Modules over operads and functors

Benoit Fresse (auth.)3540890556, 9783540890553

The notion of an operad supplies both a conceptual and effective device to handle a variety of algebraic structures in various situations. Operads were introduced 40 years ago in algebraic topology in order to model the structure of iterated loop spaces. Since then, operads have been used fruitfully in many fields of mathematics and physics.

This monograph begins with a review of the basis of operad theory. The main purpose is to study structures of modules over operads as a new device to model functors between categories of algebras as effectively as operads model categories of algebras.

Table of contents :
Front Matter….Pages 1-8
Introduction….Pages 1-13
Front Matter….Pages 17-20
Symmetric monoidal categories for operads….Pages 21-34
Symmetric objects and functors….Pages 1-18
Operads and algebras in symmetric monoidal categories….Pages 53-76
Miscellaneous structures associated to algebras over operads….Pages 77-93
Back Matter….Pages 95-96
Front Matter….Pages 98-98
Definitions and basic constructions….Pages 99-106
Tensor products….Pages 107-112
Universal constructions on right modules over operads….Pages 113-119
Adjunction and embedding properties….Pages 121-128
Algebras in right modules over operads….Pages 129-138
Miscellaneous examples….Pages 139-147
Back Matter….Pages 149-149
Front Matter….Pages 152-152
Symmetric monoidal model categories for operads….Pages 153-184
The homotopy of algebras over operads….Pages 185-202
The (co)homology of algebras over operads….Pages 203-214
Back Matter….Pages 215-216
Front Matter….Pages 218-218
The model category of right modules….Pages 219-223
Modules and homotopy invariance of functors….Pages 225-233
Extension and restriction functors and model structures….Pages 235-239
Miscellaneous applications….Pages 241-259
Back Matter….Pages 261-261
Front Matter….Pages 1-2
Shifted modules over operads and functors….Pages 267-276
Shifted functors and pushout-products….Pages 277-286
Front Matter….Pages 1-2
Applications of pushout-products of shifted functors….Pages 287-289
Back Matter….Pages 1-23