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I am in the situation of having to understand groups $[\mathbb{S},E\wedge X_+]:=\pi_0\text{Map}_{\text{Sp}(\text{Pro}(\mathcal{S}))}(\mathbb{S},E\wedge X_+)$ in $\text{Sp}(\text{Pro}(\mathcal{S}))$, the stabilization of the $\infty$-category of pro-spaces. Here, $E$ is an object of $\text{Sp}(\text{Pro}(\mathcal{S}))$, and $X$ is a pro-space. Suppose $X=\{X_{\alpha}\}$ is a projective system of spaces. Sanity check: Is it true that $\text{Map}_{\text{Sp}(\text{Pro}(\mathcal{S}))}(\mathbb{S},E\wedge X_+)=\text{holim}_{\alpha} \text{Map}_{\text{Sp}(\text{Pro}(\mathcal{S}))}(\mathbb{S},E\wedge X_{\alpha+})$? If the groups $[\mathbb{S},E\wedge X_{\alpha+}]$ are all finite, then I suspect that we would then have $[\mathbb{S},E\wedge X_+]=\text{lim}_{\alpha} [\mathbb{S},E\wedge X_{\alpha+}]$, but is there a Mittag-Lefler type of argument for when the indexing set is not sequential?

I edited the question in response to Tyler Lawson's comment.

I am in the situation of having to understand groups $[\mathbb{S},E\wedge X_+]:=\pi_0\text{Map}_{\text{Sp}(\text{Pro}(\mathcal{S}))}(\mathbb{S},E\wedge X_+)$ in $\text{Sp}(\text{Pro}(\mathcal{S}))$, the stabilization of the $\infty$-category of pro-spaces. Here, $E$ is an object of $\text{Sp}(\text{Pro}(\mathcal{S}))$, and $X$ is a pro-space. Suppose $X=\{X_{\alpha}\}$ is a projective system of spaces. It is true that $\text{Map}_{\text{Sp}(\text{Pro}(\mathcal{S}))}(\mathbb{S},E\wedge X_+)=\text{holim}_{\alpha} \text{Map}_{\text{Sp}(\text{Pro}(\mathcal{S}))}(\mathbb{S},E\wedge X_{\alpha+})$. If the groups $[\mathbb{S},E\wedge X_{\alpha+}]$ are all finite, then I suspect that we would then have $[\mathbb{S},E\wedge X_+]=\text{lim}_{\alpha} [\mathbb{S},E\wedge X_{\alpha+}]$, but is there a Mittag-Lefler type of argument for when the indexing set is not sequential?

I edited the question in response to Tyler Lawson's comment.

I am in the situation of having to understand groups $[\mathbb{S},E\wedge X_+]:=\pi_0\text{Map}_{\text{Sp}(\text{Pro}(\mathcal{S}))}(\mathbb{S},E\wedge X_+)$ in $\text{Sp}(\text{Pro}(\mathcal{S}))$, the stabilization of the $\infty$-category of pro-spaces. Here, $E$ is an object of $\text{Sp}(\text{Pro}(\mathcal{S}))$, and $X$ is a pro-space. Suppose $X=\{X_{\alpha}\}$ is a projective system of spaces. Sanity check: Is it true that $\text{Map}_{\text{Sp}(\text{Pro}(\mathcal{S}))}(\mathbb{S},E\wedge X_+)=\text{holim}_{\alpha} \text{Map}_{\text{Sp}(\text{Pro}(\mathcal{S}))}(\mathbb{S},E\wedge X_{\alpha+})$?

I edited the question in response to Tyler Lawson's comment.

I am in the situation of having to understand groups $[\mathbb{S},E\wedge X_+]:=\pi_0\text{Map}_{\text{Sp}(\text{Pro}(\mathcal{S}))}(\mathbb{S},E\wedge X_+)$ in $\text{Sp}(\text{Pro}(\mathcal{S}))$, the stabilization of the $\infty$-category of pro-spaces. Here, $E$ is an object of $\text{Sp}(\text{Pro}(\mathcal{S}))$, and $X$ is a pro-space. Suppose $X=\{X_{\alpha}\}$ is a projective system of spaces. Sanity check: Is it true that $\text{Map}_{\text{Sp}(\text{Pro}(\mathcal{S}))}(\mathbb{S},E\wedge X_+)=\text{holim}_{\alpha} \text{Map}_{\text{Sp}(\text{Pro}(\mathcal{S}))}(\mathbb{S},E\wedge X_{\alpha+})$? If the groups $[\mathbb{S},E\wedge X_{\alpha+}]$ are all finite, then I suspect that we would then have $[\mathbb{S},E\wedge X_+]=\text{lim}_{\alpha} [\mathbb{S},E\wedge X_{\alpha+}]$, but is there a Mittag-Lefler type of argument for when the indexing set is not sequential?

I edited the question in response to Tyler Lawson's comment.

I am in the situation of having to understand groups $[\mathbb{S},E\wedge X_+]:=\pi_0\text{Map}_{\text{Sp}(\text{Pro}(\mathcal{S}))}(\mathbb{S},E\wedge X_+)$ in $\text{Sp}(\text{Pro}(\mathcal{S}))$, the stabilization of the $\infty$-category of pro-spaces. Here, $E$ is an object of $\text{Sp}(\text{Pro}(\mathcal{S}))$, and $X$ is a pro-space. Suppose $X=\lim_{\alpha} X_{\alpha}$ is a representation of $X$ as a projective system of spaces. Sanity check: Is it true that $[\mathbb{S},E\wedge X_+]=\lim_{\alpha} [\mathbb{S},E\wedge X_{\alpha+}]$?

I am in the situation of having to understand groups $[\mathbb{S},E\wedge X_+]:=\pi_0\text{Map}_{\text{Sp}(\text{Pro}(\mathcal{S}))}(\mathbb{S},E\wedge X_+)$ in $\text{Sp}(\text{Pro}(\mathcal{S}))$, the stabilization of the $\infty$-category of pro-spaces. Here, $E$ is an object of $\text{Sp}(\text{Pro}(\mathcal{S}))$, and $X$ is a pro-space. Suppose $X=\{X_{\alpha}\}$ is a projective system of spaces. Sanity check: Is it true that $\text{Map}_{\text{Sp}(\text{Pro}(\mathcal{S}))}(\mathbb{S},E\wedge X_+)=\text{holim}_{\alpha} \text{Map}_{\text{Sp}(\text{Pro}(\mathcal{S}))}(\mathbb{S},E\wedge X_{\alpha+})$?

I edited the question in response to Tyler Lawson's comment.