Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > Latest Changes: n-fold category

Bottom of Page

1 to 21 of 21

  1.  
    • 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 3rd 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.

    • CommentRowNumber18.
    • CommentAuthorvarkor
    • CommentTimeOct 7th 2022

    Added original reference to nn-fold categories.

    diff, v27, current

    • CommentRowNumber19.
    • CommentAuthormattecapu
    • CommentTimeOct 26th 2022

    Added Grandis’ book as reference, together with his papers in Cahiers LVII with Parè, added redirect for ’multiple categories’. By the way, are multiple and n-fold categories exactly the same?

    diff, v28, current

    • CommentRowNumber20.
    • CommentAuthorMike Shulman
    • CommentTimeOct 27th 2022

    I always assumed that they were, but I haven’t checked myself.

    • CommentRowNumber21.
    • CommentAuthorvarkor
    • CommentTimeDec 9th 2023

    Added multiple category terminology.

    diff, v30, current

1 to 21 of 21

  1.