An embedding theorem for tangent categories

Abstract

Tangent categories were introduced by Rosicky as a categorical setting for differential structures in algebra and geometry; in recent work of Cockett, Crutwell and others, they have also been applied to the study of differential structure in computer science. In this paper, we prove that every tangent category admits an embedding into a representable tangent category---one whose tangent structure is given by exponentiating by a free-standing tangent vector, as in, for example, any model of Kock and Lawvere's synthetic differential geometry. The key step in our proof uses a coherence theorem for tangent categories due to Leung to exhibit tangent categories as a certain kind of enriched category.