Topos based homology theory

summary: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. Cha\-rac\-terizations of various types of tight cosimplicial objects in the functor topos $\text{\rm Sets}^C$ are given and, as a result, a general method for constructing non-trivial homology theories is obtained. We conclude with several explicit examples