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?
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1$\begingroup$ Isn't that just because the spheres are compact? $\endgroup$– Dave BensonCommented Aug 12 at 7:22
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3$\begingroup$ 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) $\endgroup$– Achim KrauseCommented Aug 12 at 7:32
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$\begingroup$ 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 $\mathcal{S}$. $\endgroup$– Achim KrauseCommented Aug 12 at 7:34
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1 Answer
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Here's one way to get this out of the literature:
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 $$ \mathrm{colim}_I \pi_n(X_i) \simeq \pi_n(\mathrm{colim}_I X_i). $$
(Note that the proof of 5.3.3.3 requires a model-category level description of the colimit. This fact seems to be one of the places where one really has to get their hands dirty.)
