\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{interpretation}

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

\hypertarget{model_theory}{}\paragraph*{{Model theory}}\label{model_theory}

[[!include model theory - contents]]

\hypertarget{type_theory}{}\paragraph*{{Type theory}}\label{type_theory}

[[!include type theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{interpretations_of_theories_in_each_other}{Interpretations of theories in each other}\dotfill \pageref*{interpretations_of_theories_in_each_other} \linebreak
\noindent\hyperlink{biinterpretations_of_theories}{Bi-interpretations of theories}\dotfill \pageref*{biinterpretations_of_theories} \linebreak
\noindent\hyperlink{interpretations_of_models_in_each_other}{Interpretations of models in each other}\dotfill \pageref*{interpretations_of_models_in_each_other} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{classifying_topses}{Classifying topses}\dotfill \pageref*{classifying_topses} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In [[formal logic]] and [[model theory]], \emph{interpretation} refers to equipping the [[syntax]] of some [[theory]] with a [[semantics]]. Na\"i{}vely this means finding a model (in the category of sets) for the theory. This is subsumed by treating interpretations as functors out of [[syntactic categories]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

In the language of [[categorical logic]], interpretations are representations of theories inside some category $\mathbf{C}$. Depending on the kind of theory ([[cartesian logic|cartesian]], [[regular logic|regular]], [[coherent logic|coherent]], [[first-order logic|first-order]], [[geometric logic|geometric]]), an interpretation of $T$ in $\mathbf{C}$ is just a ([[cartesian logic|cartesian]], [[regular logic|regular]], [[coherent logic|coherent]], [[first-order logic|first-order]], [[geometric logic|geometric]]) functor to $\mathbf{C}$.

When $\mathbf{C}$ is [[Set]], $T$ is first-order, and the functor (say $M$) is \emph{logical} (in the sense of Makkai-Reyes, equivalently coherent if we take the [[Morleyization]] of $T$), we get \emph{models} in the sense usually studied in [[model theory]].

\hypertarget{interpretations_of_theories_in_each_other}{}\subsection*{{Interpretations of theories in each other}}\label{interpretations_of_theories_in_each_other}

Let $T_1$ and $T_2$ be ([[cartesian logic|cartesian]], [[regular logic|regular]], [[coherent logic|coherent]], [[first-order logic|first-order]], [[geometric logic|geometric]]) theories. A ([[cartesian logic|cartesian]], [[regular logic|regular]], [[coherent logic|coherent]], [[first-order logic|first-order]], [[geometric logic|geometric]]) \emph{interpretation} $T_1 \to T_2$ is just a functor between the [[syntactic categories]] $\mathbf{Def}(T_1) \to \mathbf{Def}(T_2)$.

Elsewhere, interpretations have been defined as assignments of symbols in the language $\mathcal{L}_1$ of $T_1$ to definable sets of $T_2$ satisfying various coherence conditions (usually at least product-preserving) which amount to functoriality.

Note that via the duality between taking [[syntactic category|syntactic categories]] and [[internal logic|internal logics]], a model of $T$ in $\mathbf{Set}$ is just an interpretation of $T$ in the theory $\mathsf{Lang}(\mathbf{Set})$.

\hypertarget{biinterpretations_of_theories}{}\subsubsection*{{Bi-interpretations of theories}}\label{biinterpretations_of_theories}

We say that $T_1$ and $T_2$ are \emph{bi-interpretable} if there are functors (of appropriate logical strength) $\mathbf{Def}(T_1) \leftrightarrows \mathbf{Def}(T_2)$ forming an [[equivalence of categories]]. One direction of [[conceptual completeness]] is that bi-interpretable theories have equivalent categories of models. This follows from the fact that ([[cartesian logic|cartesian]], [[regular logic|regular]], [[coherent logic|coherent]], [[first-order logic|first-order]], [[geometric logic|geometric]]) functors are closed under composition, and that equivalent categories induce equivalences of functor categories.

