A Tangent Category Perspective on Connections in Algebraic Geometry

TL;DR

This paper explores how the abstract concept of connections in tangent categories relates to classical notions in algebraic geometry, revealing that it reproduces known concepts in schemes but not in algebras.

Contribution

It demonstrates that tangent categories can recover classical connections on modules and sheaves, and shows the absence of non-trivial connections in algebra tangent categories.

Findings

Connections in affine schemes match classical module connections

Connections in schemes correspond to quasi-coherent sheaf connections

No non-trivial connections exist in the tangent category of algebras

Abstract

There is an abstract notion of connection in any tangent category. In this paper, we show that when applied to the tangent category of affine schemes, this recreates the classical notion of a connection on a module (and similarly, in the tangent category of schemes, this recreates the notion of connection on a quasi-coherent sheaf of modules). By contrast, we also show that in the tangent category of algebras, there are no non-trivial connections.