Globular set

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 ${\displaystyle X_{0},X_{1},X_{2},\dots }$ equipped with pairs of functions ${\displaystyle s_{n},t_{n}:X_{n}\to X_{n-1}}$ such that

• ${\displaystyle s_{n}\circ s_{n+1}=s_{n}\circ t_{n+1},}$
• ${\displaystyle t_{n}\circ s_{n+1}=t_{n}\circ t_{n+1}.}$

(Equivalently, it is a presheaf on the category of “globes”.) The letters "s", "t" stand for "source" and "target" and one imagines ${\displaystyle X_{n}}$ 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, (Maltsiniotis 2010) gave a definition of a weak ∞-category in terms of globular sets.

References

• Dimitri Ara. On the homotopy theory of Grothendieck ∞ -groupoids. J. Pure Appl. Algebra, 217(7):1237–1278, 2013, arXiv:1206.2941 .
• G. Maltsiniotis. Grothendieck ∞-groupoids and still another definition of ∞-categories, preprint, 2010.