Convolution algebras and the deformation theory of infinity-morphisms

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 $\mathrm{hom} (C, A)$, 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 $\infty$-morphisms of algebras over operads and coalgebras over cooperads. We also complete the study of the compatibility between convolution algebras and $\infty$-morphisms of algebras and coalgebras. We prove that the convolution algebra bifunctor can be extended to a bifunctor that accepts $\infty$-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.

Keywords

homotopical algebra, convolution algebra, infinity-morphism

2010 Mathematics Subject Classification

Primary 18D50. Secondary 08C05, 18G55.

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The first author was supported by grants from Région Ile-de-France, and the grant ANR-14-CE25-0008-01 project SAT.

The second author acknowledges the financial support from the grant GA CR No. P201/12/G028.

This article was revised on June 29, 2022 to correct the names used for internal cross-references.

Received 13 June 2018

Received revised 2 September 2018

Accepted 7 September 2018

Published 7 November 2018