This article is about ends (and coends) in category theory. For ends in topology, see at end compactification.
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
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.
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,
are called the projection maps of the end. Then, given any extranatural transformation with components
there exists a unique map $f: x \to e$ such that
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
We unwrap the definition of an extranatural transformation to obtain a more explicit description of an end.
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:
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:
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.
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
whose left adjoint is the hom functor
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.
There is a definition of end in enriched category theory, as follows.
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
(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
In detail, the covariant action is the adjunct of the morphism
under the Hom-adjunction
in $V$. Similarly for the contravariant action.
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
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
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
from $v$ to $F$ consists of a family of arrows in $V$,
indexed over objects $c$ of $C$, such that for every pair of objects $(c, d)$ in $C$, the composites of (1) and (2) agree:
A $V$-enriched end of $F$ is an object $\int_{c: C} F(c, c)$ of $V$ equipped with a $V$-extranatural transformation
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,
for all objects $c$ of $C$.
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
That means that the end $\int_{c: C} F(c,c)$ comes equipped with an $Ob(C)$-indexed family of arrows
in $V$, such that for every object $x$ of $X$, the family of maps
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 $V$.
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
is given by the equalizer of
and
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)
If we write for the hom-set instead
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
where in components
is the adjunct of
(with the last map the adjunct of $Id_{F(c_1)}$) and where
is the adjunct of
So for definiteness, the equalizer we are looking at is that of
and
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.
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
with $\rho$ in components given by
being the adjunct of
and
being the adjunct of
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
with the parallel morphisms again induced by the two actions of $F$.
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 $F$
with weight $Hom_C : C^{op} \times C \to V$. The coend of $F$ is the colimit
of $F$ weighted by the hom functor of $C^{op}$.
If $C$ is a $V$-category, then the hom functor $C(-,-) \colon C^{op} \times C \to V$ is the coequalizer in
It is also a general fact (see e.g. Kelly, ch. 3) that weighted (co)limits are cocontinuous in their weight: that is,
and
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).
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 $W$.
(coend as colimit over category of elements)
There is a natural isomorphism in $C$
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).
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
If $W = D(-,e)$ is a representable functor, then
is the over category over the representing object $e$. This has a terminal object, namely $(e \stackrel{Id}{\to} e$). Therefore
Since this is natural in $e$, the above proposition asserts a natural isomorphism
This statement is sometimes called the co-Yoneda lemma.
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).
(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
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
if either side exists. In particular, since $\mathcal{A} \otimes \mathcal{B} \simeq \mathcal{B} \otimes \mathcal{A}$ this implies that
if either side exists.
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
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:
which is by definition a natural transformation $F \to G$.
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 $[C,D](F,G) := \int_{c \in C} D(F(c), G(c))$.
For $V = Set$ this reproduces of course the ordinary functor category.
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
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
See Kan extension for more details.
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
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.
If $S : C^\op \to D$ and $T : C \to D$ are functors, their tensor product is the coend
where the tensor product on the right hand side refers to some monoidal structure on $D$.
The formal properties of (co)ends in Propositions , and allow us to prove certain results by abstract nonsense.
Let $F: C^op \to Set$ be a functor. We prove the co-Yoneda lemma, that
We perform the following manipulations, where each isomorphism is natural:
So by the Yoneda lemma, we have
More examples can be found in Fosco.
homotopy | cohomology | homology | |
---|---|---|---|
$[S^n,-]$ | $[-,A]$ | $(-) \otimes A$ | |
category theory | covariant hom | contravariant hom | tensor product |
homological algebra | Ext | Ext | Tor |
enriched category theory | end | end | coend |
homotopy theory | derived 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
An early account with an eye towards application in geometric realization of simplicial topological spaces:
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).
See also:
Application in conformal field theory: