Filtered homotopy colimits and singular homology

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?)

• I think an argument which is much simpler than the answer by A.S. can be given as follows. Since the homotopy colimit is indexed over the filtered category, it agrees with ordinary colimit, c.f. mathoverflow.net/questions/235526/… and the homology now commutes with filtered colimit. Jul 2, 2021 at 18:22
• You could argue that, first, the singular chains functor from Spaces to Chain Complexes preserves filtered homotopy colimits because of compactness (in fact it preserves all homotopy colimits, but you don't need this), and second, homology preserves filtered homotopy colimit as a functor from chain complexes to Ab groups, which is an elementary exercise. Alternatively you can argue that homology of $X$ is the homotopy of $H\mathbb Z \wedge X$. The functor $H\mathbb Z \wedge -$ preserves arbitrary homotopy colimits, and homotopy groups preserve filtered homotopy colimits by compactness. Jul 4, 2021 at 8:08
• @GregoryArone I am looking for a reference, not a proof. Jul 4, 2021 at 18:04
• @user43326 I don't think that is true: $\text{hocolim} X_\alpha$ needn't coincide with $\text{colim} X_\alpha$ in this case. Maybe you wish to work in Simplicial Sets? If so, what is the reference there for why homology commutes with filtered colimits? Again, I am asking for a reference. I know how to prove the statement. Jul 4, 2021 at 18:11

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.