Universal constructions in homotopical algebra
Ricardo Campos, Bruno Vallette
TL;DR
This work develops a universal, Lie-graph–based framework for homotopy bialgebras governed by properads. By applying the integration theory of Lie-graph algebras, it delivers an explicit description of the Deligne groupoid as the groupoid of $\\Omega \\mathcal{C}$-algebras with $\\infty$-isotopies, bridging Maurer–Cartan theory with $\\infty$-morphisms beyond operads. It recasts the Homotopy Transfer Theorem as a gauge-action phenomenon and extends the Koszul hierarchy to properads, yielding a universal method to construct $\\mathcal{P}_\\infty$-algebras from $\\mathcal{P}^!$-algebras (with explicit, gauge-trivial genus expansions in the Frobenius case). The twisting procedure is generalized to curved and unital properadic contexts via an extended convolution algebra, leading to $\\mathcal{P}_\\infty$-Maurer–Cartan equations that unify twisting for elements and coelements and apply to a wide range of structures (e.g., Frobenius, double Poisson, pre-Calabi–Yau, quantum Airy). Overall, the paper provides a cohesive, broadly applicable toolkit for universal homotopy constructions in the properadic setting, with potential impact on deformation theory, mathematical physics, and geometry.
Abstract
We apply the effective integration theory of Lie-graph algebras, developed recently by the authors, to the deformation and homotopy theories of types of bialgebras, that is structures controlled by a properad, like associative bialgebras, (involutive) Lie bialgebras, Frobenius bialgebras, double Poisson bialgebras, pre-Calabi--Yau algebras, quantum Airy structures, etc. In these cases, we provide their associated Deligne groupoid with an explicit homotopical description. We settle the Koszul hierarchy and the twisting procedure on the properadic level. We also give a conceptual construction of the homotopy transfer theorem in terms of gauge actions. This work extends the formulas for the deformation theory of operadic algebras.