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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 $Top\to\textbf{G}$, 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 $hTop$ 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 $hTop\to\textbf{G}$. 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 $\Pi_{1}(X)$ 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 $y$.

Explicitly, the objects of $\Pi_{1}(X)$ are the points of $X$
$\mathrm{Obj}(\Pi_{1}(X))=X\,,$ 
morphisms are homotopy classes of paths “rel endpoints” that is
$\mathrm{Hom}_{{\Pi_{1}(x)}}(x,y)=\mathrm{Paths}(x,y)/\sim\,,$ where, $\sim$ denotes homotopy rel endpoints, and,

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
$\Gamma:\textbf{hU}\longrightarrow\textbf{hU},$ 
and also the construction of an approximation of an arbitrary space $X$ as the colimit $\Gamma X$ of a sequence of cellular inclusions of $CW$complexes $X_{1},...,X_{n}$ , so that one obtains $X\equiv colim[X_{i}]$.
Furthermore, the homotopy groups of the $CW$complex $\Gamma X$ are the colimits of the homotopy groups of $X_{n}$, and $\gamma_{{n+1}}:\pi_{q}(X_{{n+1}})\longmapsto\pi_{q}(X)$ 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
Mathematics Subject Classification
55P99 no label found55R10 no label found55R05 no label found55R65 no label found55R37 no label found Forums
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