# nLab end

Contents

category theory

## Applications

#### Enriched category theory

enriched category theory

## Extra stuff, structure, property

### Homotopical enrichment

#### Limits and colimits

limits and colimits

# Contents

## Idea

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

If we think of such a functor in the sense of profunctors as encoding a left and right action on the object

$\prod_{c \in C} F(c,c)$

then the end of the functor picks out the universal subobject on which the left and right action coincides. Dually, the 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 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.

## Definition

### In ordinary category theory

#### Definition via extranatural transformations

In ordinary category theory, given a functor $F: C^{op} \times C \to X$, an end of $F$ in $X$ is an object $e$ of $X$ equipped with a 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,

$\pi_c: \int_{c: C} F(c, c) \to F(c, c)$

are called the projection maps of the end. Then, given any extranatural transformation with components

$\theta_c: x \to F(c, c),$

there exists a unique map $f: x \to e$ such that

$\theta_c = \pi_c f$

for every object $c$ of $C$.

The notion of 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

$\iota_c \colon F(c,c) \to \int^{c: C} F(c,c)$

#### Explicit definition

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

###### Definition

Let $F: C^{op} \times C \to X$ be a functor. A 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:

$\array{ w & \overset{e_{c'}}{\to} & F(c', c')\\ ^\mathllap{e_c}\downarrow & & \downarrow^\mathrlap{F(f, c')}\\ F(c, c) & \underset{F(c, f)}{\to} & F(c, c') }$

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:

###### Definition

Let $F: C^{op} \times C \to X$ be a functor. An 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 $w' \to w$.

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

#### Ends as right adjoint functors

In complete analogy to how limits are right adjoint functors to the diagonal functor, ends are right adjoint functors to the hom functor.

In more detail, suppose $C$ and $X$ are categories.

If any diagram $C^{op}\times C\to X$ admits an end, then we have a functor

$end\colon Fun(C^{op}\times C,X)\to X$

whose left adjoint is the hom functor

$hom\colon X\to Fun(C^{op}\times C,X)$

that sends an object $x\in X$ to the functor $hom(x)\colon C^{op}\times C\to X$ that sends $(c,d)$ to $\coprod_{hom(c,d)}x=hom(c,d)\otimes x$. (For coends one uses $x^{hom(c,d)}$ instead.)

This immediately implies a Fubini theorem for ends and coends.

### In enriched category theory

There is a definition of end in enriched category theory, as follows.

#### End of $V$-valued functors

Let $V$ be a symmetric monoidal category, and let $C$ be a $V$-enriched category. Assuming $V$ is also 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

$\lambda_{c, d, e}: F(c, d) \otimes C(d, e) \to F(c, e),$

(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

$\rho_{c, d, e}: F(d, e) \otimes C(c, d) \to F(c, e).$

In detail, the covariant action is the adjunct of the morphism

$(F(c,-) \colon C(d,e) \to [F(c,d), F(c,e)]) \in Hom_V(C(d,e),[F(c,d), F(c,e)])$

$Hom_V(C(d,e),[F(c,d), F(c,e)]) \stackrel{\simeq}{\longrightarrow} Hom_V(C(d,e)\otimes F(c,d),F(c,e))$

in $V$. Similarly for the contravariant action.

###### Remark

Even if $V$ is not closed monoidal, we can still define a 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

$F(c) \times C(c, d) \to F(d)$

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

$C(a, b) \otimes F(b, c) \otimes C(c, d) \to F(a, d)$

with evident axioms for a bimodule structure, as a stand-in for a $V$-functor of the form $C^{op} \otimes C \to V$.

