Algebraic & Geometric Topology

Duality and small functors

Georg Biedermann and Boris Chorny

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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 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.

Article information

Source
Algebr. Geom. Topol., Volume 15, Number 5 (2015), 2609-2657.

Dates
Received: 10 September 2013
Revised: 11 December 2014
Accepted: 15 December 2014
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841027

Digital Object Identifier
doi:10.2140/agt.2015.15.2609

Mathematical Reviews number (MathSciNet)
MR3426688

Zentralblatt MATH identifier
1331.18022

Subjects
Primary: 55P25: Spanier-Whitehead duality
Secondary: 18G55: Homotopical algebra 18A25: Functor categories, comma categories

Keywords
small functors duality

Citation

Biedermann, Georg; Chorny, Boris. Duality and small functors. Algebr. Geom. Topol. 15 (2015), no. 5, 2609--2657. doi:10.2140/agt.2015.15.2609. https://projecteuclid.org/euclid.agt/1510841027


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References

  • J,F Adams, A variant of E,H Brown's representability theorem, Topology 10 (1971) 185–198
  • G Biedermann, B Chorny, O R öndigs, Calculus of functors and model categories, Adv. Math. 214 (2007) 92–115
  • F Borceux, Handbook of categorical algebra, 2: Categories and structures, Encycl. Math. Appl. 51, Cambridge Univ. Press (1994)
  • A,K Bousfield, On the telescopic homotopy theory of spaces, Trans. Amer. Math. Soc. 353 (2001) 2391–2426
  • A,K Bousfield, E,M Friedlander, Homotopy theory of $\Gamma $–spaces, spectra, and bisimplicial sets, from: “Geometric applications of homotopy theory, II”, Lecture Notes in Math. 658, Springer, Berlin (1978) 80–130
  • E,H Brown, Jr, Cohomology theories, Ann. of Math. 75 (1962) 467–484
  • B Chorny, A classification of small homotopy functors from spectra to spectra
  • B Chorny, A generalization of Quillen's small object argument, J. Pure Appl. Algebra 204 (2006) 568–583
  • B Chorny, Brown representability for space-valued functors, Israel J. Math. 194 (2013) 767–791
  • B Chorny, W,G Dwyer, Homotopy theory of small diagrams over large categories, Forum Math. 21 (2009) 167–179
  • B Chorny, J Rosický, Class-combinatorial model categories, Homology Homotopy Appl. 14 (2012) 263–280
  • B Chorny, J Rosický, Class-locally presentable and class-accessible categories, J. Pure Appl. Algebra 216 (2012) 2113–2125
  • J,D Christensen, D,C Isaksen, Duality and pro-spectra, Algebr. Geom. Topol. 4 (2004) 781–812
  • B Day, On closed categories of functors, from: “Reports of the Midwest Category Seminar, IV”, Lecture Notes in Mathematics 137, Springer, Berlin (1970) 1–38
  • B,J Day, S Lack, Limits of small functors, J. Pure Appl. Algebra 210 (2007) 651–663
  • A Dold, D Puppe, Duality, trace, and transfer, from: “Proceedings of the International Conference on Geometric Topology”, (K Borsuk, editor), PWN, Warsaw (1980) 81–102
  • D Dugger, Combinatorial model categories have presentations, Adv. Math. 164 (2001) 177–201
  • B,I Dundas, O R öndigs, P,A Østvær, Enriched functors and stable homotopy theory, Doc. Math. 8 (2003) 409–488
  • A,D Elmendorf, I Kriz, M,A Mandell, J,P May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs 47, Amer. Math. Soc. (1997)
  • P,J Freyd, Abelian categories, Harper and Row, New York (1964)
  • P Freyd, Several new concepts: Lucid and concordant functors, pre-limits, pre-completeness, the continuous and concordant completions of categories, from: “Category Theory, Homology Theory and their Applications, III”, Springer, Berlin (1969) 196–241
  • T,G Goodwillie, Calculus, III: Taylor series, Geom. Topol. 7 (2003) 645–711
  • P,S Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, Amer. Math. Soc. (2003)
  • M Hovey, B Shipley, J Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149–208
  • J,F Jardine, Representability theorems for presheaves of spectra, J. Pure Appl. Algebra 215 (2011) 77–88
  • G,M Kelly, Basic concepts of enriched category theory, London Math. Soc. Lecture Notes 64, Cambridge Univ. Press (1982)
  • J Lurie, Higher topos theory, Annals of Mathematics Studies 170, Princeton Univ. Press (2009)
  • M Lydakis, Simplicial functors and stable homotopy theory, preprint (1998) available via the Hopf archive
  • M Makkai, R Paré, Accessible categories: The foundations of categorical model theory, Contemporary Mathematics 104, Amer. Math. Soc. (1989)
  • A Neeman, Brown representability for the dual, Invent. Math. 133 (1998) 97–105
  • S Schwede, B,E Shipley, Algebras and modules in monoidal model categories, Proc. London Math. Soc. 80 (2000) 491–511
  • W,C Waterhouse, Basically bounded functors and flat sheaves, Pacific J. Math. 57 (1975) 597–610
  • G,W Whitehead, Generalized homology theories, Trans. Amer. Math. Soc. 102 (1962) 227–283