homotopy category
0.1 Homotopy category, fundamental groups and fundamental groupoids
Let us consider first the category^{} $Top$ whose objects are topological spaces^{} $X$ with a chosen basepoint $x\in X$ and whose morphisms^{} are continuous maps $X\to Y$ that associate the basepoint of $Y$ to the basepoint of $X$. The fundamental group^{} of $X$ specifies a functor^{} $Top\to \text{\mathbf{G}}$, 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 $U$, one can define the homotopy category $hU$ as the category whose objects are the same as the objects of $U$, 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 $Top$ map each basepoint to a corresponding basepoint, thus leading to the definition of the homotopy category $h\mathit{}T\mathit{}o\mathit{}p$ of based spaces. Therefore, the fundamental group is a homotopy invariant functor on $Top$, with the meaning that the latter functor factors through a functor $hTop\to \text{\mathbf{G}}$. A homotopy equivalence^{} in $U$ is an isomorphism^{} in $hTop$. 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^{} ${\mathrm{\Pi}}_{1}(X)$ of a topological space $X$ needs to be defined as the category whose objects are the base points of $X$ and whose morphisms $x\to y$ are the equivalence classes^{} of paths from $x$ to $y$.

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Explicitly, the objects of ${\mathrm{\Pi}}_{1}(X)$ are the points of $X$
$$\mathrm{Obj}({\mathrm{\Pi}}_{1}(X))=X,$$ 
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morphisms are homotopy classes of paths “rel endpoints^{}” that is
$${\mathrm{Hom}}_{{\mathrm{\Pi}}_{1}(x)}(x,y)=\mathrm{Paths}(x,y)/\sim ,$$ where, $\sim $ 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 $x$ is precisely the fundamental group $\pi (X,x)$. One can thus construct the groupoid^{} of homotopy equivalence classes; this construction can be then carried out by utilizing functors from the category $Top$, or its subcategory^{} $hU$, to the category of groupoids^{} and groupoid homomorphisms, $Grpd$. 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 CWcomplexes
As an important example, one may wish to consider the category of simplicial, or $CW$complexes and homotopy defined for $CW$complexes. Perhaps, the simplest example is that of a onedimensional $CW$complex, which is a graph. As described above, one can define a functor from the category of graphs, Grph, to $Grpd$ and then define the fundamental homotopy groupoids of graphs, hypergraphs^{}, or pseudographs^{}. The case of freely generated graphs (onedimensional $CW$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^{} $CW$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
$$\mathrm{\Gamma}:\text{\mathbf{h}\mathbf{U}}\u27f6\text{\mathbf{h}\mathbf{U}},$$ 
and also the construction of an approximation of an arbitrary space $X$ as the colimit^{} $\mathrm{\Gamma}X$ of a sequence^{} of cellular inclusions of $CW$complexes ${X}_{1},\mathrm{\dots},{X}_{n}$ , so that one obtains $X\equiv colim[{X}_{i}]$.
Furthermore, the homotopy groups^{} of the $CW$complex $\mathrm{\Gamma}X$ are the colimits of the homotopy groups of ${X}_{n}$, and ${\gamma}_{n+1}:{\pi}_{q}({X}_{n+1})\u27fc{\pi}_{q}(X)$ 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: 6380. Pdf file in arxiv: math.AT/0208211
Title  homotopy category 
Canonical name  HomotopyCategory 
Date of creation  20130322 18:17:07 
Last modified on  20130322 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 