David Roberts geometric realization

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

Concretely this is the space

$$\left(\coprod_n |\Delta^n| \times X_n\right)/\sim$$

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

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

Unless there is some control over the degeneracy maps of $X_\bullet$, 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.