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\begin{document}

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\section*{functors and comma categories}

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

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

[[!include category theory - contents]]

This entry is about special properties of [[functor]]s on [[comma category|comma categories]]. See also [[category of presheaves]].

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

\noindent\hyperlink{presheaves_on_overcategories_and_overcategories_of_presheaves}{Presheaves on over-categories and over-categories of presheaves}\dotfill \pageref*{presheaves_on_overcategories_and_overcategories_of_presheaves} \linebreak
\noindent\hyperlink{overcategories_of_presheaf_categories_and_presheaves_on_categories_of_elements}{Over-categories of presheaf categories and presheaves on categories of elements}\dotfill \pageref*{overcategories_of_presheaf_categories_and_presheaves_on_categories_of_elements} \linebreak
\noindent\hyperlink{relationship_with_the_overcategories_statement}{Relationship with the over-categories statement}\dotfill \pageref*{relationship_with_the_overcategories_statement} \linebreak
\noindent\hyperlink{in_higher_category_theory}{In higher category theory}\dotfill \pageref*{in_higher_category_theory} \linebreak


\hypertarget{presheaves_on_overcategories_and_overcategories_of_presheaves}{}\subsection*{{Presheaves on over-categories and over-categories of presheaves}}\label{presheaves_on_overcategories_and_overcategories_of_presheaves}

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 [[presheaf|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{uprop}
There is an [[equivalence]] of categories

\begin{displaymath}
e : PSh(C/c) \stackrel{\simeq}{\to} PSh(C)/Y(c)
  \,.
\end{displaymath}
\end{uprop}
\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{ulemma}
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{ulemma}
See [[over-topos]] for more.

\hypertarget{overcategories_of_presheaf_categories_and_presheaves_on_categories_of_elements}{}\subsection*{{Over-categories of presheaf categories and presheaves on categories of elements}}\label{overcategories_of_presheaf_categories_and_presheaves_on_categories_of_elements}

Generalizing the above,

\begin{uprop}
For every $P \colon \mathbf{Set}^D$, there is an [[equivalence]] of categories

\begin{displaymath}
\varphi : \mathbf{Set}^{el(P)} \stackrel{\simeq}{\to} \mathbf{Set}^D / P
  \,.
\end{displaymath}
where $el(P) = \ast / P$ is the [[category of elements]] of $P$.

\end{uprop}
\begin{proof}
The construction is completely analogous to the above; Given $F \colon \mathbf{Set}^{el(P)}$, $\varphi(F)$ is defined pointwise as a coproduct:

\begin{displaymath}
\varphi(F)(c) = \coprod_{(x,Pc)} F(x,Pc)
\end{displaymath}
where $(x,Pc) = (\ast \stackrel{x}{\to} Pc)$ is an object of $el(P)$. The action on morphisms is defined analogously. This comes equipped with a natural transformation $\alpha \colon \varphi(F) \to P$, with component

\begin{displaymath}
\itexarray{
    \alpha_c \colon \varphi(F)(c) \to Pc          \\
    \alpha_c(F(x,Pc)) = x \in Pc                  \\
  }
\end{displaymath}
Given an object $(Q \stackrel{\alpha}{\to} P)$ of $\mathbf{Set}^D / P$ the action of a weak inverse $\bar \varphi$ can be specified as $\bar{\varphi}(\alpha)(x,Pc) = \alpha_c^{-1}(x)$, that is, the wedge of the pullback:

\begin{displaymath}
\itexarray{
     \bar{\varphi}(\alpha)(x,Pc) &\to& Qc
     \\
     \downarrow && \downarrow^{\alpha_c}
     \\
     pt &\stackrel{x}{\to}& Pc
  }
  \,.
\end{displaymath}
The action of $\bar{\varphi}(\alpha)$ on arrows of $el(P)$, functoriality, etc is derived from its definition as a pullback and the def of morphisms in $el(P)$.

\end{proof}
\hypertarget{relationship_with_the_overcategories_statement}{}\subsubsection*{{Relationship with the over-categories statement}}\label{relationship_with_the_overcategories_statement}

Putting $D = C^{op}, P = Y(c)$ in the above yields:

\begin{displaymath}
\mathbf{Set}^{el(Y(c))} \simeq \mathbf{Set}^D / Y(c)
\end{displaymath}
Now it is easy to see that $el(Y(c)) \simeq (C / c)^{op}$; we get then:

\begin{displaymath}
\mathbf{Set}^{(C / c)^{op}} \simeq \mathbf{Set}^{C^{op}} / Y(c)
\end{displaymath}
\hypertarget{in_higher_category_theory}{}\subsection*{{In higher category theory}}\label{in_higher_category_theory}

For the analogous result in the context of [[(∞,1)-category]] theory see 



\end{document}