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 X 0 , X 1 , X 2 , … {\displaystyle X_{0},X_{1},X_{2},\dots } equipped with pairs of functions s n , t n : X n → X n − 1 {\displaystyle s_{n},t_{n}:X_{n}\to X_{n-1}} such that
s n ∘ s n + 1 = s n ∘ t n + 1 , {\displaystyle s_{n}\circ s_{n+1}=s_{n}\circ t_{n+1},}
t n ∘ s n + 1 = t n ∘ t n + 1 . {\displaystyle t_{n}\circ s_{n+1}=t_{n}\circ t_{n+1}.}
(Equivalently, it is a presheaf on the category of “globes”.) The letters "s", "t" stand for "source" and "target" and one imagines X n {\displaystyle X_{n}} consists of directed edges at level n. In the context of a graph, each dimension is represented as a set of k {\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. It can be viewed as a specific instance of the polygraph. In a polygraph, a source or target of a k {\displaystyle k} -cell may consist of an entire path of elements of ( k {\displaystyle k} -1)-cells, but a globular set restricts this to singular elements of ( k {\displaystyle k} -1)-cells. A variant of the notion was used by Grothendieck to introduce the notion of an ∞-groupoid. Extending Grothendieck's work, gave a definition of a weak ∞-category in terms of globular sets.
References
Further reading Dimitri Ara. On the homotopy theory of Grothendieck ∞ -groupoids. J. Pure Appl. Algebra, 217(7):1237–1278, 2013, arXiv:1206.2941 .
