Curved differential graded algebras and corings

TL;DR

This paper establishes a categorical equivalence between semi-free curved differential graded algebras and corings with surjective counits, linking their modules and comodules to various types of curved modules and connections.

Contribution

It introduces a novel equivalence between curved differential graded algebras and corings, and characterizes their modules and comodules in terms of curved modules and integrable connections.

Findings

Category of semi-free curved differential graded algebras is equivalent to corings with surjective counits

Comodules over a coring correspond to integrable connections or quasi-cohesive curved modules

Contramodules over a coring correspond to Z-divergences, a new class of curved modules

Abstract

A relationship between curved differential algebras and corings is established and explored. In particular it is shown that the category of semi-free curved differential graded algebras is equivalent to the category of corings with surjective counits. Under this equivalence, comodules over a coring correspond to integrable connections or quasi-cohesive curved modules, while contramodules over a coring correspond to a specific class of curved modules introduced and termed Z-divergences in here.