Homotopy category of topological spaces: Difference between revisions
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object = topological space}} | |||
==Definition== | ==Definition== | ||
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* Its objects are [[topological space]]s | * Its objects are [[topological space]]s | ||
* Its morphisms are homotopy classes of continuous maps | * Its morphisms are homotopy classes of continuous maps | ||
This can be viewed as the homotopy category obtained from the [[2-category of topological spaces with continuous maps and homotopies]]. | |||
==Notion of isomorphism== | |||
The isomorphisms in this category are the [[homotopy equivalence of topological spaces|homotopy equivalences]]. | |||
==Functors to and from the category== | |||
The homotopy category of topological spaces is not a concrete category, because, although its objects are sets, the morphisms are not concrete continuous maps but rather continuous maps upto homotopy. | |||
===Functor from the category of topological spaces=== | |||
This functor sends a topological space to itself, and sends a continuous map to its homotopy class of continuous maps. | |||
This functor is not conservative, because there are homotopy equivalences of topological spaces that are not homeomorphisms. | |||
Latest revision as of 20:31, 24 November 2008
This article describes a category (in the mathematical sense) whose objects are topological spaces. In other words, it gives a category structure to the collection of all topological spaces.
View other category structures on topological spaces
Definition
The homotopy category of topological spaces is defined as follows:
- Its objects are topological spaces
- Its morphisms are homotopy classes of continuous maps
This can be viewed as the homotopy category obtained from the 2-category of topological spaces with continuous maps and homotopies.
Notion of isomorphism
The isomorphisms in this category are the homotopy equivalences.
Functors to and from the category
The homotopy category of topological spaces is not a concrete category, because, although its objects are sets, the morphisms are not concrete continuous maps but rather continuous maps upto homotopy.
Functor from the category of topological spaces
This functor sends a topological space to itself, and sends a continuous map to its homotopy class of continuous maps.
This functor is not conservative, because there are homotopy equivalences of topological spaces that are not homeomorphisms.