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\begin{document}

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


\section*{geometric embedding}

\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{relation_to_localization}{Relation to localization}\dotfill \pageref*{relation_to_localization} \linebreak
\noindent\hyperlink{factorizations_and_images}{Factorizations and images}\dotfill \pageref*{factorizations_and_images} \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}

A \textbf{geometric embedding} is the right notion of embedding or inclusion of [[topoi]] $F \hookrightarrow E$, i.e. of [[subtoposes]].

Notably the inclusion $Sh(S) \hookrightarrow PSh(S)$ of a [[category of sheaves]] into its [[presheaf]] [[topos]] or more generally the inclusion $Sh_j E \hookrightarrow E$ of sheaves in a topos $E$ into $E$ itself, is a geometric embedding. Actually every geometric embedding is of this form, up to equivalence of [[topos|topoi]].

Another perspective is that a geometric embedding $F \hookrightarrow E$ is the [[localizations]] of $E$ at the class $W$ or morphisms that the [[left adjoint]] $E \to F$ sends to isomorphisms in $F$.

The induced geometric morphism of a topological immersion $X \hookrightarrow Y$ is a geometric embedding. The converse holds if $Y$ is a $T_0$ space. (Example A4.2.12(c) in (\hyperlink{Johnstone}{Johnstone}))

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

For $F$ and $E$ two [[topos|topoi]], a [[geometric morphism]]

\begin{displaymath}
F \stackrel{f}{\to}
  E
  \;\;\;\;
  F \stackrel{\stackrel{f_*}{\longrightarrow}}{\underset{f^*}{\longleftarrow}} E
\end{displaymath}
is a \textbf{geometric embedding} if the following equivalent conditions are satisfied

\begin{itemize}%
\item the [[direct image]] functor $f_*$ is [[full and faithful functor|full and faithful]] (so that $F$ is a full [[subcategory]] of $E$);


\item the counit $\epsilon :  f^* f_* \to Id_{F}$ of the [[adjoint functor|adjunction]] $(f^* \dashv f_*)$ is an [[isomorphism]]


\item there is a [[Lawvere-Tierney topology]] on $E$ and an [[equivalence of categories]] $e : F \stackrel{\simeq}{\to} Sh_j E$ such that the diagram of geometric morphisms $\itexarray{ F &\stackrel{f_*}{\to}& E \\ & {}_{e}\searrow^\simeq & \uparrow^{i} \\ && Sh_j E}$ commutes up to natural isomorphism $e^* i^* \simeq f^*$



\end{itemize}
That the first two conditions are equivalent is standard, that the third one is equivalent to the first two is for instance corollary 7 in section VII, 4 of (\hyperlink{MacLaneMoerdijk}{MacLaneMoerdijk})

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

\hypertarget{relation_to_localization}{}\subsubsection*{{Relation to localization}}\label{relation_to_localization}

There is a close relation between geometric embedding and [[localization]]: [[reflective localization]].

Let $f : F \hookrightarrow E$ be a geometric embedding and let $W \subset Mor(E)$ be the class of morphisms sent by $f^*$ to isomorphisms in $F$.

\begin{utheorem}
We have:

\begin{itemize}%
\item $F$ is equivalent to the [[localization]] $E[W]^{-1}$;


\item $F$ is equivalent to the [[full subcategory]] of $E$ on $W$-[[local objects]].



\end{itemize}
\end{utheorem}
This fact connects for instance the description of [[sheafification]] in terms of geometric embedding $Sh(S) \hookrightarrow PSh(S)$ as described for instance in

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

\end{itemize}
with that in terms of [[localization]] at [[local isomorphisms]], as described in

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

\end{itemize}
Moreover, this is the basis on which sheafification is generalized to [[(∞,1)-sheafification]] in

\begin{itemize}%
\item [[Jacob Lurie]], [[Higher Topos Theory]] .

\end{itemize}
The following gives a detailed proof of the above assertion.

Write $\eta : Id_E \to f_* f^*$ for the unit of the adjunction.

