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\begin{document}

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\section*{presheaf}

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

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

[[!include category theory - contents]]

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

[[!include topos theory - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{limits_and_colimits}{Limits and colimits}\dotfill \pageref*{limits_and_colimits} \linebreak
\noindent\hyperlink{special_cases}{Special cases}\dotfill \pageref*{special_cases} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

A \textbf{presheaf} on a [[small category]] $C$ is a [[functor]]

\begin{displaymath}
F \;\colon\; C^{op} \to Set
\end{displaymath}
from the [[opposite category]] $C^{op}$ of $C$ to the category [[Set]] of [[set]]s. Equivalently this may be thought of as a [[contravariant functor]] $F \;\colon\; C \to Set$.

More generally, given any category $S$, an \textbf{$S$-valued presheaf} on $C$ is a functor

\begin{displaymath}
F \;\colon\; C^{op} \to S.
\end{displaymath}
While, hence, presheaves are just [[functors]] (on [[small categories]]), one says ``presheaf'' to indicate a specific perspective or interest, namely interest in the \emph{[[sheafification]]} of the functor/presheaf, or at least interest in the [[functor category]] as a [[topos]] (the [[presheaf topos]]). Hence ``presheaf'' is a [[concept with an attitude]].

Historically, the initial applications of presheaves and sheaves involved cases like $S =$ [[CRing]] (the category of [[commutative ring|commutative rings]]), $S =$[[Ab]] ([[abelian groups]]), $S =$ [[Mod|RMod]] ([[modules]]), etc. Later, especially with the development of [[topos theory]], the primary importance of the [[sheaf topos|category of set-valued (pre)sheaves]] as a [[topos]] was recognized; these other cases could be considered algebraic objects which live in the topos. This article and the one on [[sheaf topos]] recognize these later developments by making the set-valued case the default (in other words, presheaf or sheaf without further qualification is understood to refer to the set-valued case).

The \textbf{[[category of presheaves]]} on $C$, usually denoted $Set^{C^{op}}$ or $[C^{op},Set]$, but often abbreviated as $\widehat{C}$, has:

\begin{itemize}%
\item functors $F : C^{op} \to Set$ as objects;


\item [[natural transformation|natural transformations]] between such functors as morphisms.



\end{itemize}
As such, it is an example of a [[functor category]].

\hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks}

\begin{itemize}%
\item Speaking of functors as presheaves indicates operations that one wants to do apply to these functors, or certain properties that one wants to check.

\begin{itemize}%
\item when $S = Set$, and especially one is interested in the [[Yoneda embedding]] of a category $C$ into its presheaf category $[C^{op}, Set]$ for purposes of studying, for instance, [[limit]]s, [[colimit]]s, [[ind-object]]s, and [[pro-object]]s of $C$;


\item or when there is the structure of a [[site]] on $C$, such that it makes sense to ask if a given presheaf is actually a [[sheaf]].



\end{itemize}

\item One generally useful way to think of presheaves is in the sense of [[space and quantity]].


\item In the case where $S = Set$ and $C$ is [[small category|small]], an important general principle is that the presheaf category $[C^{op},Set]$ is the [[free cocompletion]] of $C$; see [[Yoneda extension]]. Intuitively, it is formed by taking $C$ and `freely throwing in small colimits'. The category $C$ is contained in $[C^{op},Set]$ via the [[Yoneda embedding]]

\begin{displaymath}
Y : C \to [C^{op},Set]
\end{displaymath}
The Yoneda embedding sends each object $c \in C$ to the presheaf

