Topos based homology theory

In this paper we extend the Eilenberg-Steenrod axiomatic description of a homology theory from the category of topological spaces to an arbitrary category and, in particular, to a topos. Implicit in this extension is an extension of the notions of homotopy and excision. A general discussion of such homotopy and excision structures on a category is given along with several examples including the interval based homotopies and, for toposes, the excisions represented by “cutting out” subobjects. The existence of homology theories on toposes depends upon their internal logic. It is shown, for example, that all “reasonable” homology theories on a topos in which De Morgan’s law holds are trivial. To obtain examples on non-trivial homology theories we consider singular homology based on a cosimplicial object. For toposes singular homology satisfies all the axioms except, possibly, excision. We introduce a notion of “tightness” and show that singular homology based on a sufficiently tight cosimplicial object satisfies the excision axiom. Characterizations of various types of tight cosimplicial objects in the functor topos Sets are given and, as a result, a general method for constructing non-trivial homology theories is obtained. We conclude with several explicit examples.