Pretriangulated A ∞-categories

Preface......Page 3
Introduction......Page 5
Conventions and basic notions.......Page 6
The new features introduced in this book.......Page 13
Synopsis of the book.......Page 15
Off the principal road.......Page 20
Homotopy unital A8-categories.......Page 22
The operad of A8-algebras.......Page 23
Operadic approach to A8-categories.......Page 26
I Closed multicategories......Page 29
Conventions for finite totally ordered sets.......Page 31
Lax Monoidal categories and functors.......Page 33
Lax Monoidal V-categories.......Page 36
Algebras.......Page 48
Coherence principle.......Page 57
Multiquivers.......Page 65
Notation for sequential trees.......Page 66
Spans.......Page 69
Lax 2-categories and (co)lax 2-functors.......Page 76
V-multicategory as an algebra.......Page 78
From Monoidal categories to multicategories and back.......Page 84
Multicategories enriched in multicategories.......Page 107
A closed multicategory gives an enriched multicategory.......Page 116
Closing transformations.......Page 127
Augmented multifunctors.......Page 136
Coalgebras for a multicomonad.......Page 143
Kleisli multicategories.......Page 153
Multiplication in a closed Kleisli multicategory.......Page 154
Left and right multiplications in closed Kleisli multicategories.......Page 157
Lax Monoidal case.......Page 158
Monads for Kleisli multicategories.......Page 161
II A8-categories......Page 167
The symmetric Monoidal category of quivers.......Page 169
The tensor monad in Q/S.......Page 171
Biunital V-quivers.......Page 173
The tensor comonad T1.......Page 177
The T1-comodule T.......Page 179
Coalgebras over the comonad T1.......Page 180
Lax symmetric Monoidal comonad T1.......Page 185
Closedness of symmetric Monoidal category Qp.......Page 191
Closedness of symmetric Monoidal category Qu.......Page 194
Left multiplication in Qu.......Page 195
Right multiplication in Qu.......Page 196
Composition in the closed Monoidal category Qu.......Page 197
Closing transformation for the tensor functor.......Page 198
Various multicategories of quivers.......Page 199
Closing transformation for the multifunctor T1.......Page 203
T1-coderivations.......Page 205
Transformation theta.......Page 211
Multiplication in the closed multicategory Q.......Page 213
Differentials.......Page 215
A8-functors.......Page 221
Symmetric multicategory A8.......Page 229
Closed multicategory A8.......Page 230
Components of the differential B.......Page 235
Restriction of an A8-functor to a subset of arguments.......Page 238
A coalgebra approach to the multicategory A8.......Page 241
Differential B.......Page 243
Components of the composition M.......Page 245
Cohomological Hochschild complex.......Page 250
Strictly unital A8-categories and A8-functors.......Page 257
Multifunctor k.......Page 261
Unital A8-categories and A8-functors.......Page 267
Closedness of multicategory of unital A8-categories.......Page 269
A8u-2-functors and A8u-2-transformations.......Page 272
2-category structure of A8u.......Page 275
Unital envelopes of A8-categories.......Page 277
Free A8-categories and free dg-categories.......Page 282
An algebra in the category of differential graded categories......Page 285
The functor of shifts.......Page 289
The unit for the monad of shifts.......Page 293
Commutation between the monad of shifts and the tensor comonad.......Page 296
The multifunctor of shifts.......Page 304
The closing transformation for the multifunctor of shifts.......Page 306
Actions and the A8-multifunctor of shifts.......Page 311
Differential.......Page 312
Unit A8-2-transformation u[].......Page 314
Multiplication A8-2-transformation m[].......Page 315
Quotients and shifts.......Page 320
Embedding A8(A,C)[] into A8(A,C[]).......Page 323
Simple and multiple Maurer–Cartan quivers.......Page 325
Maurer–Cartan multifunctor.......Page 328
Closing transformation MC.......Page 330
The closed version of mC.......Page 333
Consequences of the closing transformation for multifunctor MC.......Page 334
Multifunctor Mc.......Page 336
A8-category of bounded complexes.......Page 339
The Maurer–Cartan functor on A8-functors.......Page 341
The Maurer–Cartan functor on A8-transformations.......Page 342
A8u-2-Functor mc.......Page 345
Unit for the Maurer–Cartan monad.......Page 346
The A8-2-transformation Tot.......Page 347
The A8-2-monad (mc,Tot,umc).......Page 350
Quotients and solutions to Maurer–Cartan equation.......Page 356
Embedding A8(A,B)mc into A8(A,Bmc).......Page 359
Commutation morphism between two monads.......Page 363
The natural transformation c.......Page 365
The natural A8-2-transformation c.......Page 366
Closed multicategory of pretriangulated A8-categories.......Page 375
Pretriangulated differential graded categories.......Page 376
Translation structures.......Page 379
Translation structures determined by an automorphism.......Page 383
Systems of n-triangles.......Page 391
The principal distinguished n-triangle.......Page 393
A category of strips.......Page 398
(Pre)triangulatedness and A8-modules.......Page 413
A8-bimodules and Serre A8-functors.......Page 417
The Yoneda Lemma.......Page 421
Internal Monoidal categories......Page 423
Monoidal Cat-functor from lax Monoidal categories to multicategories.......Page 428
Category C(1,D) is a Monoidal category.......Page 430
Actions of lax-Monoidal-categories.......Page 433
Action of graded categories on graded quivers.......Page 435
Actions on Kleisli multicategories.......Page 444
Intertwiner xi for action of graded categories and T1.......Page 449
An action of a symmetric-Monoidal-category.......Page 455
Another presentation of an action.......Page 456
An algebra produces a multifunctor.......Page 458
Action of differential graded categories on A8-categories.......Page 460
Action of differential graded categories on unital A8-categories.......Page 465