Brown representability for space-valued functors

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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 contravariant functors from spaces to spaces up to weak equivalence of functors. In more detail, we show that every 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 [14].

Original languageEnglish
Pages (from-to)767-791
Number of pages25
JournalIsrael Journal of Mathematics
Issue number2
StatePublished - Mar 2013

ASJC Scopus subject areas

  • General Mathematics


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