nLab globular set

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

If to the globe category we add additional generating morphisms

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

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 $[0]$ 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:

• Equipped with strictly compatible composition structure on cells in any dimension, globular sets model strict ∞-categories (originally often: “omega-categories”, see also at complicial sets), historically one of the earliest notions of higher category theory but too restrictive to be useful for most purposes (in their further restriction to strict omega-groupoids they are equivalent just to crossed complexes).

• A more general (semi-strict) notion of $n$-categories modeled on globular sets are known as associative $n$-categories, see there for more, and see a corresponding proof assistant: homotopy.io (previously: Globular).

Related concepts

• graph

• directed graph

  • directed $n$-graph

  • directed $(n,r)$-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 targets of $(n+1)$-cells.

• simplicial object

  • simplicial set

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

• semi-simplicial object

  • semisimplicial set

• strict infinity-category, strict omega-groupoid

• associative n-category, Globular

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)

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

• Denis-Charles Cisinski, Les préfaisceaux comme types d’homotopie, Astérisque 308 Soc. Math. France (2006), 392 pages [numdam:AST_2006__308__R1_0 pdf]

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

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