Ends Coends and Weighted Limits

Let $\mathcal{C}$ and $\mathcal{D}$ be categories (Category, object, morphism, domain, codomain, identity, composition, and hom-collection) and let

$P,Q:{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}⟶\mathcal{D}$

be functors on the product of ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$ with $\mathcal{C}$ (Opposite category ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$, Product category and its projection functors, Covariant functor, identity functor, composite functor, and contravariant functor). A morphism $(a,b)\to (a^{\mathrm{\prime }},b^{\mathrm{\prime }})$ of ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$ is a pair $(g,h)$ in which $h:b\to b^{\mathrm{\prime }}$ is a morphism of $\mathcal{C}$ and $g$ is a morphism of ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$ from $a$ to $a^{\mathrm{\prime }}$, that is, a morphism $a^{\mathrm{\prime }}\to a$ of $\mathcal{C}$. Thus a morphism $f:c\to c^{\mathrm{\prime }}$ of $\mathcal{C}$ supplies the four morphisms

$(f,1_c):(c^{\mathrm{\prime }},c)\to (c,c),(1_{c^{\mathrm{\prime }}},f):(c^{\mathrm{\prime }},c)\to (c^{\mathrm{\prime }},c^{\mathrm{\prime }}),(1_c,f):(c,c)\to (c,c^{\mathrm{\prime }}),(f,1_{c^{\mathrm{\prime }}}):(c^{\mathrm{\prime }},c^{\mathrm{\prime }})\to (c,c^{\mathrm{\prime }})\mathrm{.}$

A dinatural transformation $\alpha :P\to Q$ is a family of morphisms

${\alpha }_c:P(c,c)⟶Q(c,c)(c\in \mathrm{Ob}(\mathcal{C}))$

of $\mathcal{D}$ such that every morphism $f:c\to c^{\mathrm{\prime }}$ of $\mathcal{C}$ satisfies the dinaturality equation

$Q(1_c,f)\circ {\alpha }_c\circ P(f,1_c)=Q(f,1_{c^{\mathrm{\prime }}})\circ {\alpha }_{c^{\mathrm{\prime }}}\circ P(1_{c^{\mathrm{\prime }}},f)$

between morphisms $P(c^{\mathrm{\prime }},c)\to Q(c,c^{\mathrm{\prime }})$. The morphism ${\alpha }_c$ is the component of $\alpha$ at $c$.

The two sides pass through the six objects $P(c^{\mathrm{\prime }},c)$, $P(c,c)$, $Q(c,c)$, $P(c^{\mathrm{\prime }},c^{\mathrm{\prime }})$, $Q(c^{\mathrm{\prime }},c^{\mathrm{\prime }})$ and $Q(c,c^{\mathrm{\prime }})$, so the diagram expressing the dinaturality equation is a six-sided cycle and is called the hexagon:

A dinatural transformation is a family indexed by the objects of $\mathcal{C}$ and constrained by the morphisms of $\mathcal{C}$, exactly as a natural transformation is (Natural transformation and its components); what differs is which constraint. A natural transformation between two functors on ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$ is a family indexed by the pairs $(a,b)$, and its naturality equation is imposed for every morphism of the product category. A dinatural transformation has components only on the diagonal, and the dinaturality equation is imposed only for the morphisms of $\mathcal{C}$ that connect two diagonal entries through the two off-diagonal entries displayed above.

At $f=1_c$ all four displayed morphisms of the product category are identities, so the hexagon reads ${\alpha }_c={\alpha }_c$. The identity morphisms of $\mathcal{C}$ therefore impose no condition, and a dinatural transformation carries no analogue of the identity clause of a functor.

Let $P^{\mathrm{\prime }},P,Q,Q^{\mathrm{\prime }}:{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}\to \mathcal{D}$ be functors (Product category and its projection functors, Opposite category ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$), let $\sigma :P^{\mathrm{\prime }}\Rightarrow P$ and $\tau :Q\Rightarrow Q^{\mathrm{\prime }}$ be natural transformations (Natural transformation and its components), and let $\alpha :P\to Q$ be dinatural (Dinatural transformation between functors on ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$). Then composing a dinatural transformation with a natural transformation on either side gives a dinatural transformation: the families

$(\alpha \sigma {)}_c:={\alpha }_c\circ {\sigma }_{c,c}:P^{\mathrm{\prime }}(c,c)\to Q(c,c),(\tau \alpha {)}_c:={\tau }_{c,c}\circ {\alpha }_c:P(c,c)\to Q^{\mathrm{\prime }}(c,c)$

are dinatural transformations $P^{\mathrm{\prime }}\to Q$ and $P\to Q^{\mathrm{\prime }}$ respectively.

Neither half assumes anything about $\sigma$ or $\tau$ beyond naturality on the product category, and neither assumes that $\alpha$ is natural: the argument transports the hexagon for $\alpha$ along $\sigma$ or $\tau$ and never builds a new one. What it does not give is a composition rule for two dinatural transformations, and no such rule holds: Dinatural transformations do not compose in general exhibits $\alpha$ and $\beta$ both dinatural whose componentwise composite is not.

