David Roberts
geometric realization

Geometric realisation of a simplicial space $X_{•}=X:{\Delta }^{\mathrm{op}}\to \mathrm{Top}$ is the coend …

Concretely this is the space

$$(\coprod _n\mid {\Delta }^n\mid \times X_n)/\sim$$\left(\coprod_n |\Delta^n| \times X_n\right)/\sim

where the equivalence relation $\sim$ is …, and $\mid {\Delta }^n\mid$ is the topological $n$-simplex.

This clearly gives the geometric realization of simplicial sets when $X:{\Delta }^{\mathrm{op}}\to \mathrm{Set}$.

Unless there is some control over the degeneracy maps of $X_{•}$, this is not homotopically well-behaved. For example, if all the degeneracy maps are cofibrations, a level-wise (weak) homotopy equivalence of simplicial spaces induces a (weak) homotopy equivalence on geometric realization, but in general this is not the case. The fix is to pass to the fat realization.