Let us consider first the category
whose objects
are topological
spaces
with a chosen basepoint
and whose morphisms
are continuous maps
that associate the basepoint of
to the
basepoint of
. The fundamental group of
specifies a functor
, with
being the category of groups
and group homomorphisms, which is called the fundamental group functor.
where,
Therefore, the set of endomorphisms of an object
is precisely the fundamental group
. One can thus construct the groupoid
of homotopy equivalence classes; this construction can be then carried out by utilizing functors from the category
, or its subcategory
,
to the category of groupoids
and groupoid homomorphisms,
. 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
-complexes and homotopy defined
for
-complexes. Perhaps, the simplest example is that of a one-dimensional
-complex, which is a graph.
As described above, one can define a functor from the category of graphs, Grph, to
and then define the fundamental homotopy groupoids of graphs, hypergraphs, or pseudographs. The case of freely generated graphs (one-dimensional
-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
-complexes.
and also the construction of an approximation of an arbitrary space
Furthermore, the homotopy groups
of the
-complex
are the colimits of the homotopy groups of
, and
is a group epimorphism.
As of this snapshot date, this entry was owned by bci1.