Since $f_*$ is fully faithful we will identify objects and morphism of $F$ with their images in $E$. To further trim down the notation write $\bar {(-)} := f^*$ for the left adjoint.

\begin{udefn}
Write $W$ for the class of morphism that are sent to isomorphism under $f^*$,

\begin{displaymath}
W = (f^*)^{-1}\{g: c\stackrel{\simeq}{\to} d \in Mor(E)\}
  \,.
\end{displaymath}
\end{udefn}
\begin{uprop}
$E$ equipped with the class $W$ is a [[category with weak equivalences]], in that $W$ satisfies 2-out-of-3.

\end{uprop}
\begin{proof}
Follows since isomorphisms satisfy 2-out-of-3.

\end{proof}
\begin{uprop}
$W$ is a left [[calculus of fractions|multiplicative system]].

\end{uprop}
\begin{proof}
This follows using the fact that $f^*$ is left exact and hence preserves finite limits.

In more detail:

We have already seen in the previous proposition that

\begin{itemize}%
\item every isomorphism is in $W$;


\item $W$ is closed under composition.



\end{itemize}
It remains to check the following points:

Given any

\begin{displaymath}
\itexarray{
     && a
     \\
     && \downarrow^w
     \\
     b &\stackrel{h}{\to}& c 
  }
\end{displaymath}
with $w \in W$, we have to show that there is

\begin{displaymath}
\itexarray{
     d &\to& a
     \\
     \downarrow^{w'} && \downarrow^w
     \\
     b &\stackrel{h}{\to}& c 
  }
\end{displaymath}
with $w' \in W$.

To get this, take this to be the [[pullback]] diagram, $w' := h^* w$. Since $f^*$ preserves pullbacks, it follows that

\begin{displaymath}
\itexarray{
     \bar d &\to& \bar a
     \\
     \downarrow^{\bar w'} && \downarrow^{\bar w}
     \\
     \bar b &\stackrel{\bar h}{\to}& \bar c 
  }
\end{displaymath}
is a pullback diagram in $F$ with $\bar w' = \bar h^* \bar w$. But by assumption $\bar w$ is an isomorphism. Therefore $\bar w'$ is an isomorphism, therefore $w'$ is in $W$.

Finally for every

\begin{displaymath}
a \stackrel{\stackrel{r}{\to}}{\stackrel{s}{\to}} b \stackrel{w}{\to} c
\end{displaymath}
with $w \in W$ such that the two composites coincide, we need to find

\begin{displaymath}
d \stackrel{w'}{\to} a \stackrel{\stackrel{r}{\to}}{\stackrel{s}{\to}} b
\end{displaymath}
with $w' \in W$ such that the composites again coincide.

To get this, take $w'$ to be the [[equalizer]] of the two morphisms. Sending everything with $f^*$ to $F$ we find from

\begin{displaymath}
\bar a \stackrel{\stackrel{\bar r}{\to}}{\stackrel{\bar s}{\to}} b \stackrel{\bar w}{\to} c
\end{displaymath}
that $\bar r = \bar s$, since $\bar w$ is an isomorphism. This implies that $\bar w'$ is the equalizer

\begin{displaymath}
\bar d \stackrel{\bar w'}{\to} a \stackrel{\stackrel{\bar r}{\to}}{\stackrel{\bar s}{\to}} b
\end{displaymath}
of two equal morphism, hence an identity. So $w'$ is in $W$.

\end{proof}
\begin{uprop}
For every object $a \in E$

\begin{itemize}%
\item the unit $\eta_a : a \to \bar a$ is in $W$;


\item if $a$ is already in $F$ then the unit is already an isomorphism.



\end{itemize}
\end{uprop}
\begin{proof}
This follows from the [[triangle identities]] of the [[adjoint functors]].

\begin{displaymath}
\itexarray{
     & 
     \nearrow &\Downarrow^{\eta}& \searrow^{Id_E}
     \\
     E 
       &\stackrel{\bar{(-)}}{\to}& 
     F
       &\hookrightarrow&
     E
       &\stackrel{\bar{(-)}}{\to}& 
     F
     \\
     &&&
     \searrow &\Downarrow^{\simeq}& \nearrow_{Id_F}
  }
  \;\;\;\;
  =
  \;\;\;\;
  \itexarray{
     & \nearrow \searrow^{\bar{(-)}}
    \\
    E
    &\Downarrow^{Id}&
    F
    \\
    & \searrow \nearrow_{\bar{(-)}}
  }
\end{displaymath}
and

