# Tangent bundle functor

From Diffgeom

Template:Covariant bundle-valued functor

## Contents

## Definition

The **tangent bundle functor** is a functor from the category of differential manifolds with smooth maps to the category of differential manifolds with smooth maps, defined as follows:

- On
*objects*: It sends a differential manifold to its tangent bundle - On
*morphisms*: It sends a smooth map of differential manifolds to its differential,

The tangent bundle functor comes with a natural transformation to the identity functor: namely, the bundle map from to . There is also a natural transformation *from* the identity functor: namely, the zero section of in .

## Properties of the functor

### Product-preserving functor

This covariant functor is product-preserving: applying the functor to a product of objects in a category, is equivalent to taking the product after applying the functor

View other product-preserving functors

The tangent bundle functor preserves products; in other words, we have:

### Coproduct-preserving functor

This covariant functor commutes with coproducts: applying the functor after taking coproducts, has the same effect as taking the coproduct after applying the functor

View other coproduct-preserving functors

The tangent bundle functor preserves coproducts; in other words, we have: