PlanetPhysics/Homotopy Category

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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 $$Top \to G $$, with $$G $$ being the category of groups and group homomorphisms, which is called the fundamental group functor.

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.

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 $$ hTop \to G  $$. A homotopy equivalence in $$U$$ is an isomorphism in $$hTop$$. Thus, based homotopy equivalence induces an isomorphism of fundamental groups.

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.

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 \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 $$Top$$, or its subcategory $$hU$$, to the \htmladdnormallink{category of groupoids {http://planetphysics.us/encyclopedia/GroupoidCategory4.html} and groupoid homomorphisms}, $$Grpd$$. 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-complexes
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 one-dimensional $$CW$$-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 (one-dimensional $$CW$$-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.

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 $$\Gamma : hU \longrightarrow hU ,$$ 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.