\begin{displaymath}
\itexarray{
     &&& 
     \nearrow &\Downarrow^{\eta}& \searrow^{Id_E}
     \\
     F 
       &\hookrightarrow&
     E
       &\stackrel{\bar{(-)}}{\to}& 
     F
       &\hookrightarrow&
     E
     \\
     &
     \searrow &\Downarrow^{\simeq}& \nearrow_{Id_F}
  }
  \;\;\;\;
  =
  \;\;\;\;
  \itexarray{
     & \nearrow \searrow
    \\
    F
    &\Downarrow^{Id}&
    E
    \\
    & \searrow \nearrow
  }
\end{displaymath}
In components they say that

\begin{itemize}%
\item for every $a \in E$ we have $(\bar a \stackrel{\bar \eta_a}{\to} \bar{\bar a} \stackrel{\simeq}{\to} \bar a) = Id_{\bar a}$


\item for every $a \in F$ we have $(a \stackrel{\eta_a}{\to} \bar a \stackrel{\simeq}{\to} a)  = Id_a$



\end{itemize}
This implies the claim.

\end{proof}
\begin{udefn}
An object $a \in E$ is \textbf{$W$-[[local object]]} if for every $g : c \to d$ in $W$ the map

\begin{displaymath}
g^* : Hom_E(d,a) \stackrel{\simeq}{\to}
        Hom_E(c,a)
\end{displaymath}
obtained by precomposition is an isomorphism.

\end{udefn}
\begin{uprop}
Up to isomorphism, the $W$-local objects are precisely the objects of $F$ in $E$

\end{uprop}
\begin{proof}
First assume that $a \in F$. We need to show that $a$ is $W$-local.

Notice that the existence of the required isomorphism $Hom_F(d,a) \simeq Hom_F(c,a)$ is equivalent to the statement that for every diagram

\begin{displaymath}
\itexarray{
     c &\stackrel{}{\to}& d
     \\
     \downarrow^{h}
     \\
     a
  }
\end{displaymath}
there is a unique extension

\begin{displaymath}
\itexarray{
     c &\stackrel{}{\to}& d
     \\
     \downarrow^{h} & \swarrow
     \\
     a
  }
  \,.
\end{displaymath}
To see the existence of this extension, hit the original diagram with $f^*$ to get

\begin{displaymath}
\itexarray{
     \bar c &\stackrel{\simeq}{\to}& \bar d
     \\
     \downarrow^{\bar h}
     \\
     \bar a \simeq a
  }
  \,.
\end{displaymath}
By the assumption that $c \to d$ is in $W$ the morphism $\bar c \to \bar d$ here is an isomorphism. By the assumption that $a$ is already in $F$ we have $\bar a \simeq a$ since the counit is an isomorphism. Therefore this diagram clearly has a unique extension

\begin{displaymath}
\itexarray{
     \bar c &\stackrel{\simeq}{\to}& \bar d
     \\
     \downarrow^{\bar h} & \swarrow_{\exists ! k}
     \\
     \bar a \simeq a
  }
  \,.
\end{displaymath}
By the hom-isomorphism (using full faithfullness of $f_*$ to work entirely in $E$)

\begin{displaymath}
Hom_E(\bar d, a) \simeq Hom_E(d,a)
\end{displaymath}
this defines a morphism $k : d \to a$. Chasing $k$ through the naturality diagram of the hom-isomorphism

\begin{displaymath}
\itexarray{
     Hom_E(\bar d, \bar a) &\stackrel{\simeq}{\to}& Hom_E(d,\bar a)
     \\
     \downarrow && \downarrow
     \\
     Hom_E(\bar c, \bar a) &\stackrel{\simeq}{\to}& Hom_E(c,\bar a)     
  }
  \,.
\end{displaymath}
shows that $k : d \to a$ does extend the original diagram. Again by the Hom-isomorphism, it is the unique morphism with this property.

So $a \in F$ is $W$-local.

Now for the converse, assume that a given $a$ is $W$-local.

