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\section*{representable functor}

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

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

[[!include category theory - contents]]

\hypertarget{yoneda_lemma}{}\paragraph*{{Yoneda lemma}}\label{yoneda_lemma}

[[!include Yoneda lemma - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_ordinary_category_theory}{In ordinary category theory}\dotfill \pageref*{in_ordinary_category_theory} \linebreak
\noindent\hyperlink{in_enriched_category_theory}{In enriched category theory}\dotfill \pageref*{in_enriched_category_theory} \linebreak
\noindent\hyperlink{in_higher_category_theory}{In higher category theory}\dotfill \pageref*{in_higher_category_theory} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{limits}{Limits}\dotfill \pageref*{limits} \linebreak
\noindent\hyperlink{products}{Products}\dotfill \pageref*{products} \linebreak
\noindent\hyperlink{weighted_limits}{Weighted limits}\dotfill \pageref*{weighted_limits} \linebreak
\noindent\hyperlink{exponential_objects}{Exponential objects}\dotfill \pageref*{exponential_objects} \linebreak
\noindent\hyperlink{classifying_bundles}{Classifying bundles}\dotfill \pageref*{classifying_bundles} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

For a [[locally small category]] $C$, a [[presheaf]] on $C$ or equivalently a [[functor]]

\begin{displaymath}
F: C^{op} \to Set
\end{displaymath}
on the [[opposite category]] of $C$ with values in [[Set]] is \textbf{representable} if it is [[natural isomorphism|naturally isomorphic]] to a [[hom-functor]] $h_X := \hom_C(-, X): C^{op} \to Set$, which sends an object $U \in C$ to the [[hom-set]] $Hom_C(U,X)$ in $C$



and which sends a [[morphism]] $\alpha : U' \to U$ in $C$ to the [[function]] which sends each morphism $U \to X$ to the composite $U' \stackrel{\alpha}{\to} U \to X$



If we picture $Hom_C(U,X)$ as strands of morphisms as above, then the morphism $\alpha:U'\to U$ serves to ``comb'' the strands back from $Hom_C(U,X)$ to $Hom_C(U',X)$, i.e.

\begin{displaymath}
h_X\alpha: Hom_C(U,X)\to Hom_C(U',X).
\end{displaymath}
The object $X$ is determined uniquely up to [[isomorphism]] in $C$, and is called a \textbf{representing object} for $F$.

Representability is one of the most fundamental concepts of [[category theory]], with close ties to the notion of [[adjoint functor]] and to the [[Yoneda lemma]]. It is the crucial concept underlying the idea of [[universal property]]; thus for example crucial concepts such as ``[[limit]]'', ``[[colimit]]'', ``[[exponential object]]'', ``[[Kan extension]]'' and so on are naturally expressed in terms of representing objects. The concept permeates much of algebraic geometry and algebraic topology.

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

\hypertarget{in_ordinary_category_theory}{}\subsubsection*{{In ordinary category theory}}\label{in_ordinary_category_theory}

For a functor $F: C^{op} \to Set$ (also called a [[presheaf]] on $C$), a \textbf{representation} of $F$ is a specified [[natural isomorphism]]

\begin{displaymath}
\theta: \hom_C(-, c) \stackrel{\sim}{\to} F
\end{displaymath}
By the [[Yoneda lemma]], any such transformation $\theta$ (isomorphism or not) is uniquely determined by an element $\xi \in F(c)$. As above, the object $c$ is called a \textbf{representing object} (or often, \textbf{universal object}) for $F$, and the element $\xi$ is called a \textbf{[[universal element]]} for $F$. Again, it follows from the [[Yoneda lemma]] that the pair $(c, \xi)$ is determined uniquely up to unique isomorphism.

Following the proof of the Yoneda lemma, representability means precisely this: given any object $x$ of $C$ and any element $\alpha \in F(x)$, there exists a unique morphism $f: x \to c$ such that the function $F(f)$ carries the universal element $\xi \in F(c)$ to $\alpha \in F(x)$. Such a dry formulation fails to convey the remarkable power of this concept, which can really only be appreciated through the myriad examples which illustrate it.

