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\begin{document}

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\section*{over-(infinity,1)-topos}

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

\hypertarget{topos_theory}{}\paragraph*{{$(\infty,1)$-Topos Theory}}\label{topos_theory}

[[!include (infinity,1)-topos - 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}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{base_change_to_the_point}{Base change to the point}\dotfill \pageref*{base_change_to_the_point} \linebreak
\noindent\hyperlink{general_base_change}{General base change}\dotfill \pageref*{general_base_change} \linebreak
\noindent\hyperlink{SheavesOnBigSite}{As $(\infty,1)$-sheaves on the big $(\infty,1)$-site of an object}\dotfill \pageref*{SheavesOnBigSite} \linebreak
\noindent\hyperlink{ObjectClassifier}{Object classifier}\dotfill \pageref*{ObjectClassifier} \linebreak
\noindent\hyperlink{InHomotopyTypeTheory}{In homotopy type theory}\dotfill \pageref*{InHomotopyTypeTheory} \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 $\mathbf{H}$ an [[(∞,1)-topos]] and $X \in \mathbf{H}$ any [[object]], the [[over-(∞,1)-category]] $\mathbf{H}/X$ is itself an $(\infty,1)$-topos: the \emph{over-$(\infty,1)$-topos} .

If we think of $\mathbf{H}$ as a [[big topos]], then for $X \in \mathbf{H}$ we may think of $\mathbf{H}/X \in$ [[(∞,1)-topos]] as the [[little topos]]-incarnation of $X$. The [[object]]s of $\mathbf{H}/X$ we may think of as [[(∞,1)-sheaves]] on $X$.

This correspondence between objects of $X$ and their little-topos incarnation is entirly natural: $\mathbf{H}$ is equivalently recovered as the [[(∞,1)-category]] whose objects are over-$(\infty,1)$-toposes $\mathbf{H}/X$ and whose morphisms are [[(∞,1)-geometric morphism]]s \emph{over} $\mathbf{H}$.

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

\begin{prop}
\label{}\hypertarget{}{}
For $\mathbf{H}$ an [[(∞,1)-topos]] and $X \in \mathbf{H}$ an [[object]] also the [[over-(∞,1)-category]] $\mathbf{H}/X$ is an $(\infty,1)$-topos. This is the \textbf{over-$(\infty,1)$-topos} of $\mathbf{H}$ over $X$.

\end{prop}
This is [[Higher Topos Theory|HTT, prop 6.3.5.1 1)]].

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

\hypertarget{base_change_to_the_point}{}\subsubsection*{{Base change to the point}}\label{base_change_to_the_point}

\begin{prop}
\label{}\hypertarget{}{}
There is a canonical [[(∞,1)-geometric morphism]]

\begin{displaymath}
\mathbf{H}/X 
    \stackrel{\overset{X_!}{\longrightarrow}}{\stackrel{\overset{X^*}{\longleftarrow}}{\underset{X_*}{\longrightarrow}}}
  \mathbf{H}
\end{displaymath}
where the extra [[left adjoint]] $X_!$ is the obvious projection $X_! : (Y \to X) \mapsto X$, and $X_*$ is given by forming the [[(∞,1)-limit|(∞,1)-product]] with $X$.

\end{prop}
This is called an [[etale geometric morphism]]. See there for more details.

\begin{proof}
The fact that $(X_! \dashv X^*)$ follows from the universal property of the products. The fact that $X^*$ preserves [[(∞,1)-colimit]]s and hence has a further [[right adjoint]] $X_*$ by the [[adjoint (∞,1)-functor theorem]] follows from that fact that $\mathbf{H}$ has [[universal colimits]].

\end{proof}
\begin{cor}
\label{}\hypertarget{}{}
If $\mathbf{H}$ is a [[locally ∞-connected (∞,1)-topos]] then for all $X \in \mathbf{H}$ also the over-$(\infty,1)$-topos $\mathbf{H}/X$ is locally $\infty$-connected.

