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.
Morita equivalence is a category theoretic concept of equivalence that is in general weaker than isomorphism or equivalence of categories. The concept has originated in ring theory in K. Morita’s groundbreaking investigation into the equivalence relation between rings $R, S$ induced by an equivalence $Mod_R\cong Mod_S$ of their category of modules.
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
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 a weakly invertible 1-cell in the bicategory $\mathrm{Rng}$ of rings, bimodules and morphisms of bimodules.
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.
A beautiful classical exposition is in chapter II of
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)