Simplicial properadic homotopy

Eric Hoffbeck; Johan Leray; Bruno Vallette

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arXiv · 2505.22004

Simplicial properadic homotopy

Eric Hoffbeck·Johan Leray·Bruno Vallette

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Abstract

In this paper, we settle the homotopy properties of the infinity-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 infinity-morphisms, the most seminal one being the equivalence between infinity-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 infinity-morphisms, whose homotopy category provides us with the localisation with respect to infinity-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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BibTeX RIS

Eric Hoffbeck, Johan Leray, Bruno Vallette. 2025-05-28. Simplicial properadic homotopy. https://arxiv.org/abs/2505.22004

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