symmetric monoidal (∞,1)-category of spectra
equality (definitional, propositional, computational, judgemental, extensional, intensional, decidable)
isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category
Examples.
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 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 homomorphsims (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 that obtains from an ‘idempotent’ 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 category e.g. 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.
Given rings $R$ and $S$, the following properties are equivalent
The categories of left $S$-modules and left $R$-modules are equivalent;
The categories of right $S$-modules and right $R$-modules are equivalent;
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;
The ring $R$ is isomorphic to the endomorphism ring of a generator in the category of left (or right) $S$-modules;
The ring $S$ is isomorphic to the endomorphism ring of a generator in the category of left (or right) $R$-modules.
An important weakening of the Morita equivalence is Morita context (in older literature sometimes called pre-equivalence).
Two rings are Morita equivalent if the equivalent statements in the Morita theorem above are true. A Morita equivalence is an equivalence in a 2-category in the bicategory $\mathrm{Rng}$ of rings, bimodules as 1-morphisms and bimodule homomorphisms (“intertwiners” as 2-morphisms).
A theorem in ring theory says that the center of a ring is isomorphic to the center of its category of modules and that Morita equivalent rings have isomorphic centers. Especially, two commutative rings 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 $S$. Another classical example is the property of being simple. (cf. Cohn 2003)
In any homotopy theory framework a Morita equivalence between objects $C$ and $D$ is a span
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:
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.
The concept is named after Kiiti Morita.
A beautiful classical exposition is in
The concept should be covered in any decent textbook on algebra and ring theory, e.g.:
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
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:
Other references include
Ralf Meyer, Morita equivalence in algebra and geometry . (pdf)
I. Dell’Ambrogio, G. 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. Categorical Structures, 2:283–295, 1994 (pdf)
See also