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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 7th 2012
    • (edited Nov 7th 2012)

    I have edited the old entry n-fold category a little, brought the content into proper order, fixed the link to cat-n-groups and cross-linked with n-fold complete Segal space.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 27th 2018

    Last summer a “DamienC” dropped a query box at n-fold category.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeFeb 27th 2018
    • (edited Feb 27th 2018)

    Thanks for the alert (maybe this was Damien Calaque?)

    The query calls into question the statement in the entry that nn-fold complete Segal spaces are like nn-fold categories, saying that they are rather like nn-categories.

    I guess I wrote that statement. I still seem to think that it is correct, abstractly due to the iterative internalizaton nature of nn-fold Segal spaces, and concretly due to how they are represented by nn-fold simplicial sets.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 27th 2018

    But aren’t nn-fold CSS used as a model for (,n)(\infty,n)-categories? If so, then even if they “look like” nn-fold categories, it would be more correct to say that they are like nn-fold categories satisfying a “globularity” condition making them equivalent to nn-categories. There would instead be some “less complete” condition on an nn-simplicial space that would be a model for “(,n)(\infty,n)-fold categories”.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeFeb 27th 2018

    That seems reasonable.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 27th 2018

    Ok, I tried to clarify n-fold category.

    • CommentRowNumber7.
    • CommentAuthorRuneHaugseng
    • CommentTimeFeb 28th 2018
    The definition on the page "n-fold complete Segal space" is actually incorrect: If one just iterates the definition of a Segal space one indeed obtains an infinity-version of n-fold categories. (I have sometimes called these objects "n-uple Segal spaces", for lack of a better name.) To obtain (infinity,n)-categories one must impose a constancy condition (for example, for n=2 you have a bisimplicial space, and the 0th simplicial space in one direction should be constant); this gives what is usually called an n-fold Segal space. (Completeness is then a further condition on such n-fold Segal spaces, equivalent to being local for the class of fully faithful and essentially surjective morphisms.)
    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeFeb 28th 2018

    Thanks. Somebody should fix it. Probably I should do it. But it might be more efficient if you could do it.

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 28th 2018

    It would make more sense for “n-fold Segal space” to refer instead to what you call an “n-uple Segal space”.

    • CommentRowNumber10.
    • CommentAuthorRuneHaugseng
    • CommentTimeFeb 28th 2018
    I agree, but the terminology is quite well-established by now. Unfortunately it seems hard to find any terminology here that's not potentially confusing in one way or another.
    • CommentRowNumber11.
    • CommentAuthorRuneHaugseng
    • CommentTimeMar 1st 2018
    I wrote up a definition of n-fold Segal spaces. I'll try to add something on completeness later.
    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeMar 1st 2018

    Thanks!

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeMar 2nd 2018

    Thanks, indeed!

    It’s been a long time since we have been editing significantly on (,n)(\infty,n)-category theoretic issues. There is much room and will there be much appreciation for you adding more notes in this direction.

    • CommentRowNumber14.
    • CommentAuthorRuneHaugseng
    • CommentTimeMar 2nd 2018

    Now I also added a brief discussion of completeness and fully faithful and essentially surjective morphisms. (I expect I’ve broken many nlab conventions in the process - for example, am I supposed to say (,1)(\infty,1)-category instead of \infty-category?) A lot more could certainly still be written here though!

    • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeMar 2nd 2018

    Link for those coming to this thread directly: n-fold complete Segal space. (Actually we should in theory be having this discussion at the nForum thread on n-fold complete Segal spaces, where I see that the problem you just corrected was already pointed out 4 years ago.

    We do generally say (,1)(\infty,1)-category instead of \infty-category. I did a search-and-replace on that for you. It’s also permissible to use the implicit infinity-category convention if it’s declared near the top of a page, but that’s probably not appropriate for a page like this one that’s about a particular model construction.

    Can you give any more intuition for non-complete nn-fold Segal spaces? I can think of a non-complete ordinary Segal space as either an “\infty-double category with connections” whose vertical direction is all invertible, or a “rigged (,1)(\infty,1)-category” consisting of an essentially surjective functor from an \infty-groupoid to an (,1)(\infty,1)-category. How can I think of a non-complete nn-fold Segal space? It’s some kind of “\infty-(n+1)(n+1)-fold category” with some other condition – what does that condition mean intuitively in nn-fold-category language (e.g. for n=2n=2 or 33)?

    • CommentRowNumber16.
    • CommentAuthorRuneHaugseng
    • CommentTimeMar 3rd 2018

    I would rather say that Segal spaces and complete Segal spaces are both \infty-analogues of categories, with “category” used in two slightly different senses: On the one hand a category is an algebraic structure, and on the other hand a category is an object of the relative category (categories, equivalences) - or equivalently of the (2,1)-category of categories. For ordinary categories these are usually conflated, being of course very similar - since all FFES (=fully faithful and essentially surjective) functors have pseudo-inverses, you can define the (2,1)-category of categories without actually formally inverting anything.

    Segal spaces (viewed internally in the \infty-world) give precisely the algebraic structure of categories (i.e. compositions and units). They are also monadic over graphs in the \infty-category of spaces, with the monad given by the same formula as for ordinary categories.

    We can define FFES maps of Segal spaces, which produces a relative \infty-category. This relative \infty-category is arguably the most direct analogue of the way people usually work with categories. We can also invert the FFES maps, producing a new \infty-category - the “\infty-category of \infty-categories”.

    However, in the \infty-setting we are lucky enough that this localization actually corresponds to a full subcategory of local objects - the complete Segal spaces. (It is not obvious that this is possible, as it is not immediately clear from the definition that the FFES maps are generated by a set of maps.) Thus the \infty-category of complete Segal spaces is the analogue of the (2,1)-category of categories. (On the other hand, we are unlucky in that FFES maps of Segal spaces do not necessarily have pseudo-inverses, at least in the most obvious sense, so unlike for ordinary categories we can’t construct this \infty-category just by looking at natural equivalences of maps between general Segal spaces.)

    Similarly, nn-fold Segal spaces describe the algebraic structure of n-categories (compositions and units). (As do Rezk’s Θ n\Theta_n-spaces with their Segal conditions - and they are indeed equivalent to nn-fold Segal spaces (without completeness on both sides)). (They are also monadic over nn-globular objects in spaces, with the same formula for the monad as for strict nn-categories.) As before, we then consider FFES maps between nn-fold Segal spaces, with the complete objects again turning out to be the ones local for these equivalences.

    Alternatively, your second description can also be extended to (,n)(\infty,n)-categories: an nn-fold Segal space (or equivalently a Θ n\Theta_n-space) is an (,n)(\infty,n)-category with extra structure in the form of a “flag” of (,i)(\infty,i)-categories for all i<ni \lt n. This was recently proved in a preprint of Ayala and Francis.

    • CommentRowNumber17.
    • CommentAuthorMike Shulman
    • CommentTimeMar 3rd 2018

    Thanks! I understand your last paragraph. Do I understand correctly that the rest of it is saying that an nn-fold Segal space is an “nn-category object internal to \infty-groupoids”, i.e. an “n×(,0)n\times (\infty,0)-category”? As opposed to a complete nn-fold Segal space which is an (,n)(\infty,n)-category and an nn-uple Segal space which is a 1×1××1×(,0)1\times 1\times \cdots\times 1\times (\infty,0)-category.

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