Duality and small functors

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 $D:Sp\to {Sp}^{op}$ 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 ${Sp}^{op}$. In this new framework for the Spanier–Whitehead duality, $Sp$ and ${Sp}^{op}$ are full subcategories of the category of small functors and dualization becomes just a fibrant replacement in our new model structure.