nLab geometric definition of higher categories



Higher category theory

higher category theory

Basic concepts

Basic theorems





Universal constructions

Extra properties and structure

1-categorical presentations



In a geometric definition of (n,r)-categories composition of higher morphisms is not an operation with a specified outcome but a relation : the (n,r)(n,r)-category is presented much like a directed space and k-morphisms are kk-dimensional subspaces in there. When some of these kk-morphisms are suitably adjacent, there is a guarantee that there exists a kk-morphism that serves as their composite. But there may be several such. Instead of a rule for picking a specific one, subject to associativity constraints, there is a contractible space of choices of possible composites.

From a geometric presentation of an (n,r)(n,r)-category one can typically obtain an algebraic presentation by choosing composites. The contractibility of the space of choices becomes a coherence law satisfied by the collection of choices.

Conversely, one may typically think of the geometric presentation of an (n,r)(n,r)-category as being the nerve of a corresponding algebraic presentation.



When the (∞,1)-category of all (n,r)-categories is presented by a model category, then typically geometric models are cofibrant objects while algebraic models are typically fibrant objects.

For instance all in the standard model structure on simplicial sets, or the standard model structure for quasi-categories all objects are cofibrant.

There are also geometric models for operadic structures: dendroidal sets.

Last revised on February 15, 2012 at 00:30:30. See the history of this page for a list of all contributions to it.