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.
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
for all $n \in \mathbb{N}$ subject to the relations (dropping obvious subscripts)
A globular set, also called an $\omega$-graph, is a presheaf on $\mathbb{G}$. The category of globular sets is the category of presheaves
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
called the $n$-source and $n$-target maps (or similar), such that the globular identities hold: for all $n \in \mathbb{N}$
$s_n \circ s_{n+1} = s_n \circ t_{n+1}$
$t_n \circ s_{n+1} = t_n \circ t_{n+1}$.
The globular identities ensure that two sequences of boundary maps
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
For $S$ a globular set we may therefore write unambiguously
with $i_n, s_n, t_m$ the sequence of $m$ consecutive identity-assigning, source or target maps, respectively.
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:
A globular object $X$ in a category $\mathcal{C}$ is a functor $X : \mathbb{G}^{\mathrm{op}} \to \mathcal{C}$.
If to the globe category we add additional generating morphisms
satisfying the relations
we obtain the reflexive globe category, a presheaf on which is a reflexive globular set. In this case the morphism
is called the $n$th identity assigning map; it satisfies the globular identities:
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.
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. $G_n(-) := Hom_G(-,n)$.
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.
directed $n$-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.
The definition is reviewed around def. 1.4.5, p. 49 of
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.
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:
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:
$\omega$-groupoid of a filtered space“, Homotopy, Homology and Applications, 10 (2008), No. 1, pp.327-343.
Last revised on May 16, 2022 at 13:11:39. See the history of this page for a list of all contributions to it.