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\begin{document}

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\section*{end}

\begin{quote}%
This article is about ends (and coends) in [[category theory]]. For ends in [[topology]], see at [[end compactification]].


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

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

[[!include category theory - contents]]

\hypertarget{enriched_category_theory}{}\paragraph*{{Enriched category theory}}\label{enriched_category_theory}

[[!include enriched category theory contents]]

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

[[!include infinity-limits - 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{definition_via_extranatural_transformations}{Definition via extranatural transformations}\dotfill \pageref*{definition_via_extranatural_transformations} \linebreak
\noindent\hyperlink{explicit_definition}{Explicit definition}\dotfill \pageref*{explicit_definition} \linebreak
\noindent\hyperlink{in_enriched_category_theory}{In enriched category theory}\dotfill \pageref*{in_enriched_category_theory} \linebreak
\noindent\hyperlink{end_of_valued_functors}{End of $V$-valued functors}\dotfill \pageref*{end_of_valued_functors} \linebreak
\noindent\hyperlink{end_of_valued_functors_for_}{End of $C$-valued functors for $C \in V\Cat$}\dotfill \pageref*{end_of_valued_functors_for_} \linebreak
\noindent\hyperlink{end_as_an_equalizer}{End as an equalizer}\dotfill \pageref*{end_as_an_equalizer} \linebreak
\noindent\hyperlink{ordinary_ends_as_equalizers}{Ordinary ends as equalizers}\dotfill \pageref*{ordinary_ends_as_equalizers} \linebreak
\noindent\hyperlink{enriched_ends_over_valued_functors_as_equalizers}{Enriched ends over $V$-valued functors as equalizers}\dotfill \pageref*{enriched_ends_over_valued_functors_as_equalizers} \linebreak
\noindent\hyperlink{end_as_a_weighted_limit}{End as a weighted limit}\dotfill \pageref*{end_as_a_weighted_limit} \linebreak
\noindent\hyperlink{connecting_the_two_definitions}{Connecting the two definitions}\dotfill \pageref*{connecting_the_two_definitions} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{SetCoendsAsColimits}{$Set$-enriched coends as ordinary colimits}\dotfill \pageref*{SetCoendsAsColimits} \linebreak
\noindent\hyperlink{commutativity_of_ends_and_coends}{Commutativity of ends and coends}\dotfill \pageref*{commutativity_of_ends_and_coends} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{natural_transformations}{Natural transformations}\dotfill \pageref*{natural_transformations} \linebreak
\noindent\hyperlink{enriched_functor_categories}{Enriched functor categories}\dotfill \pageref*{enriched_functor_categories} \linebreak
\noindent\hyperlink{kan_extension}{Kan extension}\dotfill \pageref*{kan_extension} \linebreak
\noindent\hyperlink{geometric_realization}{Geometric realization}\dotfill \pageref*{geometric_realization} \linebreak
\noindent\hyperlink{tensor_product_of_functors}{Tensor product of functors}\dotfill \pageref*{tensor_product_of_functors} \linebreak
\noindent\hyperlink{coend_calculus}{(Co)end calculus}\dotfill \pageref*{coend_calculus} \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}

An \emph{end} is a special kind of [[limit]] over a [[functor]] of the form $F : C^{op} \times C \to D$ (sometimes called a \emph{[[bifunctor]]}).

If we think of such a functor in the sense of [[profunctor]]s as encoding a left and right [[action]] on the object

\begin{displaymath}
\prod_{c \in C} F(c,c)
\end{displaymath}
then the \emph{end} of the functor picks out the universal [[subobject]] on which the left and right action coincides. Dually, the \emph{coend} of $F$ is the universal quotient of $\coprod_{c \in C} F(c,c)$ that forces the two actions of $F$ on that object to be equal.

A classical example of an \emph{end} is the $V$-object of [[natural transformations]] between $V$-[[enriched functors]] in [[enriched category theory]]. Perhaps the most common way in which ends and coends arise is through homs and tensor products of (generalized) modules, and their close cousins, weighted limits and weighted colimits. These concepts are fundamental in enriched category theory.

