Geometric realization

Summary

The geometric realization $\abs{K}$ of a simplicial set $K$ is a topological space. It has one topological $n$-simplex for each $n$-simplex of $K$, glued as the face and degeneracy maps prescribe. It is a CW complex with one $n$-cell for each nondegenerate $n$-simplex. Realization is left adjoint to the singular simplicial set functor, and the two functors identify the homotopy theory of simplicial sets with that of spaces.