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\begin{document}

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\section*{over category}

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

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_codomain_fibration}{Relation to codomain fibration}\dotfill \pageref*{relation_to_codomain_fibration} \linebreak
\noindent\hyperlink{Adjunction}{Adjunctions on overcategories}\dotfill \pageref*{Adjunction} \linebreak
\noindent\hyperlink{RelWithPresheaves}{Presheaves on over-categories and over-categories of presheaves}\dotfill \pageref*{RelWithPresheaves} \linebreak
\noindent\hyperlink{LimitsAndColimits}{Limits and colimits}\dotfill \pageref*{LimitsAndColimits} \linebreak
\noindent\hyperlink{initial_and_terminal_objects}{Initial and terminal objects}\dotfill \pageref*{initial_and_terminal_objects} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

The \textbf{slice category} or \textbf{over category} $\mathbf{C}/c$ of a [[category]] $\mathbf{C}$ over an object $c \in \mathbf{C}$ has

\begin{itemize}%
\item objects that are all arrows $f \in \mathbf{C}$ such that $cod(f) = c$, and


\item morphisms $g: X \to X' \in \mathbf{C}$ from $f:X \to c$ to $f': X' \to c$ such that $f' \circ g = f$.



\end{itemize}
\begin{displaymath}
C/c
  =
  \left\lbrace
    \itexarray{
       X &&\stackrel{g}{\to}&& X'
       \\
       & {}_f \searrow && \swarrow_{f'}
       \\   
       && c
    }
  \right\rbrace
\end{displaymath}
The slice category is a special case of a [[comma category]].

There is a [[forgetful functor]] $U_c: \mathbf{C}/c \to \mathbf{C}$ which maps an object $f:X \to c$ to its domain $X$ and a morphism $g: X \to X' \in \mathbf{C}/c$ (from $f:X \to c$ to $f': X' \to c$ such that $f' \circ g = f$) to the morphism $g: X \to X'$.

The [[duality|dual]] notion is an [[under category]].

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

\begin{itemize}%
\item If $\mathbf{C} = \mathbf{P}$ is a [[partial order|poset]] and $p \in \mathbf{P}$, then the slice category $\mathbf{P}/p$ is the [[down set]] $\downarrow (p)$ of elements $q \in \mathbf{P}$ with $q \leq p$.


\item If $1$ is a [[terminal object]] in $\mathbf{C}$, then $\mathbf{C}/1$ is isomorphic to $\mathbf{C}$.


\item For $X$ a [[topological space]] then the [[category of covering spaces]] over $X$ is a [[full subcategory]] of the slice category $Top_{/X}$ of the [[category of topological spaces]].



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

\hypertarget{relation_to_codomain_fibration}{}\subsubsection*{{Relation to codomain fibration}}\label{relation_to_codomain_fibration}

The assignment of overcategories $C/c$ to objects $c \in C$ extends to a [[functor]]

\begin{displaymath}
C/(-) : C \to Cat
\end{displaymath}
Under the [[Grothendieck construction]] this functor corresponds to the [[codomain fibration]]

\begin{displaymath}
cod : [I,C] \to C
\end{displaymath}
from the [[arrow category]] of $C$. (Note that unless $C$ has [[pullbacks]], this functor is not actually a [[Grothendieck fibration|fibration]], though it is always an opfibration.)

\hypertarget{Adjunction}{}\subsubsection*{{Adjunctions on overcategories}}\label{Adjunction}

\begin{prop}
\label{}\hypertarget{}{}
Let

\begin{displaymath}
(L \dashv R) : D \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}}
     C
\end{displaymath}
be a pair of [[adjoint functors]], where the category $C$ has all [[pullbacks]].

Then for every object $X \in C$ there is induced a pair of adjoint functors between the slice categories

\begin{displaymath}
(L/X \dashv R/X) 
   : D/(L X) \stackrel{\overset{L/X}{\leftarrow}}{\underset{R/X}{\to}}
     C/X
\end{displaymath}
where

\begin{itemize}%
\item $L/X$ is the evident induced functor;


\item $R/X$ is the composite

\begin{displaymath}
R/X : D/{L X} \stackrel{R}{\to} C/{(R L X)} \stackrel{i_{X}^*}{\to}
   C/X
\end{displaymath}
of the evident functor induced by $R$ with the [[pullback]] along the $(L \dashv R)$-[[unit of an adjunction|unit]] at $X$.



\end{itemize}
\end{prop}
\hypertarget{RelWithPresheaves}{}\subsubsection*{{Presheaves on over-categories and over-categories of presheaves}}\label{RelWithPresheaves}

Let $C$ be a [[category]], $c$ an [[object]] of $C$ and let $C/c$ be the [[over category]] of $C$ over $c$. Write $PSh(C/c) = [(C/c)^{op}, Set]$ for the [[category of presheaves]] on $C/c$ 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]].

\begin{prop}
\label{}\hypertarget{}{}
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 $\eta_d \sqcup_{f \in C(d,c)} F(f) \to C(d,c)$.

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{example}
\label{}\hypertarget{}{}
Suppose the presheaf $F \in PSh(C/c)$ does not actually depend on the morphsims 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{example}
See also [[functors and comma categories]].

For the analogous statement in [[(∞,1)-category]] theory see 

at [[(∞,1)-category of (∞,1)-presheaves]].