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

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

\hypertarget{definition_via_extranatural_transformations}{}\paragraph*{{Definition via extranatural transformations}}\label{definition_via_extranatural_transformations}

In ordinary [[category theory]], given a [[functor]] $F: C^{op} \times C \to X$, an \textbf{end} of $F$ in $X$ is an object $e$ of $X$ equipped with a [[universal construction|universal]] [[extranatural transformation]] from $e$ to $F$. This means that given any extranatural transformation from an object $x$ of $X$ to $F$, there exists a unique map $x \to e$ which respects the extranatural transformations.

In more detail: the end of $F$ is traditionally denoted $\int_{c: C} F(c, c)$, and the components of the universal extranatural transformation,

\begin{displaymath}
\pi_c: \int_{c: C} F(c, c) \to F(c, c)
\end{displaymath}
are called the \emph{projection maps} of the end. Then, given any extranatural transformation with components

\begin{displaymath}
\theta_c: x \to F(c, c),
\end{displaymath}
there exists a unique map $f: x \to e$ such that

\begin{displaymath}
\theta_c = \pi_c f
\end{displaymath}
for every object $c$ of $C$.

The notion of \textbf{coend} is dual to the notion of end. The coend of $F$ is written $\int^{c: C} F(c, c)$, and comes equipped with a universal extranatural transformation with components

\begin{displaymath}
\iota_c \colon F(c,c) \to \int^{c: C} F(c,c)
\end{displaymath}
\hypertarget{explicit_definition}{}\paragraph*{{Explicit definition}}\label{explicit_definition}

We unwrap the definition of an extranatural transformation to obtain a more explicit description of an end.

\begin{defn}
\label{wedge}\hypertarget{wedge}{}
Let $F: C^{op} \times C \to X$ be a [[functor]]. A \textbf{wedge} $e: w \to F$ is an object $w$ and maps $e_c: w\to F(c, c)$ for each $c$, such that given any morphism $f: c \to c'$, the following diagram commutes:

\begin{displaymath}
\itexarray{
w & \overset{e_{c'}}{\to} & F(c', c')\\
^\mathllap{e_c}\downarrow & & \downarrow^\mathrlap{F(f, c')}\\
F(c, c) & \underset{F(c, f)}{\to} & F(c, c')
}
\end{displaymath}
\end{defn}
Given a wedge $e: w \to F$ and a map $f: v \to w$, we obtain a wedge $e f: v \to F$ by composition. Then we define the end as follows:

\begin{defn}
\label{}\hypertarget{}{}
Let $F: C^{op} \times C \to X$ be a functor. An \textbf{end} of $F$ is a universal wedge, ie. a wedge $e: w \to F$ such that any other wedge $e': w' \to F$ factors through $e$ via a unique map $w' \to w$.

\end{defn}
Dually, a cowedge is given by maps $F(c, c) \to w$ satisfying similar commutativity conditions, and a coend is a universal cowedge.

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

There is a definition of \emph{end} in [[enriched category theory]], as follows.

\hypertarget{end_of_valued_functors}{}\paragraph*{{End of $V$-valued functors}}\label{end_of_valued_functors}

Let $V$ be a [[symmetric monoidal category]], and let $C$ be a $V$-[[enriched category]]. Assuming $V$ is also [[closed monoidal category|closed monoidal]], $V$ may be considered as $V$-enriched; in that case, suppose $F: C^{op} \otimes C \to V$ is a $V$-[[enriched functor]].

