Hoffbeck: Obstruction theory for algebras over an operad

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Eric Hoffbeck
Obstruction theory for algebras over an operad
Annales mathématiques Blaise Pascal, 23 no. 1 (2016), p. 75-107, doi: 10.5802/ambp.355
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Class. Math.: 55S35, 18D50, 55P48
Keywords: Obstruction theory, algebras over operads

Résumé - 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 ${\mathsf {P}}$ and an algebra morphism from $H_*A$ to $H_*B$. Can we realize this morphism as a morphism of ${\mathsf {P}}$-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.

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