# 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.

They are one of the major geometric shapes for higher structures: if they satisfy a globular Segal condition then they are equivalent to strict ∞-categories.

## 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. $G_n(-) := Hom_G(-,n)$.

## Grothendieck homotopy theory

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

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.

• graph

• directed graph

• directed $n$-graph
• ribbon graph

• Also related is the notion of computad, which is similar to a globular set in some ways, but allows “formal composites” of $n$-cells to appear in the sources and targets of $(n+1)$-cells.

• simplicial object

• semi-simplicial object

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

• Sjoerd Crans, On combinatorial models for higher dimensional homotopies (web)

• R. Street, The petit topos of globular sets , JPAA 154 (2000) pp.299-315.

\bibitem{PMTH} Denis-Charles Cisinski, Les préfaisceaux comme modèles des types d’homotopie, Asterisque.

The definition of globular set, without using that term, is in 2.2 and 2.3 of the following paper:

• Ronnie Brown and Philip J. Higgins, “The equivalence of $\infty$-groupoids

and crossed complexes“, Cah. Top. G'eom. Diff. 22 (1981) 371-386.

The following paper constructs from the cubical case a strict globular $\omega$-groupoid of a filtered space:

• Ronnie Brown“A new higher homotopy groupoid: the fundamental globular

$\omega$-groupoid of a filtered space“, Homotopy, Homology and Applications, 10 (2008), No. 1, pp.327-343.