Then in particular there is a covariant [[action]] of $C$ on $F$, with components

\begin{displaymath}
\lambda_{c, d, e}: F(c, d) \otimes C(d, e) \to F(c, e),
\end{displaymath}
(where $C(d, e)$ is customary notation for the [[hom-object]] $\hom_C(d, e)$ of $C$ in $V$), and a contravariant action of $C$ on $F$, with components

\begin{displaymath}
\rho_{c, d, e}: F(d, e) \otimes C(c, d) \to F(c, e).
\end{displaymath}
In detail, the covariant action is the [[adjunct]] of the morphism

\begin{displaymath}
(F(c,-) 
    \colon 
  C(d,e) \to [F(c,d), F(c,e)]) 
    \in 
  Hom_V(C(d,e),[F(c,d), F(c,e)])
\end{displaymath}
under the [[closed monoidal category|Hom-adjunction]]

\begin{displaymath}
Hom_V(C(d,e),[F(c,d), F(c,e)]) 
    \stackrel{\simeq}{\longrightarrow} 
  Hom_V(C(d,e)\otimes F(c,d),F(c,e))
\end{displaymath}
in $V$. Similarly for the contravariant action.

\begin{uremark}
Even if $V$ is not closed monoidal, we can still define a \textbf{notion} of covariant $C$-action, sometimes called a ``left'' $C$-[[module]], as consisting of a function $F \colon Ob(C) \to Ob(V)$ together with an $Ob(V) \times Ob(V)$-indexed collection of morphisms

\begin{displaymath}
F(c) \times C(c, d) \to F(d)
\end{displaymath}
satisfying some evident unit and associativity axioms, and regard this notion as a stand-in for the notion of $V$-functor $C \to V$. Similarly we have an evident notion of contravariant $C$-action as a stand-in for a $V$-functor $C^{op} \to V$; notice that we don't even require symmetry to make sense of this. Finally, we can combine these notions into one of $C$-bimodule, where we have a function $F \colon Ob(C) \times Ob(C) \to Ob(V)$ together with a collection of morphisms

\begin{displaymath}
C(a, b) \otimes F(b, c) \otimes C(c, d) \to F(a, d)
\end{displaymath}
with evident axioms for a bimodule structure, as a stand-in for a $V$-functor of the form $C^{op} \otimes C \to V$.

\end{uremark}
A $V$- \textbf{extranatural transformation}

\begin{displaymath}
\theta: v \stackrel{\bullet}{\to} F
\end{displaymath}
from $v$ to $F$ consists of a family of arrows in $V$,

\begin{displaymath}
\theta_c: v \to F(c, c),
\end{displaymath}
indexed over objects $c$ of $C$, such that for every pair of objects $(c, d)$ in $V$, the composites of (1) and (2) agree:

\begin{displaymath}
v \otimes C(c, d) \stackrel{\theta_c \otimes 1}{\to} F(c, c) \otimes C(c, d) \stackrel{\lambda_{c, c, d}}{\to} F(c, d) \qquad (1)
\end{displaymath}
\begin{displaymath}
v \otimes C(c, d) \stackrel{\theta_d \otimes 1}{\to} F(d, d) \otimes C(c, d) \stackrel{\rho_{c, d, d}}{\to} F(c, d) \qquad (2)
\end{displaymath}
A $V$-enriched \textbf{end} of $F$ is an object $\int_{c: C} F(c, c)$ of $V$ equipped with a $V$-extranatural transformation

\begin{displaymath}
\pi: \int_{c: C} F(c, c) \stackrel{\bullet}{\to} F
\end{displaymath}
such any $V$-extranatural transformation $\theta$ from $v$ to $F$ is obtained by pulling back the components of $\pi$ along $f: v \to \int_{c: C} F(c, c)$, for some unique map $f$. That is,

\begin{displaymath}
\theta_c = \pi_c f
\end{displaymath}
for all objects $c$ of $C$.

\hypertarget{end_of_valued_functors_for_}{}\paragraph*{{End of $C$-valued functors for $C \in V\Cat$}}\label{end_of_valued_functors_for_}

If $X$ is any $V$-enriched category and $F: C^{op} \otimes C \to X$ is a $V$-enriched functor, then the \textbf{end} of $F$ in $X$ is, if it exists, an object $\int_{c: C} F(c, c)$ of $X$ that [[representable functor|represents]] the functor

\begin{displaymath}
\int_{c: C} X(-,F(c,c))\,.
\end{displaymath}
That means that the end $\int_{c: C} F(c,c)$ comes equipped with an $Ob(C)$-indexed family of arrows

\begin{displaymath}
\pi_c: I \to X(\int_{c: C} F(c, c), F(c, c))
\end{displaymath}
in $V$, such that for every object $x$ of $X$, the family of maps

\begin{displaymath}
X(x, \pi_c): X(x, \int_{c: C} F(c, c)) \to X(x, F(c, c))
\end{displaymath}
are the projection maps realizing $X(x, \int_{c: C} F(c, c))$ as the corresponding end $\int_{c: C} X(x, F(c, c))$ in $V$.

