Symmetric Monoidal Categories for Operads - Index of

More documents

Recommendations

Info

18.2 Shifted Functors and Pushouts 273 If N is a right R-module, then S[N,Y ]n forms a subobject of S[N,Y ]inthe category of right R-modules. To form the decomposition SR[M,B] = colimn SR[M,B]n, we use the sequence of reflexive coequalizers ··· SR[M,A][X ⊕ Y ]n−1 SR[M,A][X ⊕ Y ]n ··· ··· SR[M,A][Y ]n−1 SR[M,A][Y ]n ··· ··· SR[M,B]n−1 jn SR[M,B]n determined by the filtration of SR[M,A][X ⊕ Y ] and SR[M,A][Y ] (note that the morphisms d0,d1,s0 preserve filtrations). We have SR[M,B] = colimn SR[M,B]n by interchange of colimits. We have moreover SR[M,A][X ⊕ Y ]0 =SR[M,A][Y ]0 =SR[M,B]0 =SR[M,A]. The coequalizers of SR[M,B]n−1 and SR[M,B]n can be reorganized to give: 18.2.5 Lemma. The morphism jn :SR[M,B]n−1 → SR[M,B]n fits a natural pushout of the form: � p+q=n N(n + · ) ⊗Σp×Σq X q
274 18 Shifted Modules over Operads and Functors S S ⊕ U ⊕ V T T ⊕ V C D. The morphisms d0,d1 : S ⊕ U ⊕ V ⇒ T ⊕ V are the identity on V .The summand U is mapped into V by the morphism d0, intoT by the morphism d1. Accordingly, we have a commutative square d0 U d1 C V D. By an easy inspection of the categorical constructions, we check that this square forms a pushout and the lemma follows. ⊓⊔ 18.2.6 Iterated Pushout-Products. Let T0 = X, T1 = Y .Toimprove the result of lemma 18.2.5 we use the cubical diagram formed by the tensor products Tɛ1 ⊗···⊗Tɛn on vertices and the morphisms Tɛ1 ⊗···⊗T0 ⊗···⊗Tɛn Tɛ 1 ⊗···⊗i⊗···⊗Tɛn −−−−−−−−−−−−→ Tɛ1 ⊗···⊗T1 ⊗···⊗Tɛn on edges. The terminal vertex of the cube is associated to the tensor power Tn(Y/X)=T1 ⊗···⊗T1 = Y ⊗n . The latching morphism λ : Ln(Y/X) → Tn(Y/X) associated to the diagram Tɛ1 ⊗···⊗Tɛn is the canonical morphism from the colimit Ln(Y/X) = colim (ɛ1,...,ɛn)
Page 2 and 3:
Lecture Notes in Mathematics 1967 E
Page 4 and 5:
Benoit Fresse UFR de Mathématiques
Page 6 and 7:
vi Preface Γ -object, which occur
Page 8 and 9:
viii Contents 9 Algebras in Right M
Page 10 and 11:
Introduction Main Ideas and Objecti
Page 12 and 13:
Introduction 3 The category of Σ
Page 14 and 15:
Introduction 5 the bar module BC to
Page 16 and 17:
Introduction 7 modules over envelop
Page 18 and 19:
Introduction 9 to a P-algebra in ri
Page 20 and 21:
Introduction 11 DerP(A, E). There i
Page 22 and 23:
Introduction 13 In both contexts (m
Page 24 and 25:
22 1 Symmetric Monoidal Categories
Page 26 and 27:
Page 28 and 29:
Page 30 and 31:
Page 32 and 33:
Page 34 and 35:
Page 36 and 37:
Page 38 and 39:
36 2 Symmetric Objects and Functors
Page 40 and 41:
Page 42 and 43:
Page 44 and 45:
Page 46 and 47:
Page 48 and 49:
Page 50 and 51:
Page 52 and 53:
Page 54 and 55:
Page 56 and 57:
54 3 Operads and Algebras in Symmet
Page 58 and 59:
Page 60 and 61:
Page 62 and 63:
Page 64 and 65:
Page 66 and 67:
Page 68 and 69:
Page 70 and 71:
Page 72 and 73:
Page 74 and 75:
Page 76 and 77:
Page 78 and 79:
Page 80 and 81:
78 4 Miscellaneous Structures Assoc
Page 82 and 83:
Page 84 and 85:
Page 86 and 87:
Page 88 and 89:
Page 90 and 91:
Page 92 and 93:
Page 94 and 95:
Page 96 and 97:
Chapter 5 Definitions and Basic Con
Page 98 and 99:
5.1 The Functor Associated to a Rig
Page 100 and 101:
Page 102 and 103:
Page 104 and 105:
Chapter 6 Tensor Products Introduct
Page 106 and 107:
6.1 The Symmetric Monoidal Category
Page 108 and 109:
6.3 On Enriched Category Structures
Page 110 and 111:
Chapter 7 Universal Constructions o
Page 112 and 113:
7.2 Extension and Restriction of St
Page 114 and 115:
Page 116 and 117:
Page 118 and 119:
122 8 Adjunction and Embedding Prop
Page 120 and 121:
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
130 9 Algebras in Right Modules ove
Page 128 and 129:
