nLab cellular set



Higher category theory

higher category theory

Basic concepts

Basic theorems





Universal constructions

Extra properties and structure

1-categorical presentations



A cellular set is a presheaf on the Theta-category (also called the cell category or (Joyal’s) disk category), analogous to how a simplicial set is a presheaf on the simplex category.

Cellular sets provide one approach to higher categories and abstract study of homotopy types, for example in Joyal’s approach to higher quasicategories, which are a variety of (∞,n)-categories (for n=1n=1).

One detailed development of a model of (∞,n)-categories in terms of presheaves on disk categories is the notion of Theta space.


Segal condition

A cellular set that satisfies the cellular Segal condition is an omega-category.

Model category strucuture

There is a model category structure on presheaves on Θ n\Theta_n which models (∞,n)-categories. See at model structure on cellular sets and at n-quasicategory.


  • Andre Joyal, Disks, duality and θ\theta-categories , preprint (1997).

  • Mihaly Makkai, Marek Zawadowski, Duality for simple ω\omega-categories and disks, Theory and Applications of Categories, Vol. 8, 2001, No. 7, pp 114-243, link

  • Clemens Berger, Structures cellulaires en théorie d’homotopie, habilitation thesis, pdf

  • Clemens Berger, Opérades cellulaires et espaces de lacets itérés, Ann. Inst. Fourier 46 (1996), 1125-1157. MR 98c:55011, pdf

  • Clemens Berger, Cellular structures for E nE_n-operads, talk at Workshop on Operads, Bielefeld (1999), pdf

  • Clemens Berger, A cellular nerve for higher categories, Adv. Math. 169 (2002), 118-175, pdf.

A sketch of some of related combinatorics “on open boxes and prisms” is in one of the chapters in Joyal’s Barcelona course.

Last revised on March 28, 2023 at 17:33:37. See the history of this page for a list of all contributions to it.