\hypertarget{end_as_an_equalizer}{}\paragraph*{{End as an equalizer}}\label{end_as_an_equalizer}

\hypertarget{ordinary_ends_as_equalizers}{}\paragraph*{{Ordinary ends as equalizers}}\label{ordinary_ends_as_equalizers}

Now we motivate and define the \emph{end} in [[enriched category theory]] in terms of [[equalizers]].

Recall from the discussion at the end of [[limit]] that the [[limit]] over an (ordinary, i.e. not enriched) [[functor]]

\begin{displaymath}
F : C^{op} \to Set
\end{displaymath}
is given by the [[equalizer]] of

\begin{displaymath}
\prod_{c \in Obj(C)}
  F(c)
  \stackrel{\prod_{f \in Mor(c)} (F(f) \circ p_{t(f)}) }{\to}  
  \prod_{f \in Mor(C)}
  F(s(f))
\end{displaymath}
and

\begin{displaymath}
\prod_{c \in Obj(C)}
  F(c)
  \stackrel{\prod_{f \in Mor(c)} (p_{s(f)}) }{\to}  
  \prod_{f \in Mor(C)}
  F(s(f))  
  \,.
\end{displaymath}
If we want to generalize an expression like this to [[enriched category theory]] the explicit indexing over the set of morphisms has to be replaced by something that makes sense in an [[enriched category]].

To that end, observe that we have a canonical isomorphism (of sets, still)

\begin{displaymath}
\prod_{{(c_1 \stackrel{f}{\to} c_2)} \in Mor(C)} F(c_1)
  \simeq
  \prod_{c_1,c_2 \in Obj(C)} F(c_1)^{C(c_1,c_2)}
  \,.
\end{displaymath}
If we write for the [[hom-set]] instead

\begin{displaymath}
[C(c_1,c_2), F(c_1)] := F(c_1)^{C(c_1,c_2)}
\end{displaymath}
with $[-,-]$ the [[internal hom]] in [[Set]], then the expression starts to make sense in any $V$-[[enriched category]].

Still equivalently but suggestively rewriting the above, we now obtain the limit over $F$ as the [[equalizer]] of

\begin{displaymath}
\prod_{c \in Obj(C)}
  F(c)
  \stackrel{\stackrel{\rho}{\to}}{\stackrel{\lambda}{\to}}  
  \prod_{c_1,c_2 \in Obj(C)}
  [C(c_1,c_2),F(c_1)]
  \,,
\end{displaymath}
where in components

\begin{displaymath}
\rho_{c_1, c_2} : F(c_1) \to [C(c_1,c_2), F(c_1)]
\end{displaymath}
is the [[adjunct]] of

\begin{displaymath}
C(c_1, c_2) \to * \to [F(c_1), F(c_1)]
\end{displaymath}
(with the last map the [[adjunct]] of $Id_{F(c_1)}$) and where

\begin{displaymath}
\lambda_{c_1, c_2} : F(c_2) \to [C(c_1,c_2), F(c_1)]
\end{displaymath}
is the [[adjunct]] of

\begin{displaymath}
F_{c_1, c_2} : C(c_1, c_2) \to [F(c_2), F(c_1)]
  \,.
\end{displaymath}
So for definiteness, the equalizer we are looking at is that of

\begin{displaymath}
\rho := \prod_{c_1, c_2 \in C} \rho_{c_1,c_2}\circ pr_{F(c_1)}
\end{displaymath}
and

\begin{displaymath}
\lambda := \prod_{c_1, c_2 \in C} \lambda_{c_1,c_2}\circ pr_{F(c_2)}
\end{displaymath}
This way of writing the [[limit]] clearly suggests that it is more natural to have $\lambda$ and $\rho$ on equal footing. That leads to the following definition.

