Simplicial localization of homotopy algebras over a prop

Sinan Yalin (1)

1 LPP - Laboratoire Paul Painlevé - UMR 8524

We prove that a weak equivalence between two cofibrant (colored) props in chain complexes induces a Dwyer-Kan equivalence between the simplicial localizations of the associated categories of algebras. This homotopy invariance under base change implies that the homotopy category of homotopy algebras over a prop P does not depend on the choice of a cofibrant resolution of P, and gives thus a coherence to the notion of algebra up to homotopy in this setting. The result is established more generally for algebras in combinatorial monoidal dg categories.

Mots clés: homotopical algebra; simplicial localization; homotopy invariance; props; bialgebras category

Domaines: Topologie algébrique [math.AT]

Sinan Yalin. Simplicial localization of homotopy algebras over a prop. Mathematical Proceedings of the Cambridge Philosophical Society, 2014, 157 (3), pp.457-468. ⟨hal-00845807v2⟩