nLab
globular set

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 $[n]\in \mathbb{N}$ and whose morphisms are generated from

$${\sigma }_n:[n]\to [n+1]$$\sigma_n : [n] \to [n+1]

$${\tau }_n:[n]\to [n+1]$$\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 \sigma = \tau \circ \sigma

$$\sigma \circ \tau =\tau \circ \tau .$$\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

$$\mathrm{gSet}≔\mathrm{PSh}(𝔾).$$gSet \coloneqq PSh(\mathbb{G}) \,.

Remark

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

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

• $s_n\circ s_{n+1}=s_n\circ t_{n+1}$

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

Remark

The globular identities ensure that two sequences of boundary maps

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

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

$$t_n\cdots t_{n+1}\circ \cdots \circ t_{n+m}\circ i_{n+m}\circ \cdots \circ i_{n+1}\circ i_n=\mathrm{Id}.$$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 $[n]$ 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 $𝒞$ is a functor $X:{𝔾}^{\mathrm{op}}\to 𝒞$.

Reflexive globular sets

If to the globe category we add additional generating morphisms

$${\iota }_n:[n+1]\to [n]$$\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

$$i_n:=S({\iota }_n):S_n\to S_{n+1}$$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 $[0]$ through $[n]$ is called an $n$-globular set or an $n$-graph or an $n$-graph. An $n$-globular set may be identified with an $\mathrm{\infty }$-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 $k_1<k_2\in \mathbb{N}$ the induced structure on the 2-graph $C_{k_2}\overrightarrow{\to }{C}_{k_1}\overrightarrow{\to }{C}_0$ is that of a strict 2-category.

• The globular $n$-globe $G_n$ is the globular set represented by $n$, i.e. $G_n(-):={\mathrm{Hom}}_G(-,n)$.

Related concepts

• graph

• directed graph

  • directed $n$-graph

• ribbon graph

• Theta category,

• globular theory, globular operad

• 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 $(n+1)$-cells.

• simplicial object

  • simplicial set

  • simplicial object in an (∞,1)-category

• semi-simplicial object

  • semisimplicial set

References

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

• Tom Leinster: Higher operads, higher categories (arXiv:math/0305049)

See also

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