Title: Kock-Lawvere Modules in a Tangent Category
Abstract: In the category of smooth manifolds, vector spaces have the property that T(V) is isomorphic VxV. When moving to abstract settings for differential geometry, such as synthetic differential geometry, this need not be the case, especially tangent categories without any sort of ring object. Cockett and Cruttwell introduced the notion of a differential object to axiomatize this structure in tangent categories.
In this talk we will introduce tangent categories with a scalar object – a differential object satisfying a universal property – and develop the theory of Kock-Lawvere R-modules, or KL modules. This will lead to the presentation of differential objects as models of an enriched sketch.
This is joint work with Jonathan Gallagher and Rory Lucyshyn-Wright.