Duality and small functors

TL;DR

This paper develops a homotopy theory framework for small functors from spectra to spectra, linking it with Spanier-Whitehead duality and creating a new model structure that simplifies dualization as fibrant replacement.

Contribution

It introduces a new model structure on small functors from spectra to spectra, enabling a unified approach to duality and enriched representability in homotopy theory.

Findings

Constructed a new model structure on small functors

Showed Spanier-Whitehead duality factors through small functors

Dualization corresponds to fibrant replacement in the new model structure

Abstract

The homotopy theory of small functors is a useful tool for studying various questions in homotopy theory. In this paper, we develop the homotopy theory of small functors from spectra to spectra, and study its interplay with Spanier-Whitehead duality and enriched representability in the dual category of spectra. We note that the Spanier-Whitehead duality functor $D\colon \mathrm{Sp}\rightarrow \mathrm{Sp}^{\mathrm{op}}$ factors through the category of small functors from spectra to spectra and construct a new model structure on the category of small functors, which is Quillen equivalent to $\mathrm{Sp}^{\mathrm{op}}$. In this new framework for the Spanier-Whitehead duality, $\mathrm{Sp}$ and $\mathrm{Sp}^{\mathrm{op}}$ are full subcategories of the category of small functors and dualization becomes just a fibrant replacement in our new model structure.