Homotopy direct limit

Question

Suppose that $X$ is the homotopy direct limit of $\{X_i\}$. If each $X_i$ has the homotopy type of a CW-complex, then is $X$ has the homotopy type of a CW-complex?

Answer 1

Yes. This is because the construction of the homotopy colimit over CW-complexes involves taking the geometric realization of a particular coend of simplicial sets. The geometric realization of any simplicial set always has the homotopy type of a CW-complex.