\begin{displaymath}
F(-) = hom(-, c)
\end{displaymath}
Presheaves of this form, or isomorphic to those of this form, are called [[representable functors|representable]]; among their properties, representable presheaves always turn colimits into limits, in the sense that a representable functor from $C^{op}$ to $Set$ turns colimits in $C$ (i.e., limits in $C^{op}$) into limits in $Set$ (i.e., colimits in $Set^{op}$). In general, such continuity is a necessary but not sufficient criterion for representability; however, nicely enough, it \emph{is} sufficient when $C$ itself is a presheaf category. To see this, suppose $K$ is such a presheaf on $C = [D^{op}, Set]$, and let $G = K Y$, a presheaf on $D$. By the [[Yoneda lemma]], we have a natural isomorphism between $[D^{op}, Set](Y(-), G)$ and $K Y(-)$. But by the free cocompletion property of the Yoneda embedding, a colimit-preserving functor on presheaves is entirely determined by its precomposition with $Y$; accordingly, our isomorphism must extend to an identification of $[C^{op}, Set](-, G)$ with $K(-)$, thus establishing the representability of $K$.



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

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

Any [[category of presheaves]] is [[complete category|complete]] and [[cocomplete category|cocomplete]], with both [[limit|limits]] and [[colimit|colimits]] being computed \emph{pointwise}. That is, to compute the limit or colimit of a diagram $F : D \to Set^{C^op}$, we think of it as a functor $F: D \times C^{op} \to Set$ and take the limit or colimit in the $D$ variable.

\begin{uprop}
Every presheaf is a [[colimit]] of [[representable functor|representable presheaves]].

\end{uprop}
An elegant way to express this colimit for a presheaf $F : C^{op} \to Set$ is in terms of the [[coend]] identity

\begin{displaymath}
F(-) = \int^{c \in C} F(c) \times hom_C(-,c)
  \,,
\end{displaymath}
which follows by [[Yoneda reduction]]. See also [[co-Yoneda lemma]].

More concretely: let $Y : C \to [C^{op}, Set]$ denote the [[Yoneda embedding]] and let $C_F := Y/F$ be the corresponding [[comma category]], the [[category of elements]] of $F$:

\begin{displaymath}
C_F := 
  \left\lbrace
     \itexarray{
        Y(V) &&\stackrel{Y(g)}{\to}&& Y(V')
        \\
        & {}_f\searrow && \swarrow_{f'}
        \\
        && F
     }
  \right\rbrace
\end{displaymath}
and let $p : C_F \to C$ the canonical forgetful functor. Then the colimit over representables expression $F$ is

\begin{displaymath}
F \simeq colim_{(Y(V) \to F) \in C_F} (Y\circ p)
  \,.
\end{displaymath}
This is often written with some convenient abuse of notation as

\begin{displaymath}
F \simeq colim_{V \to F} V
  \,.
\end{displaymath}
Notice that these formulas can also be understood as those for the left [[Kan extension]] (see there) of $F$ along the identity functor.

\begin{proof}
Notice that for every $B \in [C^{op}, Set]$ and using the property of the [[hom-functor]] we have

\begin{displaymath}
\begin{aligned}
  Hom_{[C^{op}, Set]}(colim_{(Y(V) \to F) \in C_F} (Y\circ p),B)
  &\simeq
  lim_{(Y(V) \to F) \in C_F} Hom_{[C^{op}, Set]}(Y(V),B)
  \\
  & \simeq
  lim_{(Y(V) \to F) \in C_F} B(V)
 \end{aligned}
\end{displaymath}
by the [[Yoneda lemma]].

By the definition of [[limit]] we have that

\begin{displaymath}
\cdots=Hom_{[C_F^{op}, Set]}(pt,B),
\end{displaymath}
so for each natural transformation $\alpha \in Hom_{[C_F^{op}, Set]}(pt,B)$ and each object $h: Y(V)\to F\in C_F$, $\alpha_h$ is a map $\{*\}\to B(V)$, that is, it is an element of $B(V)$. However, by Yoneda, we know that each object $h:Y(V)\to F\in C_F$ specifies a unique element $h\in F(V)$. Then rephrasing this, $\alpha$ specifies a [[function]] $F(V)\to B(V)$. The naturality of this assignment is guaranteed by the naturality of the map $\alpha$. Then $\alpha$ induces a natural transformation $k^\alpha:F\to B$. It's easy to check that $k$ defines an isomorphism:

\begin{displaymath}
Hom_{[C_F^{op}, Set]}(pt,B) \simeq Hom_{[C^{op}, Set]}(F,B)
  \,.
\end{displaymath}
Since this holds for all $B$, the claim follows, again using the [[Yoneda lemma|Yoneda lemma]].

