homotopy category
0.1 Homotopy category, fundamental groups and fundamental groupoids
Let us consider first the category whose objects are topological spaces with a chosen basepoint and whose morphisms are continuous maps that associate the basepoint of to the basepoint of . The fundamental group of specifies a functor , with G being the category of groups and group homomorphisms, which is called the fundamental group functor.
0.2 Homotopy category
Next, when one has a suitably defined relation of homotopy between morphisms, or maps, in a category , one can define the homotopy category as the category whose objects are the same as the objects of , but with morphisms being defined by the homotopy classes of maps; this is in fact the homotopy category of unbased spaces.
0.3 Fundamental groups
We can further require that homotopies on map each basepoint to a corresponding basepoint, thus leading to the definition of the homotopy category of based spaces. Therefore, the fundamental group is a homotopy invariant functor on , with the meaning that the latter functor factors through a functor . A homotopy equivalence in is an isomorphism in . Thus, based homotopy equivalence induces an isomorphism of fundamental groups.
0.4 Fundamental groupoid
In the general case when one does not choose a basepoint, a fundamental groupoid of a topological space needs to be defined as the category whose objects are the base points of and whose morphisms are the equivalence classes of paths from to .
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Explicitly, the objects of are the points of
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morphisms are homotopy classes of paths “rel endpoints” that is
where, denotes homotopy rel endpoints, and,
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composition of morphisms is defined via piecing together, or concatenation, of paths.
0.5 Fundamental groupoid functor
Therefore, the set of endomorphisms of an object is precisely the fundamental group . One can thus construct the groupoid of homotopy equivalence classes; this construction can be then carried out by utilizing functors from the category , or its subcategory , to the category of groupoids and groupoid homomorphisms, . One such functor which associates to each topological space its fundamental (homotopy) groupoid is appropriately called the fundamental groupoid functor.
0.6 An example: the category of simplicial, or CW-complexes
As an important example, one may wish to consider the category of simplicial, or -complexes and homotopy defined for -complexes. Perhaps, the simplest example is that of a one-dimensional -complex, which is a graph. As described above, one can define a functor from the category of graphs, Grph, to and then define the fundamental homotopy groupoids of graphs, hypergraphs, or pseudographs. The case of freely generated graphs (one-dimensional -complexes) is particularly simple and can be computed with a digital computer by a finite algorithm using the finite groupoids associated with such finitely generated -complexes.
0.6.1 Remark
Related to this concept of homotopy category for unbased topological spaces, one can then prove the approximation theorem for an arbitrary space by considering a functor
and also the construction of an approximation of an arbitrary space as the colimit of a sequence of cellular inclusions of -complexes , so that one obtains .
Furthermore, the homotopy groups of the -complex are the colimits of the homotopy groups of , and is a group epimorphism.
References
- 1 May, J.P. 1999, A Concise Course in Algebraic Topology., The University of Chicago Press: Chicago
- 2 R. Brown and G. Janelidze.(2004). Galois theory and a new homotopy double groupoid of a map of spaces.(2004). Applied Categorical Structures,12: 63-80. Pdf file in arxiv: math.AT/0208211
Title | homotopy category |
Canonical name | HomotopyCategory |
Date of creation | 2013-03-22 18:17:07 |
Last modified on | 2013-03-22 18:17:07 |
Owner | bci1 (20947) |
Last modified by | bci1 (20947) |
Numerical id | 44 |
Author | bci1 (20947) |
Entry type | Topic |
Classification | msc 55P99 |
Classification | msc 55R10 |
Classification | msc 55R05 |
Classification | msc 55R65 |
Classification | msc 55R37 |
Synonym | category of homotopy equivalence classes |
Related topic | FundamentalGroupoidFunctor |
Related topic | TopologicalSpace |
Related topic | ApproximationTheoremForAnArbitrarySpace |
Related topic | FundamentalGroupoid |
Related topic | RiemannianMetric |
Related topic | CohomologyGroupTheorem |
Related topic | OmegaSpectrum |
Related topic | CategoryOfGroupoids2 |
Defines | fundamental groupoid |
Defines | fundamental group functor |
Defines | homotopy category |
Defines | fundamental groupoid of a topological space |