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\begin{document}

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


\section*{over-topos}

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

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

[[!include topos theory - contents]]

\hypertarget{bundles}{}\paragraph*{{Bundles}}\label{bundles}

[[!include bundles - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Definition}{Definition / Existence}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{EtaleGeometricMorphism}{\'E{}tale geometric morphism}\dotfill \pageref*{EtaleGeometricMorphism} \linebreak
\noindent\hyperlink{PresheafOverTopos}{In terms of sheaves on a slice site}\dotfill \pageref*{PresheafOverTopos} \linebreak
\noindent\hyperlink{GeometricMorphismBySlicing}{Geometric morphisms by slicing}\dotfill \pageref*{GeometricMorphismBySlicing} \linebreak
\noindent\hyperlink{Points}{Topos points}\dotfill \pageref*{Points} \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}

For $\mathcal{T}$ a [[topos]] and $X \in \mathcal{T}$ any [[object]] the [[over category]] $\mathcal{T}/X$ -- the \textbf{slice topos} or \textbf{over-topos} -- is itself a topos: the ``big [[little topos]]'' incarnation of $X$. This fact is sometimes called the ``Fundamental Theorem of Topos Theory''.

More generally, given a [[fibred product]]-preserving functor $u : E \to F$ between [[toposes]], the [[comma category]] $(id_F/u)$ is again a [[topos]], called the [[Artin gluing]].

\hypertarget{Definition}{}\subsection*{{Definition / Existence}}\label{Definition}

\begin{prop}
\label{}\hypertarget{}{}
For $\mathcal{T}$ a [[topos]] and $X \in \mathcal{T}$ any [[object]], the [[slice category]] $\mathcal{T}{/X}$ is itself again a [[topos]].

\end{prop}
A proof is spelled out for instance in \hyperlink{MacLaneMoerdijk}{MacLane-Moerdijk, IV.7 theorem 1}. In particular we have

\begin{prop}
\label{}\hypertarget{}{}
If $\Omega \in \mathcal{T}$ is the [[subobject classifier]] in $\mathcal{T}$, then the [[projection]] $\Omega \times X \to X$ regarded as an object in the slice over $X$ is the subobject classifier of $\mathcal{T}{/X}$.

\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
The [[power object]] of a map $f: A \to X$ is given by the equalizer of the maps $p, t$:

\begin{displaymath}
\itexarray{
    P_X f && \dashrightarrow && P A \times X && \underoverset{t}{p}{\rightrightarrows} && P A
    \\
    &&&&\downarrow^{\mathrlap{1 \times \{\cdot\}_X}} &&&& \uparrow^{\mathrlap{\wedge}}
    \\
    &&&& P A \times P X && \underset{1 \times P f}{\rightarrow} && P A \times P A
  },
\end{displaymath}
where $p$ is the projection map and $t$ is the composition $\wedge \circ (1 \times P f) \circ (1 \times \{\cdot\}_X)$. In the internal language, this says

\begin{displaymath}
P_X f = \{(B, x) \in P A \times X: (\forall b \in B) f(b) = x\}.
\end{displaymath}
The map to $X$ is given by projection onto the second factor.

\end{prop}
\begin{remark}
\label{AsExampleOfToposOfCoalgebrasOverAComonad}\hypertarget{AsExampleOfToposOfCoalgebrasOverAComonad}{}
The fact that the slice $\mathcal{T}/X$ is a topos, and particularly the construction of power objects above, can be deduced from a more general result: that the category of coalgebras of a pullback-preserving comonad $G: \mathcal{T} \to \mathcal{T}$ is a topos. See at \emph{[[topos of coalgebras over a comonad]]}. In the case of a slice topos, the comonad would be $X \times -: \mathcal{T} \to \mathcal{T}$ (with comultiplication induced by the diagonal $X \to X \times X$, and counit induced by the projection $!: X \to 1$). This result also subsumes the weaker result where $G$ is assumed to preserve finite limits. See the [[Elephant]], Section A, Remark 4.2.3. A proof of a still more general result may be found \href{https://ncatlab.org/toddtrimble/published/Three+topos+theorems+in+one}{here}.

