Simplicial properadic homotopy

TL;DR

This paper investigates the homotopy theory of infinity-morphisms in homotopy (bial)-algebras over properads, establishing foundational properties and a simplicial enrichment framework, extending operadic results to properads.

Contribution

It provides the first comprehensive characterization of infinity-morphisms for properads and introduces a simplicial enrichment for their categories, overcoming the lack of rectification methods.

Findings

Characterization of infinity-morphisms over properads

Simplicial enrichment of categories of properad algebras

Homotopy category as localization with respect to infinity-quasi-isomorphisms

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.