Homotopy theory of curved operads and curved algebras

TL;DR

This paper develops a homotopy theory for curved operads and curved algebras, introducing bar and cobar constructions, Koszul duality, and a model category structure to study these non-quasi-isomorphic algebraic objects.

Contribution

It introduces a framework for homotopical analysis of curved algebras using curved operads, including new bar/cobar constructions and a model category structure.

Findings

Established a model category structure for curved operads and algebras.

Revealed Quillen equivalence between curved associative algebras and curved Aoo-algebras.

Extended Koszul duality theory to curved operads.

Abstract

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.