Deformation theory of representations of prop(erad)s

TL;DR

This paper develops a deformation theory for prop(erad)s and properads, extending classical concepts to a non-linear setting, with applications to algebraic structures like associative bialgebras.

Contribution

It introduces a model category structure for prop(erad)s, a new method for minimal models called homotopy Koszul, and links deformation complexes to Lie algebra up to homotopy structures.

Findings

Established a model category structure for prop(erad)s

Developed a homotopy Koszul duality theory

Proved the Lie algebra up to homotopy structure on the Gerstenhaber-Schack bicomplex

Abstract

We study the deformation theory of morphisms of properads and props thereby extending to a non-linear framework Quillen's deformation theory for commutative rings. The associated chain complex is endowed with a Lie algebra up to homotopy structure. Its Maurer-Cartan elements correspond to deformed structures, which allows us to give a geometric interpretation of these results. To do so, we endow the category of prop(erad)s with a model category structure. We provide a complete study of models for prop(erad)s. A new effective method to make minimal models explicit, that extends Koszul duality theory, is introduced and the associated notion is called homotopy Koszul. As a corollary, we obtain the (co)homology theories of (al)gebras over a prop(erad) and of homotopy (al)gebras as well. Their underlying chain complex is endowed with a canonical Lie algebra up to homotopy structure in general and a Lie algebra structure only in the Koszul case. In particular, we explicit the deformation complex of morphisms from the properad of associative bialgebras. For any minimal model of this properad, the boundary map of this chain complex is shown to be the one defined by Gerstenhaber and Schack. As a corollary, this paper provides a complete proof of the existence of a Lie algebra up to homotopy structure on the Gerstenhaber-Schack bicomplex associated to the deformations of associative bialgebras.