Modules and homotopy invariance of functors

Abstract

We prove that any category of props in a symmetric monoidal model category inherits a model structure. We devote an appendix, about half the size of the paper, to the proof of the model category axioms in a general setting. We need the general argument to address the case of props in topological spaces and dg-modules over an arbitrary ring, but we give a less technical proof which applies to the category of props in simplicial sets, simplicial modules, and dg-modules over a ring of characteristic 0. We apply the model structure of props to the homotopical study of algebras over a prop. Our goal is to prove that an object X homotopy equivalent to an algebra A over a cofibrant prop P inherits a P-algebra structure so that X defines a model of A in the homotopy category of P-algebras. In the differential graded context, this result leads to a generalization of Kadeishvili's minimal model of A∞-algebras.

Realization Spaces of Algebraic Structures on Cochains

“…The conclusion follows from the fact that the endomorphism prop of F that we consider in this proof, defined by using the external hom of the diagram category Ch I K , is exactly the same prop as the endomorphism prop of diagrams in the sense of [12] which is used to encode P ∞ -algebra structures on diagrams. We apply this argument to the particular case of (X, ϕ X )…”

“…There is a free-forgetful adjunction between Σ-biobjects and props, for which we refer to [12], which transfer the cofibrantly generated model category structure of Σ-biobjects to the category of props: Theorem 1.3. (see [12], theorem 4.9) (1) Suppose that char(K) > 0.…”

“…(see [12], theorem 4.9) (1) Suppose that char(K) > 0. The category P 0 of props with non-empty inputs (or outputs) equipped with the classes of componentwise weak equivalences and componentwise fibrations forms a cofibrantly generated semi-model category.…”

“…A semi-model category structure is a slightly weakened version of model category structure: the lifting axioms work only with cofibrations with a cofibrant domain, and the factorization axioms work only on a map with a cofibrant domain (see the relevant section of [12]). The notion of a semi-model category is sufficient to apply the usual constructions of homotopical algebra.…”

“…One the one hand, we restrict ourselves to the P ∞ -algebras having X as underlying complex. On the other hand, according to [12], if two prop morphisms ϕ 1 , ϕ 2 : P ∞ → End X are homotopic then (X, φ 1 ) and (X, φ 2 ) are related by a zigzag of quasi-isomorphisms of P ∞ -algebras. But the converse is in general not true, so our realization space detects finer homotopical informations about the P ∞ -algebra structures on X than the one of [16].…”