Ambidexterity and Height

Carmeli, Shachar; Schlank, Tomer M.; Yanovski, Lior

doi:10.48550/arxiv.2007.13089

Cited by 4 publications

(8 citation statements)

References 9 publications

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“…When n = 0, we recover the classical formula of Definition 3.4, for the case m = p r (see also [CSY20,Example 4.3.3]).…”

Section: Definition and Propertiessupporting

confidence: 60%

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Chromatic Cyclotomic Extensions

Carmeli¹,

Schlank²,

Yanovski³

2021

Preprint

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We construct Galois extensions of the T (n)-local sphere, lifting all finite abelian Galois extensions of the K(n)-local sphere. This is achieved by realizing them as higher semiadditive analogues of cyclotomic extensions. Combining this with a general form of Kummer theory, we lift certain elements from the K(n)-local Picard group to the T (n)-local Picard group.Color circles from the enlarged 1708 edition of the Treatise on Miniature Painting, Claude Boutet.

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“…When n = 0, we recover the classical formula of Definition 3.4, for the case m = p r (see also [CSY20,Example 4.3.3]).…”

Section: Definition and Propertiessupporting

confidence: 60%

“…quasi-categories), and in general follow the notation of [Lur09] and [Lur]. The terminology and notation for all concepts related to higher semiadditivity and (semiadditive) height are as in [CSY20]. In addition,…”

Section: Conventionsmentioning

confidence: 99%

Chromatic Cyclotomic Extensions

Carmeli¹,

Schlank²,

Yanovski³

2021

Preprint

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“…In particular, if C is p-typically m-semiadditive, we can restrict |A| to a natural transformation of the identity functor of C ≃ CMon (p) m (C ). These natural transformations are studied systematically in [CSY20] -in particular, one can show (see [CSY20, Remark 2.1.10(2)]) that if C is further symmetric monoidal and its tensor product distributes over S (p) m -colimits, then |A|, as a natural transformation of the identity functor of C , is given by multiplication with the element…”

Section: Cardinalitiesmentioning

confidence: 99%

“…is hence again a morphism of commutative monoids, where the commutative monoid structure on Ω ∞ S K(n) is given by multiplication. Since Sp K(n) is an ∞-category of semiadditive height n (in the sense of [CSY20]), the cardinality |B n V | is invertible for every V ∈ Vect Fp (cf. [CSY20, Theorem 4.4.5]) and so (8) determines a map of spectra…”

Section: α and Symmetric Bilinear Formsmentioning

confidence: 99%

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Higher semiadditive Grothendieck-Witt theory and the $K(1)$-local sphere

Carmeli¹,

Yuan²

2021

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We develop a higher semiadditive version of Grothendieck-Witt theory. We then apply the theory in the case of a finite field to study the higher semiadditive structure of the K(1)local sphere at the prime 2. As a further application, we compute and clarify certain power operations in the homotopy of the K(1)-local sphere.Here, the first map is (the inverse of) the equivalence resulting from the higher semiadditive structure, and the second map is induced by the map A → pt.Remark 1.0.2. The collection of these ∫ A maps, together with corresponding restriction maps, endow X with a structure known as an m-commutative monoid (with 0-commutative monoids being just ordinary commutative monoids). First described systematically by Harpaz [Har20], this m-commutative monoid structure for X is captured by a functor X (−) : Span(S m ) op → Sp given by A → X A . Here, Span(S m ) is the ∞-category of m-truncated π-finite spaces and spans. In fact, we will work in a p-typical setting, where one further restricts to m-truncated π-finite spaces with homotopy groups of p-power order. We refer the reader to Section 2 for a more careful discussion of (p-typical) m-commutative monoids. Semiadditive cardinalityThis paper is focused on one part of this m-commutative monoid structure: Definition 1.1.1 ([CSY20] Definition 2.1.5, [HL13] Notation 5.1.7). Let C be an m-semiadditive ∞-category, let X ∈ C and let A be an m-truncated π-finite space. Then let |A| X : X → X be the endomorphism of X defined by the compositewhere ∆ A is the canonical diagonal map induced by A → pt. As X varies, these maps assemble to a natural transformation |A| : Id C → Id C , which we will refer to as the cardinality of A.This terminology is justified by the following example:Example 1.1.2. In the special case where A is discrete, the endomorphism |A| is given by multiplication by the cardinality of the finite set A.

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Noncommutative CW-spectra as enriched presheaves on matrix algebras

Arone¹,

Barnea²,

Schlank³

2021

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Motivated by the philosophy that C * -algebras reflect noncommutative topology, we investigate the stable homotopy theory of the (opposite) category of C * -algebras. We focus on C * -algebras which are non-commutative CW-complexes in the sense of [ELP]. We construct the stable ∞-category of noncommutative CW-spectra, which we denote by NSp. Let M be the full spectral subcategory of NSp spanned by "noncommutative suspension spectra" of matrix algebras. Our main result is that NSp is equivalent to the ∞-category of spectral presheaves on M.To prove this we first prove a general result which states that any compactly generated stable ∞-category is naturally equivalent to the ∞category of spectral presheaves on a full spectral subcategory spanned by a set of compact generators. This is an ∞-categorical version of a result by Schwede and Shipley [ScSh1]. In proving this we use the language of enriched ∞-categories as developed by Hinich [Hin2,Hin3].We end by presenting a "strict" model for M. That is, we define a category Ms strictly enriched in a certain monoidal model category of spectra Sp M . We give a direct proof that the category of Sp M -enriched presheaves M op s → Sp M with the projective model structure models NSp and conclude that Ms is a strict model for M.

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Ambidexterity and Height

Cited by 4 publications

References 9 publications

Chromatic Cyclotomic Extensions

Chromatic Cyclotomic Extensions

Higher semiadditive Grothendieck-Witt theory and the $K(1)$-local sphere

Noncommutative CW-spectra as enriched presheaves on matrix algebras

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