Delta Set - Related Functors

Related Functors

The geometric realization of a Δ-set described above defines a covariant functor from the category of Δ-sets to the category of topological spaces. Geometric realization takes a Δ-set to a topological space, and carries maps of Δ-sets to induced continuous maps between geometric realizations (which are topological spaces).

If S is a Δ-set, there is an associated free abelian chain complex, denoted, whose n-th group is the free abelian group

generated by the set, and whose n-th differential is defined by

This defines a covariant functor from the category of Δ-sets to the category of chain complexes of abelian groups. A Δ-set is carried to the chain complex just described, and a map of Δ-sets is carried to a map of chain complexes, which is defined by extending the map of Δ-sets in the standard way using the universal property of free Abelian groups.

Given any topological space X, one can construct a Δ-set as follows. A singular n-simplex in X is a continuous map

Define

to be the collection of all singular n-simplicies in X, and define

by

where again di is the i-th face map. One can check that this is in fact a Δ-set. This defines a covariant functor from the category of topological spaces to the category of Δ-sets. A topological space is carried to the Δ-set just described, and a continuous map of spaces is carried to a map of Δ-sets, which is given by composing the map with the singular n-simplices.