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 chosenbasepoint $x\in X$ and whose morphisms are continuous maps $X\to Y$ that associate the basepoint of $Y$ to thebasepoint of $X$. The fundamental group of $X$ specifies a functor $Top\to \text{𝐆}$,with G being the category of groups and group homomorphisms, which is calledthe 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 $hTop\to \text{𝐆}$. 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 arethe base points of $X$ and whose morphisms $x\to y$ are the equivalence classes of paths from $x$ to $y$.

• Explicitly, the objects of ${\mathrm{\Pi }}_1(X)$ are the points of $X$

  $$\mathrm{Obj}({\mathrm{\Pi }}_1(X))=X,$$

• 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,

• 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 functorwhich associates to each topological space its fundamental (homotopy) groupoid is appropriately called thefundamental groupoid functor.

0.6 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 definedfor $CW$-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 generatedgraphs (one-dimensional $CW$-complexes) is particularly simple and can be computed with a digital computer by a finitealgorithm 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 theapproximation theorem for an arbitrary space by considering a functor

$$\mathrm{\Gamma }:\text{𝐡𝐔}⟶\text{𝐡𝐔},$$

and also the construction of an approximation of an arbitrary space $X$ as thecolimit $\mathrm{\Gamma }X$ of a sequence of cellular inclusions of $CW$-complexes $X_1,\mathrm{\dots },X_n$ , sothat one obtains $X\equiv colim[X_i]$.

Furthermore, the homotopy groups of the $CW$-complex $\mathrm{\Gamma }X$ are the colimits of thehomotopy groups of $X_n$, and ${\gamma }_{n+1}:{\pi }_q(X_{n+1})⟼{\pi }_q(X)$ is a group epimorphism.