# PlanetPhysics/Homotopy Category

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### Homotopy category, fundamental groups and fundamental groupoidsEdit

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 being the category of groups and group homomorphisms, which is called ***the fundamental group functor* .

### Homotopy categoryEdit

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* .

### Fundamental groupsEdit

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.
**

### Fundamental groupoidEdit

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,
- composition of morphisms is defined
*via*piecing together, or concatenation, of paths.

### Fundamental groupoid functorEdit

Therefore, the set of endomorphisms of an object is precisely the fundamental group . One can thus construct the *\htmladdnormallink{groupoid* {http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html} of homotopy equivalence classes}; this construction can be then carried out by utilizing functors from the category ** , or its subcategory ,
to the ***\htmladdnormallink{category of groupoids* {http://planetphysics.us/encyclopedia/GroupoidCategory4.html} and groupoid homomorphisms}, . One such functor
which associates to each topological space its fundamental (homotopy) groupoid is appropriately called the
*fundamental groupoid functor*.

### An example: the category of simplicial, or CW-complexesEdit

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.
**

#### RemarkEdit

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.

## All SourcesEdit

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