Model Categories of Diagram Spectra

By Michael A. Mandell, Jon P. May, S. Schwede, B. Shipley. Proceedings of the London Mathematical Society, 2001

499 citations recorded by the source corpus.

Working in the category T of based spaces, we give the basic theory of diagram spaces and diagram spectra. These are functorsD→T for a suitable small topological categoryD. WhenD is symmetric monoidal, there is a smash product that gives the category of D‐spaces a symmetric monoidal structure. Examples include prespectra, as defined classically, symmetric spectra, as defined by Jeff Smith, orthogonal spectra, a coordinate‐free analogue of symmetric spectra with symmetric groups replaced by orthogonal groups in the domain category, Γ‐spaces, as defined by Graeme Segal, W‐spaces, an analogue of Γ‐spaces with finite sets replaced by finite CW complexes in the domain category. We construct and compare model structures on these categories. With the caveat that Γ‐spaces are always connective, these categories, and their simplicial analogues, are Quillen equivalent and their associated homotopy categories are equivalent to the classical stable homotopy category. Monoids in these categories are (strict) ring spectra. Often the subcategories of ring spectra, module spectra over a ring spectrum, and commutative ring spectra are also model categories. When this holds, the respective categories of ring and module spectra are Quillen equivalent and thus have equivalent homotopy categories. This allows interchangeable use of these categories in applications.

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

View Paper: https://www.semanticscholar.org/paper/bd12b2c32bef30fa7475d511b18526142b33d3a0

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Citations in the corpus

Parametrized spectra, multiplicative Thom spectra and the twisted Umkehr map (2011, methodology): “able model structure on the category SB of orthogonal spectra in (Top/B)∗ [44, 12.3.10]. This model structure is based on the qf-model structure on ex-spaces, leveraging the diagrammatic viewpoint of [42, 41]. Similarly, they construct a stable model structure on the category PB of prespectra in (To”
The Local Structure of Algebraic K-Theory (2012, background): “Since then many variants have been introduced (most notably, orthogonal spectra) and there has been some reconciliation between the different setups (see in particular [196]).”
Traces in symmetric monoidal categories (2011, background): “(Actually, there are many such categories Sp, all equivalent up to homotopy, but each having different technical advantages and disadvantages; see for instance [13, 28, 29].”
Topological Hochschild homology of Thom spectra and the free loop space (2008, methodology): “Just as there is an operadic approach to a symmetric monoidal category of spectra given by [16] and a “diagrammatic” approach given by (for example) symmetric spectra [18] [28], we have operadic and diagrammatic approaches to producing A.”
Hermitian K-theory, derived equivalences and Karoubi's fundamental theorem (2012, methodology): “Our main reference for symmetric spectra of topological spaces is [MMSS01], [Sch].”
Spectra and symmetric spectra in general model categories (2000, background): “I have mentioned [15] above, but perhaps the most ambitious attempt to generalize S-modules has been initiated by Mark Johnson [13].”
Dwyer–Kan homotopy theory of enriched categories (2012, background): “However, they are still somewhat restrictive, they do not include other examples of interest such as diagram spectra with the positive stable model structure [20] or, more generally, the underlying model category V of a homotopical algebra context in the sense of Toën and Vezzosi [32].”