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 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 $Top\to \text{𝐆}$, with G being the category of groups and group homomorphisms, which is called the 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 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 ${\mathrm{\Pi }}_1(X)$ 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 functor which associates to each topological space its fundamental (homotopy) groupoid is appropriately called the fundamental 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 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.

0.6.1 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

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

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

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

Title	homotopy category
Canonical name	HomotopyCategory
Date of creation	2013-03-22 18:17:07
Last modified on	2013-03-22 18:17:07
Owner	bci1 (20947)
Last modified by	bci1 (20947)
Numerical id	44
Author	bci1 (20947)
Entry type	Topic
Classification	msc 55P99
Classification	msc 55R10
Classification	msc 55R05
Classification	msc 55R65
Classification	msc 55R37
Synonym	category of homotopy equivalence classes
Related topic	FundamentalGroupoidFunctor
Related topic	TopologicalSpace
Related topic	ApproximationTheoremForAnArbitrarySpace
Related topic	FundamentalGroupoid
Related topic	RiemannianMetric
Related topic	CohomologyGroupTheorem
Related topic	OmegaSpectrum
Related topic	CategoryOfGroupoids2
Defines	fundamental groupoid
Defines	fundamental group functor
Defines	homotopy category
Defines	fundamental groupoid of a topological space