A classification of small linear functors

TL;DR

This paper extends Goodwillie's classification of finitary linear functors to all small functors, showing they are equivalent to filtered colimits of representables, and establishes a Quillen equivalence between related categories.

Contribution

It generalizes the classification of linear functors from finitary to arbitrary small functors and formulates this as a Quillen equivalence between model categories.

Findings

Small linear functors are weakly equivalent to filtered colimits of representable functors.

Establishes a Quillen equivalence between categories of small functors and pro-categories of spectra.

Provides a unified framework for classifying linear functors in homotopy theory.

Abstract

We extend Goodwillie's classification of finitary linear functors to arbitrary small functors. That is we show that every small linear simplicial functor from spectra to simplicial sets 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 simplicial sets equipped with the linear model structure and the opposite of the pro-category of spectra with the strict model structure.