Let \(R\) be a ring -- it is well-known that the category \(R\text{-mod}\) of (left) \(R\)-modules does not determine \(R\). For example, the functor $$-\otimes R^n: R\text{-mod}\to \text{Mat}_{n\times n}(R)\text{-mod}$$ is an equivalence of categories.

That said, there is one additional piece of data that lets us recover the ring \(R\) from the category \(R\text{-mod}\). Namely, if we let \(F\) be the functor $$F: R\text{-mod}\to \mathbb{Z}\text{-mod}$$ which sends an \(R\)-module \(M\) to its underlying abelian group, we have that $$R\simeq \text{End}(F).$$ This is a version of Tannaka duality for algebras; we call this functor \(F\) the *fiber functor*.

Let us consider a special case of this situation. Suppose \(R\) is a topological ring, complete with respect to some two-sided maximal ideal \(\mathfrak{m}\), and suppose that \(R/\mathfrak{m}=k\) is a (commutative) field. Then any continuous \(R\)-module is a (possibly infinite) iterated extension of \(k\). A little exercise shows that if \(S\) is another topological ring, complete with respect to \(\mathfrak{m}'\) and with commutative residue field \(k'\), then a (continuous) functor $$G: R\text{-mod}\to S\text{-mod}$$ sending \(k\) to \(k'\) and inducing an isomorphism $$\text{Ext}^*(k, k)\overset{\sim}{\to} \text{Ext}^*(k', k')$$ is an equivalence of categories. In particular, if such a functor commutes with the fiber functors, then $$R\simeq \text{End}(F_R)\simeq \text{End}(F_S)\simeq S.$$

In some of my work on fundamental groups, I've run into the following situation, which can be thought of as an "approximate" version of the remarks in the previous paragraph. Suppose that \(R, S\) are complete with respect to some ideals \(\mathscr{I}, \mathscr{I}'\) which are *not* maximal; rather, \(R/\mathscr{I}\simeq S/\mathscr{I}'\) are local rings whose only non-trivial ideal is the nilradical (i.e. they are direct limits of Artinian local rings). Moreover, we have a functor $$G: R\text{-mod}\to S\text{-mod}$$ which sends \(R/\mathscr{I}\) to \(S/\mathscr{I}'\). However, the induced map $$\text{Ext}^*(R/\mathscr{I}, R/\mathscr{I})\to \text{Ext}^*(S/\mathscr{I}', S/\mathscr{I}')$$ is not an isomorphism; rather, its kernel and cokernel are annihilated by some fixed element \(\beta\); moreover, there is a functor going the other way with similar properties, and both of these functors commute with fiber functors. Then one finds that \(R\), \(S\) are *approximately isomorphic*, in that there are maps \(R\to S, S\to R\) so that the induced maps $$R/\mathscr{I}^n\to S/{\mathscr{I}'}^n, S/{\mathscr{I}'}^n\to R/\mathscr{I}^n$$ have kernel and cokernel annihilated by some fixed power of \(\beta\) (in terms of \(n\).

Actually, this is not exactly the situation I am in, but a toy version which is close enough to the real thing. This kind of "approximate" Tannaka duality seems to arise naturally from certain constructions in integral \(p\)-adic Hodge theory; if someone else has thought about it, I would love to hear about it!