Stable Postnikov data of Picard 2–categories

Gurski, Nick; Johnson, Niles; Osorno, Angélica M.; Stephan, Marc

doi:10.2140/agt.2017.17.2763

Cited by 7 publications

(8 citation statements)

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“…In this section we recall from [GJO17, GJO19] the notion of permutative Gray monoid, a semi-strict type of symmetric monoidal bicategory. We refer the reader to [Gur13] for further background on the Gray tensor product and to [GJO17,GJOS17] for further background on permutative Gray monoids. Just as permutative categories provide a strict model for symmetric monoidal categories, permutative Gray monoids provide a strict model for symmetric monoidal bicategories.…”

Section: Construction Of S −1 Smentioning

confidence: 99%

2-categorical opfibrations, Quillen's Theorem B, and $S^{-1}S$

Gurski¹,

Johnson²,

Osorno³

2020

Preprint

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In this paper we show that the strict and lax pullbacks of a 2-categorical opfibration along an arbitrary 2-functor are homotopy equivalent. We give two applications. First, we show that the strict fibers of an opfibration model the homotopy fibers. This is a version of Quillen's Theorem B amenable to applications. Second, we compute the E 2 page of a homology spectral sequence associated to an opfibration and apply this machinery to a 2-categorical construction of S −1 S. We show that if S is a symmetric monoidal 2-groupoid with faithful translations then S −1 S models the group completion of S.

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Section: Construction Of S −1 Smentioning

confidence: 99%

2-categorical opfibrations, Quillen's Theorem B, and $S^{-1}S$

Gurski¹,

Johnson²,

Osorno³

2020

Preprint

Self Cite

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“…The proof follows from putting together several results in this section. To be precise, we combine Propositions 6.2 and 6.4 below, which follow easily from previous work in [GJO17,GJOS17], with Theorem 6.5, whose proof depends on the content of Sections 3 through 5. Proposition 6.2.…”

Section: Proof Of the 2-dimensional Stable Homotopy Hypothesismentioning

confidence: 99%

The 2-dimensional stable homotopy hypothesis

Gurski

Johnson

Osorno

2019

Journal of Pure and Applied Algebra

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We prove that the homotopy theory of Picard 2-categories is equivalent to that of stable 2-types.2.1. Topological spaces and nerves of 2-categories. Notation 2.1. For spaces, we work in the category of compactly-generated weak Hausdorff spaces and denote this category Top. Notation 2.2. We let sSet denote the category of simplicial sets. Notation 2.3. We let | − | and S denote, respectively, the geometric realization and singular functors between simplicial sets and topological spaces. Notation 2.4. We let Cat denote the category of categories and functors, and let 2Cat denote the category of 2-categories and 2-functors. Note that these are both 1-categories.The category of 2-categories admits a number of morphism variants, and it will be useful for us to have separate notations for these. Notation 2.5. We let 2Cat ps denote the category of 2-categories with pseudofunctors and let 2Cat nps denote the category of 2-categories with normal pseudofunctors, that is, pseudofunctors which preserve the identities strictly. We let 2Cat nop denote the category of 2-categories and normal oplax functors. Note that these are all 1-categories.The well-known nerve construction extends to 2-categories (in fact to general bicategories) in a number of different but equivalent ways [Gur09, CCG10].Notation 2.6. We let N denote the nerve functor from categories to simplicial sets. By abuse of notation, we also let N denote the 2-dimensional nerve on 2Cat ps . This nerve has 2-simplices given by 2-cells whose target is a composite of two 1-cells, as in the

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“…A proof of the stable homotopy hypothesis in dimension 2 appears in recent work of the authors [GJO19]. This uses a notion of Picard 2-category defined in [GJO17,GJOS17] that is fully algebraic and yet general enough to realize all stable 2-types.…”

Section: Slogan 23 Definitions Of Picard -Category For Which the Stamentioning

confidence: 99%

Topological Invariants from Higher Categories

Gurski¹,

Osorno²,

Johnson³

2019

Notices Amer. Math. Soc.

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While invariants of geometric or topological objects can be of any sort, most modern invariants are algebraic in nature. The goal of this article is to survey some of the ways in which categorical algebra can be utilized to measure topological phenomena. This area of mathematics is a relatively recent outcrop on a vast landscape of interactions between algebra and geometry. The critical points of a differentiable function or the Betti numbers and Euler characteristic of a topological space are two among many points of interest on this landscape, and the field of algebraic topology has its origins nearby. Venturing further, Betti numbers were recognized by Noether and Vietoris as the avatars of homology groups, a perspective that was not immediately adopted but whose advantages are now clear. Our route begins here and leads toward higher-categorical algebra and the topological applications thereof. Just as the theory of groups provides an abstraction and formalism for mathematical structures observed in many examples (e.g., symmetry or additivity), so the theory of

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Stable Postnikov data of Picard 2–categories

Cited by 7 publications

References 50 publications

2-categorical opfibrations, Quillen's Theorem B, and $S^{-1}S$

2-categorical opfibrations, Quillen's Theorem B, and $S^{-1}S$

The 2-dimensional stable homotopy hypothesis

Topological Invariants from Higher Categories

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