The category of simplicial sets has a standard model structure, where the weak equivalences are those maps whose geometric realization is a weak homotopy equivalence, the cofibrations are monomorphisms, and the fibrations are Kan fibrations.

Simplicial sets are combinatorial objects, so morally their model structure should not be dependent on topological spaces. Are there any approaches to this model structure which do not use the geometric realization functor, and do not use topological spaces?