Category of differential manifolds with cobordisms

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This article defines a category structure on manifolds (possibly with additional structure)
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Definition

The category of differential manifolds with cobordisms is defined as follows:

  • The objects of the category are differential manifolds
  • The morphisms of the category are smooth cobordisms i.e. the morphisms from M_1 to M_2 are the cobordisms from M_1 to M_2
  • Composition of morphisms is given as follows. Suppose M_1 \sqcup M_2 \to \partial N is a smooth cobordism from M_1 to M_2, and M_2 \sqcup M_3 \to \partial P is a smooth cobordism from M_2 to M_3. The cobordism from M_1 to M_3 is obtained by gluing N and P along the image of M_2, and not changing the maps from M_1 and M_3 to the respective parts.

All the morphisms in this category are isomorphisms.

A related notion is the 2-category of differential manifolds with smooth maps and smooth cobordisms.