n-category = (n,n)-category
n-groupoid = (n,0)-category
geometric definition of higher categories
In a geometric definition of (n,r)-categories composition of higher morphisms is not an operation with a specified outcome but a relation : the -category is presented much like a directed space and k-morphisms are -dimensional subspaces in there. When some of these -morphisms are suitably adjacent, there is a guarantee that there exists a -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 -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 -category as being the nerve of a corresponding algebraic presentation.
a geometric model for an (∞,n)-category is