# morita equivalence for categories

Suppose we have two categories such that presheaves on them are equivalent. What can be said in this situation?

• (I accidentally voted this down, instead of voting it up. If you edit the post, then I can correct my error! Please accept my apologies.) – Manny Reyes Jan 17 '12 at 14:50

Finn's answer, as expanded by Tom, is great. However, since the question is so broad ("what can be said?") let me point out that there is a different thing that can also be said. If $C$ and $D$ have equivalent categories of presheaves, then $C$ and $D$ are equivalent objects in the bicategory of profunctors. In other words, there are profunctors $C \to D$ and $D\to C$ whose composites in either direction are naturally isomorphic to identities. This is also a necessary and sufficient condition.
Let $k$ be a field. If $A$ and $B$ are (small) $k$-linear categories with $K$-linearly equivalent categories of $k$-linear functors to $k$-vector spaces, then $A$ can be obtained from $B$ by a sequence of "contractions and expansions» from $B$. See Theorem 4.7 in this paper by Claude Cibils and Andrea Solotar.