$\DeclareMathOperator\Sh{Sh}\DeclareMathOperator\Pt{Pt}$Maybe this is well-known or even a stupid misunderstanding of something very basic. It's well-known that the groupoid of points (i.e., groupoid completion of the category of points) of $\Sh (X_\text{ét})$, $\Pt (X_\text{ét})$, is described by geometric points as objects and paths of étale specialisations and generalisations between them (SGA4). For the corresponding (pro-)étale $\infty$-topoi, $\Sh_{\infty} (X_\text{(pro-)ét})$, the $\infty$-groupoid of points is $1$-truncated since $\Sh_{\infty} (X_\text{(pro-)ét})$ is $1$-localic and, hence, it doesn't matter if one works with the respective $\infty$-topoi.

On the other side, $\Pt (X_\text{pro-ét})$ has geometric points as points, however such collection is not even conservative (see 61.18 Points of the pro-étale site).

Now, there's also the shape of $\Sh_{\infty} (X_\text{ét})$, $\Pi_{\infty} (X_\text{ét})$, and its profinite version, $\widehat{\Pi}_{\infty} (X_\text{ét})$, which coincides with the étale homotopy type of $X$. When $X$ is qcqs, it's also known that $\Pi_{\infty} (X_\text{ét})\cong \Pi_{\infty} (X_\text{pro-ét})$ (6.1.6 in Barwick, Glasman, and Haine - Exodromy).

Let $\pi^\text{BS}_1 (X, \overline{x})$ be defined as the automorphisms of a fiber functor from étale coverings satisfying the valuative criterion for properness (Bhatt and Scholze - The pro-étale topology for schemes). In $\Sh (X_\text{pro-ét})$, there are several points that are not geometric (as mentioned above) and, in fact, they are not even a conservative family. On the other side, $\pi_1 ({\Pi}_{\infty} (X_\text{ét}), \overline{x})$, when $X$ is connected locally Noetherian, has dense image in $\pi^{BS}_1 (X)$ (Rem 7.4.12 in Bhatt and Scholze - The pro-étale topology for schemes).

1. What's the relation between $\pi^\text{BS}_1 (X, \overline{x})$ and $\Pt (X_\text{pro-ét})$? Is it just the connected component of $\overline{x}$?

2. Is every element of $\pi^\text{BS}_1 (X, \overline{x})$ given by a path of specialisations and generalisations?

3. What are the points and paths in $\Pt (X_\text{pro-ét})$? I think the points are given by connected affine $w$-contractible objects, i.e., connected $w$-strict local rings, i.e., a local ring having a section for every affine fpqc covering.

Now, let's purposefully complicate everything a little more. For $X$ qcqs, one can define a profinite stratified space or, equivalently (Hochster's duality), a spectral stratified $\infty$-topoi ${\widehat{\Pi}_{(\infty, 1)}^{X_\text{Zar}} (X_\text{(pro-)ét})}$ (Barwick, Glasman, and Haine - Exodromy). By taking the groupoid completion, one gets back $\Pi_{\infty} (X_\text{(pro-)et})$ (recall that, as mentioned above, changing from étale to pro-étale gives an equivalent shape) and, by taking the materialisation (I guess it means just taking the real limit in spaces of a profinite stratified space), one gets a $X_\text{Zar}$-stratified version of $Pt (X_\text{(pro-)ét})$ before the groupoid completion.

4. Do the materialisations coincide at first or after inverting all the maps?

5. What's really the difference between ${\widehat{\Pi}_{(\infty, 1)}^{X_\text{Zar}} (X_\text{ét})}$ and ${\widehat{\Pi}_{(\infty, 1)}^{X_\text{Zar}} (X_\text{pro-ét})}$?

Maybe I've screwed up something in my assertions. If so, please, comment below.

$\DeclareMathOperator\Sh{Sh}\DeclareMathOperator\Pt{Pt}$Maybe this is well-known or even a stupid misunderstanding of something very basic. It's well-known that the groupoid of points (i.e., groupoid completion of the category of points) of $\Sh (X_\text{ét})$, $\Pt (X_\text{ét})$, is described by geometric points as objects and paths of étale specialisations and generalisations between them (SGA4). For the corresponding (pro-)étale $\infty$-topoi, $\Sh_{\infty} (X_\text{(pro-)ét})$, the $\infty$-groupoid of points is $1$-truncated since $\Sh_{\infty} (X_\text{(pro-)ét})$ is $1$-localic and, hence, it doesn't matter if one works with the respective $\infty$-topoi.

