Tangent bundle functor

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 $M$ to its tangent bundle $TM$
• On morphisms: It sends a smooth map $f:M\to N$ of differential manifolds to its differential, $Df:TM\to TN$

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

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

$T(M\times N)\cong TM\times TN$

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

$T(M\sqcup N)=TM\sqcup TN$