\hypertarget{enriched_ends_over_valued_functors_as_equalizers}{}\paragraph*{{Enriched ends over $V$-valued functors as equalizers}}\label{enriched_ends_over_valued_functors_as_equalizers}

For $V$ a [[symmetric monoidal category]], $C$ a $V$-[[enriched category]] and $F \colon C^{op} \times C \to V$ a $V$-[[enriched functor]], the \textbf{end} of $F$ is the [[equalizer]]

\begin{equation}
\int_{c \in C}
  F(c,c)
    \longrightarrow
  \prod_{c \in Obj(C)}
  F(c,c)
   \underoverset
    {\underset{\lambda}{\longrightarrow}}  
    {\overset{\rho}{\longrightarrow}}
    {}
  \prod_{c_1,c_2 \in Obj(C)}
    [C(c_1,c_2),F(c_1,c_2)]
\label{endeq}\end{equation}
with $\rho$ in components given by

\begin{displaymath}
\rho_{c_1, c_2} \colon F(c_1,c_1) \longrightarrow [C(c_1,c_2), F(c_1,c_2)]
\end{displaymath}
being the [[adjunct]] of

\begin{displaymath}
F(c_1,-) \colon  C(c_1, c_2) \longrightarrow [F(c_1,c_1), F(c_1,c_2)]
\end{displaymath}
and

\begin{displaymath}
\lambda_{c_1, c_2} \colon F(c_2,c_2) \longrightarrow [C(c_1,c_2), F(c_1,c_2)]
\end{displaymath}
being the [[adjunct]] of

\begin{displaymath}
F(-,c_2) \colon C(c_1, c_2) \longrightarrow [F(c_2,c_2), F(c_1,c_2)]
  \,.
\end{displaymath}
This definition manifestly exhibits the \textbf{end as the equalizer of the left and right action} encoded by the [[distributor]] $F$.

Dually, the coend of $F$ is the [[coequalizer]]

\begin{equation}
\coprod_{c_1,c_2} C(c_2,c_1) \otimes F(c_1,c_2)\, \rightrightarrows\,
\coprod_c F(c,c)\,\to\,
\int^c F(c,c)
\label{coendcoeq}\end{equation}
with the parallel morphisms again induced by the two actions of $F$.

\hypertarget{end_as_a_weighted_limit}{}\paragraph*{{End as a weighted limit}}\label{end_as_a_weighted_limit}

The end for $V$-functors with values in $V$ serves, among other things, to define [[weighted limits]], and weighted limits in turn define ends of bifunctors with values in more general $V$-categories.

For $C$ and $D$ both $V$-categories and $F : C^\op \times C \to D$ an $V$-[[enriched functor]], the \textbf{end} of $F$ is the [[weighted limit]] of $F$

\begin{displaymath}
\int_{c \in C} F(c,c)
  \coloneqq
  \{Hom_C, F\} =
  lim^{Hom_C} F
  \,,
\end{displaymath}
with weight $Hom_C : C^{op} \times C \to V$. The \textbf{coend} of $F$ is the colimit

\begin{displaymath}
\int^{c \in C} F(c,c) \coloneqq Hom_{C^{op}} \ast F =  \colim^{Hom_{C^{op}}} F
\end{displaymath}
of $F$ weighted by the hom functor of $C^{op}$.

\hypertarget{connecting_the_two_definitions}{}\paragraph*{{Connecting the two definitions}}\label{connecting_the_two_definitions}

