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nLab > Latest Changes: semi-simplicial types in homotopy type theory

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  1. copied from HoTT wiki

    Anonymous

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 9th 2022

    have cross-linked with semi-simplicial object

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorHugo Herbelin
    • CommentTimeMay 7th 2023

    Fixing UF-IAS links by linking to https://ncatlab.org/ufias2012/files/

    diff, v4, current

    • CommentRowNumber4.
    • CommentAuthorHugo Herbelin
    • CommentTimeMay 7th 2023
    • (edited May 7th 2023)
    The page "semi-simplicial types in homotopy type theory" looks a bit outdated.

    If kept as it, it should indicate that it is an historical page from the IAS HoTT wiki.

    Otherwise, I guess it should clarify that:
    - Vladimir's code is only a start, leading to concluding that it is difficult (and maybe even impossible) to define semi-simplicial types in HoTT
    - two different questions overlap: semi-simplicial types compared to semi-simplicial sets and "indexed" semi-simplicial sets or types, i.e. as a sequence of families of n-semi-simplices, rather than as a presheaf to set or types
    - my formalization is not about types but (morally) sets and its technical value, besides trying to contribute to quantify the difficulty of defining semi-simplicial types, is about "indexed" semi-simplicial sets
    - Benno's note and Marc and Thierry's note are not really related to the question of semi-simplicial types

    More recent material could be added (Shulman 2015, Kraus-Sattler 2017, ...), HTS, 2LTT, ...
    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2023

    If you have the energy to edit, please do.

    • CommentRowNumber6.
    • CommentAuthorHugo Herbelin
    • CommentTimeMay 7th 2023

    Reorganization of the page: - clarify the difference between the “iterated dependent” (or “indexed”) presentation of semi-simplicial types - clarify the contributions of Voevodsky and myself in term of formalization - move historical contents not related to semi-simplicial types in a specific section - add more recent bibliography

    diff, v5, current

    • CommentRowNumber7.
    • CommentAuthorHugo Herbelin
    • CommentTimeMay 7th 2023
    I thus made a reorganization of the page. I have still a few typos/layout issues to solve (which I do blindly, not having found how to preview a page).
    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2023

    Thanks. I see, I’ll step back, I see that we were cross-editing just now.

    I have made the pointer to your and Voevodksy’s note an actual linked reference. Hope this works for you. But if your aim is visibility and claim of priority, its better to have an actual reference item than just in-line commentary.

    • CommentRowNumber9.
    • CommentAuthorHugo Herbelin
    • CommentTimeMay 7th 2023
    Thanks a lot for your updates, links, layout standardization, etc. I'll read again and tell you when I'm done.

    I have no particular claim (except that it reflects things as they go). I'll put a link to the final version of my paper.

    A side question: the family/fibration equivalence looks so central that I feel that there should be a page about it. After all, this correspondence can already be stated within the language of type theory alone, or within the language of a topos.
    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2023

    the family/fibration equivalence looks so central that I feel that there should be a page about it.

    Please feel invited to create a dedicated page!

    But we should have at least some version of the statement scattered around in some form or other. Have you seen this here:

    ?

    Even so, this might want to be on a stand-alone page, for ease of linking to it and for inviting expanding and referencing it.

    • CommentRowNumber11.
    • CommentAuthorHugo Herbelin
    • CommentTimeMay 7th 2023
    > Have you seen this here:
    >
    > [dependent type -- Relation to families of elements](https://ncatlab.org/nlab/show/dependent+type#RelationToFamiliesOfElements)

    Indeed, that's mostly what I was looking for, thanks! It then probably deserves an update. Historically, it has been the case that "In the categorical semantics of type theory, a dependent type B(x:A) is represented by a particular morphism p:B→A" but this does not seem any more specific to categorical semantics to do so (e.g. if one considers natural semantics, categories with families, etc.).

    I'm otherwise basically done with the semi-simplicial types page. For the bibliography, I tried to get closer to what seems to be the general conventions on nlab but I may have been wrong. Don't hesitate to tell if you'd like me to do further changes, or amend as you'd like.
    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2023
    • (edited May 7th 2023)

    Thanks! Looks good.

    I made some more of the in-line references into anchored pointers to the list of references.

    Also fixed this glitch:

    Vladimir Voevodsky proposed Homotopy Type Theory (HTT),

    to

    Vladimir Voevodsky proposed Homotopy Type System (HTS)

    diff, v9, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2023
    • (edited May 7th 2023)

    but this does not seem any more specific to categorical semantics

    Just to highlight that the paragraph at dependent type – Relation to families of elements does not consider any semantics at all, but stays entirely inside the type theory.

    But if you feel like writing more about this issue, either there or elsewhere, it will be most welcome!

    • CommentRowNumber14.
    • CommentAuthorHugo Herbelin
    • CommentTimeMay 7th 2023
    HTS: oh right, thanks.

    Regarding writing more on [dependent type -- Relation to families of elements](https://ncatlab.org/nlab/show/dependent+type#RelationToFamiliesOfElements), maybe one day...

    Thanks for the interaction!
    • CommentRowNumber15.
    • CommentAuthorHugo Herbelin
    • CommentTimeMay 28th 2023

    Tentatively made a comparison between defining “indexed” semi-simplicial sets and constructing effective matching object representatives.

    diff, v13, current

  16. Added the reference to Astra Kolomatskaia’s talk on semi-simplicial types

    Janos Turoczi

    diff, v14, current

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