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homotopy category
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(Topic)
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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.
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.
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.
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.
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.
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.
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.
- 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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"homotopy category" is owned by bci1.
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See Also: fundamental groupoid functors, topological space, approximation theorem for an arbitrary space, fundamental groupoid, Riemannian manifold, cohomology group theorem,  -spectrum
| Other names: |
category of homotopy equivalence classes |
| Also defines: |
fundamental groupoid, fundamental group functor, homotopy category, fundamental groupoid of a topological space |
| Keywords: |
homotopy equivalence classes of a topological space, homotopy category |
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Cross-references: group epimorphism, homotopy groups, inclusions, sequence, colimit, approximation, approximation theorem for an arbitrary space, finitely generated, algorithm, finite, computer, simple, freely generated, pseudographs, hypergraphs, homotopy groupoids, category of graphs, graph, groupoid homomorphisms, category of groupoids, subcategory, groupoid, endomorphisms, concatenation, composition, endpoints, points, paths, equivalence classes, base points, induces, isomorphism, homotopy equivalence, factors, homotopy invariant, classes, maps, homotopy, relation, group homomorphisms, groups, functor, fundamental group, associate, continuous maps, morphisms, basepoint, topological spaces, objects, category
There are 5 references to this entry.
This is version 40 of homotopy category, born on 2008-07-31, modified 2009-02-01.
Object id is 10895, canonical name is HomotopyCategory.
Accessed 1856 times total.
Classification:
| AMS MSC: | 55R37 (Algebraic topology :: Fiber spaces and bundles :: Maps between classifying spaces) | | | 55R65 (Algebraic topology :: Fiber spaces and bundles :: Generalizations of fiber spaces and bundles) | | | 55R05 (Algebraic topology :: Fiber spaces and bundles :: Fiber spaces) | | | 55R10 (Algebraic topology :: Fiber spaces and bundles :: Fiber bundles) | | | 55P99 (Algebraic topology :: Homotopy theory :: Miscellaneous) |
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Pending Errata and Addenda
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