# nLab globular set

graph theory

graph

### 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 ω-categories.

## Definition

### Basic definition

###### Definition

The globe category $𝔾$ is the category whose objects are the natural numbers, denoted here $\left[n\right]\in ℕ$ and whose morphisms are generated from

${\sigma }_{n}:\left[n\right]\to \left[n+1\right]$\sigma_n : [n] \to [n+1]
${\tau }_{n}:\left[n\right]\to \left[n+1\right]$\tau_n : [n] \to [n+1]

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

$\sigma \circ \sigma =\tau \circ \sigma$\sigma\circ \sigma = \tau \circ \sigma
$\sigma \circ \tau =\tau \circ \tau \phantom{\rule{thinmathspace}{0ex}}.$\sigma\circ \tau = \tau \circ \tau \,.
###### Definition

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

$\mathrm{gSet}≔\mathrm{PSh}\left(𝔾\right)\phantom{\rule{thinmathspace}{0ex}}.$gSet \coloneqq PSh(\mathbb{G}) \,.
###### Remark

This means that a globular set $X\in \mathrm{gSet}$ is given by a collection of sets $\left\{{X}_{n}{\right\}}_{n\in ℕ}$, called the sets of $n$-globes, equipped with functions

$\left\{{s}_{n},{t}_{n}:{X}_{n+1}\to {X}_{n}{\right\}}_{n\in ℕ}$\{s_n,t_n \colon X_{n+1} \to X_n\}_{n \in \mathbb{N}}

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

• ${s}_{n}\circ {s}_{n+1}={s}_{n}\circ {t}_{n+1}$

• ${t}_{n}\circ {s}_{n+1}={t}_{n}\circ {t}_{n+1}$.

###### Remark

The globular identities ensure that two sequences of boundary maps

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

with $n,m\in ℕ$ and for ${f}_{k},\in \left\{{s}_{k},{t}_{k}\right\}$ are equal if and only if their last term ${f}_{n}$ coincides; for all $n,m\in ℕ$ we have

${s}_{n}\cdots {s}_{n+1}\circ \cdots \circ {s}_{n+m}\circ {i}_{n+m}\circ \cdots \circ {i}_{n+1}\circ {i}_{n}=\mathrm{Id}$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}
${t}_{n}\cdots {t}_{n+1}\circ \cdots \circ {t}_{n+m}\circ {i}_{n+m}\circ \cdots \circ {i}_{n+1}\circ {i}_{n}=\mathrm{Id}\phantom{\rule{thinmathspace}{0ex}}.$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 unambigously

${s}_{n},{t}_{n}:{S}_{n+m+1}\to {S}_{n}$s_n, t_n : S_{n+m+1} \to S_n
${i}_{n}:{S}_{n}\to {S}_{n+m+1}$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 $\left[n\right]$ 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 $𝒞$ is a functor $X:{𝔾}^{\mathrm{op}}\to 𝒞$.

### Reflexive globular sets

${\iota }_{n}:\left[n+1\right]\to \left[n\right]$\iota_n : [n+1] \to [n]

satisfying the relations

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

${i}_{n}:=S\left({\iota }_{n}\right):{S}_{n}\to {S}_{n+1}$i_n := S(\iota_n) : S_{n} \to S_{n+1}

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

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

### $n$-globular sets

A presheaf on the full subcategory of the globe category containing only the integers $\left[0\right]$ through $\left[n\right]$ is called an $n$-globular set or an $n$-graph 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 ω-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 ${k}_{1}<{k}_{2}\in ℕ$ 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. ${G}_{n}\left(-\right):={\mathrm{Hom}}_{G}\left(-,n\right)$.

• 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 targest of $\left(n+1\right)$-cells.

• simplicial object

• semi-simplicial object

## References

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