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

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(New page: {{category}} ==Definition== The '''category of differential manifolds with cobordisms''' is defined as follows: * The objects of the category are differential manifolds * The morphi...)
 
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==Definition==
 
==Definition==
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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 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.