Simplicial properadic homotopy
Résumé
In this paper, we settle the homotopy properties of the ∞-morphisms of homotopy (bial)gebras over properads, i.e. algebraic structures made up of operations with several inputs and outputs. We start by providing the literature with characterizations for the various types of ∞-morphisms, the most seminal one being the equivalence between ∞-quasi-isomorphisms and zig-zags of quasi-isomorphisms which plays a key role in the study the formality property. We establish a simplicial enrichment for the categories of gebras over some cofibrant properads together with their ∞-morphisms, whose homotopy category provides us with the localisation with respect to ∞-quasi-isomorphisms. These results extend to the properadic level known properties for operads, but the lack of the rectification procedure in this setting forces us to use different methods.
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