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Can filtered colimits be computed in the homotopy category?

No, for a diagram $X: I \to \mathcal{S} \to h\mathcal{S}$ the colimit in $h\mathcal{S}$ would satisfy $[\mathrm{colim} X(i),Y] \cong \lim [X(i),Y]$ where brackets denote morphisms in $h\mathcal{S}$. …
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