homotopy category

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 $\textbf{G}$ 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 $hTop \to \textbf{G}$ . 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 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.

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

$$\Gamma : \textbf{hU} \longrightarrow \textbf{hU},$$

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.

Bibliography

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: 63-80. Pdf file in arxiv: math.AT/0208211

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