Functors of Complexes

Abstract

If T: ∂AG → ∂AG is a functor from complexes to complexes then X → TSX provides a generalization of the singular complex SX which may yield new useful topological invariants. We study this question (§§ 2–7), at least if T is the (dimension-wise) prolongation of an additive functor t: AG → AG. We find that for every abelian group G there is, essentially, one covariant and one contravariant t such that t ℤ = G. The resulting groups HTSX are the homology respectively cohomology groups of X with coefficients in G. The functors t are also useful in studying product spaces; these questions are discussed in §§ 8–12.