A $V$- extranatural transformation

$\theta: v \stackrel{\bullet}{\to} F$

from $v$ to $F$ consists of a family of arrows in $V$,

$\theta_c: v \to F(c, c),$

indexed over objects $c$ of $C$, such that for every pair of objects $(c, d)$ in $C$, the composites of (1) and (2) agree:

$v \otimes C(c, d) \stackrel{\theta_c \otimes 1}{\to} F(c, c) \otimes C(c, d) \stackrel{\lambda_{c, c, d}}{\to} F(c, d) \qquad (1)$
$v \otimes C(c, d) \stackrel{\theta_d \otimes 1}{\to} F(d, d) \otimes C(c, d) \stackrel{\rho_{c, d, d}}{\to} F(c, d) \qquad (2)$

A $V$-enriched end of $F$ is an object $\int_{c: C} F(c, c)$ of $V$ equipped with a $V$-extranatural transformation

$\pi: \int_{c: C} F(c, c) \stackrel{\bullet}{\to} F$

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,

$\theta_c = \pi_c f$

for all objects $c$ of $C$.

#### End of $C$-valued functors for $C \in V\Cat$

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

$\int_{c: C} X(-,F(c,c))\,.$

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

$\pi_c: I \to X(\int_{c: C} F(c, c), F(c, c))$

in $V$, such that for every object $x$ of $X$, the family of maps

$X(x, \pi_c): X(x, \int_{c: C} F(c, c)) \to X(x, F(c, c))$

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 $V$.

#### End as an equalizer

##### Ordinary ends as equalizers

Now we motivate and define the 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

$F : C^{op} \to Set$

is given by the equalizer of

$\prod_{c \in Obj(C)} F(c) \stackrel{\prod_{f \in Mor(c)} (F(f) \circ p_{t(f)}) }{\to} \prod_{f \in Mor(C)} F(s(f))$

and

$\prod_{c \in Obj(C)} F(c) \stackrel{\prod_{f \in Mor(c)} (p_{s(f)}) }{\to} \prod_{f \in Mor(C)} F(s(f)) \,.$

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)

$\prod_{{(c_1 \stackrel{f}{\to} c_2)} \in Mor(C)} F(c_1) \simeq \prod_{c_1,c_2 \in Obj(C)} F(c_1)^{C(c_1,c_2)} \,.$

If we write for the hom-set instead

$[C(c_1,c_2), F(c_1)] := F(c_1)^{C(c_1,c_2)}$

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

$\prod_{c \in Obj(C)} F(c) \stackrel{\stackrel{\rho}{\to}}{\stackrel{\lambda}{\to}} \prod_{c_1,c_2 \in Obj(C)} [C(c_1,c_2),F(c_1)] \,,$

where in components

$\rho_{c_1, c_2} : F(c_1) \to [C(c_1,c_2), F(c_1)]$

$C(c_1, c_2) \to * \to [F(c_1), F(c_1)]$

(with the last map the adjunct of $Id_{F(c_1)}$) and where

$\lambda_{c_1, c_2} : F(c_2) \to [C(c_1,c_2), F(c_1)]$

$F_{c_1, c_2} : C(c_1, c_2) \to [F(c_2), F(c_1)] \,.$

So for definiteness, the equalizer we are looking at is that of

$\rho := \prod_{c_1, c_2 \in C} \rho_{c_1,c_2}\circ pr_{F(c_1)}$

and

$\lambda := \prod_{c_1, c_2 \in C} \lambda_{c_1,c_2}\circ pr_{F(c_2)}$

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.

##### Enriched ends over $V$-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 end of $F$ is the equalizer

(1)$\int_{c \in C} F(c,c) \longrightarrow \prod_{c \in Obj(C)} F(c,c) \underoverset {\underset{\lambda}{\longrightarrow}} {\overset{\rho}{\longrightarrow}} {} \prod_{c_1,c_2 \in Obj(C)} [C(c_1,c_2),F(c_1,c_2)]$

with $\rho$ in components given by

$\rho_{c_1, c_2} \colon F(c_1,c_1) \longrightarrow [C(c_1,c_2), F(c_1,c_2)]$

$F(c_1,-) \colon C(c_1, c_2) \longrightarrow [F(c_1,c_1), F(c_1,c_2)]$

and

$\lambda_{c_1, c_2} \colon F(c_2,c_2) \longrightarrow [C(c_1,c_2), F(c_1,c_2)]$

$F(-,c_2) \colon C(c_1, c_2) \longrightarrow [F(c_2,c_2), F(c_1,c_2)] \,.$

This definition manifestly exhibits the end as the equalizer of the left and right action encoded by the distributor $F$.

