nLab cellular set

Idea

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 $n=1$).

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

Properties

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 $\Theta_n$ which models (∞,n)-categories. See at model structure on cellular sets and at n-quasicategory.

Related concepts

• simplicial set, globular set, cubical set, dendroidal set

• test category

• globular operad, omega-category

References

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

• Mihaly Makkai, Marek Zawadowski, Duality for simple $\omega$-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 $E_n$-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.