nLab
end

Idea

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

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

$$\prod _{c\in C}F(c,c)$$\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.

A classical example of an end is the $V$-object of natural transformations between $V$-enriched functors in enriched category theory.

In ordinary category theory

In ordinary category theory, given a functor $F:C^{\mathrm{op}}\times C\to X$, 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,

$${\pi }_c:{\int }_{c:C}F(c,c)\to F(c,c)$$\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

$${\theta }_c:x\to F(c,c),$$\theta_c: x \to F(c, c),

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

$${\theta }_c={\pi }_{c}f$$\theta_c = \pi_c f

for every object $c$ of $C$.

The notion of coend is dual to the notion of end, written ${\int }^{c:C}F(c,c)$.

Examples: Module homs and module tensor products

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.

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, let $C$ be a $V$-enriched category, and let $F:C^{\mathrm{op}}\otimes C\to V$ be a $V$-enriched functor.

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

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

where $C(d,e)$ is customary notation for the enriched hom ${\mathrm{hom}}_C(d,e)$ of $C$ in $V$, and a contravariant action of $C$ on $F$, with components

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

$$(F(c,-):C(d,e)\to [F(c,d),F(c,e)])\in {\mathrm{Hom}}_V(C(d,e),[F(c,d),F(c,e)])$$(F(c,-) : C(d,e) \to [F(c,d), F(c,e)]) \in Hom_V(C(d,e),[F(c,d), F(c,e)])

under the Hom-adjunction

$${\mathrm{Hom}}_V(C(d,e),[F(c,d),F(c,e)])\stackrel{\simeq }{\to }{\mathrm{Hom}}_V(C(d,e)\otimes F(c,d),F(c,e))$$Hom_V(C(d,e),[F(c,d), F(c,e)]) \stackrel{\simeq}{\to} Hom_V(C(d,e)\otimes F(c,d),F(c,e))

in $V$. Similarly for the contravariant action.

A $V$- extranatural transformation

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

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

$${\theta }_c:v\to F(c,c),$$\theta_c: v \to F(c, c),

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

$$v\otimes C(c,d)\stackrel{{\theta }_c\otimes 1}{\to }F(c,c)\otimes C(c,d)\stackrel{{\lambda }_{c,c,d}}{\to }F(c,d)(1)$$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)

$$v\otimes C(c,d)\stackrel{{\theta }_d\otimes 1}{\to }F(d,d)\otimes C(c,d)\stackrel{{\rho }_{c,d,d}}{\to }F(c,d)(2)$$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

$$\pi :{\int }_{c:C}F(c,c)\stackrel{•}{\to }F$$\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,

$${\theta }_c={\pi }_{c}f$$\theta_c = \pi_c f

for all objects $c$ of $C$.

End of $C$-valued functors for $C\in VCat$

If $X$ is any $V$-enriched category and $F:C^{\mathrm{op}}\otimes C\to X$ is a $V$-enriched functor, then the end of $F$ in $X$ is an object ${\int }_{c:C}F(c,c)$ of $X$ equipped with an $\mathrm{Ob}(C)$-indexed family of arrows

$${\pi }_c:I\to X({\int }_{c:C}F(c,c),F(c,c))$$\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

$$X(x,{\pi }_c):X(x,{\int }_{c:C}F(c,c))\to X(x,F(c,c))$$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 in $V$. This is an example of the general notion of weighted limit in enriched category theory.

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

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

is given by the equalizer of

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

$$\prod _{c\in \mathrm{Obj}(C)}F(c)\stackrel{\prod _{f\in \mathrm{Mor}(c)}(p_{s(f)})}{\to }\prod _{f\in \mathrm{Mor}(C)}F(s(f)).$$\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)

$$\prod _{(c_1\stackrel{f}{\to }{c}_2)\in \mathrm{Mor}(C)}F(c_1)\simeq \prod _{c_1,c_2\in \mathrm{Obj}(C)}F(c_1{)}^{C(c_1,c_2)}.$$\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

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

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

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

is the adjunct of

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

(with the last map the adjunct of ${\mathrm{Id}}_{F(c_1)}$) and where

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

is the adjunct of

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

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

and

$$\lambda :=\prod _{c_1,c_2\in C}{\lambda }_{c_1,c_2}\circ {\mathrm{pr}}_{F(c_2)}$$\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 $F:C^{\mathrm{op}}\times C\to V$ a $V$-enriched functor, the end of $F$ is the equalizer

$${\int }_{c\in C}F(c,c)\to \prod _{c\in \mathrm{Obj}(C)}F(c)\stackrel{\stackrel{\rho }{\to }}{\stackrel{\lambda }{\to }}\prod _{c_1,c_2\in \mathrm{Obj}(C)}[C(c_1,c_2),F(c_1,c_2)]$$\int_{c \in C} F(c,c) \to \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,c_2)]

with $\rho$ in components given by

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

being the adjunct of

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

and

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

being the adjunct of

$$F(-,c_2):C(c_1,c_2)\to [F(c_2,c_2),F(c_1,c_2)].$$F(-,c_2) : C(c_1, c_2) \to [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 $F$.

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

$${\int }_{c\in C}F(c,c):={\lim }^{{\mathrm{Hom}}_C}F,$$\int_{c \in C} F(c,c) := lim^{Hom_C} F \,,

where the right hand denotes the weighted limit over $F$ with weight ${\mathrm{Hom}}_C:C^{\mathrm{op}}\times C\to V$.

Examples

Enriched functor categories

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

For $V=\mathrm{Set}$ this reproduces of course the ordinary functor category.

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

$$\mathrm{Lan}F:b\mapsto {\int }_{c\in C}\mathrm{hom}(p(c),b)\cdot F(c).$$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 $Y:C\to [C^{\mathrm{op}},V]$.

The “geometric realization” of an object $X\in [C,V]$ with respect to $F$ is then the coend

$$\mid X\mid :={\int }^{c\in C}F(c)\cdot \mathrm{hom}(Y(c),X)\simeq {\int }^{c\in C}F(c)\cdot X(c),$$|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 $F:\Delta \to \mathrm{Top}$ regards the abstract $n$-simplex as the standard simplex as a topological space.

References

in

• M. 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 .