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\begin{document}

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

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

\hypertarget{category_theory}{}\paragraph*{{$(\infty,1)$-Category theory}}\label{category_theory}

[[!include quasi-category 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_over1categories}{Relation to over-1-categories}\dotfill \pageref*{relation_to_over1categories} \linebreak
\noindent\hyperlink{functoriality_of_the_slicing}{Functoriality of the slicing}\dotfill \pageref*{functoriality_of_the_slicing} \linebreak
\noindent\hyperlink{HamSpacesInASlice}{Hom-spaces in a slice}\dotfill \pageref*{HamSpacesInASlice} \linebreak
\noindent\hyperlink{LimitsAndColimits}{Limits and Colimits in a slice}\dotfill \pageref*{LimitsAndColimits} \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 $C$ an [[(∞,1)-category]] and $X \in C$ an [[object]], the \emph{over-$(\infty,1)$-category} or \emph{slice $(\infty,1)$-category} $C_{/X}$ is the $(\infty,1)$-category whose [[objects]] are [[morphism]] $p : Y \to X$ in $C$, whose morphisms $\eta : p_1 \to p_2$ are [[2-morphisms]] in $C$ of the form

\begin{displaymath}
\itexarray{
     Y_1 &&\stackrel{f}{\to}&& Y_2
     \\
     & {}_{\mathllap{p_1}}\searrow &\swArrow_{\simeq}^{\eta}& \swarrow_{\mathrlap{p_2}}
     \\
     && X
  }
  \,,
\end{displaymath}
hence 1-morphisms $f$ as indicated together with a [[homotopy]] $\eta \colon p_2 \circ f \stackrel{\simeq}{\to} p_1$; and generally whose [[n-morphisms]] are [[diagrams]]

\begin{displaymath}
\Delta[n+1] = \Delta[n] \star \Delta[0]\to C
\end{displaymath}
in $C$ of the shape of the cocone under the $n$-[[simplex]] that take the tip of the [[cocone]] to the object $X$.

This is the generalization of the notion of [[over-category]] in ordinary [[category theory]].

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

We give the definition in terms of the model of [[(∞,1)-categories]] in terms of [[quasi-categories]].

Recall from [[join of quasi-categories]] that there are two different but quasi-categorically equivalent definitions of \emph{join}, denoted $\star$ and $\diamondsuit$. Accordingly we have the following two different but quasi-categoricaly equivalent definitions of over/under quasi-categories.

\begin{prop}
\label{}\hypertarget{}{}
Let $C$ be a [[quasi-category]]. let $K$ be any [[simplicial set]] and let

\begin{displaymath}
F : K \to C
\end{displaymath}
be an [[(∞,1)-functor]] -- a morphism of simplicial sets.

\begin{enumerate}%
\item The \textbf{under-quasi-category} $C_{F/}$ is the simplicial set characterized by the property that for any other simplicial set $S$ there is a natural bijection of hom-sets

\begin{displaymath}
Hom_{sSet}(S, C_{F/})
  \cong
  (Hom_{(K\downarrow SSet)})(i_{K,S} , F)
  \,,
\end{displaymath}
where $i_{K,S} : K \to K \star S$ is the canonical inclusion of $K$ into its [[join of simplicial sets]] with $S$.

Similarly, the \textbf{over quasi-category} over $F$ is the simplicial set characterized by the property that

\begin{displaymath}
Hom_{sSet}(S, C_{/F})
  \simeq
  Hom_{(K\downarrow SSet)}(
    j_{K,S} , F
  )
\end{displaymath}
naturally in $S$, where $j_{K,S}$ is the canonical inclusion map $K\to S\star K$.


\item The functor

\begin{displaymath}
sSet \to sSet_{K/}
\end{displaymath}
\begin{displaymath}
S \mapsto K \diamondsuit S
\end{displaymath}
with $\diamondsuit$ denoting the other definition of [[join of quasi-categories]] (as described there)

has a [[right adjoint]]

\begin{displaymath}
sSet_{K/} \to sSet
\end{displaymath}
\begin{displaymath}
(F : K \to C) \mapsto C^{F/}
\end{displaymath}
and its image $C^{F/}$ is another definition of the quasi-category under $F$.



\end{enumerate}
\end{prop}
The first definition in terms of the the mapping property is due to [[Andre Joyal]]. Together with the discussion of the concrete realization it appears as [[Higher Topos Theory|HTT, prop 1.2.9.2]]. The second definition appears in [[Higher Topos Theory|HTT]] above prop. 4.2.1.5.

\begin{prop}
\label{}\hypertarget{}{}
The simplicial sets $C_{/F}$ and $C_{F/}$ are indeed themselves again [[quasi-categories]].

