Globular set

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 k-cell must be single (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.

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 X0,X1,X2,… equipped with pairs of functions sn,tn:Xn->Xn−1 such that

* sn∘sn+1=sn∘tn+1,
* tn∘sn+1=tn∘tn+1.

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

In the context of a graph, each dimension is represented as a set of 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 k-cell may consist of an entire path of elements of (k-1)-cells, but a globular set restricts this to singular elements of (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.

## References

1. ↑ 1.0 1.1 computad in nLab
2. ↑ 2.0 2.1 globular+set in nLab
3. ↑ Maltsiniotis, G (13 September 2010). "Grothendieck ∞-groupoids and still another definition of ∞-categories". arXiv:1009.2331 [18D05, 18G55, 55P15, 55Q05 18C10, 18D05, 18G55, 55P15, 55Q05].

## Further reading

* Dimitri Ara. On the homotopy theory of Grothendieck ∞ -groupoids. J. Pure Appl. Algebra, 217(7):1237–1278, 2013, arXiv:1206.2941 .

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