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\begin{document}

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\section*{category of presheaves}

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

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

\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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{functoriality}{Functoriality}\dotfill \pageref*{functoriality} \linebreak
\noindent\hyperlink{characterization}{Characterization}\dotfill \pageref*{characterization} \linebreak
\noindent\hyperlink{cartesian_closed_monoidal_structure}{Cartesian closed monoidal structure}\dotfill \pageref*{cartesian_closed_monoidal_structure} \linebreak
\noindent\hyperlink{RelWithOvercategories}{Presheaves on over-categories and over-categories of presheaves}\dotfill \pageref*{RelWithOvercategories} \linebreak
\noindent\hyperlink{finite_presheaves}{Finite presheaves}\dotfill \pageref*{finite_presheaves} \linebreak
\noindent\hyperlink{models_in_presheaf_toposes}{Models in presheaf toposes}\dotfill \pageref*{models_in_presheaf_toposes} \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}

For $C$ a [[small category]], its \textbf{category of presheaves} is the [[functor category]]

\begin{displaymath}
PSh(C) := [C^{op}, Set]
\end{displaymath}
from the [[opposite category]] of $C$ to [[Set]].

An object in this category is a [[presheaf]]. See there for more details.

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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{itemize}%
\item The category of presheaves $PSh(C)$ is the [[free cocompletion]] of $C$.


\item the [[Yoneda lemma]] says that the [[Yoneda embedding]] $j : C \to PSh(C)$ is -- in particular -- a [[full and faithful functor]].


\item A category of presheaves is a [[topos]].


\item The construction of forming (co)-presheaves extends to a [[2-functor]]

\begin{displaymath}
[-,Set] : Cat \to Topos
\end{displaymath}
from the [[2-category]] [[Cat]] to the [[2-category]] [[Topos]]. (See at [[geometric morphism]] the section \href{geometric+morphism#BetweenPresheafToposes}{Between presheaf toposes} for details.)


\item A [[reflective subcategory]] of a category of presheaves is a [[locally presentable category]] if it is closed under $\kappa$-[[directed colimit]]s for some [[regular cardinal]] $\kappa$ (the embedding is an [[accessible functor]]).


\item A [[geometric embedding|sub-topos]] of a category of presheaves is a [[Grothendieck topos]]: a [[category of sheaves]] (see there for details).



\end{itemize}
\hypertarget{functoriality}{}\subsubsection*{{Functoriality}}\label{functoriality}

See [[functoriality of categories of presheaves]].

\hypertarget{characterization}{}\subsubsection*{{Characterization}}\label{characterization}

The following [[Giraud theorem|Giraud like]] theorem stems from [[Marta Bunge|Marta Bunge's]] dissertation (1966)

\begin{theorem}
\label{bunge_theorem}\hypertarget{bunge_theorem}{}
A category $E$ is equivalent to a presheaf topos if and only if it is cocomplete, [[atomic category|atomic]], and [[regular category|regular]].

\end{theorem}
A proof as well as a second characterization using [[exact completion|exact completions]] can be found in Carboni-Vitale (\hyperlink{CarboniVitale98}{1998}) or Centazzo-Vitale (\hyperlink{Centazzo-Vitale04}{2004}). The first paper has also an interesting comparison to a classical characterization of categories [[monadic category|monadic]] over Set.

\hypertarget{cartesian_closed_monoidal_structure}{}\subsubsection*{{Cartesian closed monoidal structure}}\label{cartesian_closed_monoidal_structure}

As every [[topos]], a category of presheaves is a [[cartesian monoidal category|cartesian]] [[closed monoidal category]].

For details on the closed structure see

\begin{itemize}%
\item [[closed monoidal structure on presheaves]].

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

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

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

\begin{itemize}%
\item 

\end{itemize}
\begin{uremark}
Consider $\int_C Y(c)$ , the [[category of elements]] of $Y(c):C^{op}\to Set$. This has objects $(d_1,p_1)$ with $p_1\in Y(c)(d_1)$, hence $p_1$ is just an arrow $d_1\to c$ in $C$. A map from $(d_1, p_1)$ to $(d_2, p_2)$ is just a map $u:d_1\to d_2$ such that $p_2\circ u =p_1$ but this is just a morphism from $p_1$ to $p_2$ in $C/c$.

