2-category of topological spaces with continuous maps and homotopies
This article defines a 2-category
This article describes a category (in the mathematical sense) whose objects are topological spaces, and whose morphisms are continuous maps. In other words, it gives a category structure to the collection of all topological spaces.
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Contents
Definition
The 2-category of topological spaces with continuous maps and homotopies is defined as follows:
Aspect | Name | Definition/description |
---|---|---|
objects | topological spaces | A topological space is a set along with a collection of subsets, called open subsets, that contains the empty subset and the whole space, and is closed under taking arbitrary unions and finite intersections. |
morphisms | continuous maps between topological spaces | A continuous map of topological spaces is a set map with the property that for every open subset of , is an open subset of . |
composition law for morphisms | compose as set maps | Not needed. |
2-morphisms | homotopies between continuous maps. | Given two continuous maps between topological spaces and , a homotopy between and is a continuous map , where and for all . Here, is the closed unit interval, and is given the product topology. |
composition law for 2-morphisms | do one homotopy after the other | The composition law for homotopies is as follows: given a homotopy from to and a homotopy from to , the composite is a homotopy that does in the first half of the time, and then does in the second half of the time. |
Facts
Relation with the associated 1-category
If we forget the 2-morphisms, we get the usual category of topological spaces with continuous maps.
The homotopy category of topological spaces
The 2-category described here is very special in the sense that all its 2-morphisms are invertible. Hence, we can safely take a quotient of the 1-category modulo the equivalence relation induced by the 2-morphisms. This gives rise to the homotopy category of topological spaces, where two continuous maps and are identified if they are homotopic.
Functors to and from the 2-category
Singular chain complex
Further information: singular chain complex 2-functor
The singular chain complex can be viewed as a 2-functor from this 2-category to the 2-category of chain complexes with chain maps and chain homotopies. This gives a strong form of the homotopy-invariance theorem for singular homology.
Underlying space of a cellular space
The underlying space functor can be viewed as a 2-functor from the 2-category of cellular spaces with cellular maps and homotopies to this 2-category, which sends each cellular space to its underlying topological space, each cellular map to its underlying continuous map, and each homotopy to the corresponding homotopy.
This in particular gives a 2-functor from the 2-category of CW-complexes with cellular maps and homotopies.