\end{proof}
\hypertarget{special_cases}{}\subsection*{{Special cases}}\label{special_cases}

\begin{itemize}%
\item [[representable functor|representable presheaf]]


\item [[concrete presheaf]]



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

Examples for presheaves are abundant. Here is a non-representative selection of some examples.

\begin{itemize}%
\item For $C$ a [[locally small]] category, every object $c \in C$ gives rise to the [[representable functor|representable presheaf]] $Hom_C(-, c) : C^{op} \to Set$.


\item More generally, for $i : C \hookrightarrow D$ a [[subcategory]] of a locally small category $D$, every object $d \in D$ gives rise to the presheaf

\begin{displaymath}
Hom_D(i(-), d) : C^{op} \to Set
  \,.
\end{displaymath}
Let's spell this out in more detail: given a mophism $\phi : V \to U$ in $C$, we can take any morphism $f : i(U) \to X$ in $Hom_{D}(U,X)$ and turn it into a morphism $V \stackrel{\phi}{\to} U \stackrel{f}{\to} X$ in $Hom_{D}(i(V),X)$. This determines a map of set

\begin{displaymath}
f^* : Hom_{D}(i(U),X) \to Hom_{D}(i(V),X)
  \,.
\end{displaymath}
So we have a functorial assignment of the form

\begin{displaymath}
\itexarray{
      W && \mapsto && Hom_{Diff}(i(W),X)
    \\
      \downarrow^g &&&& \uparrow^{g^*}
      \\
    V && \mapsto && Hom_{Diff}(i(V),X)
    \\
      \downarrow^f &&&& \uparrow^{f^*}
      \\
      U && \mapsto && Hom_{Diff}(i(U),X)
   }
   \,.
\end{displaymath}
Of course $i$ here could be any functor whatsoever. Asking if such a presheaf is [[representable functor|representable]] is asking for a right [[adjoint functor]] of $i$.


\item A [[simplicial set]] is a presheaf on the [[simplex category]]

A [[globular set]] is a presheaf on the [[globe category]].

A [[cubical set]] is a presheaf on the [[cube category]].


\item A [[diffeological space]] is a [[concrete presheaf]] on [[CartSp]].


\item An important class of presheaves is those on a [[category of open subsets]] $Op(X)$ of a [[topological space]] or [[smooth manifold]] $X$.


\item Traditional standard examples include: the presheaf of [[smooth function]]s on $X$, that assigns to each $U \subset X$ the set $C^\infty(U,\mathbb{R})$ of smooth functions and to each inclusion $V \subset U$ the corresponding restriction operation of functions. This is further a sheaf.


\item Traditional standard example which is a presheaf but not a sheaf: the presheaf of exact forms on $X$, that assigns to $U \subset X$ the set $\Omega^\bullet_{exact}(U)$ of exact forms on $U$ and to each inclusion $V \subset U$ the corresponding restriction operation of functions. Here, and like above, the site is made up by open sets in $X$ with inclusions as morphisms.



\end{itemize}
\ldots{} etc. pp.

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

\begin{itemize}%
\item \textbf{presheaf}

[[constant presheaf]]


\item [[(2,1)-presheaf]]

[[presheaf of groupoids]]


\item [[(∞,1)-presheaf]]

[[simplicial presheaf]]


\item [[(∞,n)-presheaf]]


\item [[Yoneda lemma]], [[Yoneda extension]]



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



\end{document}