Homotopy theory of curved operads and curved algebras / Joan Bellier-Millès, Gabriel C. Drummond-Cole

Format:

Book

Author/Creator:

Bellier-Millès, Joan, author.

Drummond-Cole, Gabriel C., author.

Series:

Memoirs of the American Mathematical Society ; volume 312, number 1580.

Memoirs of the American Mathematical Society ; 0065-9266 am volume 312, number 1580 (August 2025) (DE-627)129556920 (DE-600)220875-1

Memoirs of the American Mathematical Society, 0065-9266 ; volume 312, number 1580 (August 2025)

Language:

English

Subjects (All):

Algebra, Homological.

Physical Description:

v, 109 pages : illustrations ; 26 cm.

Place of Publication:

Providence, RI : American Mathematical Society, 2025.

Summary:

Curved algebras are algebras endowed with a predifferential, which is an endomorphism of degree -1 whose square is not necessarily 0. This makes the usual definition of quasi-isomorphism meaningless and therefore the homotopical study of curved algebras cannot follow the same path as differential graded algebras. In this article, we propose to study curved algebras by means of curved operads. We develop the theory of bar and cobar constructions adapted to this new notion as well as Koszul duality theory. To be able to provide meaningful definitions, we work in the context of objects which are filtered and complete and become differential graded after applying the associated graded functor. This setting brings its own difficulties but it nevertheless permits us to define a combinatorial model category structure that we can transfer to the category of curved operads and to the category of algebras over a curved operad using free-forgetful adjunctions. We address the case of curved associative algebras. We recover the notion of curved Aoo-algebras, and we show that the homotopy categories of curved associative algebras and of curved Aoo-algebras are Quillen equivalent.

Contents:

Introduction

Chapter 1. The filtered framework

Chapter 2. Operads in the complete and filtered setting

Chapter 3. Complete cooperads

Chapter 4. Bar and cobar constructions in the curved setting

Chapter 5. Koszul duality for curved operads

Chapter 6. The associative case

Appendix A. Categorical filtrations and completions

Appendix B. Gr-flat objects and the associated graded

Appendix C. Model category structures

Bibliography.

Notes:

"August 2025, volume 312, number 1580 (second of 7 numbers)"

Includes bibliographic references (pages 107-109).

ISBN:

9781470475789

1470475782

OCLC:

1535710981