G. S. H. Cruttwell, Jean-Simon Pacaud Lemay, Elias Vandenberg
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A Tangent Category Perspective on Connections in Algebraic Geometry
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.