On the other side, $\Pt (X_\text{pro-ét})$ has geometric points as points, however such collection is not even conservative (see 61.18 Points of the pro-étale site).

Now, there's also the shape of $\Sh_{\infty} (X_\text{ét})$, $\Pi_{\infty} (X_\text{ét})$, and its profinite version, $\widehat{\Pi}_{\infty} (X_\text{ét})$, which coincides with the étale homotopy type of $X$. When $X$ is qcqs, it's also known that $\Pi_{\infty} (X_\text{ét})\cong \Pi_{\infty} (X_\text{pro-ét})$ (6.1.6 in Barwick, Glasman, and Haine - Exodromy).

Let $\pi^\text{BS}_1 (X, \overline{x})$ be defined as the automorphisms of a fiber functor from étale coverings satisfying the valuative criterion for properness (Bhatt and Scholze - The pro-étale topology for schemes). In $\Sh (X_\text{pro-ét})$, there are several points that are not geometric (as mentioned above) and, in fact, they are not even a conservative family. On the other side, $\pi_1 ({\Pi}_{\infty} (X_\text{ét}), \overline{x})$, when $X$ is connected locally Noetherian, has dense image in $\pi^{BS}_1 (X)$ (Rem 7.4.12 in Bhatt and Scholze - The pro-étale topology for schemes).

1. What's the relation between $\pi^\text{BS}_1 (X, \overline{x})$ and $\Pt (X_\text{pro-ét})$? Is it just the connected component of $\overline{x}$?

2. Is every element of $\pi^\text{BS}_1 (X, \overline{x})$ given by a path of specialisations and generalisations?

3. What are the points and paths in $\Pt (X_\text{pro-ét})$? I think the points are given by connected affine $w$-contractible objects, i.e., connected $w$-strict local rings, i.e., a local ring having a section for every affine fpqc covering.

Now, let's purposefully complicate everything a little more. For $X$ qcqs, one can define a profinite stratified space or, equivalently (Hochster's duality), a spectral stratified $\infty$-topoi ${\widehat{\Pi}_{(\infty, 1)}^{X_\text{Zar}} (X_\text{(pro-)ét})}$ (Barwick, Glasman, and Haine - Exodromy). By taking the groupoid completion, one gets back $\Pi_{\infty} (X_\text{(pro-)et})$ (recall that, as mentioned above, changing from étale to pro-étale gives an equivalent shape) and, by taking the materialisation (I guess it means just taking the real limit in spaces of a profinite stratified space), one gets a $X_\text{Zar}$-stratified version of $\Pt (X_\text{(pro-)ét})$ before the groupoid completion.

4. Do the materialisations coincide at first or after inverting all the maps?

5. What's really the difference between ${\widehat{\Pi}_{(\infty, 1)}^{X_\text{Zar}} (X_\text{ét})}$ and ${\widehat{\Pi}_{(\infty, 1)}^{X_\text{Zar}} (X_\text{pro-ét})}$?

Maybe I've screwed up something in my assertions. If so, please, comment below.

Maybe this is well-known or even a stupid misunderstanding of something very basic. It's well-known that the groupoid of points (i.e., groupoid completion of the category of points) of $Sh (X_{ét})$, $Pt (X_{ét})$, is described by geometric points as objects and paths of étale specialisations and generalisations between them (SGA4). For the corresponding (pro)-étale $\infty$-topoi, $Sh_{\infty} (X_{(pro)-ét})$, the $\infty$-groupoid of points is $1$-truncated since $Sh_{\infty} (X_{(pro)-ét})$ is $1$-localic and, hence, it doesn't maker if one works with the respective $\infty$-topoi.

On the other side, $Pt (X_{pro-ét})$ has geometric points as points, however such collection is not even conservative (https://stacks.math.columbia.edu/tag/0991).

Now, there's also the shape of $Sh_{\infty} (X_{ét})$, $\Pi_{\infty} (X_{ét})$, and its profinite version, $\widehat{\Pi}_{\infty} (X_{ét})$, which coincides with the étale homotopy type of $X$. When $X$ is qcqs, it's also known that $\Pi_{\infty} (X_{ét})\cong \Pi_{\infty} (X_{pro-ét})$ (6.1.6 in https://arxiv.org/pdf/1807.03281.pdf).

