A uniqueness theorem for stable homotopy theory

By S. Schwede, B. Shipley. 2000

64 citations recorded by the source corpus.

In this paper we study the global structure of the stable homotopy theory of spectra. We establish criteria for when the homotopy theory associated to a given stable model category agrees with the classical stable homotopy theory of spectra. One sufficient condition is that the associated homotopy category is equivalent to the stable homotopy category as a triangulated category with an action of the ring of stable homotopy groups of spheres. In other words, the classical stable homotopy theory, with all of its higher order information, is determined by the homotopy category as a triangulated category with an action of the stable homotopy groups of spheres. Another sufficient condition is the existence of a small generating object (corresponding to the sphere spectrum) for which a specific `unit map' from the infinite loop space QS^0 to the endomorphism space is a weak equivalence.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f87c3b1470cbf98eae0e04fbd3b814d3e3709b4c

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Citations in the corpus

Stable model categories are categories of modules (2003, background): “Another result which is very closely related to this characterization of stable model categories can be found in [57] where we give necessary and sufficient conditions for when a stable model category is Quillen equivalent to spectra, see also Example 3.2 (i).”
Triangulated categories without models (2007, background): “2]. To construct F we let X be a cofibrant-fibrant object of the model category M which is isomorphic to A in the homotopy category Ho M. The universal property of the model category of spectra [ SS02 , Thm.”
Brauer groups and étale cohomology in derived algebraic geometry (2012, methodology): “One direction is the theorem of Schwede and Shipley, in the form f und in Lurie [Lur12b, Theorem 8.1.2.1].”
Spectra and symmetric spectra in general model categories (2000, background): “See [ 23 ] for further results along these lines.”
What precisely are E∞ring spaces and E∞ring spectra? (2009, background): “Later work of Schwede and Shipley [55, 56, 59] gives ΣS a privileged role.”
The Stable Homotopy Category Has a Unique Model at the Prime 2 (2001, methodology): “Hence the result of the present paper is a strengthening of the Uniqueness Theorem of [SSa], at least 2-locally.”
Morita theory in abelian, derived and stable model categories (2003, methodology): “ a small generator P for which the unit map of ring spectra S −→End(P) is a stable equivalence. The technical conditions of being a simplicial, coﬁbrantly generated and proper can be eliminated as in [SS02] with the use of framings [Hov99, Chpt. 5]. The paper [SS02] also gives other necessary and su”