By one of the above propositions we know that the unit $\eta_a : a \to \bar a$ is in $W$, so by the $W$-locality of $a$ it follows that

\begin{displaymath}
\itexarray{
     a &\stackrel{\eta_a}{\to}& \bar a
     \\
     \downarrow^{Id_a}
     \\
     a
  }
\end{displaymath}
has an extension

\begin{displaymath}
\itexarray{
     a &\stackrel{\eta_a}{\to}& \bar a
     \\
     \downarrow^{Id_a} & \swarrow_{\rho_a}
     \\
     a
  }
  \,.
\end{displaymath}
By the 2-out-of-3 property of $W$ shown in one of the above propositions, (using that $Id_a$, being an isomorphism, is in $W$) it follows that $\rho_a : \bar a \to a$ is in $W$.

Since $\bar a$ is in $F$ and therefore $W$-local by the above, it follows that also

\begin{displaymath}
\itexarray{
     \bar a &\stackrel{\rho_a}{\to}& a
     \\
     \downarrow^{Id_{\bar a}}
     \\
     \bar a
  }
\end{displaymath}
has an extension

\begin{displaymath}
\itexarray{
     \bar a &\stackrel{\rho_a}{\to}& a
     \\
     \downarrow^{Id_{\bar a}} & \swarrow_{\lambda_a}
     \\
     \bar a
  }
  \,.
\end{displaymath}
So $\eta_a$ has a left inverse $\rho_a$ which itself has a left inverse $\lambda_a$. It follows that $\rho_a$ is also a right inverse to $\eta_a$, since

\begin{displaymath}
\begin{aligned}
    \stackrel{\rho_a}{\to}
    \stackrel{\eta_a}{\to}
    & =    
    \stackrel{\rho_a}{\to}
    \stackrel{\eta_a}{\to}
    \underset{id}{\underbrace{
      \stackrel{\rho_a}{\to} 
       \stackrel{\lambda_a}{\to}
    }}
   \\
    & = 
    \stackrel{\rho_a}{\to}
    \underset{id}{\underbrace{
      \stackrel{\eta_a}{\to}
      \stackrel{\rho_a}{\to} 
    }}
    \stackrel{\lambda_a}{\to}
    \\
    &=
    \stackrel{\rho_a}{\to}
    \stackrel{\lambda_a}{\to}
    \\
    &=
    Id
  \end{aligned}
  \,.
\end{displaymath}
So if $a$ is $W$-local we find that $\eta_a : a \to \bar a$ is an isomorphism, hence that $a$ is isomorphic to an object of $F$.

\end{proof}
\begin{ucor}
$F$ is equivalent to the full [[subcategory]] $E_{W-loc}$ of $E$ on $W$-[[local objects]].

\end{ucor}
\begin{proof}
By standard reasoning (e.g. [[Categories and Sheaves|KS lemma 1.3.11]]) there is a functor $F \to E_{W-loc}$ and a natural isomorphism

\begin{displaymath}
\itexarray{
     F &&\hookrightarrow&& E
     \\
     & \searrow &\Downarrow^{\simeq}& \nearrow
     \\
     && E_{W-loc}
  }
  \,.
\end{displaymath}
Since $F \hookrightarrow E$ and $E_{W-loc} \hookrightarrow E$ are [[full and faithful functor|full and faithful]], so is $F \to E_{W-loc}$. Since by the above it is also essentially surjective, it establishes the equivalence $F \simeq E_{W-loc}$.

\end{proof}
\begin{uprop}
$F$ is [[equivalence of categories|equivalent]] to the [[localization]] $E[W^{-1}]$ of $E$ at $W$.

\end{uprop}
\begin{proof}
By one of the above propositions we know that $W$ is a [[calculus of fractions|left multiplicative systems]].

This implies that the localization $E[W^{-1}]$ is (equivalent to) the category with the same objects as $E$, and with [[hom-sets]] given by

\begin{displaymath}
Hom_{E[W^{-1}]}(a,b)
  =
  \underset{a' \stackrel{p \in W}{\to}a}{colim} Hom_E(a',b)
  \,.
\end{displaymath}
There is an obvious candidate for a functor

\begin{displaymath}
F \to E[W^{-1}]
\end{displaymath}
given on objects by the usual embedding by $f_*$ and on morphism by the map which regards a morphism trivially as a [[span]] with left leg the identity

\begin{displaymath}
(a \to b) 
   \;\;
   \mapsto 
   \;\;
   \left(
   \itexarray{
     a &\to& b
     \\
     \downarrow^{Id_a}
     \\
     a
  }
  \right)
  \,.
\end{displaymath}
For this to be an [[equivalence of categories]] we need to show that this is a [[essentially surjective functor|essentially surjective]] and [[full and faithful functor]].

