Globular set
From HandWiki
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 [math]\displaystyle{ X_0, X_1, X_2, \dots }[/math] equipped with pairs of functions [math]\displaystyle{ s_n, t_n: X_n \to X_{n-1} }[/math] such that
- [math]\displaystyle{ s_n \circ s_{n+1} = s_n \circ t_{n+1}, }[/math]
- [math]\displaystyle{ t_n \circ s_{n+1} = t_n \circ t_{n+1}. }[/math]
(Equivalently, it is a presheaf on the category of “globes”.) The letters "s", "t" stand for "source" and "target" and one imagines [math]\displaystyle{ X_n }[/math] consists of directed edges at level n.
A variant of the notion was used by Grothendieck to introduce the notion of an ∞-groupoid. Extending Grothendieck's work,[1] gave a definition of a weak ∞-category in terms of globular sets.
References
- ↑ 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 .
External links
 |  |
