Model Categories of Diagram Spectra

Mandell, Michael A.; May, J. P.; Schwede, Stefan; Shipley, Brooke

doi:10.1112/s0024611501012692

Cited by 363 publications

(705 citation statements)

References 41 publications

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“…Some generalizations of symmetric spectra appear in [MMSS98a]. These many new symmetric monoidal categories of spectra, including S-modules and symmetric spectra, are shown to be equivalent in an appropriate sense in [MMSS98b] and [Sch98]. Another symmetric monoidal category of spectra sitting between the approaches of [EKMM97] and of this paper is developed in [DS].…”

Section: Introductionmentioning

confidence: 99%

Symmetric spectra

Hovey

Shipley

Smith³

1999

J. Amer. Math. Soc.

475

569

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The stable homotopy category, much studied by algebraic topologists, is a closed symmetric monoidal category. For many years, however, there has been no well-behaved closed symmetric monoidal category of spectra whose homotopy category is the stable homotopy category. In this paper, we present such a category of spectra; the category of symmetric spectra. Our method can be used more generally to invert a monoidal functor, up to homotopy, in a way that preserves monoidal structure. Symmetric spectra were discovered at about the same time as the category of S S -modules of Elmendorf, Kriz, Mandell, and May, a completely different symmetric monoidal category of spectra.

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Section: Introductionmentioning

confidence: 99%

Symmetric spectra

Hovey

Shipley

Smith³

1999

J. Amer. Math. Soc.

475

569

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“…Thanks to Proposition 3.4 and Theorem 3.6, for any strongly self-absorbing C * -algebra D, one might call K top Σ (D) an E ∞ -algebra object in symmetric spectra. Using Remark 0.14 of [44] (see also Theorem 1.4 of [24]) one can rectify this E ∞ -algebra structure 5.1. Semigroup C * -algebras coming from number theory.…”

Section: Strongly Self-absorbing Operadsmentioning

confidence: 99%

“…Theorem 3.6 for a more general formulation and also Example 3.7). These operadic structures up to coherent homotopy in symmetric spectra can be further rectified to strict ones (see, for instance, [44,24]). For the ∞-categorical counterparts of related results the readers may refer to [39].…”

Section: Introductionmentioning

confidence: 99%

Algebraic K-theory, K-regularity, and 𝕋 $\mathbb {T}$ -duality of 𝒪 ∞ $\mathcal {O}_{\infty }$ -stable C ∗-algebras

Mahanta

2015

Math Phys Anal Geom

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Abstract. We develop an algebraic formalism for topological T-duality. More precisely, we show that topological T-duality actually induces an isomorphism between noncommutative motives that in turn implements the well-known isomorphism between twisted K-theories (up to a shift). In order to establish this result we model topological K-theory by algebraic K-theory. We also construct an E ∞ -operad starting from any strongly self-absorbing C * -algebra D. Then we show that there is a functorial topological K-theory symmetric spectrum construction K top Σ (−) on the category of separable C * -algebras, such that K top Σ (D) is an algebra over this operad; moreover, K top Σ (A⊗D) is a module over this algebra. Along the way we obtain a new symmetric spectra valued functorial model for the (connective) topological K-theory of C * -algebras. We also show that O ∞ -stable C * -algebras are K-regular providing evidence for a conjecture of Rosenberg. We conclude with an explicit description of the algebraic K-theory of ax + b-semigroup C * -algebras coming from number theory and that of O ∞ -stabilized noncommutative tori. IntroductionWithin the category of separable C * -algebras SC * the stable ones, i.e., those A ∈ SC * satisfying A⊗K ∼ = A, play a privileged role. For instance, it is known that they satisfy the Karoubi conjecture and appear very naturally in the context of twisted K-theory. The results in this article demonstrate that O ∞ -stable separable C * -algebras, i.e., those A ∈ SC * satisfying A⊗O ∞ ∼ = A, deserve a similar prominent status. Moreover, the Cuntz algebra O ∞ is strongly self-absorbing, which has several interesting ramifications.Ever since its inception by Kontsevich [33] the homological mirror symmetry conjecture has promoted rich interaction between geometry, algebra, and (higher) category theory. Mirror symmetry is related to T-duality via the Strominger-Yau-Zaslow conjecture [61] and hence we believe that it is worthwhile to have an algebraic formalism for T-duality at our disposal. One aspect of this theory is the Bunke-Schick topological T-duality, which has received a lot of attention in the mathematical literature. It is insensitive to subtle geometric structures but its mathematical underpinnings are very well understood [8,7,9]. One of the objectives of this article is to develop an algebraic formalism for topological T-duality relating it to the theory of noncommutative motives [34,35,65,45]. Along the way we obtain several interesting applications to algebraic K-theory and K-regularity of C * -algebras. The novelty 2010 Mathematics Subject Classification. 46L85, 19Dxx, 46L80, 57R56.

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“…For additional background concerning the material in this appendix, see [1], [6], [30], [29], [4], [34], [32], [5], [23] and, especially, [24].…”

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confidence: 99%

Semi-Topological K-Homology and Thomason's Theorem

Walker¹

2002

K-Theory

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Abstract. In this paper, we introduce the "semi-topological K-homology" of complex varieties, a theory related to semi-topological K-theory much as connective topological K-homology is related to connective topological K-theory. Our main theorem is that the semi-topological K-homology of a smooth, quasiprojective complex variety Y coincides with the connective topological Khomology of the associated analytic space Y an . From this result, we deduce a pair of results relating semi-topological K-theory with connective topological K-theory. In particular, we prove that the "Bott inverted" semi-topological K-theory of a smooth, projective complex variety X coincides with the topological K-theory of X an . In combination with a result of Friedlander and the author [12, 3.8], this gives a new proof, in the special case of smooth, projective complex varieties, of Thomason's celebrated theorem that "Bott inverted" algebraic K-theory with Z/n coefficients coincides with topological K-theory with Z/n coefficients.

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Model Categories of Diagram Spectra

Cited by 363 publications

References 41 publications

Symmetric spectra

Symmetric spectra

Algebraic K-theory, K-regularity, and 𝕋 $\mathbb {T}$ -duality of 𝒪 ∞ $\mathcal {O}_{\infty }$ -stable C ∗-algebras

Semi-Topological K-Homology and Thomason's Theorem

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