\end{cor}
\begin{proof}
The composite of [[(∞,1)-geometric morphism]]s

\begin{displaymath}
\mathbf{H}/X 
    \stackrel{\overset{X_!}{\longrightarrow}}{\stackrel{\overset{X^*}{\longleftarrow}}{\underset{X_*}{\longrightarrow}}}
  \mathbf{H}
  \stackrel{\overset{\Pi_{\mathbf{H}}}{\longrightarrow}}{\stackrel{\overset{LConst_{\mathbf{H}}}{\longleftarrow}}{\underset{\Gamma_{\mathbf{H}}}{\longrightarrow}}}
  \infty Grpd
\end{displaymath}
is itself a geometric morphism. Since [[∞Grpd]] is the [[terminal object]] in [[(∞,1)Topos]] this must be the [[global section]] geometric morphism for $\mathbf{H}/X$. Since it has the extra left adjoint $\Pi \circ X_!$ it is locally $\infty$-connected.

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
Let $((\infty,1)Topos/\mathbf{H})_{et} \subset (\infty,1)Topos/\mathbf{H}$ be the full [[sub-(∞,1)-category]] on the [[etale geometric morphism]]s $\mathbf{H}/X \to \mathbf{H}$. Then there is an [[equivalence in an (∞,1)-category|equivalence]]

\begin{displaymath}
((\infty,1)Topos/\mathbf{H})_{et}
  \simeq
  \mathbf{H}
\end{displaymath}
\end{prop}
See [[etale geometric morphism]] for more details.

\hypertarget{general_base_change}{}\subsubsection*{{General base change}}\label{general_base_change}

See [[base change geometric morphism]].

\hypertarget{SheavesOnBigSite}{}\subsubsection*{{As $(\infty,1)$-sheaves on the big $(\infty,1)$-site of an object}}\label{SheavesOnBigSite}

We spell out how $\mathbf{H}/X$ is the [[(∞,1)-category of (∞,1)-sheaves]] over the [[big site]] of $X$.

\begin{prop}
\label{SlicingCommutesWithPassingToPresheaves}\hypertarget{SlicingCommutesWithPassingToPresheaves}{}
\textbf{(forming overcategories commutes with passing to presheaves)}

Let $C$ be a [[small (∞,1)-category]] and $X : K \to C$ a [[diagram]]. Write $C_{/X}$ and $PSh(C)/_{X}$ for the corresponding [[over (∞,1)-categories]], where -- notationally implicitly -- we use the [[(∞,1)-Yoneda embedding]] $C \to PSh(C)$.

Then we have an [[equivalence of (∞,1)-categories]]

\begin{displaymath}
PSh(C/X) \stackrel{\simeq}{\longrightarrow} PSh(C)/X
  \,.
\end{displaymath}
\end{prop}
This appears as [[Higher Topos Theory|HTT, 5.1.6.12]]. For more on this see [[(∞,1)-category of (∞,1)-presheaves]].

\begin{remark}
\label{}\hypertarget{}{}
Here we may think of $C/X$ as the [[big site]] of the object $c \in PSh(C)$, hence of $PSh(C/X)$ as presheaves on $X$.

\end{remark}
\begin{prop}
\label{CommaSiteForSliceTopos}\hypertarget{CommaSiteForSliceTopos}{}
Let $C$ be equipped with a [[subcanonical coverage]], let $X \in C$ and regard $C/X$ as an [[(∞,1)-site]] with the [[big site]]-[[coverage]]. Then we have

\begin{displaymath}
Sh(C/X) \simeq Sh(C)/X
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
By the discussion of  [[adjoint (∞,1)-functor]]s we have that the adjunction

\begin{displaymath}
(F \dashv i) : Sh(C) \stackrel{\overset{F}{\leftarrow}}{\hookrightarrow}
  PSh(C)
\end{displaymath}
passes to an adjunction on the [[over-(∞,1)-categories]]

\begin{displaymath}
(F/X \dashv i/X) : Sh(C)/X \stackrel{\overset{F}{\leftarrow}}{\hookrightarrow}
  PSh(C)/X
  \,,
\end{displaymath}
(where we are using that $F  i X \simeq X$ by the assumption that the coverage is subcanonical, so that the representable $X$ is a [[(∞,1)-sheaf]]), such that $i/X$ is still a [[full and faithful (∞,1)-functor]] (where we are using that the unit $X \to F i X$ is an equivalence, since $X$ is a sheaf).