Dually, the coend of $F$ is the coequalizer

(2)$\coprod_{c_1,c_2} C(c_2,c_1) \otimes F(c_1,c_2)\, \rightrightarrows\, \coprod_c F(c,c)\,\to\, \int^c F(c,c)$

with the parallel morphisms again induced by the two actions of $F$.

#### 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 end of $F$ is the weighted limit of $F$

$\int_{c \in C} F(c,c) \coloneqq \{Hom_C, F\} = lim^{Hom_C} F \,,$

with weight $Hom_C : C^{op} \times C \to V$. The coend of $F$ is the colimit

$\int^{c \in C} F(c,c) \coloneqq Hom_{C^{op}} \ast F = \colim^{Hom_{C^{op}}} F$

of $F$ weighted by the hom functor of $C^{op}$.

#### 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

$\coprod_{c,c'} C(-,c) \times C(c,c') \times C(c',-) \,\rightrightarrows\, \coprod_c C(-,c) \times C(c,-) \,\to\, C(-,-)$

It is also a general fact (see e.g. Kelly, ch. 3) that weighted (co)limits are cocontinuous in their weight: that is,

$\{W \ast V, F\} \cong \{W, \{V-, F\}\}$

and

$(W \ast V) \ast G \cong W \ast (V \ast G)$

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 (1). Similarly, $(- \ast F)$ takes the analogous coequalizer presentation of $C^{op}(-,-)$ to (2).

## Properties

### $Set$-enriched coends as ordinary colimits

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

• $C$ a $Set$-enriched category/locally small category tensored over Set;

• $D$ be a small category;

• $F : D \to C$ a functor;

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

• $el W$ the category of elements of $W$.

###### Proposition

(coend as colimit over category of elements)
There is a natural isomorphism in $C$

$\int^{d \in D} W(d) \cdot F(d) \simeq \lim_{\to}( (el W)^{op} \to D \stackrel{F}{\to} C )$

between the coend, as indicated, and the colimit over the opposite of the category of elements of $W$.

This is equation (3.34) in (Kelly) in view of (3.70).

###### Corollary

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{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}
###### Example

If $W = D(-,e)$ is a representable functor, then

$(el W)^{op} = D/e$

is the over category over the representing object $e$. This has a terminal object, namely $(e \stackrel{Id}{\to} e$). Therefore

$\lim_\to( D/e \to D \stackrel{F}{\to} C) \simeq F(e) \,.$

Since this is natural in $e$, the above proposition asserts a natural isomorphism

$F(-) \simeq \int^{k \in D} D(k,-) \cdot F(k) \,.$

This statement is sometimes called the co-Yoneda lemma.

### Commutativity of ends and coends

Ordinary limits 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 Fubini theorem for ends (for instance Kelly, p. 29).

###### Proposition

(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

$T : (\mathcal{A} \otimes \mathcal{B})^{op} \otimes (\mathcal{A} \otimes \mathcal{B}) \to \mathcal{V}$

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

$\int_{(A,B) \in \mathcal{A} \otimes \mathcal{B}} T(A,B,A,B) \simeq \int_{A \in \mathcal{A}} \int_{B \in \mathcal{B}} T(A,B,A,B)$

if either side exists. In particular, since $\mathcal{A} \otimes \mathcal{B} \simeq \mathcal{B} \otimes \mathcal{A}$ this implies that

$\int_{B \in \mathcal{B}} \int_{A \in \mathcal{A}} T(A,B,A,B) \simeq \int_{A \in \mathcal{A}} \int_{B \in \mathcal{B}} T(A,B,A,B)$

if either side exists.

