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 {\displaystyle k} -cell must be single ( k {\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.

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.[1][2]

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.[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

[edit]

1. ^ a b computad at the nLab
2. ^ a b globular+set at the 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

[edit]

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

This category theory-related article is a stub. You can help Wikipedia by expanding it.

|

* v
* t
* e

*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template