\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*{logical functor}

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

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{preserved_structures}{Preserved structures}\dotfill \pageref*{preserved_structures} \linebreak
\noindent\hyperlink{relation_to_geometric_morphisms}{Relation to geometric morphisms}\dotfill \pageref*{relation_to_geometric_morphisms} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{logical morphism} or \emph{logical functor} is a [[homomorphism]] between [[elementary topos]]es that preserves the structure of a topos as a context for [[logic]]: a functor which preserves all the elementary topos structure, including in particular [[power objects]], but not necessarily any infinitary structure (such as present additionally in a [[sheaf topos]]).

If instead a topos is regarded as a context for [[geometry]] or specifically [[geometric logic]], then the notion of [[homomorphism]] preserving this is that of a \emph{[[geometric morphism]]}.

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

Since all the elementary topos structure follows from having [[finite limits]] and [[power objects]], it suffices to define a logical functor to preserve these, up to [[isomorphism]]. It then follows that it is also a [[locally cartesian closed functor]], a [[Heyting functor]], etc.

\begin{defn}
\label{}\hypertarget{}{}
Let $\mathcal{E}$ be an [[elementary topos]]. Write $\Omega \in \mathcal{E}$ for the [[subobject classifier]]. For each [[object]] $A \in \mathcal{E}$ write

\begin{displaymath}
P A := \Omega^A
\end{displaymath}
for the [[exponential object]]. Write

\begin{displaymath}
\in_A \hookrightarrow P A \times A
\end{displaymath}
for the [[subobject]] classified by the [[evaluation map]] $ev : P A \times A \to \Omega$.

\end{defn}
This exhibits $P A$ as a [[power object]] for $A$.

\begin{defn}
\label{}\hypertarget{}{}
A [[functor]] $F : \mathcal{E} \to \mathcal{F}$ between [[elementary toposes]] is called a \textbf{logical morphism} if

\begin{enumerate}%
\item $F$ preserves finite [[limits]];


\item for every object $A \in \mathcal{E}$

\begin{itemize}%
\item the canonical morphism

\begin{displaymath}
\phi_A : F(P A) \to P (F A)
\end{displaymath}
is an [[isomorphism]]; this is the [[name of the relation]]

\begin{displaymath}
F(\in_A) \hookrightarrow F(P A \times A) \simeq F(P A) \times F A
\end{displaymath}
(using that [[cartesian functor]]s preserve both [[product]]s as well as [[monomorphism]])


\item equivalently: $F(P A)$ equipped with the [[relation]] $F(\in_A)$ is a [[power object]] for $F(A)$ in $\mathcal{F}$.



\end{itemize}


\end{enumerate}
\end{defn}
The notion of logical functors between [[topos]]es is in contrast to [[geometric morphism]]s between toposes: the former preserve the structure of an [[elementary topos]], the latter those of a [[sheaf topos]].

But both can be combined:

\begin{defn}
\label{AtomicGeometricMorphism}\hypertarget{AtomicGeometricMorphism}{}
A [[geometric morphism]] whose [[inverse image]] is a logical functor is called an [[atomic geometric morphism]].

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The other case, that the [[direct image]] of a geometric morphism is a logical functor is not of interest. See cor. \ref{LogicalMorphismsAsDirectImages} below.

\end{remark}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{prop}
\label{LeftRightAdjoint}\hypertarget{LeftRightAdjoint}{}
A logical functor has a [[left adjoint]] precisely if it has a [[right adjoint]].

\end{prop}
This appears as (\hyperlink{Johnstone}{Johnstone, cor. A2.2.10}).

\begin{proof}
For $F : \mathcal{E} \to \mathcal{F}$ a logical functor, we have by definition a [[diagram]]

\begin{displaymath}
\itexarray{
     \mathcal{E}^{op} &\stackrel{F^{op}}{\to}& \mathcal{F}^{op}
     \\
     {}^{\mathllap{P}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{P}}
     \\
     \mathcal{E} &\stackrel{F}{\to}& \mathcal{F}
  }
\end{displaymath}
in [[Cat]]. This satisfies the assumptions of the [[adjoint lifting theorem]] and hence $F$ has a right adjoint precisely if $F^{op}$ does. But a right adjoint of $F^{op}$ is a left adjoint of $F$, and vice versa.

\end{proof}
\begin{prop}
\label{PresMono}\hypertarget{PresMono}{}
If a logical functor $F : \mathcal{E} \to \mathcal{F}$ has a left adjoint $L$, then $L$ preserves [[monomorphisms]], and indeed induces a bijection between subobjects of $A\in \mathcal{F}$ and subobjects of $L A\in \mathcal{E}$.

\end{prop}
\begin{proof}
Since $F(\Omega_{\mathcal{E}}) = \Omega_{\mathcal{F}}$, we have a bijection

\begin{displaymath}
Sub_{\mathcal{F}}(A) \cong \mathcal{F}(A,\Omega_{\mathcal{F}}) \cong \mathcal{E}(L A,\Omega_{\mathcal{E}}) \cong Sub_{\mathcal{E}}(L A).
\end{displaymath}
It remains to check that this bijection is implemented by the action of $L$, which can be done with [[partial map classifiers]]; see (\hyperlink{Johnstone}{Johnstone, Lemma A2.4.8}).

\end{proof}
\begin{prop}
\label{CharPullback}\hypertarget{CharPullback}{}
For a logical functor $F : \mathcal{E} \to \mathcal{F}$ having a left adjoint $L$, the following are equivalent:

\begin{enumerate}%
\item The induced functor $L':\mathcal{F} \to \mathcal{E}/L 1$ is an [[equivalence of categories]], i.e. the adjunction $L\dashv F$ can be identified with the pullback adjunction $\mathcal{E}/B \rightleftarrows \mathcal{E}$ for some $B$ (namely $L1$).
\item $L$ is [[conservative functor|conservative]].
\item $L$ is [[faithful functor|faithful]].
\item $L$ preserves [[equalizers]].
\item $L$ preserves [[pullbacks]].

