This defines an equivalence of categories between the category of algebraic sets and the opposite category of the finitely generated reduced "k" -algebras.
For reasonably simple spaces, all of the groups will be finitely generated, whereas the singular chain groups are, in general, not even countably generated.
One is often interested in projective generators (even finitely generated projective generators, called progenerators) and minimal injective cogenerators.
It is particularly useful where finiteness assumptions are satisfied, for example finitely generated groups, or finitely presented groups (i.e. in addition the relations are finite).