Let $\mathcal E$ be an $\infty$-topos. Recall that Lurie defines the shape of $\mathcal E$ as the left-exact, accessible functor $\Gamma \Delta: Spaces \to Spaces$ where $\Delta: Spaces^\to_\leftarrow \mathcal E: \Gamma$ is the the constant / global sections adjunction, the unique geometric morphism from $\mathcal E \to Spaces$. The $\infty$-category of left-exact, accessible endofunctors of $Spaces$ is better-known as the $\infty$-category of Pro-spaces.

As a pro-space, the shape of $\mathcal E$ is like having a space, but even better, because its homotopy groups are not just groups, but pro-discrete groups. For instance, the shape of the etale site of a scheme $X$ is essentially the etale homotopy type of $X$, and its fundamental groupoid is essentially the etale fundamental groupoid, complete with its usual profinite topology (I am glossing over some additional finiteness considerations; to be honest this is largely because I don't fully understand them and have even less idea how they change the picture conceptually.)

This situation -- of homotopy groups which come with a topology -- is (from some perspective) exactly what the condensed mathematics of Clausen and Scholze / pyknotic mathematics of Barwick and Haine was developed for, and my understanding is that the example of the etale homotopy type is particularly germane, as Clausen and Scholze's first application of the theory is to study the pro-etale site and its associated fundamental group. So I think the answer to the following questions must be yes, at least modulo the finiteness conditions I haven't understood:

Question 1: Is there a condensed / pyknotic "re-interpretation" of the pro-discrete topology on the homotopy groups of the shape of an arbitrary $\infty$-topos?

Question 2: Is there a condensed / pyknotic refinement of the shape of an arbitrary $\infty$-topos, perhaps containing more information than the usual pro-space?

I think if I read Clausen and Scholze, it should all be in there. Unfortunately, I'm hampered by the fact that the etale fundamental group is usually approached differently by algebraic geometers, in a way which circumvents considering all of the data of the etale homotopy type. This approach is doubtless more practical for algebraic geometry, but unfortunately I am not an algebraic geometer and I've never dug into it to understand. It appears that Clausen and Scholze use a similar approach, which is an obstacle for me to understand it. So I'm wondering if somebody can translate these formal aspects of the situation into a language which I personally happen to understand better.

Let $\mathcal E$ be an $\infty$-topos. Recall that Lurie defines the shape of $\mathcal E$ as the left-exact, accessible functor $\Gamma \Delta: Spaces \to Spaces$ where $\Delta: Spaces^\to_\leftarrow \mathcal E: \Gamma$ is the the constant / global sections adjunction, the unique geometric morphism from $\mathcal E \to Spaces$. The $\infty$-category of left-exact, accessible endofunctors of $Spaces$ is better-known as the $\infty$-category of Pro-spaces.

As a pro-space, the shape of $\mathcal E$ is like having a space, but even better, because its homotopy groups are not just groups, but pro-discrete groups. For instance, the shape of the etale site of a scheme $X$ is essentially the etale homotopy type of $X$, and its fundamental groupoid is essentially the etale fundamental groupoid, complete with its usual profinite topology (I am glossing over some additional finiteness considerations; to be honest this is largely because I don't fully understand them and have even less idea how they change the picture conceptually.)

This situation -- of homotopy groups which come with a topology -- is (from some perspective) exactly what the condensed mathematics of Clausen and Scholze / pyknotic mathematics of Barwick and Haine was developed for, and my understanding is that the example of the etale homotopy type is particularly germane, as Bhatt and Scholze's first application of the theory is to study the pro-etale site and its associated fundamental group. So I think the answer to the following questions must be yes, at least modulo the finiteness conditions I haven't understood:

Question 1: Is there a condensed / pyknotic "re-interpretation" of the pro-discrete topology on the homotopy groups of the shape of an arbitrary $\infty$-topos?

Question 2: Is there a condensed / pyknotic refinement of the shape of an arbitrary $\infty$-topos, perhaps containing more information than the usual pro-space?

I think if I read Bhatt and Scholze, it should all be in there. Unfortunately, I'm hampered by the fact that the etale fundamental group is usually approached differently by algebraic geometers, in a way which circumvents considering all of the data of the etale homotopy type. This approach is doubtless more practical for algebraic geometry (and of course, my way of putting it is completely ahistorical, as Grothendieck constructed the etale fundamental group long before Artin and Mazur constructed the etale homotopy type), but unfortunately I am not an algebraic geometer and I've never dug into it to understand. It appears that Bhatt and Scholze use a similar approach, which is an obstacle for me to understand it. So I'm wondering if somebody can translate these formal aspects of the situation into a language which I personally happen to understand better.