There are a category $\mathcal{C}$, functors $P,Q,R:{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}\to \mathbf{S}\mathbf{e}\mathbf{t}$ (Product category and its projection functors, Opposite category ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$, Sets and functions form the large locally small category $\mathbf{S}\mathbf{e}\mathbf{t}$) and dinatural transformations $\alpha :P\to Q$ and $\beta :Q\to R$ (Dinatural transformation between functors on ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$) for which the componentwise composite $({\beta }_c\circ {\alpha }_c{)}_c$ is not a dinatural transformation $P\to R$.

Hence the dinatural transformations between functors on ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$ are not the morphisms of a category with componentwise composition.

The two empty slots are the whole mechanism. Putting $\mathrm{\varnothing }$ in the $(1,0)$ position of $Q$ and of $R$ makes the functor equation of step 1.1 vacuous there and makes every family $Q\to R$ dinatural, so the second half of the composite is unconstrained; the one-element set in the $(0,1)$ position of $Q$ makes the first half unconstrained in the other direction. The composite then has to satisfy a hexagon whose codomain $R(0,1)$ has two elements, and nothing has forced its two legs to agree.

A dinatural transformation may still be composed with a natural transformation on either side, and that composite is dinatural; this is Composing a dinatural transformation with a natural transformation on either side gives a dinatural transformation.

Let $T:{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}\to \mathcal{D}$ be a functor (Product category and its projection functors, Opposite category ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$) and let $d$ be an object of $\mathcal{D}$ (Category, object, morphism, domain, codomain, identity, composition, and hom-collection). Write $\mathrm{\Delta }d$ for the constant functor on ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$ at $d$ (Constant diagrams, cones, cocones, and their morphisms), which sends every object to $d$ and every morphism to $1_d$.

A wedge from $d$ to $T$ is a dinatural transformation from a constant functor to $T$ (Dinatural transformation between functors on ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$), that is, a family

${\omega }_c:d⟶T(c,c)(c\in \mathrm{Ob}(\mathcal{C}))$

such that every $f:c\to c^{\mathrm{\prime }}$ satisfies the wedge equation

$T(1_c,f)\circ {\omega }_c=T(f,1_{c^{\mathrm{\prime }}})\circ {\omega }_{c^{\mathrm{\prime }}}$

between morphisms $d\to T(c,c^{\mathrm{\prime }})$. Dually, a cowedge from $T$ to $d$ is a dinatural transformation from $T$ to a constant functor, that is, a family

${\rho }_c:T(c,c)⟶d$

such that every $f:c\to c^{\mathrm{\prime }}$ satisfies the cowedge equation

${\rho }_c\circ T(f,1_c)={\rho }_{c^{\mathrm{\prime }}}\circ T(1_{c^{\mathrm{\prime }}},f)$

between morphisms $T(c^{\mathrm{\prime }},c)\to d$. The object $d$ is the vertex of the wedge or cowedge, and ${\omega }_c$, ${\rho }_c$ are its components.

A morphism of wedges $(d,\omega )\to (d^{\mathrm{\prime }},{\omega }^{\mathrm{\prime }})$ is a morphism $h:d\to d^{\mathrm{\prime }}$ of $\mathcal{D}$ satisfying ${\omega }_c^{\mathrm{\prime }}\circ h={\omega }_c$ for every object $c$; a morphism of cowedges $(d,\rho )\to (d^{\mathrm{\prime }},{\rho }^{\mathrm{\prime }})$ is a morphism $h:d\to d^{\mathrm{\prime }}$ satisfying $h\circ {\rho }_c={\rho }_c^{\mathrm{\prime }}$ for every $c$. Identities of $\mathcal{D}$ are morphisms of wedges and of cowedges, and a composite of two such morphisms again satisfies the displayed equation, so wedges over $T$ and their morphisms form a category $\mathrm{Wd}(T)$, and cowedges under $T$ and their morphisms form a category $\mathrm{Cwd}(T)$; associativity and the identity laws are inherited from $\mathcal{D}$.

The wedge and cowedge equations are the hexagon of Dinatural transformation between functors on ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$ with one side made constant. For a wedge the source is $\mathrm{\Delta }d$, so both $\mathrm{\Delta }d(f,1_c)$ and $\mathrm{\Delta }d(1_{c^{\mathrm{\prime }}},f)$ are $1_d$ and drop out; for a cowedge the target is $\mathrm{\Delta }d$ and the two outer morphisms on the target side drop out instead.

A wedge is not a cone over a diagram indexed by $\mathcal{C}$: its components sit at the diagonal values $T(c,c)$ and its equation involves the two off-diagonal values $T(c,c^{\mathrm{\prime }})$, whereas a cone (Constant diagrams, cones, cocones, and their morphisms) has one component per object of the index category and one equation per morphism, with no off-diagonal term. The precise comparison between the two shapes is An end is a limit over the twisted arrow category, and a coend is a colimit over its opposite, which replaces the index category by another one.

