Convolution algebras and the deformation theory of infinity-morphisms

TL;DR

This paper develops a framework using convolution algebras to define and analyze the deformation theory of infinity-morphisms between algebras and coalgebras over operads and cooperads, extending existing results.

Contribution

It introduces a new bifunctor for convolution algebras that incorporates infinity-morphisms and proves its homotopy invariance, completing a series of related studies.

Findings

Extended convolution algebra bifunctor to include infinity-morphisms

Proved homotopy invariance of the convolution algebra bifunctor

Completed the study of compatibility between convolution algebras and infinity-morphisms

Abstract

Given a coalgebra C over a cooperad, and an algebra A over an operad, it is often possible to define a natural homotopy Lie algebra structure on hom(C,A), the space of linear maps between them, called the convolution algebra of C and A. In the present article, we use convolution algebras to define the deformation complex for infinity-morphisms of algebras over operads and coalgebras over cooperads. We also complete the study of the compatibility between convolution algebras and infinity-morphisms of algebras and coalgebras. We prove that the convolution algebra bifunctor can be extended to a bifunctor that accepts infinity-morphisms in both slots and which is well defined up to homotopy, and we generalize and take a new point of view on some other already known results. This paper concludes a series of works by the two authors dealing with the investigation of convolution algebras.