\end{prop}
This appears as [[Higher Topos Theory|HTT, prop. 1.2.9.3]]

\begin{prop}
\label{}\hypertarget{}{}
The two definitions yield equivalent results in that the canonical morphism

\begin{displaymath}
C_{/F} \to C^{/F}
  \,.
\end{displaymath}
is an equivalence of quasi-categories.

\end{prop}
This is [[Higher Topos Theory|HTT, prop. 4.2.1.5]]

From the formula

\begin{displaymath}
(C_{/F})_n = (Hom_{sSet})_F(\Delta^n \star K , C)
\end{displaymath}
we see that

\begin{itemize}%
\item an object in the over quasi-category $C_{/F}$ is a \textbf{[[cone]]} over $F$;.

For instance if $K = \Delta[1]$ then an object in $C_{/F}$ is a 2-cell

\begin{displaymath}
\itexarray{
     && v
     \\
     & \swarrow &\swArrow& \searrow
     \\
     F(0) &&\to&& F(1)
  }
\end{displaymath}
in $C$.


\item a morphism in $C_{/F}$ is a morphism of cones,


\item etc:.



\end{itemize}
So we may think of the overcategory $C_{/F}$ as the \textbf{quasi-category of cones over $F$}. Accordingly we have that

\begin{itemize}%
\item the [[terminal object in a quasi-category|terminal object]] in $C_{/F}$ is (if it exists) the [[limit in a quasi-category|limit in]] $C$ over $F$;


\item the [[terminal object in a quasi-category|initial object]] in $C_{F/}$ is (if it exists) the [[limit in a quasi-category|colimit of]] $F$ in $C$.



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

\hypertarget{relation_to_over1categories}{}\subsubsection*{{Relation to over-1-categories}}\label{relation_to_over1categories}

\begin{prop}
\label{}\hypertarget{}{}
For $C = N(\mathcal{C})$ (the [[nerve]] of) an ordinary [[category]] $\mathcal{C}$ and $K = *$, this construction coincides with the ordinary notion of [[overcategory]] $\mathcal{C}/F$ in that there is a canonical [[isomorphism]] of simplicial sets

\begin{displaymath}
N(\mathcal{C}/F) \simeq N(\mathcal{C})/F
  \,.
\end{displaymath}
\end{prop}
This appears as [[Higher Topos Theory|HTT, remark 1.2.9.6]].

\hypertarget{functoriality_of_the_slicing}{}\subsubsection*{{Functoriality of the slicing}}\label{functoriality_of_the_slicing}

\begin{prop}
\label{}\hypertarget{}{}
If $q : C \to D$ is a [[model structure for quasi-categories|categorical equivalence]] then so is the induced morphism $C_{/F} \to D_{/q F}$.

\end{prop}
This appears as [[Higher Topos Theory|HTT, prop 1.2.9.3]].

\begin{prop}
\label{}\hypertarget{}{}
For $C$ a [[quasi-category]] and $p : X \to C$ any morphism of simplicial sets, the canonical morphisms

\begin{displaymath}
C_{p/} \to C
\end{displaymath}
and

\begin{displaymath}
C^{p/} \to C
\end{displaymath}
are both [[left Kan fibrations]].

\end{prop}
This is a special case of [[Higher Topos Theory|HTT, prop 2.1.2.1]] and [[Higher Topos Theory|prop. 4.2.1.6]].

\begin{prop}
\label{FinalFunctorsInduceEquivalentSlices}\hypertarget{FinalFunctorsInduceEquivalentSlices}{}
Let $v \colon K \to \tilde K$ be a map of small [[(∞,1)-categories]], $\mathcal{C}$ an $(\infty,1)$-category, $\tilde{p}: \tilde{K} \to \mathcal{C}$ and $p = \tilde{p}v$. The induced [[(∞,1)-functor]] between slice $(\infty,1)$-categories

\begin{displaymath}
\mathcal{C}_{/ \tilde{p}} \to \mathcal{C}_{/p}
\end{displaymath}
is an [[equivalence of (∞,1)-categories]] for each diagram $\tilde{p}$ precisely if $v$ is an op-[[final (∞,1)-functor]] (hence if $v^{op}$ is final).

