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

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(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

3 Answers 3

That their Cauchy completions are equivalent.

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In a sense there's nothing to add to Finn's answer: it's a necessary and sufficient condition. But I'll add something anyway, in case the nLab page seems too daunting. Given maps p: A -> B and i: B -> A in a category such that pi = 1, the map ip is idempotent. An idempotent is said to be "split" if it is of this form. A category is said to be "Cauchy complete" if all idempotents in it are split. You can always Cauchy-complete a category, by throwing in a splitting for each idempotent. –  Tom Leinster Jan 15 '12 at 23:03
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In practice, most commonly-encountered categories are already Cauchy complete. For example, any category with finite limits or with finite colimits is Cauchy complete. For such categories, Cauchy-completion does nothing, so the categories of presheaves on them are equivalent if and only if they themselves are equivalent. –  Tom Leinster Jan 15 '12 at 23:03
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A notable exception being 1-object categories that are not groups :) –  Benjamin Steinberg Jan 16 '12 at 4:26

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.

Both Finn's and my answer generalize to enriched categories and enriched presheaves; the only difference is that the notion of "Cauchy completion" is different. In general, it means completion under all absolute colimits.

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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.

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