Weil spaces, and the embedding theorem for tangent categories

The purpose of this tutorial is to introduce the enriched perspective on tangent categories: they are precisely categories (with certain colimits) enriched in the cartesian closed category of "Weil spaces". Here a "Weil space" is more or less what an algebraic geometer would call a "formal deformation problem": a nicely-behaved functor from a category of Weil algebras (= local Artinian algebras) into Sets. We also sketch how the enriched perspective on tangent categories allows us to prove an embedding theorem: every tangent category embeds fully and faithfully into a representable tangent category.Non UBCUnreviewedAuthor affiliation: Macquarie UniversityResearche