Convolution algebras and the deformation theory of infinity-morphisms

Daniel Robert-Nicoud ; Felix Wierstra

Homology, homotopy, and applications, Tome 21 (2019) no. 1, pp. 351-373.

Résumé

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.

Détail

DOI : 10.4310/HHA.2019.v21.n1.a17

Classification : 18D50, 08C05, 18G55
Keywords: homotopical algebra, convolution algebra, infinity-morphism