Globular set

In category theory, a branch of mathematics, a globular set is a higher-dimensional generalization of a directed graph. Precisely, it is a sequence of sets ${\displaystyle X_{0},X_{1},X_{2},\dots }$ equipped with pairs of functions ${\displaystyle s_{n},t_{n}:X_{n}\to X_{n-1}}$ such that

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

A globular set with 0-cells (vertices), 1-cells (gray edges), 2-cells (red edges), and 3-cells (blue edges). The source and target of each ${\displaystyle k}$-cell must be single (${\displaystyle k}$-1)-cells. For example, the red edge A connects single 1-cells a and b, while B connects b and c, and C forms a self-connection on c.

(Equivalently, it is a presheaf on the category of “globes”.) The letters "s", "t" stand for "source" and "target" and one imagines ${\displaystyle X_{n}}$ consists of directed edges at level n.

In the context of a graph, each dimension is represented as a set of ${\displaystyle k}$-cells. Vertices would make up the 0-cells, edges connecting vertices would be 1-cells, and then each dimension higher connects groups of the dimension beneath it.^[1][2]

It can be viewed as a specific instance of the polygraph. In a polygraph, a source or target of a ${\displaystyle k}$-cell may consist of an entire path of elements of (${\displaystyle k}$-1)-cells, but a globular set restricts this to singular elements of (${\displaystyle k}$-1)-cells.^[1][2]

A variant of the notion was used by Grothendieck to introduce the notion of an ∞-groupoid. Extending Grothendieck's work,^[3] gave a definition of a weak ∞-category in terms of globular sets.