# nLab Theta-space

### Context

#### Higher category theory

higher category theory

# Contents

## Idea

The notion of ${\Theta }_{n}$-space is one model for the notion of (∞,n)-category. A ${\Theta }_{n}$-space is may be thought of as a gobular $n$-category up to coherent homotopy, an globular $n$-category internal to the (∞,1)-category ∞Grpd.

Concretely, a ${\Theta }_{n}$-space is a simplicial presheaf on the Theta_n category, a hence “cellular space”, that satisfies

1. the globular Segal condition as a weak homotopy equivalence;

2. 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, ${\mathrm{Theta}}_{n}$-spaces may be thought of as n-categories internal to the (∞,1)-category ∞Grpd, defined in the cellular way.

## Overview

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

• $0\le n\le \infty$;

• $-2\le k\le \infty$

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

${\Theta }_{n}=\Theta {\Theta }_{n-1}\phantom{\rule{thinmathspace}{0ex}}.$\Theta_n = \Theta \Theta_{n-1} \,.

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

$D-\Theta \mathrm{Sp}:=\mathrm{SPSh}\left(\Theta C{\right)}_{{S}_{\Theta }}^{\mathrm{inj}}\phantom{\rule{thinmathspace}{0ex}},$D-\Theta Sp := SPSh(\Theta C)^{inj}_{S_\Theta} \,,

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

Set

${\Theta }_{0}{\mathrm{Sp}}_{k}:={\mathrm{SSet}}_{k}\phantom{\rule{thinmathspace}{0ex}},$\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

${\Theta }_{n+1}{\mathrm{Sp}}_{k}:=\left({\Theta }_{n}{\mathrm{Sp}}_{k}\right)-\Theta \mathrm{Sp}\phantom{\rule{thinmathspace}{0ex}}.$\Theta_{n+1} Sp_k := (\Theta_n Sp_k)-\Theta Sp \,.

Unwinding this definition we see that

${\Theta }_{n}{\mathrm{Sp}}_{k}=\mathrm{SPSh}\left({\Theta }_{n}{\right)}_{{S}_{n}}^{\mathrm{inj}}\phantom{\rule{thinmathspace}{0ex}},$\Theta_{n} Sp_k = SPSh(\Theta_n)^{inj}_{S_{n}} \,,

for some set ${S}_{n}\subset \mathrm{Mor}\left(\mathrm{SPSh}\left(C\right)\right)$ of morphisms.

## Properties

### Special values of $\left(n,k\right)$

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 $\left(-2,0\right)$-category is, a triviality as you say. But for $n>0$, an $\left(n-2,n\right)$-category is the same as an $\left(n-2,n-1\right)$-category as far as I can see. (Note: I say this without having worked through your version, but just thinking about what $\left(n,r\right)$-categories should be, as at (n,r)-category.) —Toby

I would say: $\left(n-2,n\right)$-category is a trivial concept, for every $n$, though $\left(n-2,n-1\right)$ isn’t. An $\left(n+1+k,n+1\right)$-category should amount to a category enriched over $\left(n+k,n\right)$-categories. An $\left(-2,0\right)$-category is trivial (a point); an $\left(-1,1\right)$-category is a category enriched over the point, and so equivalent to the terminal category; an $\left(0,2\right)$-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 $\left(n,n+2\right)$-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 $\left(n-1\right)$-arrow.

Toby: Wait, I don't buy Charles's argument after all. Yes, a $\left(-1,1\right)$-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 $\left(-1,1\right)$-category and a $\left(-1,0\right)$-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 $\left(n-1\right)$-arrow'. Assuming I got the indexing right, I must stress. I think I grasp $\left(n,n+1\right)$-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 $\left(n-2,n\right)$-$\Theta$-space is the same as an $\left(n-1,n\right)$-$\Theta$-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.

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

Here is the idea on how to implement the notion $\left(n+1,r+1\right)$-category of all $\left(n,r\right)$-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

• $X\left[0\right]=$ collection of objects of ${\Theta }_{n}{\mathrm{Sp}}_{k}$

• $X\left(\left[m\right]\left({\theta }_{1},\cdots ,{\theta }_{m}\right)\right)={\coprod }_{{a}_{0},\cdots ,{a}_{m}}C\left({a}_{0},{a}_{1}\right)\left({\theta }_{1}\right)×\cdots ×C\left({a}_{m-1},{a}_{m}\right)\left({\theta }_{m}\right)$

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 ”$\left(n+1,k+1\right)$-$\Theta$-space of $\left(n,k\right)$-$\Theta$-spaces”.

### Homotopy hypothesis

The definition of weak $\left(n,r\right)$-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).

### Relation to cellular sets

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)

## Examples

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

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

• for $n=1$ objects in ${\Theta }_{1}{\mathrm{Sp}}_{\infty }^{\mathrm{fib}}$ 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.

Further developments are in

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

Relation to simplicial completion of the Cisinski model structure on cellular sets is in

Revised on December 3, 2012 19:39:47 by Urs Schreiber (131.174.40.163)