TITLE: "Surprising" categorical equivalences
QUESTION [8 upvotes]: This is inspired by this question about the equivalence between the category of finite sets and non-negative integers. Now this question was (rightly, I guess) closed, but the fact was surprising to the OPer. I didn't know about this, but it was easy to verify. I had a little bit of difficulty understanding the nuances between categorical equivalences and category isomorphism until I thought about the analogy with homotopic spaces and homeomorphic spaces.
I was wondering about other equivalences that might unexpected and/or not so straightforward to prove. I would appreciate answers to a general audience...things that a beginner or an expert might find of interest.

REPLY [3 votes]: I was pretty shocked by this one: if $V$ is the additive group of a finite-dimensional vector space over a $p$-adic field, the category of smooth (sometimes called algebraic or discrete) complex representations of $V$ is equivalent to the category of sheaves of complex vector spaces on the Pontryagin dual of $V$.
More generally, this is true of any Hausdorff, locally compact, and totally disconnected abelian group which is a filtered union of its compact open subgroups.