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nLab > Latest Changes: internal logic of an (oo,1)-topos

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 24th 2010
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeFeb 24th 2010

    and this goes with a new section (oo,1)-topos theory at (infinity,1)-topos

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 22nd 2015

    In view of all that’s been done, I guess this page needs some updating.

    As remarked at type theory, it is useful to distinguish between the internal type theory of a category and the internal logic which sits on top of that type theory.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeJan 22nd 2015

    Yeah!

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 18th 2018

    Three years later and I come again to the same conclusion as #3.

    It’s funny to see how little we know of HoTT eight years ago.

    • CommentRowNumber6.
    • CommentAuthorspitters
    • CommentTimeJan 18th 2018

    This is a good recent overview: https://golem.ph.utexas.edu/category/2017/11/internal_languages_of_higher_c_1.html

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 6th 2021

    This is weird, three more years later from #5, and here we still are.

    How about we just remove section 2?

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJan 6th 2021

    You mean this section?

    That looks quite okay to me. Why do you want to remove it?

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 6th 2021

    Do we need to single out one hhlevel? As it is, it sounds to me as written from a time when Mike thought to keep these levels separate, but I thought he’d shifted on this to propositions as some types.

    He wrote section 2 in 2010, but a couple of years later he’s writing on the blog here:

    But I want to argue that propositions-as-some-types, which is the approach chosen by homotopy type theory, is the most natural choice in this setting. The fundamental observation is that the “types with at most one element” which arise in propositions-as-some-types are just the first rung on an infinite ladder: they are the (−1)-truncated types (∞-groupoids), called h-props. Why, then, should they be treated specially, as they are with propositions-are-basic or logic-enriched-type-theory?

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJan 6th 2021

    But “propositions-as-some-types” just means to regard as propositions exactly the (-1)-truncated types. Which is what the section says.

    But I don’t want to get involved in another round of what, admittedly, feels like splitting hairs on trivial matters. So feel free to ignore my comments. But – while I see lots of room for improving exposition in type theory entries – I don’t see why removing that section 2 would be necessary or desireable.

    • CommentRowNumber11.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 6th 2021

    I’m just saying it’s an odd thing to emphasise a logic-enriched type-theoretic approach as the first substantial idea.

    But the whole page needs a rewrite in view of the past 10 years.

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeJan 6th 2021

    Yeah. Someone should rewrite it. (-:

    (Sorry I’ve been MIA around here for a while… it’s hard to keep up with everything.)

1 to 12 of 12

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