\end{prop}
This is (\hyperlink{Lurie}{Lurie, prop. 4.1.1.8}).

\hypertarget{HamSpacesInASlice}{}\subsubsection*{{Hom-spaces in a slice}}\label{HamSpacesInASlice}

\begin{prop}
\label{}\hypertarget{}{}
For $C$ an [[(∞,1)-category]] and $X \in C$ an [[object]] in $C$ and $f : A \to X$ and $g : B \to X$ two objects in $C/X$, the [[derived hom-space|hom-∞-groupoid]] $C/X(f,g)$ is equivalent to the [[homotopy fiber]] of $C(A,B) \stackrel{g_*}{\to} C(A,X)$ over the morphism $f$: we have an [[(∞,1)-pullback]] [[diagram]]

\begin{displaymath}
\itexarray{
     C/X(f,g) & \longrightarrow & C(A,B)
     \\
     \downarrow && \downarrow^{\mathrlap{g_*}}
     \\
     {*} & \stackrel{\vdash f}{\longrightarrow} & C(A,X)
  }
  \,.
\end{displaymath}
\end{prop}
This is [[Higher Topos Theory|HTT, prop. 5.5.5.12]].

\hypertarget{LimitsAndColimits}{}\subsubsection*{{Limits and Colimits in a slice}}\label{LimitsAndColimits}

The forgetful functor $\mathcal{C}_{/X} \to \mathcal{C}$ out of a slice ([[dependent sum]]) [[reflected limit|reflects]] [[(∞,1)-colimits]]:

\begin{prop}
\label{}\hypertarget{}{}
Let $f \colon K \to \mathcal{C}_{/X}$ be a [[diagram]] in the slice of an [[(∞,1)-category]] $\mathcal{C}$ over an object $X \in \mathcal{C}$. Then if the composite $K \stackrel{f}{\to} \mathcal{C}_{/X} \to \mathcal{C}$ has an [[(∞,1)-colimit]], then so does $f$ itself and the projection $\mathcal{C}_{/q} \to \mathcal{C}$ takes the latter to the former. Conversely, a cocone $K \star \Delta^0  \to \mathcal{C}_{/X}$ under $f$ is an [[(∞,1)-colimit]] of $f$ precisely if the composite $K \star \Delta^0 \to \mathcal{C}_{/X} \to \mathcal{C}$ is an $(\infty,1)$-colimit of the projection of $f$.

\end{prop}
This appears as (\hyperlink{Lurie}{Lurie, prop. 1.2.13.8}).

On the other hand [[(∞,1)-limits]] in the slice are computed as limits over the diagram with the slice-cocone adjoined:

\begin{prop}
\label{}\hypertarget{}{}
For $\mathcal{C}$ an [[(∞,1)-category]], $X \;\colon\; \mathcal{D} \longrightarrow \mathcal{C}$ a [[diagram]], $\mathcal{C}_{/X}$ the [[comma category]] (the over-$\infty$-category if $\mathcal{D}$ is the point) and $F \;\colon\; K \to \mathcal{C}_{/X}$ a [[diagram]] in the [[comma category]], then the [[(∞,1)-limit]] $\underset{\leftarrow}{\lim} F$ in $\mathcal{C}_{/X}$ coincides with the limit $\underset{\leftarrow}{\lim} F/X$ in $\mathcal{C}$.

\end{prop}
For a proof see at [[(∞,1)-limit]] \emph{\href{limit+in+a+quasi-category#InOvercategories}{here}}.

\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 \textbf{over-(∞,1)-category}

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


\item [[over-(∞,1)-topos]], [[etale geometric morphism]]



\end{itemize}

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


\item [[slice 2-category]]



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

Section 1.2.9 of

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

\end{itemize}
[[!redirects over-category in quasi-categories]] [[!redirects over quasi-categories]] [[!redirects over quasicategory]] [[!redirects over-quasi-category]] [[!redirects over-quasicategory]] [[!redirects overquasicategory]] [[!redirects slice quasi-category]] [[!redirects slice quasicategory]]

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

[[!redirects slice infinity-category]] [[!redirects slice infinity-categories]]

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

[[!redirects under-(∞,1)-category]] [[!redirects under-(infinity,1)-category]]

[[!redirects under-(∞,1)-categories]] [[!redirects under-(infinity,1)-categories]]

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

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

[[!redirects over quasi-category]]

[[!redirects slice-(∞,1)-category]] [[!redirects slice-(∞,1)-categories]]

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



\end{document}