nLab
geometric realization

Idea

Geometric realization denotes usually the special case of nerve and realization induced from the standard cosimplicial topological space that in degree $n$ is the standard topological $n$-simplex ${\Delta }_{\mathrm{Top}}^n$.

In this case geometric realization is the operation that reads in a simplicial set $X$ and spits out the topological space that is obtained by interpreting each element in $X_n$ – each abstract $n$-simplex in $X$ – as one copy of ${\Delta }_{\mathrm{Top}}^n$ 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 $X$ on how these simplices are supposed to be stuck together.

Details

Let $S$ 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

$\mathrm{st}:S\to$ Top

which sends the standard cellular shape $[n]$ (the standard cellular globe, simplex or cube, respectively) to the corresponding standard topological shape (for instance the standard $n$-simplex $\mathrm{st}([n]):=\{(x_1,\cdots ,x_n)\mid x_i\le x_{i+1}\}\subset {ℝ}^n$ ) with the obvious induced face and boundary maps.

Using this, in cases where $\mathrm{Top}$ can be regarded as enriched over and tensored over a base category $V$, the geometric realization of a presheaf $K^{•}:S^{\mathrm{op}}\to V$ on $S$ – e.g., of a globular set, a simplicial set or a cubical set, respectively (when $V=\mathrm{Set}$) – is the topological space given by the coend or weighted colimit

$$\mid K^{•}\mid ={\int }^{[n]\in S}\mathrm{st}([n])\cdot K^n.$$|K^\bullet| = \int^{[n] \in S} st([n]) \cdot K^n \,.

Examples

• For $G$ a group, $BG$ its one-object groupoid obtained by delooping, $N(BG)$ the corresponding simplicial nerve Kan complex, we have that the geometric realization

  $$ℬG=\mid NBG\mid$$\mathcal{B}G = |N\mathbf{B}G|

  is the topological space that is the classifying space for $G$-principal bundles (cover space?s), as long as we give $G$ the discrete topology.