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

More documents

Recommendations

Info

70In particular(49) d ϕ(CX, ∆ C X ) = αX.Corollary 4.50. In the setting of Proposition 4.47, assume that, for every ( X, C ρ X)∈C A, there exists Equ B(αX, R C ρ X). Then for every X ∈ A we haveC Ûɛ ( CX, ∆ C X ) = ϕXand henceC Ûɛ C F = ϕwhere ̂ɛ is the counit of the adjunction (K ϕ , D ϕ ).Proof. Let us calculateC Ûɛ ( CX, ∆ C X ) (47)= (ɛCX) ◦ ( Ld ϕ(CX, ∆ C X ))(48)= (ɛCX) ◦ (LαX) = Ξ (α) (X) = ϕX.Corollary 4.5<strong>1.</strong> In the setting of Proposition 4.47, assume that, for every ( X, C ρ X)∈C A, there exists Equ B(αX, R C ρ X). Then, the functor Dϕ is full and faithful if andonly if ̂ɛ is a functorial isomorphism.Proof. By Proposition 4.47, (K ϕ , D ϕ ) is an adjunction with counit ̂ɛ : K ϕ D ϕ → C A.Then we can apply Proposition <strong>2.</strong>3<strong>2.</strong>□Lemma 4.52 ([GT, Lemma <strong>2.</strong>5]). In the setting of Proposition 4.47, assume that, forevery ( X, C ρ X)∈ C A, there exists Equ B(αX, R C ρ X). Then, for every(X, C ρ X)∈C A the following diagram□LD ϕ(X, C ρ X) C Ubɛ(X, C ρ X)C U ( X, C ρ X)Ld ϕ(X, C ρ X) LR C U ( )X, C ϕ C U(X, C ρ X)ρ XC U C γ(X, C ρ X)C C U ( )X, C ρ XLα C U(X, C ρ X) LR C U C γ(X, C ρ X) ∆ C C U(X, C ρ X) C C U C γ(X, C ρ X)LRC C U ( )X, C ϕC C U(X, C ρ X)ρ X CC C U ( )X, C ρ Xserially commutes. Therefore we get( CUγ C) ◦ (C Ûɛ ) = ( ϕ C U ) ◦ (Ld ϕ ) andProof. Let us compute(∆C C U ) ◦ ( ϕ C U ) = ( ϕC C U ) ◦ ( Lα C U ) .C ρ X ◦ C Ûɛ ( X, C ρ X) (47)= C ρ X ◦ (ɛX) ◦ ( Ld ϕ(X, C ρ X))ϕmorphcomonads=C ρ X ◦ ( ε C X ) ◦ (ϕX) ◦ ( Ld ϕ(X, C ρ X))ε C = ( ε C CX ) ◦ ( C C ρ X)◦ (ϕX) ◦(Ldϕ(X, C ρ X))ϕ= ( ε C CX ) ◦ (ϕCX) ◦ ( ) ( ( ))LR C ρ X ◦ Ldϕ X, C ρ X
defd ϕ=(ε C CX ) ◦ (ϕCX) ◦ (LαX) ◦ ( Ld ϕ(X, C ρ X))defα= ( ε C CX ) ◦ (ϕCX) ◦ (LRϕX) ◦ (LηRX) ◦ ( Ld ϕ(X, C ρ X))ϕ= ( ε C CX ) ◦ (ϕϕX) ◦ (LηRX) ◦ ( ( ))Ld ϕ X, C ρ X(ε C CX ) ◦ ( ∆ C X ) ◦ (ϕX) ◦ ( ( ))Ld ϕ X, C ρ Xϕmorphcomonads=so that we deduce thatCcomonad= (ϕX) ◦ ( Ld ϕ(X, C ρ X))C ρ X ◦ (C Ûɛ ( X, C ρ X))= (ϕX) ◦(Ldϕ(X, C ρ X))71and thus( CUγ C) ◦ (C Ûɛ ) = ( ϕ C U ) ◦ Ld ϕ .Let us calculate(∆C C U ) ◦ ( ϕ C U ) ϕcomonadmorph (= ϕϕ C U ) ◦ ( LηR C U )= ( ϕC C U ) ◦ ( LRϕ C U ) ◦ ( LηR C U ) defα= ( ϕC C U ) ◦ ( Lα C U ) .Theorem 4.53 ([GT] Theorem <strong>2.