Let $\pi^{BS}_1 (X, \overline{x})$ be defined as the automorphisms of a fiber functor from étale coverings satisfying the valuative criterion for properness (https://arxiv.org/pdf/1309.1198.pdf). In $Sh (X_{pro-ét})$, there are several points that are not geometric (as mentioned above) and, in fact, they are not even a conservative family. On the other side, $\pi_1 ({\Pi}_{\infty} (X_{ét}), \overline{x})$, when $X$ is connected locally Noetherian, has dense image in $\pi^{BS}_1 (X)$ (Rem 7.4.12 https://arxiv.org/pdf/1309.1198.pdf).

1. What's the relation between $\pi^{BS}_1 (X, \overline{x})$ and $Pt (X_{pro-ét})$? Is it just the connected component of $\overline{x}$?

2. Is every element of $\pi^{BS}_1 (X, \overline{x})$ given by a path of specialisations and generalisations?

3. What are the points and paths in $Pt (X_{pro-ét})$? I think the points are given by connected affine $w$-contractible objects, i.e., connected $w$-strict local rings, i.e., a local ring having a section for every affine fpqc covering.

Now, let's purposefully complicate everything a little more. For $X$ qcqs, one can define a profinite stratified space or, equivalently (Hochster's duality), a spectral stratified $\infty$-topoi ${\widehat{\Pi}_{(\infty, 1)}^{X_{Zar}} (X_{(pro)-ét})}$ (https://arxiv.org/pdf/1807.03281.pdf). By taking the groupoid completion, one gets back $\Pi_{\infty} (X_{(pro)-et})$ (recall that, as mentioned above, changing from étale to pro-stale gives an equivalent shape) and, by taking the materialisation (I guess it means just taking the real limit in spaces of a profinite stratified space, ), one gets a $X_{Zar}$-stratified version of $Pt (X_{(pro)-et})$ before the groupoid completion.

4. Do the materialisations coincide at first or after inverting all the maps?

5. What's really the difference between ${\widehat{\Pi}_{(\infty, 1)}^{X_{Zar}} (X_{ét})}$ and ${\widehat{\Pi}_{(\infty, 1)}^{X_{Zar}} (X_{pro-ét})}$?

Maybe I've screwed up something in my assertions. If so, please, comment below.

Thanks in advance.

$\DeclareMathOperator\Sh{Sh}\DeclareMathOperator\Pt{Pt}$Maybe this is well-known or even a stupid misunderstanding of something very basic. It's well-known that the groupoid of points (i.e., groupoid completion of the category of points) of $\Sh (X_\text{ét})$, $\Pt (X_\text{ét})$, is described by geometric points as objects and paths of étale specialisations and generalisations between them (SGA4). For the corresponding (pro-)étale $\infty$-topoi, $\Sh_{\infty} (X_\text{(pro-)ét})$, the $\infty$-groupoid of points is $1$-truncated since $\Sh_{\infty} (X_\text{(pro-)ét})$ is $1$-localic and, hence, it doesn't matter if one works with the respective $\infty$-topoi.

On the other side, $\Pt (X_\text{pro-ét})$ has geometric points as points, however such collection is not even conservative (see 61.18 Points of the pro-étale site).

Now, there's also the shape of $\Sh_{\infty} (X_\text{ét})$, $\Pi_{\infty} (X_\text{ét})$, and its profinite version, $\widehat{\Pi}_{\infty} (X_\text{ét})$, which coincides with the étale homotopy type of $X$. When $X$ is qcqs, it's also known that $\Pi_{\infty} (X_\text{ét})\cong \Pi_{\infty} (X_\text{pro-ét})$ (6.1.6 in Barwick, Glasman, and Haine - Exodromy).

Let $\pi^\text{BS}_1 (X, \overline{x})$ be defined as the automorphisms of a fiber functor from étale coverings satisfying the valuative criterion for properness (Bhatt and Scholze - The pro-étale topology for schemes). In $\Sh (X_\text{pro-ét})$, there are several points that are not geometric (as mentioned above) and, in fact, they are not even a conservative family. On the other side, $\pi_1 ({\Pi}_{\infty} (X_\text{ét}), \overline{x})$, when $X$ is connected locally Noetherian, has dense image in $\pi^{BS}_1 (X)$ (Rem 7.4.12 in Bhatt and Scholze - The pro-étale topology for schemes).

