Globular set
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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 equipped with pairs of functions such that
(Equivalently, it is a presheaf on the category of “globes”.) The letters "s", "t" stand for "source" and "target" and one imagines consists of directed edges at level n.
In the context of a graph, each dimension is represented as a set of -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 -cell may consist of an entire path of elements of (-1)-cells, but a globular set restricts this to singular elements of (-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]Further reading
[edit]- Dimitri Ara. On the homotopy theory of Grothendieck ∞ -groupoids. J. Pure Appl. Algebra, 217(7):1237–1278, 2013, arXiv:1206.2941 .