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Internal (,1)(\infty,1)-Categories



The notion of Θ n\Theta_n-space is one model for the notion of (∞,n)-category. A Θ n\Theta_n-space may be thought of as a globular nn-category up to coherent homotopy, a globular nn-category internal to the (∞,1)-category ∞Grpd.

Concretely, a Θ n\Theta_n-space is a simplicial presheaf on the Theta_n category, hence a “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=1n = 1 Θ n=Δ\Theta_n = \Delta is the simplex category and a Θ 1\Theta_1-space is the same as a complete Segal space.

Noticing that a presheaf of sets on Θ n\Theta_n which satisfies the cellular Segal condition is equivalently a strict n-category, Theta nTheta_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 Θ nSp k fib\Theta_n Sp_k^{fib} of (n+k,n)(n+k,n)-Θ\Theta-spaces for all

  • 0n0 \leq n \leq \infty;

  • 2k-2 \leq k \leq \infty

as the category of fibrant objects in a model category Θ nSp k\Theta_n Sp_k,
being a left Bousfield localization of the injective model structure on simplicial presheaves on the nnth Theta category.

The weak equivalences in Θ nSp k fib\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 QuillensSet_{Quillen}.


For JJ a category, write ΘJ\Theta J for the categorical wreath product over the simplex category Δ\Delta Ber05.

Then with Θ 0:=*\Theta_0 := {*} we have inductively

Θ n=ΘΘ n1. \Theta_n = \Theta \Theta_{n-1} \,.

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

DΘSp:=SPSh(ΘC) S Θ inj, D-\Theta Sp := SPSh(\Theta C)^{inj}_{S_\Theta} \,,

where S ΘS_\Theta is the set of morphisms given by …. .


Θ 0Sp k:=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

Θ n+1Sp k:=(Θ nSp k)ΘSp. \Theta_{n+1} Sp_k := (\Theta_n Sp_k)-\Theta Sp \,.

Unwinding this definition we see that

Θ nSp k=SPSh(Θ n) S n inj, \Theta_{n} Sp_k = SPSh(\Theta_n)^{inj}_{S_{n}} \,,

for some set S nMor(SPSh(C))S_n \subset Mor(SPSh(C)) of morphisms.


Special values of (n,k)(n,k)

I would have started kk at 1-1. What does Rezk's notion do with k=2k = -2? —Toby

1-1-groupoids are spaces which are either empty or contractible. 2-2-groupoids are spaces which are contractible. So k=2k=-2 is the completely trivial case; it’s included for completeness. – Charles

I do know what a (2,0)(-2,0)-category is, a triviality as you say. But for n>0n \gt 0, an (n2,n)(n-2,n)-category is the same as an (n2,n1)(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)(n,r)-categories should be, as at (n,r)-category.) —Toby

I would say: (n2,n)(n-2,n)-category is a trivial concept, for every nn, though (n2,n1)(n-2,n-1) isn’t. An (n+1+k,n+1)(n+1+k,n+1)-category should amount to a category enriched over (n+k,n)(n+k,n)-categories. An (2,0)(-2,0)-category is trivial (a point); an (1,1)(-1,1)-category is a category enriched over the point, and so equivalent to the terminal category; an (0,2)(0,2)-category is a category enriched over categories which are equivalent to the terminal category (and so equivalent to the terminal 22-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)(n,n+2)-category CC, for all pairs of nn-arrows x,yx,y, there is a unique n+1n+1-arrow between them. This implies that xx and yy are parallel, in particular, that CC has a single (n1)(n-1)-arrow.

Toby: Wait, I don't buy Charles's argument after all. Yes, a (1,1)(-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)(-1,1)-category and a (1,0)(-1,0)-category is that every 00-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 CC has no nn-arrows at all?

David Roberts: Yes - that should then be 'Assuming CC has an object, then it has a single (n1)(n-1)-arrow'. Assuming I got the indexing right, I must stress. I think I grasp (n,n+1)(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=2k = -2, since nn might be 00; but for an (n2,n)(n-2,n)-Θ\Theta-space is the same as an (n1,n)(n-1,n)-Θ\Theta-space for n>0n \gt 0. OK, I'm happy with that; now to understand the definition! (^_^)

Cartesian monoidal and enriched structure

The model category Θ nSp k\Theta_n Sp_k is a cartesian monoidal model category.

The idea is that Θ nSp k\Theta_n Sp_k is naturally an enriched model category over itself.

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

Here is the idea on how to implement the notion (n+1,r+1)(n+1,r+1)-category of all (n,r)(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 Θ kSp n\Theta_k Sp_n as a category enriched over itself. Then define a presheaf X\mathbf{X} on Θ n+1\Theta_{n+1} by setting

  • X[0]=\mathbf{X}[0] = collection of objects of Θ nSp k\Theta_n Sp_k

  • X([m](θ 1,,θ m))= a 0,,a mC(a 0,a 1)(θ 1)××C(a m1,a m)(θ m)\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)(n+1,k+1)-Θ\Theta-space of (n,k)(n,k)-Θ\Theta-spaces”.

Homotopy hypothesis

The definition of weak (n,r)(n,r)-categories modeled by Θ\Theta-spaces does satisfy the homotopy hypothesis: there is an evident notion of groupoid objects in Θ nSp k\Theta_n 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 Θ n\Theta_n (instead of simplicial presheaves) which is Quillen equivalent to the Rezk model structure on Θ n\Theta_n-spaces.

In factm the Theta-space model structure is the simplicial completion of the Cisinski model structure on presheaves on Θ n\Theta_n (Ara)


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


The notion of Θ\Theta-spaces was introduced in

  • Charles Rezk, A cartesian presentation of weak nn-categories Geom. Topol. 14 (2010), no. 1, 521–571 (arXiv:0901.3602)

    Correction to “A cartesian presentation of weak nn-categories” Geom. Topol. 14 (2010), no. 4, 2301–2304. MR 2740648 (pdf)

  • Charles Rezk, Cartesian presentations of weak n-categories An introduction to Θ n\Theta_n-spaces (2009) (pdf)

The definition of the categories Θ n\Theta_n 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:

  • Simpson, Carlos?, A closed model structure for nn-categories, internal HomHom, nn-stacks and generalized Seifert-Van Kampen (arXiv:alg-geom/9704006)

  • Simpson, Carlos?, On the Breen-Baez-Dolan stabilization hypothesis for Tamsamani’s weak n-categories (arXiv:math/9810058)

Discussion comparing Θ n+1\Theta_{n+1}-spaces to enriched (infinity,1)-categories in Θ n\Theta_n-spaces is in

The note on the (n+1,k+1)(n+1,k+1)-Θ\Theta-space of all (n,k)(n,k)-Θ\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 July 3, 2017 07:33:07 by Damien C? (