Obstruction theory for algebras over an operad

TL;DR

This paper develops an obstruction theory for algebras over an operad within differential graded modules, addressing realization and uniqueness of algebra morphisms in the homotopy category using operadic Gamma-cohomology.

Contribution

It introduces an obstruction theory framework for algebras over operads, linking realization problems to operadic Gamma-cohomology groups.

Findings

Obstruction cocycles live in the first two operadic Gamma-cohomology groups.

Provides criteria for realizing homology morphisms as algebra morphisms.

Establishes conditions for the uniqueness of such realizations in the homotopy category.

Abstract

The goal of this paper is to set up an obstruction theory in the context of algebras over an operad and in the framework of differential graded modules over a field. Precisely, the problem we consider is the following: Suppose given two algebras A and B over an operad P and an algebra morphism from the homology of A to the homology of B. Can we realize this morphism as a morphism of P-algebras from A to B in the homotopy category? Also, if the realization exists, is it unique in the homotopy category? We identify obstruction cocycles for this problem, and notice that they live in the first two groups of operadic Gamma-cohomology.