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nLab > Latest Changes: higher Segal space

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 12th 2012

    I am starting higher Segal space (while sitting in a talk by Mikhail Kapranov about them…)

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeJan 12th 2012

    There is a similar “recursive” idea in Kontsevich’s definition of (∞,n)(\infty,n)-categories which he used at the end of 1990s. Unfortunately, it has never been published. I heard one exposition of it, but did not keep much notes.

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 17th 2012

    Added the new paper - Higher Segal spaces I.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 18th 2012
    • (edited Dec 18th 2012)

    Eventually we need to add some warning. Apparently the Dycker-Kapranov-style Segal higher spaces are equivalent to (∞,1)(\infty,1)-operads, certainly not to (∞,2)(\infty,2)-categories or similar.

    [edit: I have added a warning to the entry.]

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 18th 2012

    This seems to provide a bridge between them and Lurie:

    In analogy to the situation for (∞,1)(\infty, 1)-categories, there are various models for the notion of an (∞,2)(\infty, 2)-category. To describe the bicategorical structures appearing in this work, we will use Segal fibrations. In fact, we will also use the dual notion of a coSegal fibration. These and other models for (∞,2)(\infty, 2)-categories, as well as their relations, are studied in detail in the comprehensive treatment [Lur09b]. (p. 163)

    In 9.3 they associate an (∞,2)(\infty, 2)-category to a 2-Segal space.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeDec 18th 2012
    • (edited Dec 18th 2012)

    Thanks.

    Now I see that the statement that DK 2-Segal spaces are ∞\infty-operads is essentially in the article, in 3.6.

    (We were supposed to discuss the DK-article in our seminar this semester, but somehow we didn’t get around to. Or not yet maybe.)

    • CommentRowNumber7.
    • CommentAuthorTim Campion
    • CommentTimeSep 13th 2018

    The nlab article higher Segal Space is very unclear – there are two completely different, unrelated types of objects that are sometimes referred to as “higher Segal spaces”. One is n-fold Segal Space, a model of (∞,n)(\infty,n)-categories. The other is the Dyckerhoff-Kapranov notion of a dd-Segal space, which models something like an (∞,1)(\infty,1)-category, but without uniqueness of composites (for d≥2d\geq 2), and higher associativity only in dimension dd and above (higher associativity conditions are governed by dd-dimensional polyhedra, related to dd-dimensional field theories).

    The article as it stands refers to both notions. The “Idea” section refers to nn-fold Segal spaces.

    But we already have a separate page for n-fold Segal spaces. So I’m pretty sure the intention was that this page refer to the Dyckrhoof-Kapranov notion, and not the nn-fold Segal space notion. So I think the article needs a major cleanup.

    Does anybody object to making higher Segal spaces discuss only the Dyckerhoff-Kapranov notion (except for adding some discussion of the difference)?

    • CommentRowNumber8.
    • CommentAuthorTim Campion
    • CommentTimeSep 13th 2018

    Removed misleading references to nn-fold Segal spaces and added a bit of material.

    diff, v14, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeSep 13th 2018

    Please feel invited to work on the entry and improve it, as you see the need.

    • CommentRowNumber10.
    • CommentAuthorzskoda
    • CommentTimeSep 22nd 2023
    • Tashi Walde, On the theory of higher Segal spaces, thesis, Brexen 2020 pdf

    diff, v20, current

    • CommentRowNumber11.
    • CommentAuthorzskoda
    • CommentTimeSep 22nd 2023
    • Matthew B Young, Relative 2-Segal spaces, Algebraic & Geometric Topology 18 (2018) 975–1039 doi

    We introduce a relative version of the 2–Segal simplicial spaces defined by Dyckerhoff and Kapranov, and Gálvez-Carrillo, Kock and Tonks. Examples of relative 2–Segal spaces include the categorified unoriented cyclic nerve, real pseudoholomorphic polygons in almost complex manifolds and the ℛ •\mathcal{R}_\bullet-construction from Grothendieck–Witt theory. We show that a relative 2–Segal space defines a categorical representation of the Hall algebra associated to the base 2–Segal space. In this way, after decategorification we recover a number of known constructions of Hall algebra representations. We also describe some higher categorical interpretations of relative 2–Segal spaces.

    diff, v20, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeSep 22nd 2023

    we have an entry for Matthew B. Young

    diff, v21, current

    • CommentRowNumber13.
    • CommentAuthorperezl.alonso
    • CommentTimeJul 22nd 2024

    pointer

    • Jonte Gödicke. An ∞\infty-Category of 2-Segal Spaces (2024). (arXiv:2407.13357).

    diff, v22, current

1 to 13 of 13

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