Brown representability for space-valued functors

In this paper we prove two theorems which resemble the classical cohomological and homological Brown representability theorems. The main difference is that our results classify small contravariant functors from spaces to spaces up to weak equivalence of functors. In more detail, we show that every small contravariant functor from spaces to spaces which takes coproducts to products up to homotopy and takes homotopy pushouts to homotopy pullbacks is naturally weekly equivalent to a representable functor. The second representability theorem states: every contravariant continuous functor from the category of finite simplicial sets to simplicial sets taking homotopy pushouts to homotopy pullbacks is equivalent to the restriction of a representable functor. This theorem may be considered as a contravariant analog of Goodwillie's classification of linear functors.Comment: 19 pages, final version, accepted by the Israel Journal of Mathematic

Fibrations and nullifications

Israel Journal of Mathematics135205-22

Cellular Spaces

this paper we will use crucially several properties of L f , the localization functor with respect to a general map f : A \Gamma! D. Most of the time we will consider f : A \Gamma

Homotopy Localization Nearly Preserves Fibrations

this paper. We then prove Theorem A in the second section and (C),(D) and (E) in sections three and four. The last section concludes with the proof of (B). We work in the category of pointed CW-complexes and in particular all function complexes are spaces of pointed maps