nLab
Theta-space

Idea

$(n+k,n)$-$\Theta$-spaces are a model for (n+k,n)-categories proposed by Charles Rezk, following the idea of cellular sets? by André Joyal.

The $(n+k,n)$-$\Theta$-spaces are, roughly, presented as the fibrant objects of a certain model category structure on the category of presheaves of simplicial sets on Joyal's disk category ${\Theta }_n$. This notion is a generalization of that of complete Segal spaces, which are the $(\mathrm{\infty },1)$-$\Theta$-spaces.

Overview

There is a cartesian closed category with weak equivalences ${\Theta }_n{\mathrm{Sp}}_k^{\mathrm{fib}}$ of $(n+k,n)$-$\Theta$-spaces for all

• $0\le n\le \mathrm{\infty }$;

• $-2\le k\le \mathrm{\infty }$

as the category of fibrant objects in a model category ${\Theta }_n{\mathrm{Sp}}_k$,
being a left Bousfield localization of the injective model structure on simplicial presheaves on the $n$th Theta category.

The weak equivalences in ${\Theta }_n{\mathrm{Sp}}_k^{\mathrm{fib}}$ 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 ${\mathrm{sSet}}_{\mathrm{Quillen}}$.

Definition

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

$${\Theta }_n=\Theta {\Theta }_{n-1}.$$\Theta_n = \Theta \Theta_{n-1} \,.

For $D=\mathrm{SPSh}(C{)}_S^{\mathrm{inj}}$ a model structure on simplicial presheaves on a category $C$ obtained by left Bousfield localization at a set of morphisms $S\subset \mathrm{Mor}(\mathrm{SPSh}(C{)}^{\mathrm{inj}})$ from the global injective model structure, write

$$D-\Theta \mathrm{Sp}:=\mathrm{SPSh}(\Theta C{)}_{S_{\Theta }}^{\mathrm{inj}},$$D-\Theta Sp := SPSh(\Theta C)^{inj}_{S_\Theta} \,,

where $S_{\Theta }$ is the set of morphisms given by …. .

Set

$${\Theta }_0{\mathrm{Sp}}_k:={\mathrm{SSet}}_k,$$\Theta_0 Sp_k := SSet_k \,,

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

$${\Theta }_{n+1}{\mathrm{Sp}}_k:=({\Theta }_n{\mathrm{Sp}}_k)-\Theta \mathrm{Sp}.$$\Theta_{n+1} Sp_k := (\Theta_n Sp_k)-\Theta Sp \,.

Unwinding this definition we see that

$${\Theta }_n{\mathrm{Sp}}_k=\mathrm{SPSh}({\Theta }_n{)}_{S_n}^{\mathrm{inj}},$$\Theta_{n} Sp_k = SPSh(\Theta_n)^{inj}_{S_{n}} \,,

for some set $S_n\subset \mathrm{Mor}(\mathrm{SPSh}(C))$ of morphisms.

Properties

Special values of $(n,k)$

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 $n>0$, 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 $n>0$. OK, I'm happy with that; now to understand the definition! (^_^)

Cartesian monoidal and enriched structure

The model category ${\Theta }_n{\mathrm{Sp}}_k$ is a cartesian monoidal model category.

The idea is that ${\Theta }_n{\mathrm{Sp}}_k$ is naturally an enriched model category over itself.

$(n+1,r+1)$-$\Theta$-space of $(n,r)$-$\Theta$-spaces

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 ${\Theta }_k{\mathrm{Sp}}_n$ as a category enriched over itself. Then define a presheaf $X$ on ${\Theta }_{n+1}$ by setting

• $X[0]=$ collection of objects of ${\Theta }_n{\mathrm{Sp}}_k$

• $X([m]({\theta }_1,\cdots ,{\theta }_m))={\coprod }_{a_0,\cdots ,a_m}C(a_0,a_1)({\theta }_1)\times \cdots \times C(a_{m-1},a_m)({\theta }_m)$

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”.

Homotopy hypothesis

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 ${\Theta }_n{\mathrm{Sp}}_k$ and the full subcategory on these models homotopy n-types.

(Rez09, 11.25).

Examples

For low values of $n,k$ this reproduces the following cases:

• for $n=0$ we have ${\Theta }_0{\mathrm{Sp}}_{\mathrm{\infty }}={\mathrm{sSet}}_{\mathrm{Quillen}}$ with its standard model structure and hence ${\Theta }_0{\mathrm{Sp}}_{\mathrm{\infty }}^{\mathrm{fib}}=$ ∞Grpd.

• for $n=1$ objects in ${\Theta }_1{\mathrm{Sp}}_{\mathrm{\infty }}^{\mathrm{fib}}$ are complete Segal spaces, hence (∞,1)-categories.

References

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)

An introductory survey is in

• Charles Rezk, Cartesian presentations of weak n-categories An introduction to ${\Theta }_n$-spaces (2009) (pdf)

The definition of the categories ${\Theta }_n$ goes back to Andre Joyal who also intended to define n-categories using it.

The note on the $(n+1,k+1)$-$\Theta$-space of all $(n,k)$-$\Theta$-spaces comes from communication with Charles Rezk here.