Morita equivalence by categorical bridges

Let \mathcal Iso denote the category of 2 objects (name them 0,1) and 2 nonidentity arrows, f:0\to 1,\ g:1\to 0 which are inverses of each other.

We define a  bridge  as a category over \mathcal Iso, i.e. a category \mathcal H equipped with a functor B:\mathcal H\to\mathcal Iso. The full subcategories \mathcal A:=B^{-1}(0) and \mathcal B:=B^{-1}(1) are called banks of the bridge, and we write \mathcal H:\mathcal A \rightleftharpoons \mathcal B.
The arrows in the B-preimage of the two arrows in \mathcal Iso are referred to as through arrows or heteromorphisms in the bridge.

In other words, a category \mathcal H is a  bridge between categories \mathcal A and \mathcal B if these are disjoint full subcategories of \mathcal H and \mathcal H has no more objects.

Recall that the idempotent completion \mathcal A^{id} of a category \mathcal A has all the idempotent arrows of \mathcal A as objects (i.e. those \alpha satisfying \alpha\alpha=\alpha), and a morphism u of \mathcal A is considered as an arrow \alpha\to\beta in \mathcal A^{id} whenever \alpha u \beta=u.

Theorem 1.  Two categories are equivalent if and only if there is a bridge between them where each object has an isomorphic fellow object on the other bank (called equivalence bridge).
Proof. Given a bridge with this property, using axiom of choice, we can fix an isomorphism for each object on one bank, and use them to define an equivalence functor.
Given equivalent categories \mathcal A\simeq\mathcal B, either take an isomorphism of their skeletons \varphi:\mathcal A_0\overset\cong\to\mathcal B_0, this induces a bridge \mathcal A_0\rightleftharpoons\mathcal B_0, which can be extended on both banks, or apply the bridge construction written in A Bridge Construction.

There is a similar characterization for Morita equivalence of categories.
Theorem 2. Two categories are Morita equivalent if and only if there is a bridge between them where each (identity) morphism can be written as a composition of through arrows (called Morita bridge).

Put these together, we arrive to the well known
Theorem 3. Two categories are Morita equivalent if and only if their idempotent completions are equivalent.
Proof. Consider a Morita bridge \mathcal M:\mathcal A\rightleftharpoons\mathcal B and take its idempotent completion \mathcal M^{id}. For each object we have 1_A=fg for some through arrows f ,g. But then gf is an idempotent in \mathcal B, isomorphic to A in M^{id}. This implies that every A\in Ob\mathcal A, thus also every idempotent on A has an isomorphic fellow in \ct B, so, by symmetry, M^{id}:\mathcal A^{id}\rightleftharpoons\mathcal B^{id} is an equivalence bridge.
For the other direction, if E:\mathcal A^{id}\rightleftharpoons\mathcal B^{id} is an equivalence bridge, consider its restriction on both banks to \mathcal A\rightleftharpoons\mathcal B. Then any A\in\mathrm{Ob}\mathcal A is isomorphic to an idempotent \beta:B\to B in \mathcal B, say by \phi\psi=1_A,\ \psi\phi=\beta, and in \mathcal B^{id} we have maps \beta_B:\beta\to 1_B and {}_B\beta:1_B\to\beta that compose to \beta=1_{\beta} and thus they ensure a composition of 1_A as heteromorphisms: \phi\beta_B\cdot {}_B\beta\psi.

Leave a Reply

Your email address will not be published. Required fields are marked *