Let $\mathcal E$ be an $\infty$-topos. Recall that Lurie defines the shape of $\mathcal E$ as the left-exact, accessible functor $\Gamma \Delta: Spaces \to Spaces$ where $\Delta: Spaces^\to_\leftarrow \mathcal E: \Gamma$ is the the constant / global sections adjunction, the unique geometric morphism from $\mathcal E \to Spaces$. The $\infty$-category of left-exact, accessible endofunctors of $Spaces$ is better-known as the $\infty$-category of Pro-spaces.

As a pro-space, the shape of $\mathcal E$ is like having a space, but even better, because its homotopy groups are not just groups, but pro-discrete groups. For instance, the shape of the etale site of a scheme $X$ is essentially the etale homotopy type of $X$, and its fundamental groupoid is essentially the etale fundamental groupoid, complete with its usual profinite topology (I am glossing over some additional finiteness considerations; to be honest this is largely because I don't fully understand them and have even less idea how they change the picture conceptually.)

This situation -- of homotopy groups which come with a topology -- is exactly what the condensed mathematics of Clausen and Scholze / pyknotic mathematics of Barwick and Haine was developed for, and my understanding is that the example of the etale homotopy type is particularly germane, as Clausen and Scholze's first application of the theory is to study the pro-etale site and its associated fundamental group. So I think the answer to the following questions must be yes, at least modulo the finiteness conditions I haven't understood:

Question 1: Is there a condensed / pyknotic "re-interpretation" of the pro-discrete topology on the homotopy groups of the shape of an arbitrary $\infty$-topos?

Question 2: Is there a condensed / pyknotic refinement of the shape of an arbitrary $\infty$-topos, perhaps containing more information than the usual pro-space?

I think if I read Clausen and Scholze, it should all be in there. Unfortunately, I'm hampered by the fact that the etale fundamental group is usually approached differently by algebraic geometers, in a way which circumvents considering all of the data of the etale homotopy type. This approach is doubtless more practical for algebraic geometry, but unfortunately I am not an algebraic geometer and I've never dug into it to understand. It appears that Clausen and Scholze use a similar approach, which is an obstacle for me to understand it. So I'm wondering if somebody can translate these formal aspects of the situation into a language which I personally happen to understand better.

Let $\mathcal E$ be an $\infty$-topos. Recall that Lurie defines the shape of $\mathcal E$ as the left-exact, accessible functor $\Gamma \Delta: Spaces \to Spaces$ where $\Delta: Spaces^\to_\leftarrow \mathcal E: \Gamma$ is the the constant / global sections adjunction, the unique geometric morphism from $\mathcal E \to Spaces$. The $\infty$-category of left-exact, accessible endofunctors of $Spaces$ is better-known as the $\infty$-category of Pro-spaces.

As a pro-space, the shape of $\mathcal E$ is like having a space, but even better, because its homotopy groups are not just groups, but pro-discrete groups. For instance, the shape of the etale site of a scheme $X$ is essentially the etale homotopy type of $X$, and its fundamental groupoid is essentially the etale fundamental groupoid, complete with its usual profinite topology (I am glossing over some additional finiteness considerations; to be honest this is largely because I don't fully understand them and have even less idea how they change the picture conceptually.)

This situation -- of homotopy groups which come with a topology -- is (from some perspective) exactly what the condensed mathematics of Clausen and Scholze / pyknotic mathematics of Barwick and Haine was developed for, and my understanding is that the example of the etale homotopy type is particularly germane, as Clausen and Scholze's first application of the theory is to study the pro-etale site and its associated fundamental group. So I think the answer to the following questions must be yes, at least modulo the finiteness conditions I haven't understood:

Question 1: Is there a condensed / pyknotic "re-interpretation" of the pro-discrete topology on the homotopy groups of the shape of an arbitrary $\infty$-topos?

Question 2: Is there a condensed / pyknotic refinement of the shape of an arbitrary $\infty$-topos, perhaps containing more information than the usual pro-space?

I think if I read Clausen and Scholze, it should all be in there. Unfortunately, I'm hampered by the fact that the etale fundamental group is usually approached differently by algebraic geometers, in a way which circumvents considering all of the data of the etale homotopy type. This approach is doubtless more practical for algebraic geometry, but unfortunately I am not an algebraic geometer and I've never dug into it to understand. It appears that Clausen and Scholze use a similar approach, which is an obstacle for me to understand it. So I'm wondering if somebody can translate these formal aspects of the situation into a language which I personally happen to understand better.