nLab
globular set

Idea

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

Definition

A globular set, also called an $\omega$-graph or $\omega$-quiver, is a presheaf on the globe category (described below), one of the geometric shapes for higher structures.

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.

The globe category $G$ is the category whose objects are the integers 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

A globular object $S$ in a category $K$ is a functor $S:G^{\mathrm{op}}\to K$. In particular, for $K=\mathrm{Sets}$ we say $S$ is a globular set.

For $n\in \mathbb{N}$ we write

$$S_n≔S(n)$$S_n \coloneqq S(n)

$$s_n≔S({\sigma }_n):S_{n+1}\to S_n$$s_n \coloneqq S(\sigma_n) : S_{n+1} \to S_{n}

$$t_n≔S({\tau }_n):S_{n+1}\to S_n$$t_n \coloneqq S(\tau_n) : S_{n+1} \to S_{n}

and call $S_n$ the collection of $n$-cells; $s_n$ the $(n+1)$st source map and $t_n$ the $(n+1)$st target map.

The relations (dropping obvious subscripts)

$$s\circ s=s\circ t$$s \circ s = s \circ t

$$t\circ s=t\circ t$$t \circ s = t \circ t

are called the globular identities.

A morphism of globular objects is a natural transformation of the corresponding functors. For the resulting category of globular objects in $K$ we write

$$\mathrm{GlobularObjects}(K):=K^{G^{\mathrm{op}}}$$GlobularObjects(K) := K^{G^{\mathrm{op}}}

The category $\mathrm{GlobularObjects}(\mathrm{Set})$ of globular sets is often written $\mathrm{GSet}$ or $\mathrm{GlobSet}$.

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 $n$-quiver. 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 quiver, and a $0$-globular set is just a set.

Notation for composite globular maps

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} \,.

We therefore write

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

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.

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

Remarks

• Globular sets are based on one of the three major geometric shapes for higher structures.

• 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.

Related concepts

• graph

• directed graph

  • directed $n$-graph

• ribbon graph

References

Tom Leinster: higher operads, higher categories