\hypertarget{in_enriched_category_theory}{}\subsubsection*{{In enriched category theory}}\label{in_enriched_category_theory}

The above definition generalizes straightforwardly to [[enriched category theory]].

Let $V$ be a [[closed monoidal category]] and $C$ a $V$-[[enriched category]].

Then for every object $c \in C$ there is a $V$-[[enriched functor]]

\begin{displaymath}
C(c,-) : C \to V
\end{displaymath}
from $C$ to $V$ regarded canonically as a $V$-[[enriched category]].

This is defined

\begin{itemize}%
\item on objects by $C(c,-) : d \mapsto C(c,d) \in V$


\item on morphisms between $d$ and $d'$ by



\end{itemize}
\begin{displaymath}
C(c,-)_{d,d'} : C(d,d') \to [C(c,d), C(c,d')]
  \,,
\end{displaymath}
being the [[adjunct]] of the composition morphism

\begin{displaymath}
\circ_{c,d,d'} : C(d,d') \otimes C(c,d) \to C(c,d')
  \,.
\end{displaymath}
A $V$-enriched functor $F : C \to V$ is \textbf{representable} if there is $c \in C$ and a $V$-[[enriched natural transformation]] $\eta : F \to C(c,-)$.

If $V$ is [[symmetric monoidal category|symmetric monoidal]] one can form the [[opposite category]] $C^{op}$ and have the analogous definition for representable functors $F : C^{op} \to V$.

\hypertarget{in_higher_category_theory}{}\subsubsection*{{In higher category theory}}\label{in_higher_category_theory}

The notion of representable functors has its straightforward analogs also in [[higher category theory]].

\begin{itemize}%
\item For [[2-category theory]] see \ldots{} .


\item For [[(∞,1)-category theory]] theory see [[(∞,1)-presheaf]]



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

The central point about examples of representable functors is:

\emph{Representable functors are ubiquitous} .

To a fair extent, [[category theory]] is all about representable functors and the other [[universal construction]]s: [[Kan extension]]s, [[adjoint functor]]s, [[limit]]s, which are all special cases of representable functors -- and representable functors are special cases of these.

Listing examples of representable functors in [[category theory]] is much like listing examples of [[integral]]s in [[analysis]]: one can and does fill books with these. (In fact, that analogy has more to it than meets the casual eye: see [[coend]] for more).

Keeping that in mind, we do list some special cases and special classes of examples that are useful to know. But any list is necessarily wildly incomplete.

\hypertarget{limits}{}\subsubsection*{{Limits}}\label{limits}

If $F:J\to C$ is a diagram in $C$, we can construct a diagram $\hom_C(-,F):J\to Set^{C^{op}}$ in the [[functor category]] $Set^{C^{op}}$ as the composite of $F$ with the curried [[hom-functor]] $C\to\Set^{C^{op}}$ (the [[Yoneda embedding]]). The object-wise [[limit]] of this diagram in [[Set]], that is, the functor $C^{op}\to\Set$ sending an object $x$ to the set which is the limit of the diagram $\hom_C(x,F):J\to\Set$, is representable iff the diagram $F$ has a limit in $C$; in fact, a representing object for that limit functor is exactly $\lim F$, and we obtain a natural isomorphism

\begin{displaymath}
\lim \hom_C(-,F)\cong\hom_C(-,\lim F).
\end{displaymath}
\hypertarget{products}{}\paragraph*{{Products}}\label{products}

For an example in the case of the [[product]], let $c, d$ be objects of $C$, and consider the presheaf given by a product of [[hom-functor|hom-functors]]

\begin{displaymath}
\hom_C(-, c) \times \hom_C(-, d): C^{op} \to Set;
\end{displaymath}
that is, the functor which takes an object $x$ of $C$ to the set $\hom_C(x, c) \times \hom_C(x, d)$. A product $c \times d$ is precisely a representing or universal object for this presheaf, where the universal element is precisely the pair of projection maps

\begin{displaymath}
(\pi_c, \pi_d) \in \hom(c \times d, c) \times \hom(c \times d, d)
\end{displaymath}
We leave it to the reader to check that the representability here means precisely that given a pair of maps

\begin{displaymath}
(f, g) \in \hom(x, c) \times \hom(x, d)
\end{displaymath}
there exists a unique element in $\hom(x, c \times d)$, denoted $\langle f, g \rangle$, such that

\begin{displaymath}
\pi_c \langle f, g \rangle = f \qquad \pi_d \langle f, g \rangle = g.
\end{displaymath}
\hypertarget{weighted_limits}{}\paragraph*{{Weighted limits}}\label{weighted_limits}

The above example has an important straightforward generalization.

