nLab Morita equivalence

Idea

In the original sense, a Morita equivalence between two rings is two bimodules between them that behave as inverses to each other under tensor product of modules, up to isomorphism of bimodules. This is just the 2-categorical concept of equivalence for the two rings regarded as objects in the 2-category of rings, with bimodules as 1-morphisms and bimodule homomorphisms (intertwiners) as 2-morphisms (see here). By the Eilenberg-Watts theorem any such pair of bimodules equivalently induces a suitably “linear” equivalence of categories between the categories of modules over the given rings.

This is the original concept equivalence of rings due to Kiiti Morita (see below).

Nowadays, the term is applied in different but closely related senses in a wide range of mathematical fields, and one speaks of Morita equivalent categories, algebraic theories, geometric theories and so on.

Typically, such Morita situations involve three ingredients: a ‘syntactic’ ground level to which the respective concept of Morita equivalence applies, a ‘hypersyntactic’ level obtained via Cauchy completion, and a second process of completion to a ‘semantic’ level where the equivalence relation for the syntactic ground level is defined by plain equivalence of categories. For example Morita equivalence for small categories is defined as equivalence of their presheaf categories, with Cauchy completion as intermediate hypersyntactic level.

So the broad intuition is that Morita equivalence is a coarse grained semantic equivalence that obtains between syntactic gadgets - basically two theories that have up to equivalence the same category of models. The role of the intermediate hypersyntactic level in this analogy is that of an ‘ideal syntax’ (syntax classifier) that already reflects the relations at the semantic level. The categorical equivalence (via bimodules) from the semantic level then shows up at the intermediate level as a (‘Cauchy convergent’$\sim$ ‘fgp-module’) bidirectional translation from one syntax into another.

An important weakening of Morita equivalence is Morita context (in older literature sometimes called pre-equivalence).

Classical Morita theorem

Given (associative unital) algebras $R$ and $S$ over a commutative ring $k$, 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 the functors $- \otimes_R M$ and $- \otimes_S N$ form an adjoint equivalence between the category of right $S$-modules and the category of right $R$-modules;

4. The algebra $R$ is isomorphic to the endomorphism algebra of a finitely generated projective generator in the category of left (or right) $S$-modules;

5. The algebra $S$ is isomorphic to the endomorphism algebra of a finitely generated projective generator in the category of left (or right) $R$-modules.

As usual, these results give results for rings when specialized to the case $k = \mathbb{Z}$, with an algebra over $\mathbb{Z}$ being a ring.

Definitions

In algebra

Two algebras over a commutative ring $k$ are Morita equivalent if the equivalent statements in the Morita theorem above are true. A Morita equivalence is an equivalence in the bicategory $\mathrm{Alg}_k$ with

• associative unital algebras over $k$ as objects,

• bimodules as 1-morphisms, and

• bimodule homomorphisms (“intertwiners” as 2-morphisms).

A theorem in ring theory says that the center of an algebra is isomorphic to the center of its category of modules and that Morita equivalent algebras have isomorphic centers. In particular, two commutative algebras are Morita equivalent precisely when they are isomorphic!

This shows that the property of having center $Z$ up to isomorphism is stable within Morita equivalence classes. Properties of this kind are sufficiently important to deserve a special name:

A property $P$ of rings is called a Morita invariant iff whenever $P$ holds for a ring $R$, and $R$ and $S$ are Morita equivalent then $P$ also holds for $S$. Other Morita invariant properties are the property of being simple, semisimple, left or right Noetherian, and left or right Artinian (cf. Anderson and Fuller (1992)).

The appearance of finitely generated projective modules in the classical Morita theorem may seem unexpected, but it emerges from the following

Lemma

Suppose $R,S$ are algebras over a commutative ring $k$ and $M$ is an $R,S$-bimodule. Then $M$ is a left adjoint in the bicategory $Alg_k$ of $k$-algebras, bimodules and bimodule morphisms if and only if $M$ is finitely generated projective as an $S$-module.

