# An embedding theorem for tangent categories

## TL;DR

This paper proves that every tangent category can be embedded into a representable tangent category, linking abstract differential structures to concrete models like synthetic differential geometry.

## Contribution

It establishes an embedding theorem for tangent categories, showing their equivalence to a class of representable tangent categories, thus connecting abstract and concrete differential frameworks.

## Key findings

- Every tangent category admits an embedding into a representable tangent category.
- The proof uses a coherence theorem and enrichment techniques.
- Connects tangent categories to models of synthetic differential geometry.

## 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.