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 .
Explicitly, the objects of are the points of
morphisms are homotopy classes of paths “rel endpoints” that is
where, denotes homotopy rel endpoints, and,
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.
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
Furthermore, the homotopy groups of the -complex are the colimits of the homotopy groups of , and is a group epimorphism.
|Date of creation||2013-03-22 18:17:07|
|Last modified on||2013-03-22 18:17:07|
|Last modified by||bci1 (20947)|
|Synonym||category of homotopy equivalence classes|
|Defines||fundamental group functor|
|Defines||fundamental groupoid of a topological space|