Homotopical algebra of Lie-Rinehart pairs

TL;DR

This paper establishes an equivalence between the homotopical categories of dg Lie-Rinehart pairs and strong homotopy Lie-Rinehart pairs, introducing cofibrations, factorizations, and proving homotopy uniqueness of resolutions.

Contribution

It develops a homotopical framework for Lie-Rinehart pairs, including cofibrations, fibrations, and resolutions, and compares different cofibrancy conditions within the $alculus of homotopical algebra.

Findings

Dwyer-Kan localization of dg Lie-Rinehart pairs is equivalent to that of strong homotopy Lie-Rinehart pairs.

Constructed factorizations and proved lifting properties for SH LR pairs.

Showed the functor from pairs to cdga is a Cartesian fibration with presentable fibers.

Abstract

Dwyer-Kan localization at pairs of quasi-isomorphisms of the category of dg Lie-Rinehart pairs $(A,M)$, where $A$ is a semi-free cdga over a field $k$ of characteristic zero and $M$ a cell complex in $A$-modules, is shown to be equivalent to that of strong homotopy Lie-Rinehart (SH LR) pairs satisfying the same cofibrancy condition. Latter is a category of fibrant objects. We introduce cofibrations of SH LR pairs, construct factorizations, and prove lifting properties. Applying them, we show uniqueness up to homotopy of certain BV-type resolutions. Restricting to dg LR pairs whose underlying cdga is of finite type, and using a different (co)fibrancy condition, we show that the functor $(A,M)\mapsto A$ is a Cartesian fibration with presentable fibers. The two (co)fibrancy conditions yield equivalent $\infty$-categories under Dwyer-Kan localization.