K-theory for 2-categories

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

doi:10.1016/j.aim.2017.10.011

Cited by 12 publications

(32 citation statements)

References 32 publications

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“…Despite considerable emphasis by different investigators about the role of temporally overlapping contacts, we still lack a general definition of concurrency in that slightly different definitions have been used across these studies. For example, Gursky and Hoffman [24] assumed concurrency based on the lifetime of an edge, following the definition in [25].…”

Section: Previous Studies On Concurrency and Reachabilitymentioning

confidence: 99%

Exploring Concurrency and Reachability in the Presence of High Temporal Resolution

Lee¹,

Moody

Mucha³

2019

Computational Social Sciences

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Network properties govern the rate and extent of spreading processes on networks, from simple contagions to complex cascades. Recent advances have extended the study of spreading processes from static networks to temporal networks, where nodes and links appear and disappear. We review previous studies on the effects of temporal connectivity for understanding the spreading rate and outbreak size of model infection processes. We focus on the effects of accessibility, whether there is a temporally consistent path from one node to another, and reachability, the density of the corresponding accessibility graph representation of the temporal network. We study reachability in terms of the overall level of temporal concurrency between edges, quantifying the overlap of edges in time. We explore the role of temporal resolution of contacts by calculating reachability with the full temporal information as well as with a simplified interval representation approximation that demands less computation. We demonstrate the extent to which the computed reachability changes due to this simplified interval representation.

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Section: Previous Studies On Concurrency and Reachabilitymentioning

confidence: 99%

Exploring Concurrency and Reachability in the Presence of High Temporal Resolution

Lee¹,

Moody

Mucha³

2019

Computational Social Sciences

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“…Note that this is a stronger result than what we were able to achieve using direct methods in [GJO17], namely it strengthens Theorem 4.37 and Corollary 4.47 of loc. cit.…”

Section: Proposition 225 Let K Be a Complete And Cocomplete 2-categmentioning

confidence: 69%

“…Let I be a skeleton of the category of finite based sets considered as a discrete 2-category and let K be either Cat or 2Cat 2 , the 2-category of 2-categories, 2-functors, and 2-natural transformations. Then [I, Cat ] red is the 2-category of Γ-categories, Γ-functors, and Γ-transformations [Seg74], while [I, 2Cat 2 ] red is the 2-category of Γ-2-categories, Γ-2-functors, and Γ-transformations studied in [GJO17].…”

Section: Proposition 225 Let K Be a Complete And Cocomplete 2-categmentioning

confidence: 99%

Extending homotopy theories across adjunctions

Gurski

Johnson

Osorno

2017

Homology, Homotopy and Applications

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ABSTRACT. Constructions of spectra from symmetric monoidal categories are typically functorial with respect to strict structure-preserving maps, but often the maps of interest are merely lax monoidal. We describe conditions under which one can transport the weak equivalences from one category to another with the same objects and a broader class of maps. Under mild hypotheses this process produces an equivalence of homotopy theories. We describe examples including algebras over an operad, such as symmetric monoidal categories and n-fold monoidal categories; and diagram categories, such as Γ-categories.

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“…Our proof of the 2-dimensional stable homotopy hypothesis is a culmination of previous work in [GJO17] and [GJOS17]. Although we have attempted to make the current account as self-contained as possible, we rely heavily on this and other previous work.…”

Section: Stable Homotopy Hypothesis There Is An Equivalence Of Homotmentioning

confidence: 99%

“…In both SMBicat s and SM2Cat s , we consider strictly functorial morphisms which commute strictly with this multiplication pseudofunctor. The work in [GJO17] shows that, relative to all stable equivalences, it is possible to restrict the structure further and still represent every stable homotopy type. Relative only to the categorical equivalences, however, we must retain some pseudofunctoriality in the multiplication.…”

Section: Is a Symmetric Monoidal Equivalence If And Only If It Is An mentioning

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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K-theory for 2-categories

Cited by 12 publications

References 32 publications

Exploring Concurrency and Reachability in the Presence of High Temporal Resolution

Exploring Concurrency and Reachability in the Presence of High Temporal Resolution

Extending homotopy theories across adjunctions

The 2-dimensional stable homotopy hypothesis

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