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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 $(\infty,n)$-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 “$(\infty,n)$-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 $(\infty,n)$-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 $(\infty,1)$-category instead of $\infty$-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 $(\infty,1)$-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 $n$-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 $(\infty,1)$-category” consisting of an essentially surjective functor from an $\infty$-groupoid to an $(\infty,1)$-category. How can I think of a non-complete $n$-fold Segal space? It’s some kind of “$\infty$-$(n+1)$-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 $\infty$-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, $n$-fold Segal spaces describe the algebraic structure of n-categories (compositions and units). (As do Rezk’s $\Theta_n$-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 $(\infty,n)$-categories: an $n$-fold Segal space (or equivalently a $\Theta_n$-space) is an $(\infty,n)$-category with extra structure in the form of a “flag” of $(\infty,i)$-categories for all $i \lt n$. 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 $\infty$-groupoids”, i.e. an “$n\times (\infty,0)$-category”? As opposed to a complete $n$-fold Segal space which is an $(\infty,n)$-category and an $n$-uple Segal space which is a $1\times 1\times \cdots\times 1\times (\infty,0)$-category.
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