nLab
Theta category

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Idea

For nn \in \mathbb{N} the category Θ n\Theta_nJoyal’s disk category or cell category – may be thought of as the full subcategory of the category StrnCatStr n Cat of strict n-categories on those nn-categories that are free on pasting diagrams of nn-globes.

For instance Θ 2\Theta_2 contains an object that is depicted as

a b c b , \array{ & \nearrow &\Downarrow& \searrow && && \nearrow && \searrow \\ a && \to && b &\to & c && \Downarrow && b \\ & \searrow &\Downarrow& \nearrow && && \searrow && \nearrow && } \,,

being the pasting diagram of two 2-globes along a common 1-globe and of the result with a 1-globe and another 2-globe along common 0-globes.

Such pasting diagrams may be alternatively be encoded in planar trees, the above one corresponds to the tree:

2 1 *. \array{ \nwarrow \nearrow & & & \uparrow &&& 2 \\ & \nwarrow & \uparrow & \nearrow &&& 1 \\ && {*} } \,.

Accordingly, Θ n\Theta_n is also the category of planar rooted trees of level n\leq n.

In low degree we have

  • Θ 0=*\Theta_0 = * is the point.

  • Θ 1=Δ\Theta_1 = \Delta is the simplex category: the nn-simplex [n][n] is thought of as a linear quiver and as such the pasting diagram of nn 1-morphisms

    01n. 0 \to 1 \to \cdots \to n \,.

    Dually, this is the planar rooted tree of the form

    * \array{ \nwarrow &\uparrow & \cdots \nearrow \\ &{*} }

    with nn-branches.

Definition

We discuss two equivalent definitions

Via the free strict ω-category

Let T(1)T(1) denote the free strict ω-category generated from the terminal globular set 11.

Notice that this terminal globular set consists of precisely one kk-globe for each kk \in \mathbb{N}: one point, one edge from the point to itself, one disk from the edge to itself, and so on.

So T(1)T(1) is freely generated under composition from these cells. As described above, this means that each element of the underlying globular set of T(1)T(1) may be depicted by a pasting diagram made out of globes, and such a pasting diagram itself may be considered as a globular set whose kk-cells are instances of the kk-globes appearing in the diagram.

We now describe this formally.

The n-cells of T(1)T(1) may be identified with planar trees τ\tau of height nn, which by definition are functors

τ:[n] opΔ\tau: [n]^{op} \to \Delta

(Δ\Delta is the category of simplices and [n]Δ[n] \in \Delta is a simplex, i.e., ordered set {0<1<<n}\{0 \lt 1 \lt \ldots \lt n\}, regarded as a category) such that τ(0)=1\tau(0) = 1. Such a τ\tau is exhibited as a chain of morphisms in Δ\Delta,

τ(n)τ(n1)τ(0)=1,\tau(n) \to \tau(n-1) \to \ldots \to \tau(0) = 1,

and we will denote each of the maps in the chain by ii. Thus, for each xτ(k)x \in \tau(k), there is a fiber i 1(x)i^{-1}(x) which is a linearly ordered set. (Need to fill in how j\circ_j composition of such trees is defined.)

To each planar tree τ\tau we associate an underlying globular set [τ][\tau], as follows. Given τ\tau, define a new tree τ\tau' where we adjoin a new bottom and top x 0x_0, x 1x_1 to every fiber i 1(x)i^{-1}(x) of τ\tau, for every xτ(k)x \in \tau(k):

i τ 1(x)={x 0}i τ 1(x){x 1}i_{\tau'}^{-1}(x) = \{x_0\} \cup i_{\tau}^{-1}(x) \cup \{x_1\}

Now define a τ\tau-sector to be a triple (x,y,z)(x, y, z) where i(y)=x=i(z)i(y) = x = i(z) and y,zy, z are consecutive edges of i τ 1(x)i_{\tau'}^{-1}(x). A kk-cell of the globular set [τ][\tau] is a τ\tau-sector (x,y,z)(x, y, z) where xτ(k)x \in \tau(k). If k1k \geq 1, the source of a kk-cell (x,y,z)(x, y, z) is the (k1)(k-1)-cell (i(x),u,x)(i(x), u, x) and the target is the (k1)(k-1)-cell (i(x),x,v)(i(x), x, v) where u<x<vu \lt x \lt v are consecutive elements in i τ 1(i(x))i_{\tau'}^{-1}(i(x)). It is trivial to check that the globular axioms are satisfied.

Now let T([τ])T([\tau]) denote the free strict ω\omega-category generated by the globular set [τ][\tau].

Definition

Θ\Theta is the full subcategory of StrωCatStr \omega Cat on the strict ω-categories T([τ])T([\tau]), as τ\tau ranges over cells in the underlying globular set of T(1)T(1).

Via iterated wreath product

Proposition/Definition

Θ n\Theta_n is the nn-fold categorical wreath product of the simplex category with itself

Θ nΔ n. \Theta_n \simeq \Delta^{\wr n} \,.

(Berger, section 3)

Examples

So

Θ 1=Δ \Theta_1 = \Delta
Θ 2=ΔΔ \Theta_2 = \Delta \wr \Delta

etc.

