Let $R$ be a ring, and $\mathfrak{p}$ be a prime ideal. The stalk at $\mathfrak{p}$ with respect to the etale topology is $(R_{\mathfrak{p}})^{sh}$ (the strict henselization of $R_{\mathfrak{p}}$). The stalk at $\mathfrak{p}$ with respect to the Nisnevich topology is $(R_{\mathfrak{p}})^h$ (the henselization of $R_{\mathfrak{p}}$).

Grothendieck also spoke of formal neighborhoods, and I wonder if this fits into the pattern above. To be precise: is there some Grothendieck topology for which the stalk at $\mathfrak{p}$ would be the completion of $R_{\mathfrak{p}}$ with respect to $\mathfrak{p}$? If so, what is it?

no. – Charles Rezk Dec 19 '10 at 3:07