1. What's the relation between $\pi^\text{BS}_1 (X, \overline{x})$ and $\Pt (X_\text{pro-ét})$? Is it just the connected component of $\overline{x}$?

2. Is every element of $\pi^\text{BS}_1 (X, \overline{x})$ given by a path of specialisations and generalisations?

3. What are the points and paths in $\Pt (X_\text{pro-ét})$? I think the points are given by connected affine $w$-contractible objects, i.e., connected $w$-strict local rings, i.e., a local ring having a section for every affine fpqc covering.

Now, let's purposefully complicate everything a little more. For $X$ qcqs, one can define a profinite stratified space or, equivalently (Hochster's duality), a spectral stratified $\infty$-topoi ${\widehat{\Pi}_{(\infty, 1)}^{X_\text{Zar}} (X_\text{(pro-)ét})}$ (Barwick, Glasman, and Haine - Exodromy). By taking the groupoid completion, one gets back $\Pi_{\infty} (X_\text{(pro-)et})$ (recall that, as mentioned above, changing from étale to pro-étale gives an equivalent shape) and, by taking the materialisation (I guess it means just taking the real limit in spaces of a profinite stratified space), one gets a $X_\text{Zar}$-stratified version of $Pt (X_\text{(pro-)ét})$ before the groupoid completion.

4. Do the materialisations coincide at first or after inverting all the maps?

5. What's really the difference between ${\widehat{\Pi}_{(\infty, 1)}^{X_\text{Zar}} (X_\text{ét})}$ and ${\widehat{\Pi}_{(\infty, 1)}^{X_\text{Zar}} (X_\text{pro-ét})}$?

Maybe I've screwed up something in my assertions. If so, please, comment below.

Maybe this is well-known or even a stupid misunderstanding of something very basic. It's well-known that the groupoid of points (i.e., groupoid completion of the category of points) of $Sh (X_{ét})$, $Pt (X_{ét})$, is described by geometric points as objects and paths of étale specialisations and generalisations between them (SGA4). For the corresponding (pro)-étale $\infty$-topoi, $Sh_{\infty} (X_{(pro)-ét})$, the $\infty$-groupoid of points is $1$-truncated since $Sh_{\infty} (X_{(pro)-ét})$ is $1$-localic and, hence, it doesn't maker if one works with the respective $\infty$-topoi.

On the other side, $Pt (X_{pro-ét})$ has geometric points as points, however such collection is not even conservative (https://stacks.math.columbia.edu/tag/0991).

Now, there's also the shape of $Sh_{\infty} (X_{ét})$, $\Pi_{\infty} (X_{ét})$, and its profinite version, $\widehat{\Pi}_{\infty} (X_{ét})$, which coincides with the étale homotopy type of $X$. When $X$ is qcqs, it's also known that $\Pi_{\infty} (X_{ét})\cong \Pi_{\infty} (X_{pro-ét})$ (6.1.6 in https://arxiv.org/pdf/1807.03281.pdf).

Let $\pi^{BS}_1 (X, \overline{x})$ be defined as the automorphisms of a fiber functor from étale coverings satisfying the valuative criterion for properness (https://arxiv.org/pdf/1309.1198.pdf). In $Sh (X_{pro-ét})$, there are several points that are not geometric (as mentioned above) and, in fact, they are not even a conservative family. On the other side, $\pi_1 (\widehat{\Pi}_{\infty} (X_{ét}), \overline{x})$ is the usual profinite étale $\widehat{\pi}_1 (X, \overline{x})$ and, when $X$ is connected locally Noetherian, it has dense image in $\pi^{BS}_1 (X)$ (Rem 7.4.12 https://arxiv.org/pdf/1309.1198.pdf).

1. What's the relation between $\pi^{BS}_1 (X, \overline{x})$ and $Pt (X_{pro-ét})$? Is it just the connected component of $\overline{x}$?

2. Is every element of $\pi^{BS}_1 (X, \overline{x})$ given by a path of specialisations and generalisations?

3. What are the points and paths in $Pt (X_{pro-ét})$? I think the points are given by connected affine $w$-contractible objects, i.e., connected $w$-strict local rings, i.e., a local ring having a section for every affine fpqc covering.

