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\begin{document}

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\section*{model structure on an over category}

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

\hypertarget{model_category_theory}{}\paragraph*{{Model category theory}}\label{model_category_theory}

[[!include model category theory - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{cofibrant_generation_properness_combinatoriality}{Cofibrant generation, properness, combinatoriality}\dotfill \pageref*{cofibrant_generation_properness_combinatoriality} \linebreak
\noindent\hyperlink{derived_homspaces}{Derived hom-spaces}\dotfill \pageref*{derived_homspaces} \linebreak
\noindent\hyperlink{quillen_adjunctions_between_slice_categories}{Quillen adjunctions between slice categories}\dotfill \pageref*{quillen_adjunctions_between_slice_categories} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\begin{prop}
\label{}\hypertarget{}{}
For $C$ a [[model category]] and $X \in C$ an [[object]], the [[over category]] $C/X$ as well as the [[undercategory]] $X/C$ inherit themselves structures of model categories whose [[fibrations]], [[cofibration]]s and [[weak equivalences]] are precisely the morphism that become fibrations, cofibrations and weak equivalences in $C$ under the forgetful functor $C/X \to C$ or $X/C \to C$.

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

\hypertarget{cofibrant_generation_properness_combinatoriality}{}\subsubsection*{{Cofibrant generation, properness, combinatoriality}}\label{cofibrant_generation_properness_combinatoriality}

\begin{prop}
\label{ModelStructureInheritsGoodProperties}\hypertarget{ModelStructureInheritsGoodProperties}{}
If $\mathcal{C}$ is

\begin{itemize}%
\item a [[cofibrantly generated model category]]


\item or a [[proper model category]]


\item or a [[cellular model category]]



\end{itemize}
then so are $\mathcal{C}_{/X}$ and $\mathcal{C}^{X/}$.

More in detail, if $I,J \subset Mor(\mathcal{C})$ are the classes of [[generating cofibrations]] and of generating acylic cofibrations of $\mathcal{C}$, respectively, then

\begin{itemize}%
\item the generating (acyclic) cofibrations of $\mathcal{C}^{X/}$ are the image under $X \sqcup(-)$ of those of $\mathcal{C}$.

\end{itemize}
\end{prop}
This is spelled out in (\hyperlink{Hirschhorn05}{Hirschhorn 05}).

\begin{prop}
\label{ModelStructureInheritsCombinatorial}\hypertarget{ModelStructureInheritsCombinatorial}{}
If $\mathcal{C}$ is a [[combinatorial model category]], then so is $\mathcal{C}_{/X}$.

\end{prop}
\begin{proof}
By \href{locally+presentable+category#Properties}{basic properties} of [[locally presentable categories]] they are stable under slicing. Hence with $\mathcal{C}$ locally presentable also $\mathcal{C}_{/X}$ is, and by prop. \ref{ModelStructureInheritsGoodProperties} with $\mathcal{C}$ cofibrantly generated also $\mathcal{C}_{/X}$ is.

\end{proof}
\begin{prop}
\label{ModelStructureInheritsEnriched}\hypertarget{ModelStructureInheritsEnriched}{}
If $\mathcal{C}$ is an [[cartesian enriched model category]], then so is $\mathcal{C}_{/X}$.

\end{prop}
\begin{proof}
By basic properties of [[cartesian enriched categories]] they are stable under slicing, where tensoring is computed in $\mathcal{C}$. Hence with $\mathcal{C}$ enriched also $\mathcal{C}_{/X}$ is. The [[pushout product axiom]] now follows from the fact that in overcategories pushouts can be computed in the underlying category $\mathcal{C}$. The [[unit axiom]] follows from the unit axiom of $\mathcal{C}$ using the fact that tensorings are computed in $\mathcal{C}$.

\end{proof}
\hypertarget{derived_homspaces}{}\subsubsection*{{Derived hom-spaces}}\label{derived_homspaces}

\begin{prop}
\label{PresentationOfSliceInfinityCat}\hypertarget{PresentationOfSliceInfinityCat}{}
If $C$ is a [[simplicial model category]] and $X \in C$ is fibrant, then the [[overcategory]] $C/X$ with the above slice model structure is a [[presentable (infinity,1)-category|presentation]] of the [[over-(∞,1)-category]] $C^\circ / X$: we have an [[equivalence of (∞,1)-categories]]

\begin{displaymath}
(C/X)^\circ \simeq C^\circ / X
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
It is clear that we have an [[essentially surjective (∞,1)-functor]] $C^\circ/X \to (C/X)^\circ$. What has to be shown is that this is a [[full and faithful (∞,1)-functor]] in that it is an [[equivalence in an (∞,1)-category|equivalence]] on all [[hom-object|hom]]-[[∞-groupoid]]s $C^\circ/X(a,b) \simeq (C/X)^\circ(a,b)$.

To see this, notice that the hom-space in an [[over-(∞,1)-category]] $C^\circ/X$ between objects $a : A \to X$ and $b : B \to X$ is given (as discussed there) by the [[(∞,1)-pullback]]

\begin{displaymath}
\itexarray{
    C^\circ/X(A \stackrel{a}{\to} X, B \stackrel{b}{\to} X) 
     &\to& C^\circ(A,B)
    \\
    \downarrow && \downarrow^{\mathrlap{b_*}}
    \\
    {*} &\stackrel{a}{\to}& C^\circ(A,X)
  }
\end{displaymath}
in [[∞Grpd]].

