Simplicial properadic homotopy
Eric Hoffbeck, Johan Leray, Bruno Vallette
TL;DR
This work generalizes the homotopy theory of infinity-morphisms from operads to properads by developing a complete framework for Gebras over cofibrant properads. It constructs a cofibrant 2-colored dg properad encoding pairs of $\Omega\mathcal{C}$-gebras linked by an $\infty$-morphism and pairs this with a convolution $\mathrm{L}_\infty$-algebra to describe morphisms via Maurer–Cartan elements. A central result extends the zig-zag versus $\infty$-quasi-isomorphism correspondence to the properadic setting, enabling formality analyses for a broad class of algebraic structures (including (involutive) bialgebras, Frobenius and Calabi–Yau analogues, and pre-Calabi–Yau-type algebras). The authors then empower a simplicial enrichment of the $\mathcal{P}_\infty$-gebras category through the Deligne–Hinich MC$_{\bullet}$ construction, showing the resulting homotopy category localizes precisely with respect to both quasi-isomorphisms and $\infty$-quasi-isomorphisms, and characterizes $\infty$-qi morphisms via weak equivalences of Kan complexes. By addressing the lack of rectification in the properadic context with model-category techniques and novel graph filtrations, the paper provides a versatile, broadly applicable framework for deformation theory and homotopy transfer in properadic settings.
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.