To see essential surjectivity, let $a$ be any object in $E$ and let $\eta_a : a \to \bar a$ be the component of the unit of our adjunction on $a$, as above. By one of the above propositons, $\eta_a$ is in $W$. This means that the span

\begin{displaymath}
\itexarray{
    a &\stackrel{Id_a}{\to}& a
    \\
    \downarrow^{\eta_a}
    \\
    \bar a
  }
\end{displaymath}
represents an element in $Hom_{E[W^{-1}]}(\bar a,a)$, and this element is clearly an isomorphism: the inverse is represented by

\begin{displaymath}
\itexarray{
    a &\stackrel{\eta_a}{\to}& \bar a
    \\
    \downarrow^{Id_a}
    \\
    a
  }
  \,.
\end{displaymath}
Since every $\bar a$ is in the image of our functor, this shows that it is essentially surjective.

To see fullness and faithfulness, let $a, b\in F$ be any two objects. By one of the above propositions this means in particular that $b$ is a $W$-[[local object]]. As discussed above, this means that every span

\begin{displaymath}
\itexarray{
    a' &\to& b
    \\
    \downarrow^w
    \\
    a
  }
\end{displaymath}
with $w \in W$ has a unique extension

\begin{displaymath}
\itexarray{
    a' &\to& b
    \\
    \downarrow^w & \nearrow
    \\
    a
  }
  \,.
\end{displaymath}
But this implies that in the colimit that defines the [[hom-set]] of $E[W^{-1}]$ all these spans are identified with spans whose left leg is the identiy. And these are clearly in bijection with the morphisms in $Hom_E(a,b) \simeq Hom_F(a,b)$ so that indeed

\begin{displaymath}
Hom_{E[W^{-1}]}(a,b)
  \simeq
  Hom_{F}(a,b)
\end{displaymath}
for all $a,b \in F$. Hence our functor is also full and faithful and therefore define an [[equivalence of categories]]

\begin{displaymath}
F \stackrel{\simeq}{\to} E[W^{-1}]
  \,.
\end{displaymath}
\end{proof}
\hypertarget{factorizations_and_images}{}\subsubsection*{{Factorizations and images}}\label{factorizations_and_images}

There is a [[factorization system on a 2-category|factorization system]] on the [[2-category]] [[Topos]] whose left class is the [[surjective geometric morphisms]] and whose right class is the geometric embeddings. The factorization of a geometric morphism can be said to construct its [[image]] in the topos-theoretic sense.

See [[geometric surjection/embedding factorization]].

Moreover, each geometric embedding itself has a [[(dense,closed)-factorization]].

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

In the more general context of [[(∞,1)-topos theory]] an \textbf{$(\infty,1)$-geometric embedding} is an [[(∞,1)-geometric morphism]]

\begin{displaymath}
(f^* \dashv f_*) : \mathcal{X} \stackrel{\leftarrow}{\hookrightarrow}
   \mathcal{Y}
\end{displaymath}
such that the [[right adjoint]] [[direct image]] $f_*$ is a [[full and faithful (∞,1)-functor]].

See [[reflective sub-(∞,1)-category]] for more details.

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

Section VII, 4 of

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

\end{itemize}
and section A4.2 of

\begin{itemize}%
\item [[Peter Johnstone]], \emph{[[Elephant|Sketches of an elephant: a topos theory compendium]]}.

\end{itemize}
[[!redirects geometric embeddings]]

[[!redirects (∞,1)-geometric embedding]] [[!redirects (infinity,1)-geometric embedding]]

[[!redirects sub-(∞,1)-topos]] [[!redirects sub-(∞,1)-toposes]] [[!redirects sub-(infinity,1)-topos]] [[!redirects sub-(infinity,1)-toposes]]

[[!redirects sub-topos]] [[!redirects sub-toposes]] [[!redirects sub-topoi]]

[[!redirects subtopos]] [[!redirects subtoposes]] [[!redirects subtopoi]]

[[!redirects geometric inclusion]] [[!redirects geometric inclusions]]



\end{document}