# nLab globular set

Globular sets

### Context

#### Graph theory

graph theory

graph

category of simple graphs

### Extra structure

#### Higher category theory

higher category theory

# Globular sets

## Idea

Globular sets are to simplicial sets as globes are to simplices.

Much like simplicial sets underly common geometric definitions of higher categories, so globular sets underly some algebraic definitions of higher categories, see below.

## Definition

### Basic definition

###### Definition

The globe category $\mathbb{G}$ is the category whose objects are the natural numbers, denoted here $[n] \in \mathbb{N}$ (N.B. not to be confused with ordinals in any structural sense) and whose morphisms are generated from

$\sigma_n : [n] \to [n+1]$
$\tau_n : [n] \to [n+1]$

for all $n \in \mathbb{N}$ subject to the relations (dropping obvious subscripts)

$\sigma\circ \sigma = \tau \circ \sigma$
$\sigma\circ \tau = \tau \circ \tau \,.$
###### Definition

A globular set, also called an $\omega$-graph, is a presheaf on $\mathbb{G}$. The category of globular sets is the category of presheaves

$gSet \coloneqq PSh(\mathbb{G}) \,.$
###### Remark

This means that a globular set $X \in gSet$ is given by a collection of sets $\{X_n\}_{n \in \mathbb{N}}$ (the set of $n$-globes) equipped with functions

$\{s_n,t_n \colon X_{n+1} \to X_n\}_{n \in \mathbb{N}}$

called the $n$-source and $n$-target maps (or similar), such that the globular identities hold: for all $n \in \mathbb{N}$

• $s_n \circ s_{n+1} = s_n \circ t_{n+1}$

• $t_n \circ s_{n+1} = t_n \circ t_{n+1}$.

###### Remark

The globular identities ensure that two sequences of boundary maps

$f_n \circ \cdots \circ f_{n+m-1} \circ f_{n+m} : S_{n+m+1} \to S_n$

with $n,m \in \mathbb{N}$ and for $f_k, \in \{s_k, t_k\}$ are equal if and only if their last term $f_n$ coincides; for all $n,m \in \mathbb{N}$ we have

$s_n \cdots s_{n+1} \circ \cdots \circ s_{n+m} \circ i_{n+m} \circ \cdots \circ i_{n+1} \circ i_n = \mathrm{Id}$
$t_n \cdots t_{n+1} \circ \cdots \circ t_{n+m} \circ i_{n+m} \circ \cdots \circ i_{n+1} \circ i_n = \mathrm{Id} \,.$

For $S$ a globular set we may therefore write unambiguously

$s_n, t_n : S_{n+m+1} \to S_n$
$i_n : S_n \to S_{n+m+1}$

with $i_n, s_n, t_m$ the sequence of $m$ consecutive identity-assigning, source or target maps, respectively.

###### Remark

The presheaf definition can understood from the point of view of space and quantity: a globular set is a space characterized by the fact that and how it may be probed by mapping standard globes into it: the set $S_n$ assigned by a globular set to the standard $n$-globe $[n]$ is the set of $n$-globes in this space, hence the way of mapping a standard $n$-globe into this spaces.

More generally:

###### Definition

A globular object $X$ in a category $\mathcal{C}$ is a functor $X : \mathbb{G}^{\mathrm{op}} \to \mathcal{C}$.

### Reflexive globular sets

$\iota_n : [n+1] \to [n]$

satisfying the relations

$\iota \circ \sigma = \mathrm{Id}$
$\iota \circ \tau = \mathrm{Id}$

we obtain the reflexive globe category, a presheaf on which is a reflexive globular set. In this case the morphism

$i_n := S(\iota_n) : S_{n} \to S_{n+1}$

is called the $n$th identity assigning map; it satisfies the globular identities:

$s \circ i = \mathrm{Id}$
$t \circ i = \mathrm{Id}$

### $n$-globular sets

A presheaf on the full subcategory of the globe category containing only the integers $$ through $[n]$ is called an $n$-globular set or an $n$-graph. An $n$-globular set may be identified with an $\infty$-globular set which is empty above dimension $n$.

Note that a $1$-globular set is just a directed graph, and a $0$-globular set is just a set.

## Examples

• Any strict 2-category or bicategory has an underlying 2-globular set. Likewise, any tricategory has an underlying 3-globular set. Globular sets can be used as underlying data for n-categories as well; see for instance Batanin ∞-category.

• A strict omega-category is a globular set $C$ equipped in each degree with the structure of a category such that for every pair $k_1 \lt k_2 \in \mathbb{N}$ the induced structure on the 2-graph $C_{k_2} \stackrel{\to}{\to} C_{k_1} \stackrel{\to}{\to} C_0$ is that of a strict 2-category.

• The globular $n$-globe $G_n$ is the globular set represented by $n$, i.e. $G_n(-) := Hom_G(-,n)$.

## Properties

### Grothendieck homotopy theory

The category of globes is not a weak test category according to Scholium 8.4.14 in Cisinski 06.

However, if we construct the free strict monoidal category on the category of globes, while ensuring that the terminal object becomes the monoidal unit, then the resulting category of polyglobes is a test category.

### As shapes for higher categories

Globular sets may be used as a geometric shape for algebraic definitions of higher categories:

The definition is reviewed around def. 1.4.5, p. 49 of

• Ronnie Brown, Philip J. Higgins, The equivalence of $\infty$-groupoids and crossed complexes, Cah. Top. G'eom. Diff. 22 (1981) 371-386.