homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
category object in an (∞,1)-category, groupoid object
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.
Concretely, a -space is a simplicial presheaf on the Theta_n category, hence a “cellular space” that satisfies
the globular Segal condition as a weak homotopy equivalence;
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.
There is a cartesian closed category with weak equivalences of --spaces for all
;
as the category of fibrant objects in a model category ,
being a left Bousfield localization of the injective model structure on simplicial presheaves on the th Theta category.
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 .
For a category, write for the categorical wreath product over the simplex category Ber05.
Then with we have inductively
For a model structure on simplicial presheaves on a category obtained by left Bousfield localization at a set of morphisms from the global injective model structure, write
where is the set of morphisms given by …. .
Set
the left Bousfield localization of the standard model structure on simplicial sets such that fibrant objects are the Kan complexes that are homotopy k-types. Then finally define inductively
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.
Thanks for joining, in, Charles. Toby is, by the way, our esteemed expert for – if not the inventor of – negative thinking. :-) - USc
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 model category is a cartesian monoidal model category.
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.
(Rez09, 11.25).
There is a model structure on cellular sets (see there), hence on set-valued presheaves on (instead of simplicial presheaves) which is Quillen equivalent to the Rezk model structure on -spaces.
In fact the Theta-space model structure is the simplicial completion of the Cisinski model structure on presheaves on (Ara)
For low values of this reproduces the following cases:
for we have with its standard model structure and hence ∞Grpd.
for objects in are complete Segal spaces, hence (∞,1)-categories.
The notion of -spaces was introduced in
Charles Rezk, A cartesian presentation of weak -categories Geom. Topol. 14 (2010), no. 1, 521–571 (arXiv:0901.3602)
Correction to “A cartesian presentation of weak -categories” Geom. Topol. 14 (2010), no. 4, 2301–2304. MR 2740648 (pdf)
An introduction to -spaces_ (2009) (pdf)
The definition of the categories goes back to Andre Joyal who also intended to define n-categories using it. This has been achieved at about the same time by Simpson:
Carlos Simpson, A closed model structure for -categories, internal , -stacks and generalized Seifert-Van Kampen (arXiv:alg-geom/9704006)
Carlos Simpson, On the Breen-Baez-Dolan stabilization hypothesis for Tamsamani’s weak n-categories (arXiv:math/9810058)
Discussion comparing -spaces to enriched (infinity,1)-categories in -spaces is in
Julie Bergner, Charles Rezk, Comparison of models for -categories (arXiv:1204.2013)
Julie Bergner, Charles Rezk, Comparison of models for -categories II (arXiv:1406.4182)
The note on the --space of all --spaces comes from communication with Charles Rezk here.
Relation to simplicial completion of the Cisinski model structure on cellular sets is in
Last revised on November 1, 2023 at 16:52:42. See the history of this page for a list of all contributions to it.