Globular set

File:Globular set.svg

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 $X\_0,\; X\_1,\; X\_2,\; \backslash dots$ equipped with pairs of functions $s\_n,\; t\_n:\; X\_n\; \backslash to\; X\_\{n-1\}$ such that

• $s\_n\; \backslash circ\; s\_\{n+1\}\; =\; s\_n\; \backslash circ\; t\_\{n+1\},$
• $t\_n\; \backslash circ\; s\_\{n+1\}\; =\; t\_n\; \backslash circ\; t\_\{n+1\}.$

(Equivalently, it is a presheaf on the category of “globes”.) The letters "s", "t" stand for "source" and "target" and one imagines $X\_n$ 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.

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.{{nlab|id=computad}}{{nlab|id=globular+set}}

A variant of the notion was used by Grothendieck to introduce the notion of an ∞-groupoid. Extending Grothendieck's work,{{cite arXiv |last=Maltsiniotis |first=G |author-link= |eprint= |title=Grothendieck ∞-groupoids and still another definition of ∞-categories |class=18C10, 18D05, 18G55, 55P15, 55Q05 |date=13 September 2010 |arxiv=1009.2331 }} gave a definition of a weak ∞-category in terms of globular sets.