Page 130 and 131:
Page 132 and 133:
Page 134 and 135:
Page 136 and 137:
140 10 Miscellaneous Examples 10.1.
Page 138 and 139:
142 10 Miscellaneous Examples Let
Page 140 and 141:
144 10 Miscellaneous Examples Proof
Page 142 and 143:
146 10 Miscellaneous Examples In th
Page 144 and 145:
Chapter 11 Symmetric Monoidal Model
Page 146 and 147:
11.1 Recollections: The Language of
Page 148 and 149:
Page 150 and 151:
Page 152 and 153:
Page 154 and 155:
Page 156 and 157:
11.2 Examples of Model Categories i
Page 158 and 159:
Page 160 and 161:
Page 162 and 163:
11.3 Symmetric Monoidal Model Categ
Page 164 and 165:
11.4 The Model Category of Σ∗-Ob
Page 166 and 167:
11.4 The Model Category of Σ∗-Ob
Page 168 and 169:
11.5 The Pushout-Product Property f
Page 170 and 171:
11.6 Symmetric Monoidal Categories
Page 172 and 173:
Page 174 and 175:
Page 176 and 177:
Chapter 12 The Homotopy of Algebras
Page 178 and 179:
12.1 Semi-Model Categories 187 Simi
Page 180 and 181:
12.1 Semi-Model Categories 189 The
Page 182 and 183:
12.1 Semi-Model Categories 191 Say
Page 184 and 185:
12.2 The Semi-Model Category of Ope
Page 186 and 187:
12.2 The Semi-Model Category of Ope
Page 188 and 189:
12.3 The Semi-Model Categories of A
Page 190 and 191:
12.3 The Semi-Model Categories of A
Page 192 and 193:
12.5 The Homotopy of Extension and
Page 194 and 195:
Chapter 13 The (Co)homology of Alge
Page 196 and 197:
13.1 The Construction 205 13.1.2 Pr
Page 198 and 199:
13.1 The Construction 207 associate
Page 200 and 201: 13.2 Universal Coefficient Spectral
Page 202 and 203: 13.3 The Operadic Cotriple Construc
Page 204 and 205: 13.3 The Operadic Cotriple Construc
Page 206 and 207: Chapter 14 The Model Category of Ri
Page 208 and 209: 14.1 The Symmetric Monoidal Model C
Page 210 and 211: 14.3 Model Categories of Algebras i
Page 212 and 213: 226 15 Modules and Homotopy Invaria
Page 220 and 221: Chapter 16 Extension and Restrictio
Page 222 and 223: 16.1 Proofs 237 In the remainder of
Page 224 and 225: 16.2 Applications to Bimodules over
Page 226 and 227: 242 17 Miscellaneous Applications c
Page 228 and 229: 244 17 Miscellaneous Applications N
Page 230 and 231: 246 17 Miscellaneous Applications n
Page 232 and 233: 248 17 Miscellaneous Applications B
Page 234 and 235: 250 17 Miscellaneous Applications h
Page 236 and 237: 252 17 Miscellaneous Applications r
Page 238 and 239: 254 17 Miscellaneous Applications N
Page 240 and 241: 256 17 Miscellaneous Applications a
Page 242 and 243: 258 17 Miscellaneous Applications T
Page 244 and 245: Chapter 18 Shifted Modules over Ope
Page 246 and 247: 18.2 Shifted Functors and Pushouts
Page 254 and 255: Chapter 19 Shifted Functors and Pus
Page 256 and 257: 19.1 Preliminary Step 279 Proof. By
Page 258 and 259: 19.2 Pushouts 281 In the next proof
Page 260 and 261: 19.2 Pushouts 283 and morphism (1)
Page 262 and 263: 19.3 Third Step: Composites 285 19.
Page 264 and 265: Chapter 20 Applications of the Push
Page 266 and 267: 20.2 Applications: Homotopy Invaria
Page 268 and 269: 292 References cohomology, and alge
Page 270 and 271: 294 References [57] V. Smirnov, On
Page 272 and 273: Index adjunction of symmetric monoi
Page 274 and 275: Index 299 differential graded modul
Page 276 and 277: Index 301 of simplicial sets, §11.
Page 278 and 279: Index 303 semi-model category, §12
Page 280 and 281: Glossary of Notation A: the associa
Page 282 and 283: Glossary of Notation 307 M 0 : the
Page 284 and 285: Lecture Notes in Mathematics For in
Page 286 and 287: Vol. 1874: M. Yor, M. Émery (Eds.)
Page 288 and 289: LECTURE NOTES IN MATHEMATICS Edited
show all

Symmetric Monoidal Categories for Operads - Index of

Create successful ePaper yourself

Delete template?

Save as template?