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 H * A to H * 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?
Highlights
We identify obstruction cocycles for this problem, and notice that they live in the first two groups of operadic Γ-cohomology
In this paper we study a question of realizability of morphisms in a category of algebras over an operad
The obstructions live in some Harrison cohomology groups
Summary
(3) The category of algebras over a cofibrant operad inherits a model structure where fibrations When we are given an operad morphism Bc(D) → Q, we have a functor which, to any D-coalgebra C, associates a quasi-free Q-algebra RQ(C) = (Q(C), ∂) for some twisting differential ∂ (cf [GJ] or [F2, Section 4.2.1]) We apply this construction to D = B(P ⊠ E), the morphism id : Bc(D) → Bc(D) = Pand the coalgebra C = (D(A), ∂α) associated to a P-algebra A (the action is denoted by α). The augmentation ǫ : RP(D(A), ∂α) = (P(D(A), ∂α), ∂) → A defines a weak equivalence and (P(D(A), ∂α), ∂) forms a cofibrant replacement of A in the category of P-algebras In this context, to study morphisms in the homotopy category of P-algebras, we just have to study morphisms of quasi-cofree D-coalgebras. An easy way to understand this definition is the following: the Γ-cohomology of a P-algebra A is the usual Andre-Quillen cohomology of A seen as an algebra over a Σ∗-cofibrant replacement of P
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