Hence, the above proposition \ref{representable_case} can be rephrased as $PSh(\int_C Y(c))\simeq PSh(C)/Y(c)$ which is an instance of the following formula:

\end{uremark}
\begin{prop}
\label{}\hypertarget{}{}
Let $P:C^{op}\to Set$ be a presheaf. Then there is an [[equivalence of categories]]

\begin{displaymath}
PSh(\int_C P) \simeq PSh(C)/P
  \,.
\end{displaymath}
\end{prop}
On objects this takes $F : (\int_C P)^{op} \to Set$ to

\begin{displaymath}
i(F)(A \in C) = \{ (p,a) | p \in P(A), a \in F(A,p) \} = \Sigma_{p \in P(A)} F(A,p)
\end{displaymath}
with obvious projection to $P$. The inverse takes $f : Q \to P$ to

\begin{displaymath}
i^{-1}(f)(A, p \in P(A)) = f_A^{-1}(p)\;.
\end{displaymath}
For a proof see \hyperlink{KS06}{Kashiwara-Schapira (2006, Lemma 1.4.12, p. 26)}. For a more general statement involving slices of Grothendieck toposes see \hyperlink{MacLaneMoerdijk}{Mac Lane-Moerdijk (1994, p.157)}.

In particular, this equivalence shows that \emph{slices of presheaf toposes are presheaf toposes}.

\hypertarget{finite_presheaves}{}\subsubsection*{{Finite presheaves}}\label{finite_presheaves}

A \emph{finite presheaf} on a category $C$ is a functor $C^{op}\to FinSet$ valued in the [[FinSet|category of finite sets]]. Categories of finite presheaves will hardly be [[Grothendieck toposes]] for want of infinite limits but they still can turn out to be [[elementary toposes]] as e.g. in the case of $FinSet$ itself.

By going through the proof that ordinary categories of presheaves are toposes, one observes that the constructions stay within finite presheaves when applied to a finite category $C$ i.e. one with only a finite set of morphisms. Hence, one has the following

\begin{prop}
\label{}\hypertarget{}{}
Let $C$ a finite category. Then the category of finite presheaves $[C^{op},FinSet]$ is a topos. $\qed$

\end{prop}
Note, that the category $[G,FinSet]$ of finite $G$-sets is a topos even when the group $G$ is infinite! In this case it is crucial that $\Omega =\{\emptyset , G\}$ in $[G,Set]$ is a finite set.

(Cf. \hyperlink{Borceux3}{Borceux (1994, p.299)})

\hypertarget{models_in_presheaf_toposes}{}\subsubsection*{{Models in presheaf toposes}}\label{models_in_presheaf_toposes}

See at \emph{[[models in presheaf toposes]]}.

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

For [[(∞,1)-category]] theory see [[(∞,1)-category of (∞,1)-presheaves]].

[[!include locally presentable categories - table]]

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

A classical (advanced) reference is expos\'e{} 1 of

\begin{itemize}%
\item [[Michael Artin|M.Artin]], [[Alexander Grothendieck|A.Grothendieck]], [[J. L. Verdier]] (eds.), \emph{Th\'e{}orie des Topos et Cohomologie Etale des Sch\'e{}mas - [[SGA 4]]} , LNM \textbf{269} Springer Heidelberg 1972.

\end{itemize}
An elementary introduction to presheaf toposes emphasizing finite underlying categories $C$ is

\begin{itemize}%
\item M. La Palme Reyes, [[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari, \emph{Generic Figures and their Glueings} , Polimetrica Milano 2004.

\end{itemize}
Standard references are

\begin{itemize}%
\item [[Francis Borceux]], \emph{Handbook of Categorical Algebra 3 : Categories of Sheaves} , Cambridge UP 1994.


\item [[Masaki Kashiwara]], [[Pierre Schapira]], \emph{Categories and Sheaves} , Springer Heidelberg 2006.


\item [[Saunders Mac Lane]], [[Ieke Moerdijk]], \emph{[[Sheaves in Geometry and Logic]]} , Springer Heidelberg 1994.



\end{itemize}
The characterizations of categories of presheaves are discussed in

\begin{itemize}%
\item [[Aurelio Carboni|A. Carboni]], [[Enrico Vitale|E. M. Vitale]], \emph{Regular and exact completions} , JPAA \textbf{125} (1998) pp.79-116.


\item C. Centazzo, [[Enrico Vitale|E. M. Vitale]], \emph{Sheaf theory} , pp.311-358 in Pedicchio, Tholen (eds.), \emph{Categorical Foundations} , Cambridge UP 2004. (\href{https://perso.uclouvain.be/enrico.vitale/chapter7.pdf}{draft})



\end{itemize}
[[!redirects categories of presheaves]] [[!redirects presheaf category]] [[!redirects presheaf categories]] [[!redirects presheaf-category]] [[!redirects presheaf-categories]]

[[!redirects categories of presheaves]]

[[!redirects presheaf topos]] [[!redirects presheaf toposes]] [[!redirects presheaf topoi]]



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