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

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

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

External links

• https://ncatlab.org/nlab/show/globular+set