Pro-nilpotently extended dgca-s and SH Lie-Rinehart pairs

TL;DR

This paper establishes an equivalence between pro-nilpotently extended dgca-s and strong homotopy Lie-Rinehart pairs, introducing a categorical framework that links these algebraic structures through Chevalley-Eilenberg constructions.

Contribution

It introduces the category of pro-nilpotently extended dgca-s and proves an equivalence with strong homotopy Lie-Rinehart pairs, expanding the understanding of their homotopical properties.

Findings

Chevalley-Eilenberg construction yields an equivalence of categories.

Pairs with semi-free dgca and cell complex modules form a fibrant category.

The Chevalley-Eilenberg complexes form a cofibrant category.

Abstract

Category of pro-nilpotently extended differential graded commutative algebras is introduced. Chevalley-Eilenberg construction provides an equivalence between its certain full subcategory and the opposite to the full subcategory of strong homotopy Lie Rinehart pairs with strong homotopy morphisms, consisting of pairs $(A,M)$ where $M$ is flat as a graded $A$-module. It is shown that pairs $(A,M)$, where $A$ is a semi-free dgca and $M$ a cell complex in $\op{Mod}(A)$, form a category of fibrant objects by proving that their Chevalley-Eilenberg complexes form a category of cofibrant objects.