Props in model categories and homotopy invariance of structures

Details Props in model categories and homotopy invariance of structures Props in model categories and homotopy invariance of structures

2008-12-15

arxiv.org/abs/0812.2738v4

Mathematics

Algebraic Topology

63 pages. Confusing typos (in section 8) fixed in v2. Minor updates in v3. Last typo corrections in v4. To appear in the Georg

Scientific paper

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-infinity algebras.

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