Manchester Geometry Seminar 2005/2006

A kind of unstable homotopy theory on the category of associative rings (without unit) is developed. There are the notions of fibrations, homotopy (in the sense of Karoubi), path spaces, Puppe sequences, etc. One introduces the notion of a quasi-isomorphism (or weak equivalence) for rings and shows that - similar to spaces - the derived category obtained by inverting the quasi-isomorphisms is naturally left triangulated. Also, homology theories on rings are studied. These must be homotopy invariant in the algebraic sense, meet the Mayer-Vietoris property and plus some minor natural axioms. To any functor X from rings to pointed simplicial sets a homology theory is associated in a natural way. If X=GL and fibrations are the GL-fibrations, one recovers Karoubi-Villamayor's functors KV_i, i>0. If X is Quillen's K-theory functor and fibrations are the surjective homomorphisms, one recovers the (non-negative) homotopy K-theory in the sense of Weibel. Technical tools we use are the homotopy information for the category of simplicial functors on rings and the Bousfield localization theory for model categories.