homotopy category

0.1 Homotopy category, fundamental groups and fundamental groupoids

Let us consider first the categoryMathworldPlanetmath Top whose objects are topological spacesMathworldPlanetmath X with a chosen basepoint xX and whose morphismsMathworldPlanetmath are continuous maps XY that associate the basepoint of Y to the basepoint of X. The fundamental groupMathworldPlanetmathPlanetmath of X specifies a functorMathworldPlanetmath Top𝐆, with G being the category of groups and group homomorphismsMathworldPlanetmath, which is called the fundamental group functor.

0.2 Homotopy category

Next, when one has a suitably defined relationMathworldPlanetmathPlanetmathPlanetmath of homotopyMathworldPlanetmathPlanetmath 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 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 hTop𝐆. A homotopy equivalenceMathworldPlanetmathPlanetmath in U is an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath 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 groupoidMathworldPlanetmath Π1(X) of a topological space X needs to be defined as the category whose objects are the base points of X and whose morphisms xy are the equivalence classesMathworldPlanetmathPlanetmath of paths from x to y.

  • Explicitly, the objects of Π1(X) are the points of X

  • morphisms are homotopy classes of paths “rel endpointsMathworldPlanetmath” that is


    where, denotes homotopy rel endpoints, and,

  • compositionMathworldPlanetmathPlanetmath 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 π(X,x). One can thus construct the groupoidPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of homotopy equivalence classes; this construction can be then carried out by utilizing functors from the category Top, or its subcategoryMathworldPlanetmath hU, to the category of groupoidsPlanetmathPlanetmath 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, hypergraphsMathworldPlanetmath, or pseudographsMathworldPlanetmath. 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 generatedMathworldPlanetmath 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


and also the construction of an approximation of an arbitrary space X as the colimitMathworldPlanetmath ΓX of a sequencePlanetmathPlanetmath of cellular inclusions of CW-complexes X1,,Xn , so that one obtains Xcolim[Xi].

Furthermore, the homotopy groupsMathworldPlanetmath of the CW-complex ΓX are the colimits of the homotopy groups of Xn, and γn+1:πq(Xn+1)πq(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 groupoidPlanetmathPlanetmath of a map of spaces.(2004). Applied Categorical StructuresMathworldPlanetmath,12: 63-80. Pdf file in arxiv: math.AT/0208211
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