category object in an (∞,1)-category, groupoid object
The notion of $\Theta_n$-space is one model for the notion of (∞,n)-category. A $\Theta_n$-space may be thought of as a globular $n$-category up to coherent homotopy, a globular $n$-category internal to the (∞,1)-category ∞Grpd.
Concretely, a $\Theta_n$-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 $n = 1$ $\Theta_n = \Delta$ is the simplex category and a $\Theta_1$-space is the same as a complete Segal space.
Noticing that a presheaf of sets on $\Theta_n$ which satisfies the cellular Segal condition is equivalently a strict n-category, $Theta_n$-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 $\Theta_n Sp_k^{fib}$ of $(n+k,n)$-$\Theta$-spaces for all
$0 \leq n \leq \infty$;
$-2 \leq k \leq \infty$
as the category of fibrant objects in a model category $\Theta_n Sp_k$,
being a left Bousfield localization of the injective model structure on simplicial presheaves on the $n$th Theta category.
The weak equivalences in $\Theta_n Sp_k^{fib}$ 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 $sSet_{Quillen}$.
For $J$ a category, write $\Theta J$ for the categorical wreath product over the simplex category $\Delta$ Ber05.
Then with $\Theta_0 := {*}$ we have inductively
For $D = SPSh(C)^{inj}_{S}$ a model structure on simplicial presheaves on a category $C$ obtained by left Bousfield localization at a set of morphisms $S \subset Mor(SPSh(C)^{inj})$ from the global injective model structure, write
where $S_\Theta$ 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 $S_n \subset Mor(SPSh(C))$ of morphisms.
I would have started $k$ at $-1$. What does Rezk's notion do with $k = -2$? —Toby
$-1$-groupoids are spaces which are either empty or contractible. $-2$-groupoids are spaces which are contractible. So $k=-2$ is the completely trivial case; it’s included for completeness. – Charles
I do know what a $(-2,0)$-category is, a triviality as you say. But for $n \gt 0$, an $(n-2,n)$-category is the same as an $(n-2,n-1)$-category as far as I can see. (Note: I say this without having worked through your version, but just thinking about what $(n,r)$-categories should be, as at (n,r)-category.) —Toby
I would say: $(n-2,n)$-category is a trivial concept, for every $n$, though $(n-2,n-1)$ isn’t. An $(n+1+k,n+1)$-category should amount to a category enriched over $(n+k,n)$-categories. An $(-2,0)$-category is trivial (a point); an $(-1,1)$-category is a category enriched over the point, and so equivalent to the terminal category; an $(0,2)$-category is a category enriched over categories which are equivalent to the terminal category (and so equivalent to the terminal $2$-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 $(n,n+2)$-category $C$, for all pairs of $n$-arrows $x,y$, there is a unique $n+1$-arrow between them. This implies that $x$ and $y$ are parallel, in particular, that $C$ has a single $(n-1)$-arrow.
Toby: Wait, I don't buy Charles's argument after all. Yes, a $(-1,1)$-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 $(-1,1)$-category and a $(-1,0)$-category is that every $0$-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 $C$ has no $n$-arrows at all?
David Roberts: Yes - that should then be 'Assuming $C$ has an object, then it has a single $(n-1)$-arrow'. Assuming I got the indexing right, I must stress. I think I grasp $(n,n+1)$-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 $k = -2$, since $n$ might be $0$; but for an $(n-2,n)$-$\Theta$-space is the same as an $(n-1,n)$-$\Theta$-space for $n \gt 0$. OK, I'm happy with that; now to understand the definition! (^_^)
The model category $\Theta_n Sp_k$ is a cartesian monoidal model category.
The idea is that $\Theta_n Sp_k$ is naturally an enriched model category over itself.
Here is the idea on how to implement the notion $(n+1,r+1)$-category of all $(n,r)$-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 $\Theta_k Sp_n$ as a category enriched over itself. Then define a presheaf $\mathbf{X}$ on $\Theta_{n+1}$ by setting
$\mathbf{X}[0] =$ collection of objects of $\Theta_n Sp_k$
$\mathbf{X}([m](\theta_1, \cdots, \theta_m)) = \coprod_{a_0, \cdots, a_m} C(a_0,a_1)(\theta_1) \times \cdots \times C(a_{m-1},a_m)(\theta_m)$
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 “$(n+1,k+1)$-$\Theta$-space of $(n,k)$-$\Theta$-spaces”.
The definition of weak $(n,r)$-categories modeled by $\Theta$-spaces does satisfy the homotopy hypothesis: there is an evident notion of groupoid objects in $\Theta_n Sp_k$ 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 $\Theta_n$ (instead of simplicial presheaves) which is Quillen equivalent to the Rezk model structure on $\Theta_n$-spaces.
In factm the Theta-space model structure is the simplicial completion of the Cisinski model structure on presheaves on $\Theta_n$ (Ara)
For low values of $n,k$ this reproduces the following cases:
for $n=0$ we have $\Theta_0 Sp_\infty = sSet_{Quillen}$ with its standard model structure and hence $\Theta_0 Sp_\infty^{fib} =$ ∞Grpd.
for $n=1$ objects in $\Theta_1 Sp_\infty^{fib}$ are complete Segal spaces, hence (∞,1)-categories.
The notion of $\Theta$-spaces was introduced in
Charles Rezk, A cartesian presentation of weak $n$-categories Geom. Topol. 14 (2010), no. 1, 521–571 (arXiv:0901.3602)
Correction to “A cartesian presentation of weak $n$-categories” Geom. Topol. 14 (2010), no. 4, 2301–2304. MR 2740648 (pdf)
The definition of the categories $\Theta_n$ goes back to Andre Joyal who also intended to define n-categories using it.
Discussion comparing $\Theta_{n+1}$-spaces to enriched (infinity,1)-categories in $\Theta_n$-spaces is in
Julie Bergner, Charles Rezk, Comparison of models for $(\infty,n)$-categories (arXiv:1204.2013)
Julie Bergner, Charles Rezk, Comparison of models for $(\infty,n)$-categories II (arXiv:1406.4182)
The note on the $(n+1,k+1)$-$\Theta$-space of all $(n,k)$-$\Theta$-spaces comes from communication with Charles Rezk here.
Relation to simplicial completion of the Cisinski model structure on cellular sets is in