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 ∈ X and whose morphisms are continuous maps X → Y that associate the basepoint of Y to the basepoint of X. The fundamental group of X specifies a functor Top → 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 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 → 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 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 x → y are the equivalence classes of paths from x to y.

• Explicitly, the objects of Π₁(X) are the points of X
  
• morphisms are homotopy classes of paths “rel endpoints” that is
  

  where, ∼ 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 π(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

and also the construction of an approximation of an arbitrary space X as the colimit ΓX of a sequence of cellular inclusions of CW-complexes X₁,...,X_n , so that one obtains X ≡ colim[X_i].

Furthermore, the homotopy groups of the CW-complex ΓX are the colimits of the homotopy groups of X_n, and γ_n+1 : π_q(X_n+1)π_q(X) is a group epimorphism.

References