Tangent bundle functor

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Template:Covariant bundle-valued 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:

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
View other coproduct-preserving functors

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

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