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

\hypertarget{EtaleGeometricMorphism}{}\subsubsection*{{\'E{}tale geometric morphism}}\label{EtaleGeometricMorphism}

\begin{prop}
\label{}\hypertarget{}{}
For $\mathcal{T}$ a [[Grothendieck topos]] and $X \in \mathcal{T}$ any object, the canonical projection functor $X_! : \mathcal{T}/X \to \mathcal{T}$ is part of an [[essential geometric morphism]]

\begin{displaymath}
(X_! \dashv X^* \dashv X_*)
  : 
  \mathcal{T}/X \stackrel{\overset{X_!}{\to}}{\stackrel{\overset{X^*}{\leftarrow}}{\underset{X_*}{\to}}}
  \mathcal{T}
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
The functor $X^*$ is given by taking the [[product]] with $X$:

\begin{displaymath}
X^* : K \mapsto (p_2 : K \times X \to X)
  \,,
\end{displaymath}
since [[commuting diagram]]s

\begin{displaymath}
\itexarray{
    A &&\to&& K \times X
    \\
    & \searrow && \swarrow_{\mathrlap{p_2}}
    \\
    && X
  }
\end{displaymath}
are evidently uniquely specified by their components $A \to K$.

Moreover, since in the Grothendieck topos $\mathcal{T}$ we have [[universal colimits]], it follows that $(-) \times X$ preserves all [[colimit]]s. Therefore by the [[adjoint functor theorem]] a further [[right adjoint]] $X_*$ exists.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
One also says that $X_!$ is the \emph{[[dependent sum]]} operation and $X_*$ the [[dependent product]] operation. As discussed there, this can be seen to compute spaces of [[section]]s of [[bundles]] over $X$.

Moreover, in terms of the [[internal logic]] of $\mathcal{T}$ the functor $X_!$ is the \emph{[[existential quantifier]]} $\exists$ and $X_*$ is the \emph{[[universal quantifier]]} $\forall$.

\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
A [[geometric morphism]] $\mathcal{E} \to \mathcal{T}$ equivalent to one of the form $(X_! \dashv X^* \dashv X_*)$ is called an \textbf{[[etale geometric morphism]]}.

\end{defn}
More generally:

\begin{prop}
\label{GeneralEtaleGeometricMorphism}\hypertarget{GeneralEtaleGeometricMorphism}{}
For $\mathcal{E}$ a [[Grothendieck topos]] and $f : X \to Y$ a [[morphism]] in $\mathcal{E}$, there is an induced [[essential geometric morphism]]

\begin{displaymath}
(f_! \dashv  f^* \dashv f_*)
  :
  \mathcal{E}/X
    \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}}
  \mathcal{E}/Y
  \,,
\end{displaymath}
where $f_!$ is given by postcomposition with $f$ and $f^*$ by [[pullback]] along $X$.

\end{prop}
\begin{proof}
By [[universal colimit]]s in $\mathcal{E}$ the [[pullback]] functor $f^*$ preserves both [[limit]]s and [[colimit]]s. By the [[adjoint functor theorem]] and using that the over-toposes are [[locally presentable categories]], this already implies that it has a [[left adjoint]] and a [[right adjoint]]. That the left adjoint is given by postcomposition with $f$ follows from the universality of the pullback: for $(a : A \to X)$ in $\mathcal{E}/X$ and $(b : B \to Y)$ in $\mathcal{E}/Y$ we have unique factorizations

\begin{displaymath}
\itexarray{
    A &\to& X \times_X B &\to& B
    \\
    &{}_{\mathllap{a}}\searrow& \downarrow^{\mathrlap{f^*(b)}} && \downarrow^{\mathrlap{b}}
    \\
    && X &\stackrel{f}{\to}& Y
  }
\end{displaymath}
in $\mathcal{E}$, hence an isomorphism

\begin{displaymath}
\mathcal{E}/Y(f_*(A \to X), (B \to Y)) \simeq
  \mathcal{E}/X((A \to X), f^*(B \to Y))
  \,.
\end{displaymath}
\end{proof}
\hypertarget{PresheafOverTopos}{}\subsubsection*{{In terms of sheaves on a slice site}}\label{PresheafOverTopos}

