  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.
