Higher category theory
higher category theory
Extra properties and structure
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 ).
One detailed development of a model of (∞,n)-categories in terms of presheaves on disk categories is the notion of Theta space.
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 which models (∞,n)-categories. See at model structure on cellular sets.
Andre Joyal, Disks, duality and -categories , preprint (1997).
Mihaly Makkai, Marek Zawadowski, Duality for simple -categories and disks, Theory and Applications of Categories, Vol. 8, 2001, No. 7, pp 114-243, link
Clemens Berger, STRUCTURES CELLULAIRES EN TH´EORIE D0HOMOTOPIE, 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 -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.
Revised on December 3, 2012 18:17:02
by Urs Schreiber