# 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.) Commented Jan 17, 2012 at 14:50

That their Cauchy completions are equivalent.

• 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. Commented Jan 15, 2012 at 23:03
• 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. Commented Jan 15, 2012 at 23:03
• A notable exception being 1-object categories that are not groups :) Commented Jan 16, 2012 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.

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.

In short: it happens exactly when there is a bigger category and disjoint full embeddings of the two categories $$\mathcal A,\mathcal B$$ into it, such that every arrow is a composition of arrows of the form $$a\to b$$ or $$b\to a$$.

It's easy to see that the Eilenberg-Watts theorem holds for monoid actions, moreover, it also holds for categories, presheaves and profunctors in place of rings, modules and bimodules:

Theorem.
Let $$\mathcal A$$ and $$\mathcal B$$ be categories, and $$F:\mathcal{Set}^{\mathcal A}\to\mathcal{Set}^{\mathcal B}$$ has a right adjoint, then there is a profunctor $$U:\mathcal A\not\to\mathcal B$$, such that $$F\simeq (H\mapsto H\otimes_{\mathcal A}U)$$.

In other words, in the realm of modules (even over categories), the only adjunctions are the tensor-hom adjunctions.
[As a note, $$U$$ can be obtained from $$F$$ by setting $$U(a,b)=F(\hom_{\mathcal A}(a,-))\,(b)\$$.]

If $$F$$ is an equivalence of categories, then its inverse $$G$$ is both its left and right adjoint, so by the above theorem, $$F\simeq(-\otimes U)$$ and $$G\simeq(-\otimes V)$$ for some profunctors $$U:\mathcal A\not\to\mathcal B$$ and $$V:\mathcal B\not\to\mathcal A$$.
And, by $$F$$ and $$G$$ being inverses to each other, we obtain $$U\otimes V\simeq \hom_{\mathcal A}$$ and $$V\otimes U\simeq\hom_{\mathcal B}$$, as the hom functors are the identites for profunctor composition $$\otimes$$, meaning that $$U$$ and $$V$$ form an equivalence in the bicategory of categories and profunctors.
As for such, there exist an adjoint equivalence with arrows $$U$$ and $$V$$ in $$\mathcal{Prof}$$, that is, profunctor isomorphisms $$\eta:\hom_{\mathcal A}\to U\otimes V$$ and $$\varepsilon:V\otimes U\to\hom_{\mathcal B}$$ satisfying the zigzag identities.
Writing up these identities for the inverse of $$\eta$$, we receive a structure called Morita context: $$1_U\otimes\varepsilon=\eta^{-1}\otimes 1_U \quad \text{ and }\quad\varepsilon\otimes 1_V=1_V\otimes\eta^{-1}\quad\ (*)\,,$$ which can be very neatly interpreted as follows:

First, the elements of the sets $$U(a,b)$$ for a profunctor $$U:\mathcal A^{op}\times\mathcal B\to\mathcal{Set}$$ can be regarded as heteromorphisms $$a\to b$$ which we can add to the disjoint union $$\mathcal A+\mathcal B$$, and their compositions is described by the action of $$U$$ on the morphisms.
Thus, we obtain a bigger category $$\mathcal U$$ containing both $$\mathcal A$$ and $$\mathcal B$$ and no arrows of the form $$b\to a$$.
Composition ($$\otimes$$) of profunctors can be viewed as having formal compositions $$\langle u,v\rangle$$ of these heteromorphisms $$u\in U(a,b),\ v\in V(b,a')$$, quotienting out by $$\langle u\beta,\,v\rangle\sim\langle u,\,\beta v\rangle$$ for arrows $$\beta\in\mathcal B$$.

Thus, the above maps $$\varepsilon:U\otimes V\to\hom_{\mathcal A}$$ and $$\eta^{-1}:V\otimes U\to\hom_{\mathcal B}$$ extend the composition operations to $$\mathcal U\cup\mathcal V$$, assigning certain arrow $$a\to a'\,\in\mathcal A$$ to every $$\langle u,v\rangle\in(U\otimes V)(a,a')$$ and similarly for compositions $$b\to b'$$ in $$\mathcal B$$, and the above two equations $$(*)$$ state exactly the altogether associativity of these operations.
In other words, we obtain a bigger category, which is called a bridge $$\mathcal A\rightleftharpoons\mathcal B$$, that is a category containing $$\mathcal A$$ and $$\mathcal B$$ as disjoint full subcategories and no more objects (thus, it is a symmetric version of a profunctor, in a sense).

One direction of my initial statement then follows, since the composition maps $$\varepsilon$$ and $$\eta^{-1}$$ are in particular surjective.
For the other direction, their surjectivity in the presence of equations $$(*)$$ implies invertibility.

If $$\mathcal M:\mathcal A\rightleftharpoons\mathcal B$$ is a bridge with surjective zig-zag compositions, then in particular, for every object $$a\in Ob\mathcal A$$, its identity $$1_a$$ also factors as $$uv$$ for some $$u:a\to b,\ v:b\to a,\ b\in Ob\mathcal B$$, but then $$vu:b\to b$$ is an idempotent,
so if we take the idempotent completion $$\mathcal M^{id}$$ of $$\mathcal M$$, that will be a bridge $$\mathcal A^{id}\rightleftharpoons\mathcal B^{id}$$, in which every object of $$\mathcal A$$, and thus also each of its idempotents, is isomorphic to some object of $$\mathcal B^{id}$$ and vice-versa.
The existence of such a bridge $$\mathcal C\rightleftharpoons\mathcal D$$ is actually equivalent to the equivalence of $$\mathcal C$$ and $$\mathcal D$$ (at least in the presence of choice): fixing isomorphism for each $$a$$ yields an equivalence functor $$\mathcal C\to\mathcal D$$, and the other direction can be proved by a similar argument as above.

So we obtain that $$\mathcal A^{id}\simeq\mathcal B^{id}$$.

For more on the subject, refer to this paper.