External Spanier-Whitehead duality for C-spectra and applications to C-homology theories

Abstract

The thesis constructs a Spanier-Whitehead type duality functor relating finite C-spectra to finite Cop-spectra, where C is a topological index category. We prove that every C-homology theory is given by taking the homotopy groups of a balanced smash product with a fixed Cop-spectrum. As an application, we construct Chern characters for certain rational C-homology theories. The existence of Chern characters leads to the algebraic question of the hereditarity of category algebras for which a complete characterisation is given. A particular focus is set on the case of orbit categories.