Generally, for $C$ a [[site]], $c \in C$ an [[object]], and $L(y(c)) \in Sh(C)$ the [[sheafification]] of its image under the [[Yoneda embedding]], there is an [[equivalence of categories]]

\begin{displaymath}
Sh\big(
    C /c
  \big)
  \;\simeq\;
  Sh(C)/(L(y(c)))
\end{displaymath}
between the [[category of sheaves]] on the [[slice category]] $C/c$ with its evident induced [[structure]] of a [[site]], and the [[slice topos]] of the category of sheaves on $C$, sliced over $L(y(c))$.

This is for instance in Verdier's exposé III.5 prop.5.4 (\hyperlink{SGA4}{SGA4}, p.295).

$\backslash$linebreak

We now discuss this in more detail for the special case of over-[[presheaf toposes]].

Let $C$ be a [[small category]], $c$ an [[object]] of $C$ and let $C/c$ be the [[over category]] of $C$ over $c$.

Write

\begin{itemize}%
\item $PSh(C/c) = [(C/c)^{op}, Set]$ for the [[category of presheaves]] on $C/c$


\item and write $PSh(C)/Y(c)$ for the [[over category]] of [[presheaf|presheaves]] on $C$ over the presheaf $Y(c)$, where $Y : C \to PSh(C)$ is the [[Yoneda embedding]].



\end{itemize}
\begin{prop}
\label{representable_case}\hypertarget{representable_case}{}
There is an [[equivalence of categories]]

\begin{displaymath}
e : PSh(C/c) \stackrel{\simeq}{\to} PSh(C)/Y(c)
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
The functor $e$ takes $F \in PSh(C/c)$ to the presheaf $F' : d \mapsto \sqcup_{f \in C(d,c)} F(f)$ which is equipped with the natural transformation $\eta : F' \to Y(c)$ with component map

\begin{displaymath}
\eta_d : \sqcup_{f \in C(d,c)} F(f) \to C(d,c)
  : ((f \in C(d,c), \theta \in F(f)) \mapsto f
  \,.
\end{displaymath}
A weak inverse of $e$ is given by the functor

\begin{displaymath}
\bar e : PSh(C)/Y(c) \to PSh(C/c)
\end{displaymath}
which sends $\eta : F' \to Y(c)$ to $F \in PSh(C/c)$ given by

\begin{displaymath}
F : (f : d \to c) \mapsto F'(d)|_c
  \,,
\end{displaymath}
where $F'(d)|_c$ is the [[pullback]]

\begin{displaymath}
\itexarray{
     F'(d)|_c &\to& F'(d)
     \\
     \downarrow && \downarrow^{\eta_d}
     \\
     pt &\stackrel{f}{\to}& C(d,c)
  }
  \,.
\end{displaymath}
\end{proof}
\begin{lemma}
\label{}\hypertarget{}{}
Suppose the presheaf $F \in PSh(C/c)$ does not actually depend on the morphisms to $c$, i.e. suppose that it factors through the forgetful functor from the [[over category]] to $C$:

\begin{displaymath}
F : (C/c)^{op} \to C^{op} \to Set
  \,.
\end{displaymath}
Then $F'(d) = \sqcup_{f \in C(d,c)} F(f)
  = \sqcup_{f \in C(d,c)} F(d)
  \simeq
  C(d,c) \times F(d)$ and hence $F ' = Y(c) \times F$ with respect to the [[closed monoidal structure on presheaves]].

