# 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.