If $C$ is a $V$-category, then the hom functor $C(-,-) \colon C^{op} \times C \to V$ is the [[coequalizer]] in

\begin{displaymath}
\coprod_{c,c'} C(-,c) \times C(c,c') \times C(c',-)
\,\rightrightarrows\,
\coprod_c C(-,c) \times C(c,-) \,\to\, C(-,-)
\end{displaymath}
It is also a general fact (see e.g. \hyperlink{Kelly}{Kelly, ch. 3}) that weighted (co)limits are cocontinuous in their weight: that is,

\begin{displaymath}
\{W \ast V, F\} \cong \{W, \{V-, F\}\}
\end{displaymath}
and

\begin{displaymath}
(W \ast V) \ast G \cong W \ast (V \ast G)
\end{displaymath}
This implies that $\{-,F\}$ takes the coequalizer above to an equalizer, which, after some fiddling with the [[Yoneda lemma]], turns out to be isomorphic to \eqref{endeq}. Similarly, $(- \ast F)$ takes the analogous coequalizer presentation of $C^{op}(-,-)$ to \eqref{coendcoeq}.

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

\hypertarget{SetCoendsAsColimits}{}\subsubsection*{{$Set$-enriched coends as ordinary colimits}}\label{SetCoendsAsColimits}

Let the enriching category be $\mathcal{V} =$ [[Set]]. We describe a special way in this case to express ends/coends that give [[weighted limits]]/colimits in terms of ordinary (co)limits over categories of elements.

Consider

\begin{itemize}%
\item $C$ a $Set$-enriched category/[[locally small category]] [[tensoring|tensored]] over [[Set]];


\item $D$ be a [[small category]];


\item $F : D \to C$ a functor;


\item $W : D^{op} \to Set$ another functor;


\item $el W \to D^{op}$ the [[category of elements]] of $W$.



\end{itemize}
\begin{prop}
\label{}\hypertarget{}{}
We have a natural isomorphism in $C$

\begin{displaymath}
\int^{d \in D} W(d) \cdot F(d)
  \simeq
  \lim_{\to}(
    (el W)^{op} \to D \stackrel{F}{\to} C
  )
\end{displaymath}
between the coend as indicated and the [[colimit]] over the opposite of the category of elements of $W$.

\end{prop}
This is equation (3.34) in (\hyperlink{Kelly}{Kelly}) in view of (3.70).

\begin{cor}
\label{ConPres}\hypertarget{ConPres}{}
Any [[continuous functor]] preserves ends, and any cocontinuous functor preserves coends. In particular, for functors $F: D^{op} \times D \to C$ and $c \in C$, we have the isomorphisms

\begin{displaymath}
\begin{aligned}
  C(\int^x F(x, x), c) &\cong \int_x C(F(x, x), c)\\
  C(c, \int_x F(x, x)) &\cong \int_x C(c, F(x, x))
\end{aligned}
\end{displaymath}
\end{cor}
\begin{example}
\label{}\hypertarget{}{}
If $W = D(-,e)$ is a [[representable functor]], then

\begin{displaymath}
(el W)^{op} = D/e
\end{displaymath}
is the [[over category]] over the representing object $e$. This has a [[terminal object]], namely $(e \stackrel{Id}{\to} e$). Therefore

\begin{displaymath}
\lim_\to( D/e \to D \stackrel{F}{\to} C) \simeq F(e)
  \,.
\end{displaymath}
Since this is natural in $e$, the above proposition asserts a [[natural isomorphism]]

\begin{displaymath}
F(-) \simeq \int^{k \in D} D(k,-) \cdot F(k)
  \,.
\end{displaymath}
This statement is sometimes called the [[co-Yoneda lemma]].

\end{example}
\hypertarget{commutativity_of_ends_and_coends}{}\subsubsection*{{Commutativity of ends and coends}}\label{commutativity_of_ends_and_coends}

Ordinary [[limit]]s commute with each other, if both limits exist separately. The analogous statement does hold for ends and coends. Since there it looks like the commutativity of two integrals, it is called the \emph{Fubini theorem} for ends (for instance \href{http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf#page=29}{Kelly, p. 29}).

\begin{prop}
\label{Fubini}\hypertarget{Fubini}{}
\textbf{(Fubini theorem for ends)}

Let $\mathcal{V}$ be a [[symmetric monoidal category]]. Let $\mathcal{A}$ and $\mathcal{B}$ be small $\mathcal{V}$-[[enriched categories]].