Since moreover the [[(∞,1)-limits]] in $Sh(C)/X$ are computed as limits in $Sh(C)$ over diagrams with a bottom element adjoined (as discussed at ) it follows that with $F$ preserving all finite limits, so does $F/X$.

In summary we have that $(F/X \dashv i/X)$ is a [[reflective sub-(∞,1)-category]] of $PSh(C/X)$ hence is the [[(∞,1)-category of (∞,1)-sheaves]] on the category $C/X$ for \emph{some} [[(∞,1)-site]]-structure. But since $F/X$ inverts precisely those morphisms that are inverted by $F$ under the projection $PSh(C)/X \to PSh(C)$, it follows that this is the [[big site]] structure on $C/X$ (this is the defining property of the big site).

\end{proof}
Specifically for the $(\infty,1)$-topos $\mathbf{H} =$ [[∞Grpd]] we also have the following characterization.

\begin{prop}
\label{}\hypertarget{}{}
For $\mathbf{H} =$ [[∞Grpd]] we have that for $X \in \infty Grp$ any [[∞-groupoid]] the corresponding over-$(\infty,1)$-topos is equivalent to the [[(∞,1)-category of (∞,1)-presheaves]] on $X$:

\begin{displaymath}
\infty Grpd/X \simeq PSh(X) \simeq [X, \infty Grpd]
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
This is a special case of the [[(∞,1)-Grothendieck construction]]. See the section .

\end{proof}
The following proposotion asserts that the over-$(\infty,1)$-topos over an $n$-[[truncated]] object indeed behaves like a generalized [[n-groupoid]]

\begin{prop}
\label{}\hypertarget{}{}
For $n \in \mathbb{N}$ and $\mathcal{X}$ an [[n-localic (∞,1)-topos]], then the over-$(\infty,1)$-topos $\mathcal{X}/U$ is $n$-localic precisely if the object $U$ is $n$-[[truncated]].

\end{prop}
This is (\hyperlink{StrSp}{StrSp, lemma 2.3.16}).

\hypertarget{ObjectClassifier}{}\subsubsection*{{Object classifier}}\label{ObjectClassifier}

If $Obj_\kappa \in \mathbf{H}$ is an [[object classifier]] for $\kappa$-[[small objects]], then the [[projection]] $Obj_\Kappa \times X \to X$ regarded as an object in the slice is a $\kappa$-small object classifier in $\mathbf{H}_{/X}$.

\hypertarget{InHomotopyTypeTheory}{}\subsubsection*{{In homotopy type theory}}\label{InHomotopyTypeTheory}

If a [[homotopy type theory]] is the [[internal language]] of $\mathbf{H}$, then then theory in [[context]] $x : X \vdash \cdots$ is the internal language of $\mathbf{H}_{/X}$.

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

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


\item [[over-category]]

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


\item [[under category]]


\item [[over topos]]



\end{itemize}

\item [[over (∞,1)-category]],

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


\item \textbf{over-(∞,1)-topos}



\end{itemize}

\item [[arrow (∞,1)-category]], [[arrow (∞,1)-topos]]



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

The general notion is discussed in section 6.3.5 of

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

\end{itemize}
Some related remarks are in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Structured Spaces]]}

\end{itemize}
[[!redirects over-(∞,1)-topos]]

[[!redirects over-(∞,1)-toposes]] [[!redirects over-(∞,1)-topoi]] [[!redirects over-(infinity,1)-toposes]] [[!redirects over-(infinity,1)-topoi]]

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

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

[[!redirects slice homotopy-topos]] [[!redirects slice homotopy-toposes]] [[!redirects slice homotopy-topoi]]



\end{document}