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$ .

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

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

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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