Geometric realization denotes usually the special case of nerve and realization induced from the standard cosimplicial topological space that in degree is the standard topological -simplex .
In this case geometric realization is the operation that reads in a simplicial set and spits out the topological space that is obtained by interpreting each element in – each abstract -simplex in – as one copy of and then guing together all these along their boundaries to a big topological space, using the information encoded in the face and degeneracy maps of on how these simplices are supposed to be stuck together.
Let be one of the categories of geometric shapes for higher structures, such as the globe category or the simplex category or the cube category.
There is an obvious functor
which sends the standard cellular shape (the standard cellular globe, simplex or cube, respectively) to the corresponding standard topological shape (for instance the standard -simplex ) with the obvious induced face and boundary maps.
Using this, in cases where can be regarded as enriched over and tensored over a base category , the geometric realization of a presheaf on – e.g., of a globular set, a simplicial set or a cubical set, respectively (when ) – is the topological space given by the coend or weighted colimit
For a group, its one-object groupoid obtained by delooping, the corresponding simplicial nerve Kan complex, we have that the geometric realization
is the topological space that is the classifying space for -principal bundles (cover space?s), as long as we give the discrete topology.