Now, let's purposefully complicate everything a little more. For $X$ qcqs, one can define a profinite stratified space or, equivalently (Hochster's duality), a spectral stratified $\infty$-topoi ${\widehat{\Pi}_{(\infty, 1)}^{X_{Zar}} (X_{(pro)-ét})}$ (https://arxiv.org/pdf/1807.03281.pdf). By taking the groupoid completion, one gets back $\Pi_{\infty} (X_{(pro)-et})$ (recall that, as mentioned above, changing from étale to pro-stale gives an equivalent shape) and, by taking the materialisation (I guess it means just taking the real limit in spaces of a profinite stratified space, ), one gets a $X_{Zar}$-stratified version of $Pt (X_{(pro)-et})$ before the groupoid completion.

4. Do the materialisations coincide at first or after inverting all the maps?

5. What's really the difference between ${\widehat{\Pi}_{(\infty, 1)}^{X_{Zar}} (X_{ét})}$ and ${\widehat{\Pi}_{(\infty, 1)}^{X_{Zar}} (X_{pro-ét})}$?

Maybe I've screwed up something in my assertions. If so, please, comment below.

Thanks in advance.

Maybe this is well-known or even a stupid misunderstanding of something very basic. It's well-known that the groupoid of points (i.e., groupoid completion of the category of points) of $Sh (X_{ét})$, $Pt (X_{ét})$, is described by geometric points as objects and paths of étale specialisations and generalisations between them (SGA4). For the corresponding (pro)-étale $\infty$-topoi, $Sh_{\infty} (X_{(pro)-ét})$, the $\infty$-groupoid of points is $1$-truncated since $Sh_{\infty} (X_{(pro)-ét})$ is $1$-localic and, hence, it doesn't maker if one works with the respective $\infty$-topoi.

On the other side, $Pt (X_{pro-ét})$ has geometric points as points, however such collection is not even conservative (https://stacks.math.columbia.edu/tag/0991).

Now, there's also the shape of $Sh_{\infty} (X_{ét})$, $\Pi_{\infty} (X_{ét})$, and its profinite version, $\widehat{\Pi}_{\infty} (X_{ét})$, which coincides with the étale homotopy type of $X$. When $X$ is qcqs, it's also known that $\Pi_{\infty} (X_{ét})\cong \Pi_{\infty} (X_{pro-ét})$ (6.1.6 in https://arxiv.org/pdf/1807.03281.pdf).

Let $\pi^{BS}_1 (X, \overline{x})$ be defined as the automorphisms of a fiber functor from étale coverings satisfying the valuative criterion for properness (https://arxiv.org/pdf/1309.1198.pdf). In $Sh (X_{pro-ét})$, there are several points that are not geometric (as mentioned above) and, in fact, they are not even a conservative family. On the other side, $\pi_1 ({\Pi}_{\infty} (X_{ét}), \overline{x})$, when $X$ is connected locally Noetherian, has dense image in $\pi^{BS}_1 (X)$ (Rem 7.4.12 https://arxiv.org/pdf/1309.1198.pdf).

1. What's the relation between $\pi^{BS}_1 (X, \overline{x})$ and $Pt (X_{pro-ét})$? Is it just the connected component of $\overline{x}$?

2. Is every element of $\pi^{BS}_1 (X, \overline{x})$ given by a path of specialisations and generalisations?

3. What are the points and paths in $Pt (X_{pro-ét})$? I think the points are given by connected affine $w$-contractible objects, i.e., connected $w$-strict local rings, i.e., a local ring having a section for every affine fpqc covering.

Now, let's purposefully complicate everything a little more. For $X$ qcqs, one can define a profinite stratified space or, equivalently (Hochster's duality), a spectral stratified $\infty$-topoi ${\widehat{\Pi}_{(\infty, 1)}^{X_{Zar}} (X_{(pro)-ét})}$ (https://arxiv.org/pdf/1807.03281.pdf). By taking the groupoid completion, one gets back $\Pi_{\infty} (X_{(pro)-et})$ (recall that, as mentioned above, changing from étale to pro-stale gives an equivalent shape) and, by taking the materialisation (I guess it means just taking the real limit in spaces of a profinite stratified space, ), one gets a $X_{Zar}$-stratified version of $Pt (X_{(pro)-et})$ before the groupoid completion.

4. Do the materialisations coincide at first or after inverting all the maps?

5. What's really the difference between ${\widehat{\Pi}_{(\infty, 1)}^{X_{Zar}} (X_{ét})}$ and ${\widehat{\Pi}_{(\infty, 1)}^{X_{Zar}} (X_{pro-ét})}$?

Maybe I've screwed up something in my assertions. If so, please, comment below.

Thanks in advance.