A classification of small homotopy functors from spectra to spectra

TL;DR

This paper classifies small homotopy functors from spectra to spectra, showing they are equivalent to colimits of representable functors and establishing a Quillen equivalence between relevant categories.

Contribution

It provides a new classification of small homotopy functors via a Quillen equivalence, connecting functor categories and pro-categories of spectra.

Findings

Small homotopy functors are equivalent to colimits of representable functors.

Established a Quillen equivalence between functor and pro-category models.

Clarified the structure of small homotopy functors in spectral categories.

Abstract

We show that every small homotopy functor from spectra to spectra is weakly equivalent to a filtered colimit of representable functors represented in cofibrant spectra. Moreover, we present this classification as a Quillen equivalence of the category of small functors from spectra to spectra equipped with the homotopy model structure and the opposite of the pro-category of spectra with the strict model structure.