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In category theory, a branch of mathematics, a globular set is a higher-dimensional generalization of a
directed graph In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pai ...
. 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_ such that * s_n \circ s_ = s_n \circ t_, * t_n \circ s_ = t_n \circ t_. (Equivalently, it is a presheaf on the category of β€œglobes”.) 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 In category theory, a branch of mathematics, an ∞-groupoid is an abstract homotopical model for topological spaces. One model uses Kan complexes which are fibrant objects in the category of simplicial sets (with the standard model structure). I ...
. Extending Grothendieck's work, gave a definition of a
weak ∞-category In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a category. Th ...
in terms of globular sets.


References


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

*https://ncatlab.org/nlab/show/globular+set {{categorytheory-stub Category theory