Bar and cobar constructions for curved algebras and coalgebras

TL;DR

This paper introduces bar and cobar constructions as functors between categories of curved algebras and coalgebras, establishing their adjoint relationship to deepen understanding of curved algebraic structures.

Contribution

It develops functorial bar and cobar constructions for curved algebras and coalgebras, expanding classical homological tools to curved settings.

Findings

Bar and cobar constructions are adjoint functors.

The constructions work over graded commutative rings.

New framework for curved algebraic structures.

Abstract

We provide bar and cobar constructions as functors between some categories of curved algebras and curved augmented coalgebras over a graded commutative ring. These functors are adjoint to each other.