Title: Brown representability for space-valued functors

Author(s): Boris Chorny

Author's e-mail address: Boris.Chorny@maths.anu.edu.au

Author's mailing address: Centre for Mathematics and its Applications,
Mathematical Sciences Institute, Australian National University,
Canberra, ACT 0200, Australia
AMS classification number: 55P60, 55P65, 18G55
Other useful information (such as arXive submission number): arXiv:0707.0904

Abstract: 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 weakly
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.