\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*{inverse image}

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

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

[[!include topos theory - contents]]

\begin{quote}%
This page is about \textbf{inverse images of sheaves} and related subjects. For the set-theoretic operation, see [[preimage]].


\end{quote}

\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{inverse_images}{}\section*{{Inverse images}}\label{inverse_images}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{on_presheaves}{on presheaves}\dotfill \pageref*{on_presheaves} \linebreak
\noindent\hyperlink{on_sheaves}{on sheaves}\dotfill \pageref*{on_sheaves} \linebreak
\noindent\hyperlink{on_sheaves_on_topological_spaces}{on sheaves on topological spaces}\dotfill \pageref*{on_sheaves_on_topological_spaces} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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}

An \emph{inverse image} operation is the [[left adjoint]] part $f^*$ of a [[geometric morphism]] $(f^* \dashv f_*) : E \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} F$ of [[topos]].

Given a morphism $f : X \to Y$ of [[site]]s, the \emph{inverse image} operation of the induced geometric morphism $Sh(X) \to Sh(Y)$ on [[categories of sheaves]] is a [[functor]]

\begin{displaymath}
f^{-1} : Sh(Y) \to Sh(X)
\end{displaymath}
that may be interpreted as encoding the idea of \emph{pulling back along $f$} the ``bundle of which the sheaf is the sheaf of sections''.

In the case that $X$ and $Y$ are (the [[site]]s corresponding to) [[topological space]]s this interpretation becomes literally true: the inverse image of a sheaf on topological spaces is the pullback operation on the corresponding [[etale space]]s.

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

Given a [[site|morphisms of sites]] $f : X \to Y$ coming from a [[functor]] $f^t : S_Y \to S_X$ of the underlying [[category|categories]].

\hypertarget{on_presheaves}{}\subsubsection*{{on presheaves}}\label{on_presheaves}

The [[direct image]] operation $f_* : PSh(X) \to PSh(Y)$ on [[presheaf|presheaves]] is just precomposition with $f^t$

\begin{displaymath}
\itexarray{
    S_Y^{op}
    &\stackrel{f_* F}{\to}&
    Set
    \\
    \downarrow^{f^t} & \nearrow_{F}
    \\
    S_X^{op}
  }
  \,.
\end{displaymath}
The \textbf{inverse image} operation

\begin{displaymath}
f^{-1} : PSh(Y) \to PSh(X)
\end{displaymath}
on [[presheaf|presheaves]] is the [[adjoint functor|left adjoint]] to the direct image operation on presheaves, hence the left [[Kan extension]]

\begin{displaymath}
f^{-1} F := Lan_{f^t} F
\end{displaymath}
of a [[presheaf]] $F$ along $f^t$.

\hypertarget{on_sheaves}{}\subsubsection*{{on sheaves}}\label{on_sheaves}

The inverse image operation on the [[category of sheaves]] $Sh(Y) \subset PSh(Y)$ inside the category of presheaves involves [[Kan extension]] followed by [[sheafification]].

First notice that

\begin{ulemma}
The [[direct image]] operation $f_* : PSh(X) \to PSh(Y)$ restricts to a functor $f_* : Sh(X) \to Sh(Y)$ that sends sheaves to sheaves.

\end{ulemma}
\begin{proof}
The direct image $f_* : PSh(X) \to PSh(Y)$ is more generally characterized by

\begin{displaymath}
Hom_{PSh(Y)}(A, f_* F)
  \simeq
  Hom_{PSh(X)}(\hat {f^t} A, F)
\end{displaymath}
where $\hat f^t$ is the [[Yoneda extension]] of $Y \circ f^t : Y \to PSh(X)$ to a functor $\hat {f^t} : PSh(Y) \to PSh(X)$, because using the [[co-Yoneda lemma]] and the colim expression for the [[Yoneda extension]] we have

\begin{displaymath}
\begin{aligned}
   Hom(A, f_* F) & \simeq 
   Hom(colim_{Y(U) \to A}) U, f_* F)
   \\
   & \simeq
    \lim_{Y(U) \to A} Hom(U, f_* F)
   \\
   & \simeq
    \lim_{Y(U) \to A} F(f^t(U))
   \\
   & \simeq
    Hom( colim_{Y(U) \to A} f^t(U), F )
   \\
   & \simeq Hom(\hat {f^t}(A), F)
   \,.
 \end{aligned}
\end{displaymath}
Let now $\pi : B \to A$ be a [[local isomorphism]] in $PSh(Y)$. By definition of morphism of [[site]]s we have that

\begin{displaymath}
\hat {f^t}(\pi) : \hat{f^t}(B) \to \hat{f^t}(A)
\end{displaymath}
is a [[local isomorphism]] in $X$. From this and the above we obtain the isomorphism

\begin{displaymath}
Hom(B, f_* F)
  \simeq
  Hom(\hat {f^t}(B), F)
  \stackrel{\simeq}{\to}
  Hom(\hat {f^t}(A), F)  
  \simeq
  Hom(A, f_* F)
  \,,
\end{displaymath}
where the isomorphism in the middle is due to the fact that $F$ is a sheaf on $X$. Since this holds for all local isomorphism $\pi : B \to A$ in $PSh(Y)$, $f_* F$ is a sheaf on $Y$.