\end{lemma}
\begin{lemma}
\label{}\hypertarget{}{}
Consider $\int_C Y(c)$ , the [[category of elements]] of $Y(c):C^{op}\to Set$. This has objects $(d_1,p_1)$ with $p_1\in Y(c)(d_1)$, hence $p_1$ is just an arrow $d_1\to c$ in $C$. A map from $(d_1, p_1)$ to $(d_2, p_2)$ is just a map $u:d_1\to d_2$ such that $p_2\circ u =p_1$ but this is just a morphism from $p_1$ to $p_2$ in $C/c$.

Hence, the above proposition \ref{representable_case} can be rephrased as $PSh(\int_C Y(c))\simeq PSh(C)/Y(c)$ which is an instance of the following formula:

\end{lemma}
\begin{prop}
\label{}\hypertarget{}{}
Let $P:C^{op}\to Set$ be a presheaf. Then there is an [[equivalence of categories]]

\begin{displaymath}
PSh(\int_C P) \simeq PSh(C)/P
  \,.
\end{displaymath}
\end{prop}
On objects this takes $F : (\int_C P)^{op} \to Set$ to

\begin{displaymath}
i(F)(A \in C) = \{ (p,a) | p \in P(A), a \in F(A,p) \} = \Sigma_{p \in P(A)} F(A,p)
\end{displaymath}
with obvious projection to $P$. The inverse takes $f : Q \to P$ to

\begin{displaymath}
i^{-1}(f)(A, p \in P(A)) = f_A^{-1}(p)\;.
\end{displaymath}
For a proof see \hyperlink{KS06}{Kashiwara-Schapira (2006, Lemma 1.4.12, p. 26)}. For a more general statement involving slices of Grothendieck toposes see \hyperlink{MacLaneMoerdijk}{Mac Lane-Moerdijk (1994, p.157)}.

In particular, this equivalence shows that \emph{slices of presheaf toposes are presheaf toposes}.

\hypertarget{GeometricMorphismBySlicing}{}\subsubsection*{{Geometric morphisms by slicing}}\label{GeometricMorphismBySlicing}

\begin{prop}
\label{SliceGeometricMorphism}\hypertarget{SliceGeometricMorphism}{}
For $(f^* \dashv f_*) : \mathcal{T} \to \mathcal{E}$ a [[geometric morphism]] of toposes and $X \in \mathcal{E}$ any [[object]], there is an induced geometric morphism between the slice-toposes

\begin{displaymath}
(f^*/X \dashv f_*) : \mathcal{T}/f^*X \to \mathcal{E}/X
  \,,
\end{displaymath}
where the [[inverse image]] $f^*/X$ is the evident application of $f^*$ to diagrams in $\mathcal{E}$.

\end{prop}
\begin{proof}
The slice adjunction $(f^*/X \dashv f_*/X)$ is discussed : the [[left adjoint]] $f^*/X$ is the evident induced functor. Since  $\mathcal{E}/X$ are computed as limits in $\mathcal{E}$ of diagrams with a single bottom element $X$ adjoined, $f^*/X$ preserves finite limits, since $f^*$ does, so that $(f^*/X \dashv f_*/X)$ is indeed a [[geometric morphism]].

\end{proof}
\hypertarget{Points}{}\subsubsection*{{Topos points}}\label{Points}

We discuss [[point of a topos|topos points]] of over-toposes.