There is a slick proof of this using enriched category theory, working with categories enriched over $V = k Mod$. It makes use of these facts: a $k$-algebra $R$ is a one-object $V$-enriched category, a bimodule of $k$-algebras is a $V$-enriched profunctor, and the Cauchy completion of $R$, viewed as a one-object $V$-enriched category, is the category of finitely generated projective $R$-modules.

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.

In tensor category theory

A fusion category $\mathcal{C}$ is Morita equivalent to another fusion category $\mathcal{D}$ if there exist a $\mathcal{C}$-module category $\mathcal{M}$ such that $\mathcal{D}$ is equivalent to the reverse category of $\mathcal{C}$-module functors from $\mathcal{M}$ to itself, namely

$$\mathcal{D}\simeq (\mathcal{C}_{\mathcal{M}}^{\vee})^{rev}:=\mathsf{Fun}_{\mathcal{C}}(\mathcal{M},\mathcal{M})^{rev}.$$

By the ‘reverse category’ of a monoidal category $\mathcal{C}$ we mean the monoidal category with reversed tensor product $x\otimes^{rev}y:=y \otimes x$.

Related Concepts

• derived Morita equivalence
• Morita context
• Eilenberg-Watts theorem
• generator
• bimodule
• projective module
• Picard group
• bicategory
• Cauchy completion, Karoubi envelope

References and Literature

The concept is named after Kiiti Morita.

A beautiful classical exposition is in

• Hyman Bass, chapter II of Algebraic K-theory, Benjamin 1968.

The concept should be covered in any decent textbook on algebra and ring theory, e.g.:

• F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Graduate Texts in Mathematics Vol. 13, Springer, New York, 1992.

• P. M. Cohn, Further Algebra and Applications, Springer Heidelberg 2003. (sec. 4.4-4.5 pp.148ff)

• Ross Street, Quantum Groups - A Path to Current Algebra , Cambridge UP 2007. (ps-draft)

For an early extension to domains other than ring theory see

• H. Lindner, Morita equivalences of enriched categories , Cah. Top. Géom. Diff. Cat 15 no.4 (1974) pp.377-397. (pdf)

The generalizations to graded rings, Hopf algebras and corings are studied in references

• A. Marcus, Equivalences induced by graded bimodules, Comm. Algebra 26 (1998) 713–731; Homology of fully graded algebras, Morita and derived equivalences, J. Pure Appl. Alg. 133:1–2 (1998) 209-218 doi
• S. Caenepeel, J.Vercruysse, Shuanhong Wang, Morita theory for corings and cleft entwining structures, J. Algebra 2761 (2004) 210-235 doi
• Stefaan Caenepeel, Septimiu Crivei, Andrei Marcus, Mitsuhiro Takeuchi, Morita equivalences induced by bimodules over Hopf–Galois extensions, J. Algebra 314 (2007) 267–302 pdf
• Gabriella Böhm, Joost Vercruysse, Morita theory for comodules over corings, 3207-3247 (2009) Commun. Alg. 37:9 (2009) doi

The case of algebraic theories is covered in

• F. Borceux, Handbook of Categorical Algebra 2 , CUP 1994. (sec. 3.12)

• J. Adámek, M. Sobral, L. Sousa, Morita equivalence of many-sorted algebraic theories , JA 297 (2006) pp.361-371. (preprint)

For the use in O. Caramello’s ‘toposes as bridges’- approach that brings out the logical side of the concept:

• O. Caramello, Topos-theoretic background , ms. 2014. (pdf)

Other references include

• Ralf Meyer, Morita equivalence in algebra and geometry . (PDF)

• I. Dell'Ambrogio, Goncalo Tabuada, A Quillen model structure for classical Morita theory and a tensor categorification of the Brauer group , arXiv:1211.2309 (2012). (pdf)

• Hans Porst, Generalized Morita Theories: The power of categorical algebra, (pdf)

• Francis Borceux and Enrico Vitale, On the notion of bimodel for functorial semantics, Appl. Categ. Structures, 2:283–295 (1994) pdf

See also

• Wikipedia, Morita equivalence