Tangent Categories from the Coalgebras of Differential Categories

TL;DR

This paper establishes that the coEilenberg-Moore category of a differential category is a tangent category, linking algebraic structures with differential geometric concepts and broadening understanding in algebraic geometry and computer science.

Contribution

It proves that the coEilenberg-Moore category of a differential category is a tangent category, extending the connection between differential categories and tangent structures.

Findings

The coEilenberg-Moore category of a differential category is a tangent category.

The opposite of the category of commutative rings forms a tangent category.

The results apply to categories without negatives, relevant to combinatorics and computer science.

Abstract

Following the pattern from linear logic, the coKleisli category of a differential category is a Cartesian differential category. What then is the coEilenberg-Moore category of a differential category? The answer is a tangent category! A key example arises from the opposite of the category of Abelian groups with the free exponential modality. The coEilenberg-Moore category, in this case, is the opposite of the category of commutative rings. That the latter is a tangent category captures a fundamental aspect of both algebraic geometry and Synthetic Differential Geometry. The general result applies when there are no negatives and thus encompasses examples arising from combinatorics and computer science. This is an extended version of a conference paper for CSL2020.