nLab
geometric definition of higher categories

There are two general types of strategies for defining and handling higher categories. In the geometric approach a higher category is defined to be a simplicial set (or more generally a presheaf over a suitable category of shapes, such as opetopes) with extra properties.

From the alternative point of view of algebraic definition of higher category the simplicial set here is the nerve of the $\mathrm{\infty }$-category which it encodes.

The simplest example is the concept of Kan complex. Slightly more general is the notion of quasi-category. If one keeps relaxing conditions this way while keeping the required coherence, one arrives at the definition of weak $\mathrm{\infty }$-category given by Ross Street, developed at simplicial model for weak omega-categories.