Let

\begin{displaymath}
T : (\mathcal{A} \otimes \mathcal{B})^{op} \otimes 
      (\mathcal{A} \otimes \mathcal{B})
      \to 
      \mathcal{V}
\end{displaymath}
be a $\mathcal{V}$-[[enriched functor]]. Then:

If for all object $B,B' \in \mathcal{B}$ the end $\int_{A \in \mathcal{A}} T(A,B,A,B')$ exists, then

\begin{displaymath}
\int_{(A,B) \in \mathcal{A} \otimes \mathcal{B}}
   T(A,B,A,B)
  \simeq
  \int_{A \in \mathcal{A}}
  \int_{B \in \mathcal{B}}
  T(A,B,A,B)
\end{displaymath}
if either side exists. In particular, since $\mathcal{A} \otimes \mathcal{B} \simeq \mathcal{B} \otimes \mathcal{A}$ this implies that

\begin{displaymath}
\int_{B \in \mathcal{B}}
  \int_{A \in \mathcal{A}}
  T(A,B,A,B)
  \simeq
  \int_{A \in \mathcal{A}}
  \int_{B \in \mathcal{B}}
  T(A,B,A,B)
\end{displaymath}
if either side exists.

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

\hypertarget{natural_transformations}{}\subsubsection*{{Natural transformations}}\label{natural_transformations}

\begin{prop}
\label{NatTrans}\hypertarget{NatTrans}{}
Let $F, G: C \to D$ be [[functors]] between two categories, and let $Nat (F, G)$ be the set of [[natural transformations]] between them. Then we have

\begin{displaymath}
[C, D] (F, G) = \int_{c \in C} D(F(c), G(c)).
\end{displaymath}
\end{prop}
\begin{proof}
An element of $\int_{c \in C} D(F(c), G(c))$ is by definition a collection $\tau_c: F(c) \to G(c)$ of morphisms in $D$ such that for any morphism $f: c \to d$ in $C$, the following square commutes:

\begin{displaymath}
\itexarray{
F(c) & \overset{F(f)}{\to} & F(d)\\
^\mathllap{\tau_c}\downarrow & & \downarrow^\mathrlap{\tau_d}\\
G(c) & \underset{G(f)}{\to} & G(d)
}
\end{displaymath}
which is by definition a natural transformation $F \to G$.

\end{proof}
\hypertarget{enriched_functor_categories}{}\subsubsection*{{Enriched functor categories}}\label{enriched_functor_categories}

In light of Proposition \ref{NatTrans}, we can define the natural transformations object for [[enriched functors]] as an end:

For $C$ and $D$ both $V$-[[enriched category|enriched categories]], the $V$-[[enriched functor category]] $[C,D]$ is the $V$-[[enriched category]] whose

\begin{itemize}%
\item objects are $V$-[[enriched functors]] $F : C \to D$;


\item [[hom-objects]] in $V$ are given by the end-formula $[C,D](F,G) := \int_{c \in C} D(F(c), G(c))$.



\end{itemize}
\begin{example}
\label{}\hypertarget{}{}
For $V = Set$ this reproduces of course the ordinary [[functor category]].

\end{example}
\begin{example}
\label{}\hypertarget{}{}
For $V = \mathbb{R}_{\geq 0}\cup \{\infty\}$ with the monoidal product given by addition, a $V$-enriched category $X$ is a [[metric space]], with the distance between points $x, y \in X$ given by $X(x, y)$. Given two metric spaces $X, Y$ and maps $f, g: X \to Y$, the distance between the maps is

\begin{displaymath}
[X, Y](f, g) = \int_{x \in X} Y(f(x), g(x)) = \sup_{x \in X} Y(f(x), g(x)).
\end{displaymath}
\end{example}
\hypertarget{kan_extension}{}\subsubsection*{{Kan extension}}\label{kan_extension}

If the $V$-[[enriched category]] $D$ is [[copower|tensor]]ed over $V$, then the (left) [[Kan extension]] of a [[functor]] $F : C \to D$ along a functor $p : C \to B$ is given by the coend

\begin{displaymath}
Lan F : b \mapsto  \int^{c \in C} hom(p(c),b) \cdot F(c)
  \,.
\end{displaymath}
See [[Kan extension]] for more details.

