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Suppose $E\simeq\text{hocolim}_iE_i$ is a filtered homotopy colimit. Suppose $X=\{X_j\}_j$ is a pro-space (assume some finiteness conditions on the spaces or spectra if you have to...for example, assume $X_i$ are simplicial sets with finitely many simplicies in each degree). Is it true that in the $\infty$-category of pro-spectra, we have $\text{hocolim}^{pro}_i\{E_i\wedge X_j\}_j\simeq \{\text{hocolim}_i(E_i\wedge X_j)\}_j$? More generally, is it true that if $E_i$ are pro-spectra with some mild finiteness condition and all indexed by the same cofiltered diagram, then I can compute their homotopy colimit degreewise?

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This is not true, unfortunately. Here's an example.

Let $E_i$ be the directed system $$ \DeclareMathOperator{\hocolim}{hocolim} \DeclareMathOperator{\holim}{holim} S^0 \to S^0 \vee S^1 \to S^0 \vee S^1 \vee S^2 \to \dots $$ of wedge inclusions, and let $X_j$ be the inverse system $$ S^0 \leftarrow S^0 \vee S^1 \leftarrow S^0 \vee S^1 \vee S^2 \leftarrow \dots $$ of projections. Both are systems of objects of finite type, and both the homotopy limit and homotopy colimit are also of finite type.

We also have a natural isomorphism $$ Map(E_i \wedge X_j, F) \cong \bigoplus_{k=0}^i \bigoplus_{l = 0}^j Map(S^{i+j}, F). $$

Let $F$ be any spectrum, viewed as a constant pro-object. Then we can show that the two spectra you describe are different by calculating homotopy classes of maps out to $F$: $$ \begin{align*} Map_{pro}(\{\hocolim_i(E_i\wedge X_j)\}_j, F) &= \hocolim_j \holim_i Map(E_i \wedge X_j, F)\\ Map_{pro}(\hocolim^{pro}_i\{E_i\wedge X_j\}_j, F) &= \holim_i \hocolim_j Map(E_i \wedge X_j, F) \end{align*} $$ Thus, we have isomorphisms $$ \begin{align*} [\{\hocolim_i(E_i\wedge X_j)\}_j, F]_{pro} &= \bigoplus_l \prod_k \pi_{k+l}(F)\\ [\hocolim^{pro}_i\{E_i\wedge X_j\}_j, F]_{pro} &= \prod_k \bigoplus_l \pi_{k+l}(F) \end{align*} $$ In particular, the natural map from the top to the bottom is not an isomorphism unless the homotopy groups of $F$ are zero above a certain degree.

As you can see, finiteness properties on the $E_i$ or $X_j$ don't buy you what you need. If you are taking a hocolim over a finite diagram instead (pushouts, finite wedges, things built from those by finitely many steps), then the fact that the category of pro-spectra is stable allows you to move the homotopy colimit inside the inverse system.

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  • $\begingroup$ (There are some applications that localize the category of pro-spectra to ensure that they have convergent Postnikov towers. In that case there might be some assumptions that might work for you.) $\endgroup$ – Tyler Lawson Apr 10 '17 at 0:47
  • $\begingroup$ Basically, the situation is that the $E_i$ are cohomologically Bousfield localized with respect to $H\mathbb{Z}/p$, and $X$ is $p$-profinite, meaning that it is a pro-object in the category of $p$-finite spaces. The final weak equivalence that I want may also be after cohomologically Bousfield localizing with respect to $H\mathbb{Z}/p$. $\endgroup$ – user108170 Apr 10 '17 at 10:32
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    $\begingroup$ @user108170, unfortunately that's not true either. If you let $M = \Bbb{RP}^2$ viewed as a based space, replace $S^n$ with $S^n \wedge M$ throughout, and take $F = \prod_{i=0}^\infty H\Bbb Z/2$, then you can get a contradiction satisfying those conditions too. $\endgroup$ – Tyler Lawson Apr 10 '17 at 17:26

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