Homotopical algebra of Lie-Rinehart pairs
Damjan Pištalo
TL;DR
The paper develops a robust homotopy theory for dg Lie-Rinehart pairs by comparing them to strong-homotopy LR (SH LR) pairs and establishing an equivalence of their ∞-categories under cofibrancy. It constructs and analyzes cofibrations, factorizations, and lifting properties for SH LR[1] objects, and uses these to obtain BV-type resolutions whose uniqueness is up to homotopy. Koszul duality and bar–cobar arguments relate SH LR structures to dg LR algebras, yielding a DK-localization equivalence between the two formalisms; for finite-type bases, the DG-LR framework forms a Cartesian fibration with presentable fibers, and a further equivalence shows nice finite-type models capture the same ∞-categorical content. Collectively, the results provide a coherent, flexible homotopical framework for Lie-Rinehart algebras in algebraic and derived-geometry contexts, including a BV-BRST perspective and a Cartesian-fibration view over base cdgas.
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.