Morita Equivalence

The notion of Morita equivalence, ubiquitous in mathematics and even logic, is a family of simultaneously loosely and closely related equivalences. Broadly speaking, Morita equivalent ’theories’ have same category of ‘models’.

Some algebraic examples are:

• Two algebras $\mathcal{A}$ and $\mathcal{B}$ over a fixed group ring $R$ are Morita equivalent if the categories of left (or right) $\mathcal{A}$-modules $_{\mathcal{A}}\mathfrak{M}$ and left (or right) $\mathcal{B}$-modules $_{\mathcal{B}}\mathfrak{M}$ are equivalent. This means we have bimodules $X = _{\mathcal{A}}X_{\mathcal{B}}$ and $Y = _{\mathcal{B}}Y_\mathcal{A}$ with $X\otimes_\mathcal{B} Y \cong \mathcal{A}$ and $Y\otimes_\mathcal{A}X\cong \mathcal{B}$.
• Two $C^\ast$-algebras are Morita equivalent if they have equivalent categories of Hermitian modules and if the equivalence functors $T$ are $\ast$-functors. This is too weak for most applications, since the category of Hermitian modules over a $C^\ast$ algebra $A$ is equivalent to the category of normal modules over the enveloping von Neumann algebra $n(A)$ and so this is really a von Neumann algebra concept.
• Two $W^\ast$-algebras are Morita equivalent if they have equivalent categories of normal modules and if the equivalence functors $T$ are $\ast$-functors.
• Two $C^\ast$-algebras $A$ and $B$ are strongly Morita equivalent if there is an $A$-$B$-equivalence bimodule. This notion was introduced to improve the understanding of induced representations of locally compact groups.

Now for geometry, one usually replaces the notion of “module” by a manifold that is acted upon by a groupoid action.

• Two locally compact topological groupoids $\mathcal{G}$ and $\mathcal{H}$ are Morita equivalent if there is a $\mathcal{G}$-$\mathcal{H}$-equivalence bimodule, i.e., a locally compact space $X$ that is free and proper as a left $\mathcal{G}$ and right $\mathcal{H}$ space, with the momentum map (that define the action) $\rho_\mathcal{G}: X\to \mathcal{G}_0$ inducing a bijection $X/\mathcal{H}\leftrightarrow \mathcal{G}_0$ and same for $\mathcal{H}$.
• Two Lie groupoids $\mathcal{G}$ and $\mathcal{H}$ are equivalent if there is an invertible anafunctor between them. That is to say, there is a smooth manifold $F$ with commuting actions of $\mathcal{G}$ and $\mathcal{H}$ where $\mathcal{G}$ acts on the left and $\mathcal{Y}$ acts on the right, and the left anchor (or momentum) map $F\to \mathcal{G}_0$ is a principal $\mathcal{H}$-bundle over $\mathcal{G}_0$, which is invertible (composition with the inverse gives identity functor).
• Similarly, for symplectic groupoids, the manifold that act as a morphism should be a symplectic manifold, the actions should be symplectic and the momentum maps should induce diffemorphisms.

We also have Morita equivalence for algebraic theories, etc.