Symmetric spectra

Hovey, Mark; Shipley, Brooke; Smith, Jessica M.

doi:10.1090/s0894-0347-99-00320-3

Cited by 477 publications

(576 citation statements)

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Section: P42; 55u35mentioning

confidence: 99%

“…We have chosen to let the spheres act from the right on the spaces in a symmetric spectrum, and not from the left as in [3]. The reason for this is that we find the M -action on the homotopy groups more transparent this way.…”

Section: Remarkmentioning

confidence: 99%

“…Around 1993, Jeff Smith made the crucial observation that the "FSPs on spheres" are the monoids in a category of "symmetric spectra" with respect to an associative and commutative smash product, and he suspected compatible model category structures so that one obtains as homotopy categories "the" stable homotopy category (for symmetric spectra), the homotopy category of A 1 ring spectra (for symmetric ring spectra), respectively the homotopy category of E 1 ring spectra (for commutative symmetric ring spectra). The details of various model structures were worked out by Hovey, Shipley and Smith in [3].Maybe the only tricky point with symmetric spectra is that the stable equivalences can not be defined by looking at stable homotopy groups (defined as the classical sequential colimit of the unstable homotopy groups of the terms in a symmetric spectrum). Formally inverting the -isomorphisms, ie, those morphisms which induce isomorphisms of stable homotopy groups, leaves too many homotopy types.…”

mentioning

confidence: 99%

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On the homotopy groups of symmetric spectra

Schwede

2008

Geom. Topol.

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We construct a natural, tame action of the monoid of injective self-maps of the set of natural numbers on the homotopy groups of a symmetric spectrum. This extra algebraic structure allows a conceptual and uniform understanding of various phenomena related to -isomorphisms, semistability and the relationship between naive and true homotopy groups for symmetric spectra. 55P42; 55U35Symmetric spectra are an easy-to-define and convenient model for the stable homotopy category with a nice smash product. Symmetric ring spectra first showed up under the name "FSP on spheres" in the context of algebraic K -theory and topological Hochschild homology. Around 1993, Jeff Smith made the crucial observation that the "FSPs on spheres" are the monoids in a category of "symmetric spectra" with respect to an associative and commutative smash product, and he suspected compatible model category structures so that one obtains as homotopy categories "the" stable homotopy category (for symmetric spectra), the homotopy category of A 1 ring spectra (for symmetric ring spectra), respectively the homotopy category of E 1 ring spectra (for commutative symmetric ring spectra). The details of various model structures were worked out by Hovey, Shipley and Smith in [3].Maybe the only tricky point with symmetric spectra is that the stable equivalences can not be defined by looking at stable homotopy groups (defined as the classical sequential colimit of the unstable homotopy groups of the terms in a symmetric spectrum). Formally inverting the -isomorphisms, ie, those morphisms which induce isomorphisms of stable homotopy groups, leaves too many homotopy types. Instead, Hovey, Shipley and Smith introduce a strictly larger class of stable equivalences, defined as the morphisms which induce isomorphisms on all cohomology theories. The difference between -isomorphisms and stable equivalences has previously confused at least the present author. The precise relationship between the naively defined homotopy groups and the "true" homotopy groups (morphisms from sphere in the stable homotopy category) has largely been mysterious (although Shipley's detection functor [7, Section 3] sheds considerable light on this).In this paper we advertise and systematically exploit extra algebraic structure on the (classical) homotopy groups of a symmetric spectrum which, in the authors opinion, clarifies several otherwise ad hoc observations and illuminates various mysterious points in the theory of symmetric spectra. This extra structure is an action of the monoid M of injective self-maps of the set of natural numbers. The M -modules that come up, however, have a special property which we call tameness; see Definition 1.4. Tameness has strong algebraic consequences and severely restricts the kinds of M -modules which can arise as homotopy groups of symmetric spectra.Here is a first example of the use of the M -action. An important class of symmetric spectra is formed by the semistable symmetric spectra. Within this class, stable equivalences coincide with -isomorp...

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Section: P42; 55u35mentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

mentioning

confidence: 99%

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On the homotopy groups of symmetric spectra

Schwede

2008

Geom. Topol.

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“…For the following corollary one can use any symmetric monoidal category of R-modules satisfying the conditions of Proposition 2.20 and condition (a) of Proposition 4.11. These conditions are easily verified in the standard cases such as those of [18,26,35]. Corollary 4.13.…”

Section: Algebras Over Operadsmentioning

confidence: 75%

Lifting homotopy T-algebra maps to strict maps

Johnson

Noel

2014

Advances in Mathematics

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The settings for homotopical algebra-categories such as simplicial groups, simplicial rings, A ∞ spaces, E ∞ ring spectra, etc.-are often equivalent to categories of algebras over some monad or triple T. In such cases, T is acting on a nice simplicial model category in such a way that T descends to a monad on the homotopy category and defines a category of homotopy T-algebras. In this setting there is a forgetful functor from the homotopy category of T-algebras to the category of homotopy T-algebras.Under suitable hypotheses we provide an obstruction theory, in the form of a Bousfield-Kan spectral sequence, for lifting a homotopy T-algebra map to a strict map of T-algebras. Once we have a map of T-algebras to serve as a basepoint, the spectral sequence computes the homotopy groups of the space of T-algebra maps and the edge homomorphism on π 0 is the aforementioned forgetful functor. We discuss a variety of settings in which the required hypotheses are satisfied, including monads arising from algebraic theories and operads. We also give sufficient conditions for the E 2 -term to be calculable in terms of Quillen cohomology groups.We provide worked examples in G-spaces, G-spectra, rational E ∞ algebras, and A ∞ algebras. Explicit calculations, connected to rational unstable homotopy theory, show that the forgetful functor from the homotopy category of E ∞ ring spectra to the category of H ∞ ring spectra is generally neither full nor faithful. We also apply a result of the second named author and Nick Kuhn to compute the homotopy type of the space E ∞ (Σ ∞ + Coker J, L K(2) R).

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“…We work in one of the modern symmetric monoidal categories of spectra constructed by Elmendorf-Kriz-Mandell-May [14], Hovey-Shipley-Smith [19] or Mandell-MaySchwede-Shipley [24], which we shall refer to as S -modules. The monoids (resp.…”

Section: S -Algebrasmentioning

confidence: 99%