I think I believe in two facts that seem to be contradictory. They are about the $\infty-$category of pointed spaces $S_*$, hence all the colimits of spaces are to be intended as homotopy colimits.
The compact objects of $S_*$ are the finite spaces. Recall that an object $X$ is said to be compact if $\mathrm{Map}(X, \mathrm{colim}_I X_i) \simeq \mathrm{colim}_I \mathrm{Map}(X, X_i)$ for every filtered category $I$.
$\pi_*\mathrm{colim}_I X_i \simeq \mathrm{colim}_I \pi_* X_i$ for every filtered category $I$.
I say that they seem to be contradictory because I think that from (2) one can deduce that $S^1$ is a compact object of $S_*$. In fact $\pi_*\mathrm{Map}(S^1, \mathrm{colim}_I X_i) \simeq \pi_{*+1}\mathrm{colim}_I X_i \simeq \mathrm{colim}_I \pi_{*+1} X_i \simeq \mathrm{colim}_I \pi_{*}\mathrm{Map}(S^1, X_i) \simeq \pi_{*} \mathrm{colim}_I \mathrm{Map}(S^1, X_i)$ applying (2) twice. I'd bet that this equivalence is induced by the natural map $\mathrm{colim}_I \mathrm{Map}(S^1, X_i) \to \mathrm{Map}(S^1, \mathrm{colim}_I X_i)$ and so it is a weak equivalence, hence an equivalence. But this is in contradiction with (1) since $S^1$ is not discrete.
What's wrong then: (1), (2) or my reasoning?