# Morita equivalence by categorical bridges

Let denote the category of 2 objects (name them ) and 2 nonidentity arrows, which are inverses of each other.

We define a  bridge  as a category over , i.e. a category equipped with a functor . The full subcategories and are called banks of the bridge, and we write .
The arrows in the -preimage of the two arrows in are referred to as through arrows or heteromorphisms in the bridge.

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

Recall that the idempotent completion of a category has all the idempotent arrows of as objects (i.e. those satisfying ), and a morphism of is considered as an arrow in whenever .

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 , either take an isomorphism of their skeletons , this induces a bridge , 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 and take its idempotent completion . For each object we have for some through arrows . But then is an idempotent in , isomorphic to in . This implies that every , thus also every idempotent on has an isomorphic fellow in , so, by symmetry, is an equivalence bridge.
For the other direction, if is an equivalence bridge, consider its restriction on both banks to . Then any is isomorphic to an idempotent in , say by , and in we have maps and that compose to and thus they ensure a composition of as heteromorphisms: .