PlanetPhysics/Homotopy Category

\newcommand{\sqdiagram}[9]{Failed to parse (unknown function "\diagram"): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }

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.