# nLab geometric definition of higher categories

### Context

#### Higher category theory

higher category theory

# Contents

## Idea

In a geometric definition of (n,r)-categories composition of higher morphisms is not an operation with a specified outcome but a relation : the $(n,r)$-category is presented much like a directed space and k-morphisms are $k$-dimensional subspaces in there. When some of these $k$-morphisms are suitably adjacent, there is a guarantee that there exists a $k$-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 $(n,r)$-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 $(n,r)$-category as being the nerve of a corresponding algebraic presentation.

## Properties

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.

Revised on February 15, 2012 00:30:30 by Urs Schreiber (82.169.65.155)