# Homotopy direct limit

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?

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Obviously not true if you require finite CW complexes. Are you letting your CW complexes have infinitely many cells? Arbitrarily high dimension? –  you Apr 9 '12 at 15:46

## 1 Answer

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.

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