## Examples

### Natural transformations

###### Proposition

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

$[C, D] (F, G) = \int_{c \in C} D(F(c), G(c)).$
###### 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:

$\array{ F(c) & \overset{F(f)}{\to} & F(d)\\ ^\mathllap{\tau_c}\downarrow & & \downarrow^\mathrlap{\tau_d}\\ G(c) & \underset{G(f)}{\to} & G(d) }$

which is by definition a natural transformation $F \to G$.

### Enriched functor categories

In light of Proposition , we can define the natural transformations object for enriched functors as an end:

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

• objects are $V$-enriched functors $F : C \to D$;

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

###### Example

For $V = Set$ this reproduces of course the ordinary functor category.

###### Example

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

$[X, Y](f, g) = \int_{x \in X} Y(f(x), g(x)) = \sup_{x \in X} Y(f(x), g(x)).$

### Kan extension

If the $V$-enriched category $D$ is tensored 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

$Lan F : b \mapsto \int^{c \in C} hom(p(c),b) \cdot F(c) \,.$

See Kan extension for more details.

### 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

$|X| := \int^{c \in C} F(c) \cdot hom(Y(c),X) \simeq \int^{c \in C} F(c) \cdot X(c) \,,$

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.

### Tensor product of functors

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

$S \otimes_C T = \int^c S(c) \otimes T(c),$

where the tensor product on the right hand side refers to some monoidal structure on $D$.

## (Co)end calculus

The formal properties of (co)ends in Propositions , and allow us to prove certain results by abstract nonsense.

###### Example

Let $F: C^op \to Set$ be a functor. We prove the co-Yoneda lemma, that

$F(c) \simeq \int^{c' \in C} C(c,c')\times F(c')$

We perform the following manipulations, where each isomorphism is natural:

\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}

So by the Yoneda lemma, we have

$F(c) \simeq \int^{c' \in C} C(c,c')\times F(c').$

More examples can be found in Fosco.

homotopycohomologyhomology
$[S^n,-]$$[-,A]$$(-) \otimes A$
category theorycovariant homcontravariant homtensor product
homological algebraExtExtTor
enriched category theoryendendcoend
homotopy theoryderived hom space $\mathbb{R}Hom(S^n,-)$cocycles $\mathbb{R}Hom(-,A)$derived tensor product $(-) \otimes^{\mathbb{L}} A$

Coends and ends were introduced by Nobuo Yoneda (of the Yoneda lemma) in the paper

• Nobuo Yoneda, On Ext and exact sequences, Journal of the Faculty of Science. University of Tokyo. Section I. Volume 8 (1960), 507–576.

An early account with an eye towards application in geometric realization of simplicial topological spaces:

• Saunders MacLane, Section 2 of: The Milgram bar construction as a tensor product of functors, In: F.P. Peterson (eds.) The Steenrod Algebra and Its Applications: A Conference to Celebrate N.E. Steenrod’s Sixtieth Birthday, Lecture Notes in Mathematics 168, Springer 1970 (doi:10.1007/BFb0058523, pdf)

Textbook accounts:

• Max Kelly, Basic concepts in enriched category theory

(pdf)

• ends of $V$-valued bifunctors are discussed in section 2.1

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

• enriched weighted limits in terms of enriched functor categories are in section 3.1

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

• Francis Borceux, Def. 6.6.8 in: Handbook of Categorical Algebra, Vol. 2: Categories and Structures, Encyclopedia of Mathematics and its Applications 50 Cambridge University Press (1994) (doi:10.1017/CBO9780511525865)

• Fosco Loregian, Coend calculus, Cambridge University Press 2021 (arXiv:1501.02503, doi:10.1017/9781108778657, ISBN:9781108778657).

• Ends, $n$-Category Café discussion.