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nLab > Latest Changes: straightening functor

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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeNov 3rd 2021

    Added:

    Related concepts

    References

    The original result is due to Lurie:

    A considerably simplified presentation is available in

    diff, v4, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 4th 2021

    I have added an “Idea”-section header and one sentence to the very top, in the spirit of starting out with a single sentence or two that expresses the whole idea, before going into more details.

    With this, the line “This is a subentry of…” could now be dropped, I think. But I leave that to you.

    diff, v5, current

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 4th 2021

    Probably being dim, but what is CC in

    (St ϕ⊣Un ϕ):sSet/S→St[C op,sSet] (St_\phi\dashv Un_\phi) \;\colon\; sSet/S \stackrel{St}{\to} [C^{op}, sSet]
    • CommentRowNumber4.
    • CommentAuthorHurkyl
    • CommentTimeNov 4th 2021

    I’ve added in what CC and ϕ\phi are.

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 4th 2021

    Thanks. So ϕ\phi is the simplicial functor ℭ[S]→C\mathfrak{C}[S] \to C?

    Dare I ask further what operation ℭ\mathfrak{C} is?

    • CommentRowNumber6.
    • CommentAuthorHurkyl
    • CommentTimeNov 4th 2021
    • (edited Nov 4th 2021)

    It’s the left part of the Quillen equivalence between the model structures on quasicategories and simplicially enriched categories. I don’t know what page mentions it off hand. (too distracted to do a search atm)

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeNov 5th 2021

    Yes, we have a page for this (which would deserve improvement, as always): relation between quasi-categories and simplicial categories.

    I am adding a pointer to this entry here.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeAug 19th 2022

    added more references:

    diff, v9, current

    • CommentRowNumber9.
    • CommentAuthorvarkor
    • CommentTimeSep 19th 2022

    The page contains a justification for the “straightening” terminology:

    These names have been chosen due to the fact that objects in the left hand category are defined by existential assertions and choices where on the right side these properties become coherence laws being part of the structure.

    But this is not an explanation, since it is not obvious why coherence laws should be “straighter” than existential assertions. I looked in HTT, but could not see an explanation there.

    • CommentRowNumber10.
    • CommentAuthorHurkyl
    • CommentTimeSep 20th 2022

    I think it’s a general metaphor for converting things defined “up to homotopy” into things defined strictly.

    For example, the fact that the category of simplicially enriched categories is a model for (∞,1)Cat(\infty,1)Cat I’ve heard described as saying “every (∞,1)(\infty,1)-category can be straightened into a simplicially enriched category”. Analogously to how every 2-category is equivalent to a strict 2-category.

    • CommentRowNumber11.
    • CommentAuthorTim_Porter
    • CommentTimeSep 21st 2022

    The older term was ’rectification’ I believe and was used in categorical circles for years.

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 17th 2023

    Added

    diff, v12, current

  13. added a related Mathoverflow discussion

    Kensuke Arakawa

    diff, v13, current

1 to 13 of 13

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