# Category of differential manifolds with cobordisms

From Diffgeom

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 to are the cobordisms from to
- Composition of morphisms is given as follows. Suppose is a smooth cobordism from to , and is a smooth cobordism from to . The cobordism from to is obtained by gluing and along the image of , and not changing the maps from and 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.