</strong>6). Let (L, R) be an adjunction where L : B → Aand R : A → B, let C = ( C, ∆ C , ε C) be a comonad on a category A and let ϕ : LR =(LR, LηR, ɛ) → C = ( C, ∆ C , ε C) be a comonad morphism. Let α = Θ (ϕ) = (Rϕ) ◦(ηR) and assume that, for every ( ) ( )X, C ρ X ∈ C A, there exists Equ B αX, R C ρ X .Then we can consider the functor K ϕ = Υ (ϕ) : B → C A and its right adjointD ϕ : C A → B. D ϕ is full and faithful if and only if1) L preserves the equalizer(D ϕ , d ϕ ) = Equ Fun(α C U, R C Uγ C) .2) ϕ : LR → C is a comonad isomorphism.Proof. Recall that, by Corollary 4.50,(50)C Ûɛ C F = ϕ.By Corollary 4.51, D ϕ is full and faithful if and only if ̂ɛ is a functorial isomorphism.Let us assume that ̂ɛ is a functorial isomorphism, hence ϕ is an isomorphism too.Recall that, by Lemma4.52, we have((51)CUγ C) ◦ (C Ûɛ ) = ( ϕ C U ) ◦ (Ld ϕ )so that(52)Let us consider the diagramC Uγ C = ( ϕ C U ) ◦ (Ld ϕ ) ◦ (C Ûɛ −1) .d ϕLD ϕLR C U LRC Uγ C LRC C ULα C U□
Page 1 and 2:
Contents1.
Page 3 and 4:
linearity and compatibility conditi
Page 5 and 6:
5and since g ◦ f is an epimorphis
Page 7 and 8:
Proof. Clearly (qP )◦(αP ) = (qP
Page 9 and 10:
Lemma 2.13 ([BM, L
Page 11 and 12:
i.e. Hom B (Y, iX) equalizes Hom B
Page 13 and 14:
13such thatd 0 ◦ v = Id Yd 1 ◦
Page 15 and 16:
f ↦→ Rfis bijective for every X
Page 17 and 18:
Remark 3.10. Let A = (A, m A , u A
Page 19 and 20: 19and fromµ A P ◦ ( µ A P A ) =
Page 21 and 22: and thusk ◦ (u A QZ) ◦ (Qz) = h
Page 23 and 24: and since A preserves equalizers, A
Page 25 and 26: Conversely, let Φ be a functorial
Page 27 and 28: Proof. Apply Proposition 3.24 to th
Page 29 and 30: Since Q is a left A-module functor,
Page 31 and 32: (Q BB F, p QB F ) = Coequ Fun(µBQ
Page 33 and 34: Theorem 3.37. Let B = (B, m B , u B
Page 35 and 36: where A UG B F : B → A is such th
Page 37 and 38: Proposition 3.44. Let A = (A, m A ,
Page 39 and 40: Note that, since f and g are A-bili
Page 41 and 42: Proposition 3.54. Let (L, R) be an
Page 43 and 44: Corollary 3.58. Let (L, R) be an ad
Page 45 and 46: Definition 4.2. A
Page 47 and 48: Proposition 4.13. Let C = ( C, ∆
Page 49 and 50: Then we have(P Cx) ◦ ( ρ C P X )
Page 51 and 52: and since C preserves coequalizers,
Page 53 and 54: Proof. Apply Corollary 4.24 to the
Page 55 and 56: Let( (CQ ) ()D, ι Q) C = Equ Fun
Page 58 and 59: 58F D right D-comodule functors Q :
Page 60 and 61: 60prove that C ν D : C F D → (C
Page 62 and 63: 624.2. The compari
Page 64 and 65: 64and[(Ω ◦ Γ) (ϕ)] (Y ) = (LY,
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 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 Ã : 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 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◦
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
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 220 and 221:
220• 2-cells: monad functor trans
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
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?