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 $X$ $\mathrm{O}\mathrm{b}\mathrm{j}({\Pi }_1(X))=X,$
• morphisms are homotopy classes of paths "rel endpoints" that is ${\mathrm{H}\mathrm{o}\mathrm{m}}_{{\Pi }_1(x)}(x,y)=\mathrm{P}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{s}(x,y)/\sim ,$ 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 $\Gamma {:}^{‴}h{U}^{‴}{⟶}^{‴}h{U}^{‴},$ 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 $X_1,\mathrm{...},X_n$ , 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 ${\gamma }_{n+1}:{\pi }_q(X_{n+1})⟼{\pi }_q(X)$ is a group epimorphism.