Reference for homotopy groups of filtered homotopy colimits

It seems to be well known that for a filtered category $$I$$ and a functor to the category of pointed spaces $$X:I \to \mathcal{S}_*$$ the homotopy groups of the filtered homotopy colimit are colimits of the homotopy groups $$\pi_*( {\rm hocolim}\ X_i ) \cong {\rm colim}\ \pi_*(X_i).$$ But for some reason I can’t find this statement in the literature. Where can I find this?

• Isn't that just because the spheres are compact? Commented Aug 12 at 7:22
• Compactness of spheres gives you a statement on mapping spaces, to get the statement about homotopy groups you still need that $\pi_0:\mathcal{S} \to\mathrm{Set}$ commutes with filtered colimits (but this is easier, it commutes with all colimits since it is a left adjoint) Commented Aug 12 at 7:32
• But I guess compactness of spheres (in the categorical sense!) is basically equivalent to the statement here, so at some point you need to invest something about filtered colimits in $\mathcal{S}$. Commented Aug 12 at 7:34

By [Lurie, Higher topos theory, Prop. 5.3.3.3], for filtered $$I$$ we have that the colimit functor $$\mathrm{Fun}(I,\mathcal{S})\to \mathcal{S}$$ commutes with finite limits.
We have $$\mathrm{Map}(S^0,X) \simeq X\times X$$, $$\mathrm{Map}(S^n,X) \simeq X\times_{\mathrm{Map}(S^{n-1},X)} X$$, and $$\mathrm{Map}_*(S^n,X) = \mathrm{Map}(S^n,X)\times_X \mathrm{pt}$$, all of which are finite limits. It follows that for a filtered diagram $$X_i: I\to \mathcal{S}$$, we have $$\mathrm{colim}_I \mathrm{Map}_*(S^n,X_i) \simeq \mathrm{Map}_*(S^n, \mathrm{colim}_I X_i).$$
Now finally, $$\pi_0: \mathcal{S}\to \mathrm{Set}$$ preserves all colimits, since it is left adjoint to the "discrete space" functor $$\mathrm{Set}\to \mathcal{S}$$. Applying this to the above equivalence, we learn $$\mathrm{colim}_I [S^n,X_i]_* \simeq [S^n,\mathrm{colim}_I X_i]_*,$$ or $$\mathrm{colim}_I \pi_n(X_i) \simeq \pi_n(\mathrm{colim}_I X_i).$$