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# Homology, Homotopy and Applications

## Volume 21 (2019)

### Number 1

### Convolution algebras and the deformation theory of infinity-morphisms

Pages: 351 – 373

DOI: http://dx.doi.org/10.4310/HHA.2019.v21.n1.a17

#### Authors

#### 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 $\mathrm{hom} (C, A)$, 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.

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.

Received 13 June 2018

Received revised 2 September 2018

Published 7 November 2018