TITLE: Filtered homotopy colimits and singular homology
QUESTION [6 upvotes]: Suppose I have a functor
$$
X_\bullet: I \to \text{Spaces}
$$
where $I$ is a small filtered category.
It seems to be a "folk theorem" that the homomorphism
$$
\underset{\alpha\in I}{\text{colim }}  H_\ast(X_\alpha) \to 
H_\ast(\underset{\alpha\in I}{\text{hocolim }} X_\alpha)
$$
is an isomorphism, where $H_\ast$ denotes singular homology.
Is there a reference for this?
(Preferably standard?)

REPLY [6 votes]: I think the usual reference here is chapter XII section 5 of the Bousfield-Kan book "Homotopy limits, completions, and localizations," where they build a spectral sequence $E_2^{*,*} \cong L_*colim_i H_*(X_i) \Rightarrow H_*(hocolim_i X_i)$. The rest of the argument is not explicit in the book (but it's very straightforward so I think it's still correct to cite their book for your result): since your diagram is filtered, the derived functors $L_*colim_i H_*(X_i)$ vanish in positive degrees, so the spectral sequence collapses to the $L_0colim_i H_*(X_i)$ line, yielding your isomorphism.