\hypertarget{geometric_realization}{}\subsubsection*{{Geometric realization}}\label{geometric_realization}

A special case of the example of Kan extension is that of [[geometric realization]].

Very generally, geometric realization is the left Kan extension of a functor $F : C \to D$ along the [[Yoneda embedding]] $Y : C \to [C^{op},V]$.

The ``geometric realization'' of an object $X \in [C^{op},V]$ with respect to $F$ is then the coend

\begin{displaymath}
|X| := \int^{c \in C} F(c) \cdot hom(Y(c),X)
  \simeq \int^{c \in C} F(c) \cdot X(c)
  \,,
\end{displaymath}
where the last step on the right uses the [[Yoneda lemma]].

More specifically, traditionally this is thought of as applying to the case where $C = \Delta$ is the [[simplex category]] and where $F : \Delta \to Top$ regards the abstract $n$-[[simplex]] as the standard simplex as a [[topological space]].

\hypertarget{tensor_product_of_functors}{}\subsubsection*{{Tensor product of functors}}\label{tensor_product_of_functors}

If $S : C^\op \to D$ and $T : C \to D$ are functors, their [[tensor product of functors|tensor product]] is the coend

\begin{displaymath}
S \otimes_C T = \int^c S(c) \otimes T(c),
\end{displaymath}
where the tensor product on the right hand side refers to some [[monoidal structure]] on $D$.

\hypertarget{coend_calculus}{}\subsection*{{(Co)end calculus}}\label{coend_calculus}

The formal properties of (co)ends in Propositions \ref{ConPres}, \ref{Fubini} and \ref{NatTrans} allow us to prove certain results by [[abstract nonsense]].

\begin{example}
\label{}\hypertarget{}{}
Let $F: C^op \to Set$ be a functor. We prove the [[co-Yoneda lemma]], that

\begin{displaymath}
F(c) \simeq \int^{c' \in C} C(c,c')\times F(c')
\end{displaymath}
We perform the following manipluations, where each isomorphism is natural:

\begin{displaymath}
\begin{aligned}
  Set (\int^{c' \in C} C(c,c')\times F(c'), y) &\simeq \int_{c' \in C} Set (C(c,c')\times F(c'), y)\\
  &\simeq \int_{c' \in C} Set (C(c, c'), Set(F(c'), y))\\
  &\simeq [C, Set] (C(c, -), Set(F(-), y))\\
  &\simeq Set(F(c), y).
\end{aligned}
\end{displaymath}
So by the [[Yoneda lemma]], we have

\begin{displaymath}
F(c) \simeq \int^{c' \in C} C(c,c')\times F(c').
\end{displaymath}
\end{example}
More examples can be found in \hyperlink{Fosco}{Fosco}.

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

[[!include homotopy-homology-cohomology]]

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

The standard reference is

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

\begin{itemize}%
\item ends of $V$-valued bifunctors are discussed in section 2.1


\item the enriched functor category that they give rise to is discussed in section 2.2;


\item enriched [[weighted limits]] in terms of enriched functor categories are in section 3.1


\item the end of general $V$-enriched functors in terms of weighted limits is in section 3.10 .



\end{itemize}

\item \href{http://golem.ph.utexas.edu/category/2014/01/ends.html}{Ends}, $n$-Category Caf\'e{} discussion.


\item [[Fosco Loregian]], \emph{This is the (co)end, my only (co)friend} (\href{http://arxiv.org/abs/1501.02503}{arXiv}).



\end{itemize}
Application in [[conformal field theory]]:

\begin{itemize}%
\item [[Jürgen Fuchs]], [[Christoph Schweigert]], \emph{Coends in conformal field theory} (\href{https://arxiv.org/abs/1604.01670}{arXiv:1604.01670})

\end{itemize}
[[!redirects end]] [[!redirects ends]] [[!redirects coend]] [[!redirects coends]]

[[!redirects Fubini theorem for ends]] [[!redirects Fubini theorem for coends]]



\end{document}