Brown representability for space-valued functors

TL;DR

This paper proves two theorems classifying certain space-valued contravariant functors up to weak equivalence, extending Brown representability to a homotopical setting for functors from spaces to spaces.

Contribution

It introduces two new Brown representability theorems for space-valued functors, generalizing classical results to a homotopical and contravariant context.

Findings

Classifies small contravariant functors from spaces to spaces up to weak equivalence.

Shows such functors are naturally weakly equivalent to representable functors under certain conditions.

Establishes a contravariant analog of Goodwillie's classification of linear functors.

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 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.