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\section*{Morita equivalence}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{equality_and_equivalence}{}\paragraph*{{Equality and Equivalence}}\label{equality_and_equivalence}

[[!include equality and equivalence - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{ClassicalMoritaTheorem}{Classical Morita theorem}\dotfill \pageref*{ClassicalMoritaTheorem} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{in_algebra}{In algebra}\dotfill \pageref*{in_algebra} \linebreak
\noindent\hyperlink{in_homotopy_theory}{In homotopy theory}\dotfill \pageref*{in_homotopy_theory} \linebreak
\noindent\hyperlink{in_lie_groupoid_theory}{In Lie groupoid theory}\dotfill \pageref*{in_lie_groupoid_theory} \linebreak
\noindent\hyperlink{related_concepts}{Related Concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In the original sense, a \emph{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 [[equivalence in a 2-category|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 \href{bimodule#AsMorphismsInA2Category}{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 \hyperlink{ClassicalMoritaTheorem}{below}).

Nowadays, the term is applied in different but closely related senses in a wide range of mathematical fields, and one speaks of \emph{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 \emph{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 \emph{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 \emph{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 \emph{translation} from one syntax into another.

\hypertarget{ClassicalMoritaTheorem}{}\subsection*{{Classical Morita theorem}}\label{ClassicalMoritaTheorem}

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

\begin{enumerate}%
\item The categories of left $S$-[[modules]] and left $R$-modules are [[equivalence of categories|equivalent]];


\item The categories of right $S$-modules and right $R$-modules are equivalent;


\item 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;


\item The ring $R$ is isomorphic to the endomorphism ring of a finitely generated projective [[generator]] in the category of left (or right) $S$-modules;


\item The ring $S$ is isomorphic to the endomorphism ring of a finitely generated projective [[generator]] in the category of left (or right) $R$-modules.



\end{enumerate}
An important weakening of the Morita equivalence is [[Morita context]] (in older literature sometimes called pre-equivalence).

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\hypertarget{in_algebra}{}\subsubsection*{{In algebra}}\label{in_algebra}

Two rings are \textbf{Morita equivalent} if the equivalent statements in the Morita theorem above are true. A \textbf{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 \emph{commutative} rings are Morita equivalent precisely when they are isomorphic!

This shows that the property of \emph{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 \textbf{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 ring|simple]]. (cf. Cohn 2003)

\hypertarget{in_homotopy_theory}{}\subsubsection*{{In homotopy theory}}\label{in_homotopy_theory}

In any [[homotopy theory]] framework a \textbf{Morita equivalence} between objects $C$ and $D$ is a span

\begin{displaymath}
C \lt \stackrel{\simeq}{\leftarrow}
  \hat C
  \stackrel{\simeq}{\to} \gt
  D
\end{displaymath}
where both legs are acyclic fibrations.

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

Important fibrant objects are in particular [[infinity-groupoid]]s (for instance [[Kan complex]]es are fibrant in the standard [[model structure on simplicial sets]] and [[omega-groupoid]]s 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:

\hypertarget{in_lie_groupoid_theory}{}\subsubsection*{{In Lie groupoid theory}}\label{in_lie_groupoid_theory}

A \textbf{[[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 stack]]s. 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.

\hypertarget{related_concepts}{}\subsection*{{Related Concepts}}\label{related_concepts}

\begin{itemize}%
\item [[derived Morita equivalence]]
\item [[Morita context]]
\item [[Eilenberg-Watts theorem]]
\item [[generator]]
\item [[bimodule]]
\item [[projective module]]
\item [[Picard group]]
\item [[bicategory]]
\item [[Cauchy completion]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The concept is named after [[Kiiti Morita]].

A beautiful classical exposition is in

\begin{itemize}%
\item [[Hyman Bass]], chapter II of \emph{Algebraic K-theory}, Benjamin 1968.

\end{itemize}
The concept should be covered in any decent textbook on algebra and ring theory, e.g.:

\begin{itemize}%
\item P. M. Cohn, \emph{Further algebra and applications} , Springer Heidelberg 2003. (sec. 4.4-4.5 pp.148ff)


\item [[Ross Street]], \emph{Quantum Groups - A Path to Current Algebra} , Cambridge UP 2007. (\href{http://www-texdev.ics.mq.edu.au/Quantum/Quantum.ps}{ps-draft})



\end{itemize}
For an early extension to domains other than ring theory see

\begin{itemize}%
\item H. Lindner, \emph{Morita equivalences of enriched categories} , Cah. Top. G\'e{}om. Diff. Cat \textbf{15} no.4 (1974) pp.377-397. (\href{http://archive.numdam.org/article/CTGDC_1974__15_4_377_0.pdf}{pdf})

\end{itemize}
The case of algebraic theories is covered in

\begin{itemize}%
\item [[Francis Borceux|F. Borceux]], \emph{Handbook of Categorical Algebra 2} , CUP 1994. (sec. 3.12)


\item [[Jiri Adamek|J. Adámek]], M. Sobral, L. Sousa, \emph{Morita equivalence of many-sorted algebraic theories} , JA \textbf{297} (2006) pp.361-371. (\href{https://estudogeral.sib.uc.pt/bitstream/10316/4616/1/filee8df8c9585d34e1a8d56fdaf0460d008.pdf}{preprint})



\end{itemize}
For the use in O. Caramello's `toposes as bridges'- approach that brings out the logical side of the concept:

\begin{itemize}%
\item [[Olivia Caramello|O. Caramello]], \emph{Topos-theoretic background} , ms. 2014. (\href{http://www.oliviacaramello.com/Unification/ToposTheoreticPreliminariesOliviaCaramello.pdf}{pdf})

\end{itemize}
Other references include

\begin{itemize}%
\item [[Ralf Meyer]], \emph{Morita equivalence in algebra and geometry} . ([[MeyerMoritaEquivalence.pdf:file]])


\item I. Dell'Ambrogio, G. Tabuada, \emph{A Quillen Model Structure for Classical Morita Theory and a Tensor Categorification of the Brauer Group} , arXiv:1211.2309 (2012). (\href{http://arxiv.org/pdf/1211.2309v1}{pdf})


\item [[Hans Porst]], \emph{Generalized Morita Theories: The power of categorical algebra}, (\href{http://www.math.uni-bremen.de/~porst/dvis/SAMSnoticesCorr.pdf}{pdf})


\item [[Francis Borceux]] and [[Enrico Vitale]], \emph{On the Notion of Bimodel for Functorial Semantics}, Appl. Categorical Structures, 2:283--295, 1994 (\href{http://perso.uclouvain.be/enrico.vitale/BimodAPCS.pdf}{pdf})



\end{itemize}
See also

\begin{itemize}%
\item wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Morita_equivalence}{Morita equivalence}}

\end{itemize}
[[!redirects Morita equivalences]]



\end{document}