The following type of statements is often bandied about around the mathematical watering hole: "etale close is closer than formally close". Namely, if one imagines some sort of 'absolute topology' encompassing all the usual Grothendieck topologies on schemes (fppf, fpqc, etale, Nisnevich,...) that zooming in 'formally close' is a coarser zoom than zooming in 'etale close'.

Can someone give meat to these statements? An actual comparison statement of some form?

Here is a sampler of the sort of things I'm curious about:

1. Is there a natural Grothendieck topology on schemes whose 'local rings' are completions? Namely, in the etale topology the local ring of a scheme $X$ at a point $x$ is $\mathcal{O}_{X,x}^\text{sh}$. Is there a natural Grothendieck topology whose local ring of $X$ at $x$ is $\widehat{\mathcal{O}_{X,x}}$?
2. Does this topology have a natural notion of fundamental groups? Is there some category of 'finite ____ covers" which forms a reasonable Galois category? How do the fundamental groups in this category compare?
3. A literal question: Let $R$ be any local ring. How does, $\pi_1^{\acute{et}}(\text{Spec}(\widehat{R}))$ relate to $\pi_1^{\acute{e}t}(\text{Spec}(R^\ast))$ for $\ast\in\{\text{h},\text{sh}\}$? Similarly, how does $G_{\text{Frac}(\widehat{R})}$ (if $R$ is a domain) relate to $G_{\text{Frac}(R^\text{h})}$ and $G_{\text{Frac}(R^{\text{sh}})}$? What if $R$ is a DVR?

One reason I'm curious is that it's often easy to make nice intuitive relationships between geometry and the formal local geometry of a scheme (say over $\mathbb{C}$) but less obvious to me how to make analogies using henselizations.

For example, when one wants to intuit $G_{\mathbb{Q}_p}$ as a 'group classifying the geometry of a punctured $\text{Spec}(\mathbb{Z}$) ' one often times looks to $G_{\mathbb{C}(T)}$. There one sees that the analogue of $G_{\mathbb{Q}_p}$ is $G_{\mathbb{C}((T))}$ which one can imagine as being the (profinite) fundamental group of a small punctured disk around a point $p$ which makes total sense, and is beautiful. That said, it's a small punctured formal disk around the point whereas, for me, it would have made more sense to consider small etale disks (whatever this means) around the point.

This is further backed up by the point that if $D^\ast$ is a small punctured disk around $p$ then $\widehat{\pi_1(D_p)}=\widehat{\mathbb{Z}}=G_{\mathbb{C}((T))}$. This should still be true if $\mathbb{C}((T))$ is replaced by $\mathbb{C}[T]_{(t)}^{\text{sh}}$, but it's then less clear that this equivalence of fundamental groups is not a mistake.

Thanks!

The following type of statements is often bandied about around the mathematical watering hole: "etale close is closer than formally close". Namely, if one imagines some sort of 'absolute topology' encompassing all the usual Grothendieck topologies on schemes (fppf, fpqc, etale, Nisnevich,...) that zooming in 'formally close' is a coarser zoom than zooming in 'etale close'.

Can someone give meat to these statements? An actual comparison statement of some form? Also, perhaps more pressingly, are they 'intuitively true' (I've never really seen a rigorous formalization of these statements made)?

Here is a sampler of the sort of things I'm curious about:

1. Is there a natural Grothendieck topology on schemes whose 'local rings' are completions? Namely, in the etale topology the local ring of a scheme $X$ at a point $x$ is $\mathcal{O}_{X,x}^\text{sh}$. Is there a natural Grothendieck topology whose local ring of $X$ at $x$ is $\widehat{\mathcal{O}_{X,x}}$?
2. Does this topology have a natural notion of fundamental groups? Is there some category of 'finite ____ covers" which forms a reasonable Galois category? How do the fundamental groups in this category compare?
3. A literal question: Let $R$ be any local ring. How does, $\pi_1^{\acute{et}}(\text{Spec}(\widehat{R}))$ relate to $\pi_1^{\acute{e}t}(\text{Spec}(R^\ast))$ for $\ast\in\{\text{h},\text{sh}\}$? Similarly, how does $G_{\text{Frac}(\widehat{R})}$ (if $R$ is a domain) relate to $G_{\text{Frac}(R^\text{h})}$ and $G_{\text{Frac}(R^{\text{sh}})}$? What if $R$ is a DVR?
4. I'm almost positive that Artin approximation enters this picture, but not totally sure how so.

One reason I'm curious is that it's often easy to make nice intuitive relationships between geometry and the formal local geometry of a scheme (say over $\mathbb{C}$) but less obvious to me how to make analogies using henselizations.

For example, when one wants to intuit $G_{\mathbb{Q}_p}$ as a 'group classifying the geometry of a punctured $\text{Spec}(\mathbb{Z}$) near $p$ ' one often times looks to $G_{\mathbb{C}(T)}$. There one sees that the analogue of $G_{\mathbb{Q}_p}$ is $G_{\mathbb{C}((T))}$ which one can imagine as being the (profinite) fundamental group of a small punctured disk around a point $p$ which makes total sense, and is beautiful. That said, it's a small punctured formal disk around the point whereas, for me, it would have made more sense to consider small etale disks (whatever this means) around the point.

This is further backed up by the point that if $D^\ast$ is a small punctured disk around $p$ then $\widehat{\pi_1(D^\ast)}=\widehat{\mathbb{Z}}=G_{\mathbb{C}((T))}$. This should still be true if $\mathbb{C}((T))$ is replaced by $\mathbb{C}[T]_{(t)}^{\text{sh}}$, but it's then less clear that this equivalence of fundamental groups is not a mistake.

Thanks!