Noticing that the limit over the functor $H : J \to Set$ is just the collection of [[cone]]s over $H$ whose tip is the point

\begin{displaymath}
lim H = [J,Set](\Delta pt, H)
\end{displaymath}
the above expression $\lim\hom_C(-,F)$ can be rewritten equivalently as $[J,Set](\Delta pt, C(-,F(-)))$. Replacing in this expression the constant terminal functor $\Delta pt : J \to Set$ by any other functor leads to the notion of [[weighted limit]], as described there.

\hypertarget{exponential_objects}{}\subsubsection*{{Exponential objects}}\label{exponential_objects}

Suppose $C$ is a category which admits finite products; given objects $c, d$, consider the presheaf

\begin{displaymath}
\hom_C(- \times c, d): C^{op} \to Set.
\end{displaymath}
A representing or universal object for this presheaf is an [[exponential object]] $d^c$; the universal element

\begin{displaymath}
e \in \hom_C(d^c \times c, d)
\end{displaymath}
is a morphism called the \textbf{evaluation} map $eval: d^c \times c \to d$.

\hypertarget{classifying_bundles}{}\subsubsection*{{Classifying bundles}}\label{classifying_bundles}

Consider a category $Top$ of `nice' spaces (just to fix the discussion, let's say paracompact spaces, although this is a technical point), and a topological group $G$ therein, i.e., a group internal to $Top$. There is a presheaf

\begin{displaymath}
G\Bund: Top^{op} \to Set
\end{displaymath}
which assigns to each space $X$ the set of isomorphism classes of $G$-bundles over $X$, and assigns to each continuous map $f: X \to Y$ the function

\begin{displaymath}
G\Bund(f): G\Bund(Y) \to G\Bund(X)
\end{displaymath}
which carries a (class of a) $G$-bundle $E \to Y$ to the (class of the) pullback bundle $f^*E \to X$. It is well-known that the pullback construction is invariant with respect to homotopic deformations; that is, this presheaf descends to a functor on the [[homotopy category]],

\begin{displaymath}
G\Bund: Ho_{Top}^{op} \to Set.
\end{displaymath}
A \textbf{[[classifying space]]} $\mathcal{B}G$ is precisely a representing object for this functor; the universal element is the (isomorphism class of the) classifying $G$-bundle $[\pi: \mathcal{E}G \to \mathcal{B}G]$.

These general considerations are quite commonplace in algebraic topology, where they crop up for example in the connection between generalized cohomology theories and spectra; cf. Brown's representability theorem.

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

\begin{itemize}%
\item [[representable functor theorem]]


\item [[Yoneda lemma]]


\item [[classifying space]], [[classifying stack]], [[moduli space]], [[moduli stack]], [[derived moduli space]]


\item [[classifying morphism]]


\item [[representable morphism of stacks]]


\item [[Artin representability theorem]], [[Artin-Lurie representability theorem]]



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

A discussion of representable functors in the context of enriched category theory is in section 1.6 and section 1.10 of

\begin{itemize}%
\item [[Max Kelly]], \emph{Basic concepts of enriched category theory} (\href{http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf}{pdf})

\end{itemize}
A query discussion on differences between representable functor and representation of a functor is archived \href{https://nforum.ncatlab.org/discussion/4497/representable-functor/}{here}.

[[!redirects representable functor]] [[!redirects representable functors]] [[!redirects represented functor]] [[!redirects represented functors]] [[!redirects representable]] [[!redirects representables]] [[!redirects representable presheaf]] [[!redirects representable presheaves]] [[!redirects representing object]] [[!redirects representing objects]]



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