The notion of -space is one model for the notion of (∞,n)-category. A -space may be thought of as a globular -category up to coherent homotopy, a globular -category internal to the (∞,1)-category ∞Grpd.
and a completeness condition analogous to that of complete Segal spaces.
In fact for is the simplex category and a -space is the same as a complete Segal space.
Noticing that a presheaf of sets on which satisfies the cellular Segal condition is equivalently a strict n-category, -spaces may be thought of as n-categories internal to the (∞,1)-category ∞Grpd, defined in the cellular way.
The weak equivalences in are then (by the standard result discussed at Bousfield localization of model categories) just the objectwise weak equivalences in the standard model structure on simplicial sets .
Then with we have inductively
where is the set of morphisms given by …. .
Unwinding this definition we see that
for some set of morphisms.
I would have started at . What does Rezk's notion do with ? —Toby
-groupoids are spaces which are either empty or contractible. -groupoids are spaces which are contractible. So is the completely trivial case; it’s included for completeness. – Charles
I do know what a -category is, a triviality as you say. But for , an -category is the same as an -category as far as I can see. (Note: I say this without having worked through your version, but just thinking about what -categories should be, as at (n,r)-category.) —Toby
I would say: -category is a trivial concept, for every , though isn’t. An -category should amount to a category enriched over -categories. An -category is trivial (a point); an -category is a category enriched over the point, and so equivalent to the terminal category; an -category is a category enriched over categories which are equivalent to the terminal category (and so equivalent to the terminal -category, etc.) – Charles R.
H\'m, that is a good argument.
(Sorry for not noticing before that you are Charles Rezk; for some reason I though of Charles Wells.) —Toby
David Roberts: I’m a little confused. The way I think about it, and I may have the indexing wrong, is that in an -category , for all pairs of -arrows , there is a unique -arrow between them. This implies that and are parallel, in particular, that has a single -arrow.
Toby: Wait, I don't buy Charles's argument after all. Yes, a -category is a category enriched over the point, but that doesn't make it necessarily the terminal category; it makes it a truth value. If it has an object, then it's trivial, but it might be empty instead. The difference between a -category and a -category is that every -morphism in the latter must be invertible, which is no difference at all; that's why we have this repetition. (And thereafter it propagates indefinitely.)
Similarly, with David's argument, what if has no -arrows at all?
David Roberts: Yes - that should then be 'Assuming has an object, then it has a single -arrow'. Assuming I got the indexing right, I must stress. I think I grasp -categories, but I’m not solid on these new beasties.
Toby, I guess you are right. I don’t know what I was thinking. – Charles R.
Toby: All right, so we allow , since might be ; but for an --space is the same as an --space for . OK, I'm happy with that; now to understand the definition! (^_^)
The idea is that is naturally an enriched model category over itself.
Here is the idea on how to implement the notion -category of all -categories in the context of Theta-spaces. At the time of this writing, this hasn’t been spelled out in total.
As mentioned above regard as a category enriched over itself. Then define a presheaf on by setting
collection of objects of
This object satisfies the Segal conditions (its descent conditions) in all degrees except degree 0. A suitable localization operation ca-n fix this. The resulting object should be the “--space of --spaces”.
The definition of weak -categories modeled by -spaces does satisfy the homotopy hypothesis: there is an evident notion of groupoid objects in and the full subcategory on these models homotopy n-types.
For low values of this reproduces the following cases:
The notion of -spaces was introduced in
Correction to “A cartesian presentation of weak -categories” Geom. Topol. 14 (2010), no. 4, 2301–2304. MR 2740648 (pdf)
Discussion comparing -spaces to enriched (infinity,1)-categories in -spaces is in