On higher structure on the operadic deformation complexes ${Def}(e_n\to \mathcal{P})$

TL;DR

This paper establishes a canonical homotopy $(n+1)$-algebra structure on the shifted operadic deformation complex for any operad and operad map, generalizing previous results and introducing a categorical algebra approach.

Contribution

It introduces a new categorical algebra method to prove higher algebraic structures on deformation complexes for operad maps, extending prior work.

Findings

Proves the existence of a homotopy $(n+1)$-algebra structure on the deformation complex.

Generalizes previous results to arbitrary operads and maps.

Develops a categorical algebra framework for deformation theory.

Abstract

In this paper, we prove that there is a canonical homotopy $(n+1)$-algebra structure on the shifted operadic deformation complex $Def(e_n\to\mathcal{P})[-n]$ for any operad $\mathcal{P}$ and a map of operads $f\colon e_n\to\mathcal{P}$. This result generalizes the result of [T2], where the case $\mathcal{P}=\mathrm{End}_{Op}(X)$ was considered. Another more computational proof of the same statement was recently sketched in [CW]. Our method combines the one of [T2] with the categorical algebra on the category of symmetric sequences, introduced in [R] and further developed in [KM] and [Fr1]. We define suitable deformation functors on $n$-coalgebras, which are considered as the "non-commutative" base of deformation, prove their representability, and translate properties of the functors to the corresponding properties of the representing objects. A new point, which makes the method more powerful, is to consider the argument of our deformation theory as an object of the category of symmetric sequences of dg vector spaces, not as just a single dg vector space.