Corollary

For all nn \in \mathbb{N} there is a canonical embedding

σ:Θ nΘ n+1 \sigma : \Theta_n \hookrightarrow \Theta_{n+1}

given by σ:a([1],a)\sigma : a \mapsto ([1], a).

Via duals of disks

In analogy to how the simplex category is equivalent to the opposite category of finite strict linear intervals, Δ𝕀 op\Delta \simeq \mathbb{I}^{op}, so the Θ\Theta-category is equivalent to the opposite of the category of Joyal’s combinatorial finite disks.

Θ𝔻 op. \Theta \coloneqq \mathbb{D}^{op} \,.

(…)

Properties

Embedding of grids (products of the simplex category)

Definition

For any small category AA there is a canonical functor

δ A:Δ×AΔA \delta_A : \Delta \times A \to \Delta \wr A

given by

δ A([n],a)=([n],(A,A,,A)). \delta_A([n], a) = ([n], (A, A, \cdots, A)) \,.

(Berger, def. 3.8)

Remark

By iteration, this induces a canonical functor

δ n:Δ ×nΘ n. \delta_n : \Delta^{\times n} \to \Theta_n \,.

Embedding into strict nn-categories

Write StrnCatStr n Cat for the category of strict n-categories.

Proposition

There is a dense full embedding

Θ nStrnCat. \Theta_n \hookrightarrow Str n Cat \,.

This was conjectured in (Batanin-Street) and shown in terms of free nn-categories on nn-graphs in (Makkai-Zawadowsky, theorem 5.10) and (Berger 02, prop. 2.2). In terms of the wreath product presentation, prop. 1 this is (Berger 05, theorem 3.7).

Proposition

Under this embedding an object ([k],(a 1,,a k))ΔΔ (n1)([k], (a_1, \cdots, a_k)) \in \Delta \wr \Delta^{\wr (n-1)} is identified with the kk-fold horizontal composition of the pasting composition of the (n1)(n-1)-morphisms a ia_i:

([k],(a 1,,a k))=a 1a 2a k. ([k], (a_1, \cdots, a_k)) = a_1 \cdot a_2 \cdot \cdots \cdot a_k \,.
Example

The pasting diagram

a b c b \array{ & \nearrow &\Downarrow& \searrow && && \nearrow && \searrow \\ a && \to && b &\to & c && \Downarrow && b \\ & \searrow &\Downarrow& \nearrow && && \searrow && \nearrow && }

corresponds to the objects of Θ 2=ΔΔ\Theta_2 = \Delta \wr \Delta given by

([3],(a 1,a 2,a 3)), ([3], (a_1, a_2, a_3)) \,,

where in turn

  • a 1=[2]a_1 = [2]

  • a 2=[0]a_2 = [0]

  • a 3=[1]a_3 = [1].

Example

Composing with the functor δ n\delta_n from remark 1 we obtain an embedding of nn-fold simplices into strict nn-categories

Δ ×nδ nΘ nStrnCat. \Delta^{\times n} \stackrel{\delta_n}{\to} \Theta_n \hookrightarrow Str n Cat \,.

Under this embedding an object ([k 1],[k 2],,[k n])([k_1], [k_2], \cdots, [k_n]) is sent to the nn-category which looks like (a globular version of) a k 1×k 2××k nk_1 \times k_2 \times \cdots \times k_n grid of nn-cells.

Write

StrnCat gauntStrnCat Str n Cat_{gaunt} \hookrightarrow Str n Cat

for the inclusion of the gaunt strict nn-categoeries into all strict n-categories.

Proposition

Θ n\Theta_n is the smallest full subcategory of StrnCat gauntStr n Cat_{gaunt} containing the grids, the image of δ n:Δ ×nStrnCat\delta_n : \Delta^{\times n} \to Str n Cat, example 3, and closed under retracts.

(B-SP, prop. 10.5)

Groupoidal version

The groupoidal version Θ˜\tilde \Theta of Θ\Theta is a test category (Ara).

Examples

In Θ 0\Theta_0 write O 0O_0 for the unique object. Then write in Θ n\Theta_n

O n:=[1](O n1). O_n := [1](O_{n-1}) \,.

This is the strict n-category free on a single nn-globe.

References

The Θ\Theta-categories were introduced in

A discussion with lots of pictures is in chapter 7 of

More discussion is in

  • David Oury, On the duality between trees and disks, TAC vol. 24 (pdf)

Discussion of its embedding into strict nn-categories is in

  • Michael Makkai, M. Zawadowsky, Duality for simple ω\omega-categories and disks, Theory Appl. Categories 8 (2001), 114–243
  • Clemens Berger, A cellular nerve for higher categories, Adv. Math. 169 (2002), 118–175.

The characterization in terms of nn-fold categorical wreath products is in

see also section 3 of

there leading over to the notion of Theta space.

The groupoidal version Θ˜\tilde \Theta is discussed in

The relation of Θ n\Theta_n to configuration spaces of points in the Euclidean space n\mathbb{R}^n is discussed in

Related discussion in the context of (infinity,n)-categories is also in

Revised on December 3, 2012 19:05:39 by Urs Schreiber (131.174.40.163)