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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.
Last summer a “DamienC” dropped a query box at n-fold category.
Thanks for the alert (maybe this was Damien Calaque?)
The query calls into question the statement in the entry that n-fold complete Segal spaces are like n-fold categories, saying that they are rather like n-categories.
I guess I wrote that statement. I still seem to think that it is correct, abstractly due to the iterative internalizaton nature of n-fold Segal spaces, and concretly due to how they are represented by n-fold simplicial sets.
But aren’t n-fold CSS used as a model for (∞,n)-categories? If so, then even if they “look like” n-fold categories, it would be more correct to say that they are like n-fold categories satisfying a “globularity” condition making them equivalent to n-categories. There would instead be some “less complete” condition on an n-simplicial space that would be a model for “(∞,n)-fold categories”.
That seems reasonable.
Ok, I tried to clarify n-fold category.
Thanks. Somebody should fix it. Probably I should do it. But it might be more efficient if you could do it.
It would make more sense for “n-fold Segal space” to refer instead to what you call an “n-uple Segal space”.
Thanks!
Thanks, indeed!
It’s been a long time since we have been editing significantly on (∞,n)-category theoretic issues. There is much room and will there be much appreciation for you adding more notes in this direction.
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)-category instead of ∞-category?) A lot more could certainly still be written here though!
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)-category instead of ∞-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 n-fold Segal spaces? I can think of a non-complete ordinary Segal space as either an “∞-double category with connections” whose vertical direction is all invertible, or a “rigged (∞,1)-category” consisting of an essentially surjective functor from an ∞-groupoid to an (∞,1)-category. How can I think of a non-complete n-fold Segal space? It’s some kind of “∞-(n+1)-fold category” with some other condition – what does that condition mean intuitively in n-fold-category language (e.g. for n=2 or 3)?
I would rather say that Segal spaces and complete Segal spaces are both ∞-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 ∞-world) give precisely the algebraic structure of categories (i.e. compositions and units). They are also monadic over graphs in the ∞-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 ∞-category. This relative ∞-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 ∞-category - the “∞-category of ∞-categories”.
However, in the ∞-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 ∞-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 ∞-category just by looking at natural equivalences of maps between general Segal spaces.)
Similarly, n-fold Segal spaces describe the algebraic structure of n-categories (compositions and units). (As do Rezk’s Θn-spaces with their Segal conditions - and they are indeed equivalent to n-fold Segal spaces (without completeness on both sides)). (They are also monadic over n-globular objects in spaces, with the same formula for the monad as for strict n-categories.) As before, we then consider FFES maps between n-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)-categories: an n-fold Segal space (or equivalently a Θn-space) is an (∞,n)-category with extra structure in the form of a “flag” of (∞,i)-categories for all i<n. This was recently proved in a preprint of Ayala and Francis.
Thanks! I understand your last paragraph. Do I understand correctly that the rest of it is saying that an n-fold Segal space is an “n-category object internal to ∞-groupoids”, i.e. an “n×(∞,0)-category”? As opposed to a complete n-fold Segal space which is an (∞,n)-category and an n-uple Segal space which is a 1×1×⋯×1×(∞,0)-category.
I always assumed that they were, but I haven’t checked myself.
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