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

More documents

Recommendations

Info

126Now, since cocan 1 : AC → QP is a functorial morphism and by formula (138) , theright square serially commutes. By formula (137) also the left square commutes.Moreover, by definition, ι P and C Uγ C are monomorphisms. Since QD preservesequalizers, by Lemma 4.18 also Q preserves equalizers. Since C, Q preserve equalizers,Qι P and A C Uγ C are also monomorphisms. Then, if cocan 1A U is an isomorphism,also cocan C is an isomorphism. Since C U C cocan C = cocan C , by 4.17, also C cocan Cis an isomorphism.Conversely, assume that C cocan C is an isomorphism. Then alsococan C = C U C cocan C is an isomorphism. Then we haveC U C cocan C C F = cocan C C F = (A µ Q P C C F ) ◦ ( Aδ C C C F )Pro4.32,Lem6.39=( Aµ Q P ) ◦ (Aδ C ) = cocan 1so that also cocan 1 is an isomorphism.□6.1<strong>2.</strong> The cotame case. The following subsection is presented without proofs,which can be obtained as the dual versions of <strong>results</strong> of the tame case (see Subsection6.6).Definition 6.4<strong>2.</strong> A formal codual structure X = (C, D, P, Q, δ C , δ D ) is called acoMorita context on the categories A and B if it satisfies also the balanced conditions((139) ρDQ P ) ◦ δ C = ( )Q D ρ P ◦ δC and ( ρ C P Q ) ◦ δ D = (P C ρ Q ) ◦ δ D .Lemma 6.43. Let X = (C, D, P, Q, δ C , δ D ) be a coMorita context on the categories Aand B and assume that C, D, P, Q preserve equalizers. Hence, there exist functorialmorphisms(140)(141)(142)(143)• CD δ DCC: IdC A → C Q DD P C such thatC U CD δCDC = D δCDC( )where D δC DC is uniquely determined by Q D ι D P◦ D δC DC = (D δC DC U ) ◦ (C Uγ C)and(ι QD P ) ◦ D δC D = δ C• DC δ CDD: IdD B → D P C C Q D such thatD U DC δDCD = C δDCD( )where C δD CD is uniquely determined by P C ι C Q◦ C δD CD = (C δD C D U ) ◦ (D Uγ D)and(ιP C Q ) ◦ C δD C = δ DDefinitions 6.44. Let X = (C, D, P, Q, δ C , δ D ) be a coMorita context. We will saythat X is cotame if the lifted functorial morphisms CD δCDC : IdC A → C Q DD P C andDC δDCD : IdD B → D P C C Q D are isomorphisms so that the lifted functors C Q D : D B →C A and D P C : C A → D B yield a category equivalence. In this case, if χ : QP Q → Qis a coherd for X, we will say that χ is a cotame coherd.
Proposition 6.45. Let X = (C, D, P, Q, δ C , δ D ) be a cotame coMorita context.Then unit and counit of the adjunction (D P C , C Q D) are given byη ( D P C , C Q D ) = CD δC DC and ɛ ( D P C , C Q D ) = ( ) (DC δDCD −1◦ D P ( ) )C CD δCDC −1 CQ D ◦( DCδDCDDP C C Q D) so that(C )(η (P C , C Q) = Q D γ DD P C ◦ CD δC DC and ɛ (P C , C Q) = ε D ◦D U ( ) )DC δDCD −1 DF ◦(D U D P ( ) )C CD δCDC −1 CQ DD F ◦ (D U DC δD CDDP C C Q DD F ) .Corollary 6.46. Let X = (C, D, P, Q, δ C , δ D ) be a cotame coMorita context. Assumethat the functors A, B, P, Q preserve equalizers. Then the units of the adjunctions( P C , C Q ) and ( Q D , D P ) are given by ɛ (P C , C Q) = C δ C C and ɛ (Q D , D P ) = D δ D D .Lemma 6.47. Let X = (C, D, P, Q, δ C , δ D ) be a formal codual structure where theunderlying functors are C : A → A, D : B → B, P : A → B and Q : B →A. Assume that both categories A and B have equalizers and the functors C, QDpreserve them. Assume that• A = (A, m A , u A ) is a monad on the category A such that A preserves equalizers)•(Ã, Ã = mA e, u A e is a lifting of the monad A to the category C A( )•C Q, eA µC Q is a left Ã-module functor• X is a cotame coMorita context.Then cocan 1 is an isomorphism if and only if C cocan C is an isomorphism if andonly if C Q is a left Ã-coGalois functor.The following Theorem is a formulation, in pure categorical terms, for the coherdversion of [BV, Theorem <strong>2.</strong>18].Theorem 6.48. Let X = (C, D, P, Q, δ C , δ D ) be a regular cotame coMorita context.Assume that• both categories A and B have equalizers and coequalizers,• the functors C and D preserve coequalizers,• the functors C, D, P, Q preserve equalizers.Then the existence of the following structures are equivalent:(a) A coherd χ : QP Q → Q for X(b) A monad A = (A, m A , u A ) on the category A such that the functor A preservescoequalizers and a mixed distributive law Λ : AC → CA such that C Qis a coGalois module functor over Ã (where Ã is the lifting of A)(c) A monad B = (B, m B , u B ) on the category B such that the functor B preservescoequalizers and an opposite mixed distributive law Γ : DB → BDsuch that D P is a coGalois module functor over ˜B (where ˜B is the lifting ofB).127
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 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 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 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?