Classifying spaces and moduli spaces of algebras over a prop

Sinan Yalin (1)

1 LPP - Laboratoire Paul Painlevé - UMR 8524

The purpose of this article is two-fold. First we show that a weak equivalence between cofibrant props induces a weak equivalence between the associated classifying spaces of bialgebras. This statement generalizes to the prop setting a homotopy invariance result which is well known in the case of algebras over operads. The absence of model category structure on algebras over a prop leads us to introduce new methods to overcome this difficulty. We also explain how our result can be extended to algebras over colored props in any symmetric monoidal model category tensored over the chain complexes. Then we provide a generalization of a theorem of Charles Rezk in the setting of algebras over a (colored) prop: we introduce the notion of moduli space of algebra structures over a prop, and prove that under certain conditions such a moduli space is the homotopy fiber of a map between classifying spaces.

Mots clés: prop; classifying space; bialgebras category; homotopy invariance

Domaines: Topologie algébrique [math.AT]

Sinan Yalin. Classifying spaces and moduli spaces of algebras over a prop. 2012. ⟨hal-00717281v2⟩