦ ( QP ε C C )Since (AC, xC) = Coequ Fun(w l C, w r C ) ,by the universal property of coequalizers,there exists a unique functorial morphism Λ : AC → CA such thatΛ ◦ (xC) = λ = (Cx) ◦ (C ρ Q P ) ◦ (χP ) ◦ (QP δ C ) .We want to prove that Λ is a mixed distributive law. We compute(Cm A ) ◦ (ΛA) ◦ (AΛ) ◦ (xxC) x = (Cm A ) ◦ (ΛA) ◦ (AΛ) ◦ (xAC) ◦ (QP xC)x= (Cm A ) ◦ (ΛA) ◦ (xCA) ◦ (QP Λ) ◦ (QP xC)defλ= (Cm A ) ◦ (CxA) ◦ (C ρ Q P A ) ◦ (χP A) ◦ (QP δ C A) ◦ (QP Cx)◦ ( QP C ρ Q P ) ◦ (QP χP ) ◦ (QP QP δ C )δ C= (Cm A ) ◦ (CxA) ◦ (C ρ Q P A ) ◦ (χP A) ◦ (QP QP x) ◦ (QP δ C QP )◦ ( QP C ρ Q P ) ◦ (QP χP ) ◦ (QP QP δ C )χ= (Cm A ) ◦ (CxA) ◦ (C ρ Q P A ) ◦ (QP x) ◦ (χP QP ) ◦ (QP δ C QP )◦ ( QP C ρ Q P ) ◦ (QP χP ) ◦ (QP QP δ C )C ρ Q= (CmA ) ◦ (CxA) ◦ (CQP x) ◦ (C ρ Q P QP ) ◦ (χP QP ) ◦ (QP δ C QP )◦ ( QP C ρ Q P ) ◦ (QP χP ) ◦ (QP QP δ C )x,(127)= (Cm A ) ◦ (Cxx) ◦ (C ρ Q P QP ) ◦ (χP QP ) ◦ (QP Qδ D P )◦ ( QP ρ D QP ) ◦ (QP χP ) ◦ (QP QP δ C )(102)= (Cx) ◦ (CχP ) ◦ (C ρ Q P QP ) ◦ (χP QP ) ◦ (QP Qδ D P )◦ ( QP ρ D QP ) ◦ (QP χP ) ◦ (QP QP δ C )χ= (Cx) ◦ (CχP ) ◦ (C ρ Q P QP ) ◦ (Qδ D P ) ◦ (χDP ) ◦ ( QP ρ D QP ) ◦ (QP χP )◦ (QP QP δ C )C ρ Q= (Cx) ◦ (CχP ) ◦ (CQδD P ) ◦ (C ρ Q DP ) ◦ (χDP ) ◦ ( QP ρ D QP ) ◦ (QP χP )◦ (QP QP δ C )(113)= (Cx) ◦ ( CQε D P ) ◦ (C ρ Q DP ) ◦ (χDP ) ◦ ( QP ρ D QP ) ◦ (QP χP )◦ (QP QP δ C )C ρ Q= (Cx) ◦( Cρ Q P ) ◦ ( Qε D P ) ◦ (χDP ) ◦ ( QP ρ D QP ) ◦ (QP χP ) ◦ (QP QP δ C )χ= (Cx) ◦ (C ρ Q P ) ◦ (χP ) ◦ ( QP Qε D P ) ◦ ( QP ρ D QP ) ◦ (QP χP ) ◦ (QP QP δ C )Qcomfun= (Cx) ◦ (C ρ Q P ) ◦ (χP ) ◦ (QP χP ) ◦ (QP QP δ C )(111)= (Cx) ◦ (C ρ Q P ) ◦ (χP ) ◦ (χP QP ) ◦ (QP QP δ C )
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