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Duality and small functors

Georg Biedermann and Boris Chorny

Algebraic & Geometric Topology 15 (2015) 2609–2657

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 Spop 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 Spop. In this new framework for the Spanier–Whitehead duality, Sp and Spop are full subcategories of the category of small functors and dualization becomes just a fibrant replacement in our new model structure.

small functors, duality
Mathematical Subject Classification 2010
Primary: 55P25
Secondary: 18G55, 18A25
Received: 10 September 2013
Revised: 11 December 2014
Accepted: 15 December 2014
Published: 10 December 2015
Georg Biedermann
LAGA – Institut Galilée
Université Paris 13
99 avenue Jean Baptiste Clément
93430 Villetaneuse
Boris Chorny
Department of Mathematics, Physics and Computer Science
University of Haifa at Oranim
36006 Tivon