\hypertarget{LimitsAndColimits}{}\subsubsection*{{Limits and colimits}}\label{LimitsAndColimits}

\begin{prop}
\label{}\hypertarget{}{}
A [[limit]] in an [[under category]] is computed as a limit in the underlying category.

Precisely: let $C$ be a [[category]], $t \in C$ an [[object]], and $t/C$ the corresponding [[under category]], and $p : t/C \to C$ the obvious projection.

Let $F : D \to t/C$ be any [[functor]]. Then, if it exists, the [[limit]] over $p \circ F$ in $C$ is the image under $p$ of the limit over $F$:

\begin{displaymath}
p(\lim F)  \simeq \lim (p F)
\end{displaymath}
and $\lim F$ is uniquely characterized by $\lim (p F)$.

\end{prop}
\begin{proof}
Over a morphism $\gamma : d \to d'$ in $D$ the limiting cone over $p F$ (which exists by assumption) looks like

\begin{displaymath}
\itexarray{
    && \lim p F
    \\
    & \swarrow && \searrow
    \\
    p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d') 
  }
\end{displaymath}
By the universal property of the limit this has a unique lift to a cone in the [[under category]] $t/C$ over $F$:

\begin{displaymath}
\itexarray{
    && t 
    \\
    & \swarrow &\downarrow & \searrow
    \\
    && \lim p F
    \\
    \downarrow & \swarrow && \searrow & \downarrow
    \\
    p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d') 
  }
\end{displaymath}
It therefore remains to show that this is indeed a limiting cone over $F$. Again, this is immediate from the universal property of the limit in $C$. For let $t \to Q$ be another cone over $F$ in $t/C$, then $Q$ is another cone over $p F$ in $C$ and we get in $C$ a universal morphism $Q \to \lim p F$

\begin{displaymath}
\itexarray{
    && t
    \\
    & \swarrow & \downarrow & \searrow
    \\
    && Q
    \\
    \downarrow & \swarrow &\downarrow & \searrow & \downarrow
    \\
    && \lim p F
    \\
    \downarrow & \swarrow && \searrow & \downarrow
    \\
    p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d') 
  }
\end{displaymath}
A glance at the diagram above shows that the composite $t \to Q \to \lim p F$ constitutes a morphism of cones in $C$ into the limiting cone over $p F$. Hence it must equal our morphism $t \to \lim p F$, by the universal property of $\lim p F$, and hence the above diagram does commute as indicated.

This shows that the morphism $Q \to \lim p F$ which was the unique one giving a cone morphism on $C$ does lift to a cone morphism in $t/C$, which is then necessarily unique, too. This demonstrates the required universal property of $t \to \lim p F$ and thus identifies it with $\lim F$.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
One often says ``$p$ [[reflected limit|reflects limits]]'' to express the conclusion of this proposition. A conceptual way to consider this result is by appeal to a more general one: if $U: A \to C$ is [[monadic functor|monadic]] (i.e., has a left adjoint $F$ such that the canonical comparison functor $A \to (U F)-Alg$ is an equivalence), then $U$ both reflects and preserves limits. In the present case, the projection $p: A = t/C \to C$ is monadic, is essentially the category of algebras for the monad $T(-) = t + (-)$, at least if $C$ admits binary coproducts. (Added later: the proof is even simpler: if $U: A \to C$ is the underlying functor for the category of algebras of an \emph{endofunctor} on $C$ (as opposed to algebras of a monad), then $U$ reflects and preserves limits; then apply this to the endofunctor $T$ above.)

\end{remark}
\begin{prop}
\label{LimitsInSliceViaLimitsOfCoconedDiagram}\hypertarget{LimitsInSliceViaLimitsOfCoconedDiagram}{}
For $\mathcal{C}$ a [[category]], $X \;\colon\; \mathcal{D} \longrightarrow \mathcal{C}$ a [[diagram]], $\mathcal{C}_{/X}$ the [[comma category]] (the over-category if $\mathcal{D}$ is the point) and $F \;\colon\; K \to \mathcal{C}_{/X}$ a [[diagram]] in the [[comma category]], then the [[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{initial_and_terminal_objects}{}\subsubsection*{{Initial and terminal objects}}\label{initial_and_terminal_objects}

As a special case of the above discussion of limits and colimits in a slice $\mathcal{C}_{/X}$ we obtain the following statement, which of course is also immediately checked explicitly.

\begin{cor}
\label{}\hypertarget{}{}
\begin{itemize}%
\item If $\mathcal{C}$ has an initial object $\emptyset$, then $\mathcal{C}_{/X}$ has an [[initial object]], given by $\langle \emptyset \to X\rangle$.


\item The [[terminal object]] of $\mathcal{C}_{/X}$ is $\mathrm{id}_X$.



\end{itemize}
\end{cor}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item \textbf{over-category}

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


\item [[under category]]


\item [[over topos]]



\end{itemize}

\item [[slice 2-category]]


\item [[over (∞,1)-category]],

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


\item [[over-(∞,1)-topos]]



\end{itemize}


\end{itemize}
[[!redirects overcategory]] [[!redirects over categories]] [[!redirects over-category]] [[!redirects over-categories]] [[!redirects overcategories]] [[!redirects slice category]] [[!redirects slice categories]]



\end{document}