# Difference between revisions of "Category of differential manifolds with cobordisms"

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* The morphisms of the category are [[smooth cobordism]]s i.e. the morphisms from <math>M_1</math> to <math>M_2</math> are the cobordisms from <math>M_1</math> to <math>M_2</math> | * The morphisms of the category are [[smooth cobordism]]s i.e. the morphisms from <math>M_1</math> to <math>M_2</math> are the cobordisms from <math>M_1</math> to <math>M_2</math> | ||

* Composition of morphisms is given as follows. Suppose <math>M_1 \sqcup M_2 \to \partial N</math> is a smooth cobordism from <math>M_1</math> to <math>M_2</math>, and <math>M_2 \sqcup M_3 \to \partial P</math> is a smooth cobordism from <math>M_2</math> to <math>M_3</math>. The cobordism from <math>M_1</math> to <math>M_3</math> is obtained by gluing <math>N</math> and <math>P</math> along the image of <math>M_2</math>, and not changing the maps from <math>M_1</math> and <math>M_3</math> to the respective parts. | * Composition of morphisms is given as follows. Suppose <math>M_1 \sqcup M_2 \to \partial N</math> is a smooth cobordism from <math>M_1</math> to <math>M_2</math>, and <math>M_2 \sqcup M_3 \to \partial P</math> is a smooth cobordism from <math>M_2</math> to <math>M_3</math>. The cobordism from <math>M_1</math> to <math>M_3</math> is obtained by gluing <math>N</math> and <math>P</math> along the image of <math>M_2</math>, and not changing the maps from <math>M_1</math> and <math>M_3</math> to the respective parts. | ||

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+ | All the morphisms in this category are isomorphisms. | ||

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

## Latest revision as of 19:34, 18 May 2008

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

View other category structures on manifolds

## 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.