Many notions from [[geometric stability theory]] and [[classification theory]] are invariant under bi-interpretability, e.g. [[stability in model theory|stability]], [[quantifier elimination]], [[elimination of imaginaries]], etc.

Since bi-interpretations induce equivalences of categories of models, monsters of bi-interpretable first-order theories $T_1, T_2$ will have isomorphic automorphism groups, with the isomorphism induced by restrictions along reducts. This indicates the bi-interpretation extends to the [[imaginaries]] of $T_1$ and $T_2$ also, so that $T_1^{\operatorname{eq}} \simeq T_2^{\operatorname{eq}}$, in fact uniquely---which is the universal property of the [[pretopos completion]].

\hypertarget{interpretations_of_models_in_each_other}{}\subsection*{{Interpretations of models in each other}}\label{interpretations_of_models_in_each_other}

This notion has appeared in the model-theoretic literature, and is what some model theorists mean when they say ``interpretation.'' Specialize to first-order logic and models in [[Set]], and fix models $M_1 \models T_1, M_2 \models T_2$. An \emph{interpretation} of $M_1$ in $M_2$ is a surjection $U \overset{f}{\twoheadrightarrow} M_1$ for $U$ some subset of $M_2^k$, some $k \in \mathbb{N}$, such that the pullback of $f^*X$ of any definable (with parameters) set $X$ of $M_1$ along $f$ is again definable in $Y$. This is enough to induce a logical functor $\mathbf{Def}(T_1) \to \mathbf{Def}(T_2)$ (surjectivity implies witnessed existentials continue to be witnessed, and the functor being induced by pullback implies logicalness), in fact a logical functor $\mathbf{Def}(T_1(M_1)) \to \mathbf{Def}(T_2(M_2))$ of $T_1$ and $T_2$ enriched with the elementary diagrams of $M_1$ and $M_2$.

Facts:

\begin{itemize}%
\item Every interpretation between theories can be realized as being induced by a concrete interpretation between sufficiently saturated models of those theories.


\item Given an $f : U \twoheadrightarrow M_1$, any other $g : U \twoheadrightarrow M_1$ which is also an interpretation $M_1 \to M_2$ is of the form $\sigma \circ f$ for some $\sigma \in \operatorname{Aut}(M_1)$.



\end{itemize}
\hypertarget{examples}{}\subsubsection*{{Examples}}\label{examples}

\begin{itemize}%
\item The construction of the [[Grothendieck group of a commutative monoid]] implements a canonical interpretation of the ring $\mathbb{Z}$ in the semiring $\mathbb{N}$ (in fact inside $\mathbb{N}$ viewed as the standard model of [[Peano arithmetic]]).


\item The complex field $\mathbb{C}$ is canonically interpreted in the real field $\mathbb{R}$ by identifying $\mathbb{C}$ with $\mathbb{R}^2$ and noting that the multiplication of complex numbers is definable from multiplication on the reals.



\end{itemize}
\hypertarget{classifying_topses}{}\subsection*{{Classifying topses}}\label{classifying_topses}

An interpretation $T \to T'$ is precisely the inverse image part of a [[geometric morphism]] of [[classifying topos|classifying toposes]] $\mathcal{E}(T') \to \mathcal{E}(T)$. In particular, [[structure in model theory|models]] of $T$ are [[points of a topos | points]] of [[Set]] in $\mathcal{E}(T)$.

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[formula]]


\item [[model theory]], [[structure (model theory)]]


\item [[geometric morphism]]



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

\begin{itemize}%
\item wikipedia \href{http://en.wikipedia.org/wiki/Interpretation_%28model_theory%29}{interpretation (model theory)}


\item Wilfrid Hodges, \emph{A shorter model theory}, Cambridge Univ. Press 1997


\item Olivia Caramello, \emph{Topos-theoretic preliminaries}.



\end{itemize}


\end{document}