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homotopy category (Definition)

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 $X$

    $\displaystyle \mathrm{Obj}(\Pi_1(X))=X\,,$
  • morphisms are homotopy classes of paths “rel endpoints” that is

    $\displaystyle \mathrm{Hom}_{\Pi_1(x)}(x,y)=\mathrm{Paths}(x,y)/\sim\, ,$
    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

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



"homotopy category" is owned by bci1.

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Also defines:  fundamental group functor, homotopy classes of maps, fundamental groups
Keywords:  homotopy category

Cross-references: epimorphism, homotopy groups, approximation theorem for an arbitrary space, concept, algorithm, computer, hypergraphs, graph, simplicial, fundamental groupoid functor, groupoid homomorphisms, category of groupoids, groupoid, composition, fundamental groupoid, isomorphism, homotopy, relation, homomorphisms, groups, functor, fundamental group, morphisms, topological, objects, category
There are 15 references to this object.

This is version 1 of homotopy category, born on 2009-05-02.
Object id is 719, canonical name is HomotopyCategory.
Accessed 997 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
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