\end{enumerate}
\end{prop}
This appears as (\hyperlink{Johnstone}{Johnstone, Prop. A2.3.8}).

\begin{proof}
The left adjoint of pullback has all the other properties. Any pullback-preserving functor preserves equalizers. An equalizer-preserving functor is faithful as soon as it reflects invertibility of monomorphisms, which follows from Proposition \ref{PresMono} above. A faithful functor reflects monos and epis, hence is conservative once its domain is [[balanced category|balanced]].

Finally, if $L$ is conservative then so is $L'$, and its right adjoint $F'$ (given by $F$ followed by pullback along $\eta : 1 \to F L 1$) is also logical, hence cartesian closed. Thus, $L'\dashv F'$ is a [[Hopf adjunction]] for the cartesian monoidal structures, and $L'$ preserves the terminal object. Now the counit of $L'\dashv F'$ can be factored as

\begin{displaymath}
L' F' A \cong L'(1\times F' A) \xrightarrow{\cong} L' 1 \times A \cong 1\times A\cong A
\end{displaymath}
so it is an isomorphism. By the triangle identity, $L'(\eta)$ is an isomorphism, but $L'$ is conservative, hence the unit $\eta$ is also an isomorphism, and $L'\dashv F'$ is an equivalence.

\end{proof}
This appears as (\hyperlink{Johnstone}{Johnstone, cor. A2.2.10}).

\begin{cor}
\label{LogicalMorphismsRightAdjointToCartesianFunctors}\hypertarget{LogicalMorphismsRightAdjointToCartesianFunctors}{}
If a logical functor is [[right adjoint]] to a [[left exact functor]], then it is an [[equivalence of categories]].

\end{cor}
This appears as (\hyperlink{Johnstone}{Johnstone, scholium 2.3.9}).

\hypertarget{preserved_structures}{}\subsubsection*{{Preserved structures}}\label{preserved_structures}

In particular, a logical functor preserves the truth of all sentences in the [[internal logic]]. If it is moreover [[conservative functor|conservative]], then it also \emph{reflects} the truth of such sentences. For example, the [[transfer principle]] of [[nonstandard analysis]] can be stated as the fact that a certain functor is logical and conservative.

\hypertarget{relation_to_geometric_morphisms}{}\subsubsection*{{Relation to geometric morphisms}}\label{relation_to_geometric_morphisms}

The difference between geometric and logical functors between toposes is, in a certain sense, a [[categorification]] of the difference between a [[homomorphism]] of [[frames]] and a homomorphism of [[Heyting algebras]]. When the latter are [[complete lattice|complete]], these are the same objects with the same [[isomorphisms]] but different morphisms.

However, while frame homomorphisms naturally categorified by geometric functors, a more precise categorification of Heyting algebra homomorphisms would be [[Heyting functors]], which preserve the internal first-order logic, but not the higher-order logic as logical functors do.

\begin{prop}
\label{LogicalMorphismsAsDirectImages}\hypertarget{LogicalMorphismsAsDirectImages}{}
A logical functor is the [[direct image]] of a [[geometric morphism]] precisely if it is an [[equivalence of categories|equivalence]].

\end{prop}
\begin{proof}
Since by definition the direct image of a geometric morphism has a [[left adjoint]] that preserves [[finite limits]].

\end{proof}
But logical inverse images are of interest. Recall from def. \ref{AtomicGeometricMorphism} above that a [[geometric morphism]] with logical inverse image is called an [[atomic geometric morphism]].

\begin{cor}
\label{}\hypertarget{}{}
Every [[atomic geometric morphism]] is an [[essential geometric morphism]].

\end{cor}
\begin{proof}
By prop. \ref{LeftRightAdjoint}.

\end{proof}
The following is the main source of examples of atomic geometric morphisms.

\begin{prop}
\label{}\hypertarget{}{}
The [[inverse image]] of any [[base change geometric morphism]], hence in particular of any [[etale geometric morphism]], is a [[logical morphism]].

\end{prop}
\begin{proof}
The inverse image is given by [[pullback]] along the given morphism.

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
When considering the [[internal logic]] of a given [[topos]] $\mathcal{E}$ relations, [[predicate]]s/[[proposition]]s about [[variable]]s of [[type]] $A \in ob \mathcal{E}$ are [[subobject]]s of $A$. Application of function symbols to such expressions corresponds to pullback along the morphism representing the function symbol. The above says that this is, indeed, a \emph{logical operation}.

\end{prop}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item The inclusion [[FinSet]] $\hookrightarrow$ [[Set]] is logical.


\item More generally, for any [[small category]] $C$ the inclusion

\begin{displaymath}
[C^{op}, FinSet] \hookrightarrow [C^{op}, Set]
\end{displaymath}
into the [[presheaf topos]] is logical.


\item For $G$ a [[group]] and $\mathbf{B}G$ its [[delooping]] [[groupoid]], the [[forgetful functor]]

\begin{displaymath}
[\mathbf{B}G, Set] \to Set
\end{displaymath}
from [[permutation representation]]s to [[Set]] is logical.



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

Section A2.1 in

\begin{itemize}%
\item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]]}

\end{itemize}
Section IV.2, page 170 of

\begin{itemize}%
\item [[Saunders MacLane]], [[Ieke Moerdijk]], \emph{[[Sheaves in Geometry and Logic]]}

\end{itemize}
[[!redirects logical functor]] [[!redirects logical functors]] [[!redirects logical morphism]] [[!redirects logical morphisms]]



\end{document}