Let $A \in C$ be a cofibrant representative and $b : B \to X$ be a fibration representative in $C$ (which always exists) of the objects of these names in $C^\circ$, respectively. In terms of these we have a cofibration

\begin{displaymath}
\itexarray{
    \emptyset &&\hookrightarrow&& A
    \\
    & \searrow && \swarrow_{\mathrlap{a}}
    \\ 
    && X
  }
\end{displaymath}
in $C/X$, exhibiting $a$ as a cofibrant object of $C/X$; and a fibration

\begin{displaymath}
\itexarray{
    B &&\stackrel{b}{\to}&& X
    \\
    & {}_{\mathllap{b}}\searrow && \swarrow_{\mathrlap{Id}}
    \\ 
    && X
  }
\end{displaymath}
in $C/X$, exhibiting $b$ as a fibrant object in $C/X$.

Moreover, the diagram in [[sSet]] given by

\begin{displaymath}
\itexarray{
    C/X(a, b) &\to& C(A,B)
    \\
    \downarrow && \downarrow^{\mathrlap{b_*}}
    \\
    {*} &\stackrel{a}{\to}& C(A,X)
  }
\end{displaymath}
is

\begin{enumerate}%
\item a [[pullback]] diagram in [[sSet]] (by the definition of morphism in an ordinary [[overcategory]]);


\item a [[homotopy pullback]] in the [[model structure on simplicial sets]], because by the axioms on the [[sSet]]${}_{Quillen}$ [[enriched model category]] $C$ and the above (co)fibrancy assumptions, all objects are [[Kan complex]]es and the right vertical morphism is a [[Kan fibration]].


\item has in the top left the correct [[derived hom-space]] in $C/X$ (since $a$ is cofibrant and $b$ fibrant).



\end{enumerate}
This means that this correct hom-space $C/X(a,b) \simeq (C/X)^\circ(a,b) \in sSet$ is indeed a model for $C^\circ/X(a,b) \in \infty Grpd$.

\end{proof}
\hypertarget{quillen_adjunctions_between_slice_categories}{}\subsection*{{Quillen adjunctions between slice categories}}\label{quillen_adjunctions_between_slice_categories}

\begin{prop}
\label{AdjunctionSliceCat}\hypertarget{AdjunctionSliceCat}{}
Given an adjunction $L\dashv R$ with $L\colon A\to B$ and $R\colon B\to A$, the following compositions define two Quillen ajdunctions between associated slice categories. If $X\in A$, then

\begin{displaymath}
L:A/X\leftrightarrows B/L X:R
\end{displaymath}
is an adjunction, where is the composition $R\colon B/L X\to A/R L X\to A/X$, the second arrow is the base change functor along the unit $X\to R L X$. If $Y\in B$, then

\begin{displaymath}
L:A/R Y\leftrightarrows B/Y:R
\end{displaymath}
is an adjunction, where $L\colon A/R Y\to B/L R Y\to B/Y$. The first adjunction is a Quillen equivalence if $X$ is cofibrant and $L X$ is fibrant. The second adjunction is a Quillen equivalence if $Y$ is fibrant.

\end{prop}
\begin{proof}
These adjunctions are Quillen adjunctions because their left (respectively right) adjoints are left (respectively right) Quillen functors: in the model structures on slice categories (co)fibrations and weak equivalences are created by the forgetful functor to $A$ or $B$, see Hirschhorn's note (\hyperlink{Hirschhorn05}{Hirschhorn 05}). An object in $A/X$ given by an arrow $Z\to X$ is cofibrant if and only if $Z$ is cofibrant and fibrant if and only if $Z\to X$ is a fibration. Quillen's criterion for Quillen equivalences now yields the statements about equivalences.

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

\begin{itemize}%
\item the [[classical model structure on pointed topological spaces]] is the model structure on the undercategory under the point of the [[classical model structure on topological spaces]].

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

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

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


\item [[under category]]


\item [[over topos]]



\end{itemize}

\item [[over (∞,1)-category]],

\begin{itemize}%
\item \textbf{model structure on an over-category}


\item [[over-(∞,1)-topos]]



\end{itemize}

\item [[opposite model structure]]



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

\begin{itemize}%
\item [[Philip Hirschhorn]], \emph{Overcategories and undercategories of model categories}, 2005 (\href{http://www-math.mit.edu/~psh/undercat.pdf}{pdf})

\end{itemize}
[[!redirects model structure on an overcategory]] [[!redirects model structure on an under category]] [[!redirects model structure on an undercategory]]

[[!redirects model structure on a slice category]] [[!redirects model structure on a coslice category]]

[[!redirects model structures on a slice category]] [[!redirects model structures on a coslice category]]

[[!redirects slice model category]] [[!redirects slice model categories]]

[[!redirects slice model structure]] [[!redirects slice model structures]]

[[!redirects coslice model category]] [[!redirects coslice model categories]]

[[!redirects coslice model structure]] [[!redirects coslice model structures]]

[[!redirects co-slice model category]] [[!redirects co-slice model categories]]

[[!redirects co-slice model structure]] [[!redirects co-slice model structures]]



\end{document}