# nLab Morita equivalence

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Classical Morita theorem

Given rings $R$ and $S$, the following properties are equivalent

1. The categories of left $S$-modules and left $R$-modules are equivalent;
2. The categories of right $S$-modules and right $R$-modules are equivalent;
3. There are bimodules ${}_R M_S$ and ${}_S N_R$ such that $\otimes_R M$ and $\otimes_S N$ form an adjoint equivalence between the category of right $S$- and the category of right $R$-modules;
4. The ring $R$ is isomorphic to the endomorphism ring of a generator in the category of left (or right) $S$-modules;
5. The ring $S$ is isomorphic to the endomorphism ring of a generator in the category of left (or right) $R$-modules.

Dmitri Pavlov: Tsit-Yuen Lam in his book “Lectures on modules and rings” on pages 488 and 489 states the Morita equivalence theorem using progenerators (i.e., finitely generated projective generators) instead of just generators. Are these two versions equivalent?

Dmitri Pavlov: I would like to state the Morita equivalence theorem as a 2-equivalence between two bicategories: The bicategory of rings, bimodules and their intertwiners and the bicategory of abelian categories that are equivalent to the category of modules over some ring (i.e., abelian categories that have all small coproducts and a compact projective generator), Eilenberg-Watts functors between these categories (i.e., right exact functors that commute with direct sums) and natural transformations. Is it possible to do this and what is the precise statement then?

## Definitions

### In algebra

Two rings are Morita equivalent if the equivalent statements in Morita theorem above are true. A Morita equivalence is a weakly invertible 1-cell in the bicategory $\mathrm{Rng}$ of rings, bimodules and morphisms of bimodules.

### In homotopy theory

In any homotopy theory framework a Morita equivalence between objects $C$ and $D$ is a span

$C \lt \stackrel{\simeq}{\leftarrow} \hat C \stackrel{\simeq}{\to} \gt D$

where both legs are acyclic fibrations.

In particular, if the ambient homotopical category is a category of fibrant objects, then the factorization lemma (see there) ensures that every weak equivalence can be factored as a span of acyclic fibrations as above.

Important fibrant objects are in particular infinity-groupoids (for instance Kan complexes are fibrant in the standard model structure on simplicial sets and omega-groupoids are fibrant with respect to the Brown-Golasinski folk model structure). And indeed, Morita equivalences play an important role in the theory of groupoids with extra structure:

### In Lie groupoid theory

A Morita morphism equivalence of Lie groupoids is an anafunctor that is invertible, equivalently an invertible Hilsum-Skandalis morphism/bibundle.

Lie groupoids up to Morita equivalence are equivalent to differentiable stacks. This relation between Lie groupoids and their stacks of torsors is analogous to the relation between algebras and their categories of modules, which is probably the reason for the choice of terminology.

## References

Revised on April 3, 2013 17:47:50 by Urs Schreiber (82.169.65.155)