\begin{prop}
\label{}\hypertarget{}{}
Let $\mathcal{E}$ be a [[topos]], $X \in \mathcal{E}$ an [[object]] and

\begin{displaymath}
(e^* \dashv e_*)  : Set \to \mathcal{E}
\end{displaymath}
a [[point of a topos|point]] of $\mathcal{E}$. Then for every element $x \in e^*(X)$ there is a point of the slice topos $\mathcal{E}/X$ given by the composite

\begin{displaymath}
(e,x)
  :
  Set \stackrel{\overset{x^*}{\leftarrow}}{\underset{x_*}{\to}}
  Set/e^*(X) 
  \stackrel{\overset{e^*/X}{\leftarrow}}{\underset{e_*/X}{\to}}
  \mathcal{E}/X
  \,.
\end{displaymath}
\end{prop}
Here $(e^*/X \dashv e_*/X)$ is the slice geometric morphism of $e$ over $X$ discussed \href{SliceGeometricMorphism}{above} and $(x^* \dashv x_*)$ is the \'e{}tale geometric morphism discussed \href{GeneralEtaleGeometricMorphism}{above} induced from the morphism $* \stackrel{x}{\to} e^*(X)$.

Hence the [[inverse image]] of $(e,x)$ sends $A \to X$ to the [[fiber]] of $e^*(A) \to e^*(X)$ over $x$.

\begin{prop}
\label{}\hypertarget{}{}
If $\mathcal{E}$ has [[point of a topos|enough point]]s then so does the slice topos $\mathcal{E}/X$ for every $X \in \mathcal{E}$.

\end{prop}
\begin{proof}
That $\mathcal{E}$ has enough points means that a morphism $f : A \to B$ in $\mathcal{E}$ is an [[isomorphism]] precisely if for every [[point of a topos|point]] $e : Set \to \mathcal{E}$ the function $e^*(f) : e^*(A) \to e^*(B)$ is an isomorphism.

A morphism in the slice topos, given by a diagram

\begin{displaymath}
\itexarray{
     A &&\stackrel{f}{\to}&& B
     \\
     & \searrow && \swarrow
     \\
     && X
  }
\end{displaymath}
in $\mathcal{E}$ is an isomorphism precisely if $f$ is. By the above observation we have that under the [[inverse image]]s of the slice topos points $(e,x \in e^*(X))$ this maps to the fibers of

\begin{displaymath}
\itexarray{
     e^*(A) &&\stackrel{e^*(f)}{\to}&& e^*(B)
     \\
     & \searrow && \swarrow
     \\
     && e^*(X)
  }
\end{displaymath}
over all points $* \stackrel{x}{\to} e^*(X)$. Since in [[Set]] every object $S$ is a [[coproduct]] of the point indexed over $S$, $S \simeq \coprod_S *$ and using [[universal colimits]] in $S$, we have that if $x^* e^*(f)$ is an isomorphism for all $x \in e^*(X)$ then $e^*(f)$ was already an isomorphism.

The claim then follows with the assumption that $\mathcal{E}$ has enough points.

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

\begin{itemize}%
\item [[over-category]]

\begin{itemize}%
\item [[slice category]]


\item [[under category]]


\item \textbf{over-topos}


\item [[Artin gluing]]



\end{itemize}

\item [[over (∞,1)-category]],

\begin{itemize}%
\item [[model structure on an over category]]


\item [[over-(∞,1)-topos]]



\end{itemize}


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

\begin{itemize}%
\item [[Michael Artin]], [[Alexander Grothendieck]], [[Jean-Louis Verdier]], \emph{Théorie des Topos et Cohomologie Etale des Schémas ([[SGA4]])}, Springer \textbf{LNM} vol.269 (1972). (In particular, exposé III.5 and exposé IV.5 on the ``induced topos'' - \emph{topos induit} = slice topos)


\item [[Masaki Kashiwara]], [[Pierre Schapira]], \emph{Categories and Sheaves} , Springer Heidelberg 2006.


\item [[Saunders Mac Lane]], [[Ieke Moerdijk]], \emph{[[Sheaves in Geometry and Logic]]} , Springer Heidelberg 1994. (Especially section IV.7)



\end{itemize}
[[!redirects over topos]] [[!redirects over toposes]] [[!redirects over-toposes]] [[!redirects over topoi]] [[!redirects over-topoi]] [[!redirects slice topos]] [[!redirects slice toposes]] [[!redirects slice topoi]]



\end{document}