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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 X 0 , X 1 , X 2 , … {\displaystyle X_{0},X_{1},X_{2},\dots } equipped with pairs of functions s n , t n : X n → X n − 1 {\displaystyle s_{n},t_{n}:X_{n}\to X_{n-1}} such that

  • s n ∘ s n + 1 = s n ∘ t n + 1 , {\displaystyle s_{n}\circ s_{n+1}=s_{n}\circ t_{n+1},}
  • t n ∘ s n + 1 = t n ∘ 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 X n {\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, gave a definition of a weak ∞-category in terms of globular sets.

References

  1. Maltsiniotis, G (13 September 2010). "Grothendieck ∞-groupoids and still another definition of ∞-categories". arXiv:1009.2331 .

Further reading

  • Dimitri Ara. On the homotopy theory of Grothendieck ∞ -groupoids. J. Pure Appl. Algebra, 217(7):1237–1278, 2013, arXiv:1206.2941 .

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