\end{proof}
\begin{udefn}
For $f : X \to Y$ a morphism of [[site]]s, the \textbf{inverse image of sheaves} is the functor

\begin{displaymath}
f^{-1} : Sh(Y) \to Sh(X)
\end{displaymath}
defined as the inverse image on presheaves followed by [[sheafification]]

\begin{displaymath}
f^{-1} : Sh(Y) \hookrightarrow PSh(Y) \stackrel{Lan_{f^t}}{\to} PSh(X) \stackrel{\bar{-}}{\to}
  Sh(X)
  \,.
\end{displaymath}
\end{udefn}
\begin{uproposition}
The inverse image $f^{-1} : Sh(Y) \to Sh(X)$ of sheaves has the following properties:

\begin{itemize}%
\item it is [[left adjoint]] to the [[direct image]] $(f^{-1} \dashv f_*)$;


\item it therefore commutes with small [[colimit]]s but is in addition left [[exact functor|exact]] in that it commutes with finite [[limit]]s.



\end{itemize}
\end{uproposition}
\begin{proof}
The left-adjointness is obtained by the following computation, for any two $F \in Sh(X)$ and $G \in Sh(Y)$ and using the above facts as well as the fact that [[sheafification]] $\bar {(-)} : PSh(X) \to Sh(X)$ is [[left adjoint]] to the inclusion $Sh(X) \hookrightarrow PSh(X)$:

\begin{displaymath}
\begin{aligned}
    Hom_{Sh(Y)}(G, f_*F)
     & \simeq
    Hom_{PSh(Y)}(G, f_* F)
     \\
     & \simeq
       Hom_{PSh(X)}(Lan_{f^t} G, F)
     \\
     & \simeq
      Hom_{Sh(X)}( \bar{(Lan_{f^t} G)}, F)
     \\
     & =:
      Hom_{Sh(X)}(f^{-1}G, F)
  \end{aligned}
  \,.
\end{displaymath}
The proof of left-exactness requires more technology and work.

\end{proof}
\hypertarget{on_sheaves_on_topological_spaces}{}\subsubsection*{{on sheaves on topological spaces}}\label{on_sheaves_on_topological_spaces}

In the case where the [[site]]s $X$ and $Y$ in question are given by [[category of open subsets|categories of open subsets]] of [[topological space]]s denoted, by an abuse of symbols, also by $X$ and $Y$, one can identify sheaves with their corresponding [[etale space]]s over $X$ and $Y$. In that case the inverse image is simply obtained by the pullback along the continuous map $f : X \to Y$ of the corresponding [[etale space]]s.

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

\begin{itemize}%
\item See also [[restriction and extension of sheaves]].


\item It follows that direct image and inverse image of sheaves define a [[geometric morphism]] $f : Sh(X) \to Sh(Y)$ of [[sheaf]] [[topos|topoi]]


\item Generally, therefore, the left adjoint partner in the adjoint pair defining a [[geometric morphism]] of [[topos|topoi]] (or abelian categories of quasicoherent sheaves) is called the \textbf{inverse image functor}. In fact more general in geometry, including [[noncommutative geometry|noncommutative]] morphisms often induce or are defined via pairs of adjoint functors among some associated categories of objects over a geometric space; then the left adjoint part is called the inverse image part. Geometers also often say inverse image for an arbitrary functor of the form $f^*$ in a [[fibered category]]. For abelian categories of sheaf-like objects, the corresponding higher derived functors of inverse image functors are sometimes called higher (derived) inverse image functors.


\item The other adjoint to the [[direct image]], the [[adjoint functor|right adjoint]], is (if it exists) the [[restriction and extension of sheaves|extension]] of sheaves.



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

The standard example is that where $X$ and $Y$ are [[topological space]]s and $S_X = Op(X)$, $S_Y = Op(Y)$ are their [[category of open subsets|categories of open subsets.]]

A [[continuous map]] $f : X \to Y$ induces the obvious functor $f^{-1} : Op(Y) \to Op(X)$, since [[preimages]] of open subsets under continuous maps are open.

Hence presheaves canonically push-forward

\begin{displaymath}
f_* : PSh(X) \to PSh(Y)
\end{displaymath}
They do not in the same simple way pull back, since images of open subsets need not be open. The Kan extension computes the best possible approximation:

The inverse image $(f^{-1})^\dagger : PSh(Y) \to PSh(X)$ sends $F \in PSh(Y)$ to

\begin{displaymath}
f^\dagger F : U \mapsto colim_{(U \to f^{-1}(V)) \in (const_U, f^{-1})} F(V)
  \,.
\end{displaymath}
This approximates the possibly non-open subset $f^{-1}(V)$ by all open subsets $U$ \emph{inside} it.

On the other hand, the extension

$(f^{-1})^\ddagger : PSh(Y) \to PSh(X)$ sends $F \in PSh(Y)$ to

\begin{displaymath}
f^\dagger F : U \mapsto colim_{(f^{-1}(V) \to U) \in (f^{-1},const_U)} F(V)
  \,.
\end{displaymath}
This approximates the possibly non-open subset $f^{-1}(V)$ by all open subsets $U$ \emph{containing} it.

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

\begin{itemize}%
\item [[derived inverse image]]


\item [[exceptional inverse image]]



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

for the general description in terms of Kan extension and sheafification see section 17.5 of

\begin{itemize}%
\item Kashiwara-Schapira, [[Categories and Sheaves]]

\end{itemize}
For the description in terms of pullback of etale spaces see secton VII.1 of

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

\end{itemize}
[[!redirects inverse image]] [[!redirects inverse images]] [[!redirects inverse image functor]]



\end{document}