Let $T:{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}\to \mathcal{D}$ be a functor and let $\mathrm{Wd}(T)$ and $\mathrm{Cwd}(T)$ be the categories of wedges over $T$ and of cowedges under $T$ (Wedges and cowedges, and the categories they form).

An end of $T$ is a terminal object of $\mathrm{Wd}(T)$, and a coend of $T$ is an initial object of $\mathrm{Cwd}(T)$ (Initial object, terminal object, and zero object). In short: an end is a terminal wedge and a coend an initial cowedge. Neither need exist.

Written out, an end is a pair $(e,\omega )$ in which ${\omega }_c:e\to T(c,c)$ is a wedge and, for every wedge $(x,\xi )$ over $T$, there is exactly one morphism $u:x\to e$ with

${\omega }_c\circ u={\xi }_c\text{for every object }c\text{ of }\mathcal{C}\mathrm{.}$

A coend is a pair $(q,\rho )$ in which ${\rho }_c:T(c,c)\to q$ is a cowedge and, for every cowedge $(x,\xi )$ under $T$, there is exactly one morphism $v:q\to x$ with $v\circ {\rho }_c={\xi }_c$ for every $c$.

The vertex of an end is written ${\int }_{c}T(c,c)$ and the vertex of a coend ${\int }^{c}T(c,c)$, the subscripted integral denoting the end and the superscripted one the coend. The components ${\omega }_c$ of the terminal wedge are the projections of the end and the components ${\rho }_c$ of the initial cowedge the injections of the coend. The variable $c$ in the integral notation is bound: it names the dinatural variable and nothing else.

Because an end is a dinatural transformation (Dinatural transformation between functors on ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$) with a universal property rather than an element-level construction, the definition applies to any target category $\mathcal{D}$ whatever, and it never asserts existence. Which functors have ends, and in which targets, is a separate question answered by the comparison theorems on this page and by the hypotheses they carry.

The projections ${\omega }_c$ are indexed by the objects of $\mathcal{C}$, while the wedge equation they satisfy is indexed by its morphisms. Thus an end records the family of diagonal values together with every tie imposed through the off-diagonal values $T(c,c^{\mathrm{\prime }})$. For a discrete category there are no nonidentity ties and the end reduces to the product of the diagonal values.

Let $T:{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}\to \mathcal{D}$ be a functor.

If $(e,\omega )$ and $(e^{\mathrm{\prime }},{\omega }^{\mathrm{\prime }})$ are ends of $T$ (The end and the coend of a functor ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}\to \mathcal{D}$), there is exactly one isomorphism $u:e\to e^{\mathrm{\prime }}$ satisfying ${\omega }_c^{\mathrm{\prime }}\circ u={\omega }_c$ for every object $c$ of $\mathcal{C}$. If $(q,\rho )$ and $(q^{\mathrm{\prime }},{\rho }^{\mathrm{\prime }})$ are coends of $T$, there is exactly one isomorphism $v:q\to q^{\mathrm{\prime }}$ satisfying $v\circ {\rho }_c={\rho }_c^{\mathrm{\prime }}$ for every $c$.

An end and a coend of $T$ are therefore unique up to a unique isomorphism compatible with every component.

The compatibility clause is not an extra verification: it is what being a morphism in $\mathrm{Wd}(T)$ or $\mathrm{Cwd}(T)$ means, so the published uniqueness of terminal and initial objects delivers it already. This is why the definition of an end is stated as a universal property in a category of wedges rather than as a family of morphisms with an ad hoc uniqueness clause.

An isomorphism of wedges is in particular an isomorphism of $\mathcal{D}$: its inverse in $\mathrm{Wd}(T)$ is a morphism of $\mathcal{D}$ satisfying the displayed equation, and the two composites are the identities of $e$ and $e^{\mathrm{\prime }}$.

Let $P,P^{\mathrm{\prime }}:{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}\to \mathcal{D}$ be functors and let $\eta :P\Rightarrow P^{\mathrm{\prime }}$ be a natural transformation (Natural transformation and its components); a wedge is a dinatural transformation from a constant functor (Dinatural transformation between functors on ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$).

If $P$ and $P^{\mathrm{\prime }}$ have ends $(e,\omega )$ and $(e^{\mathrm{\prime }},{\omega }^{\mathrm{\prime }})$ (The end and the coend of a functor ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}\to \mathcal{D}$), then a natural transformation induces a unique morphism of ends: there is exactly one morphism $\int \eta :e\to e^{\mathrm{\prime }}$ satisfying

${\omega }_c^{\mathrm{\prime }}\circ \int \eta ={\eta }_{c,c}\circ {\omega }_c\text{for every object }c\mathrm{.}$

If $P$ and $P^{\mathrm{\prime }}$ have coends $(q,\rho )$ and $(q^{\mathrm{\prime }},{\rho }^{\mathrm{\prime }})$, there is exactly one morphism ${\int }^{\eta }:q\to q^{\mathrm{\prime }}$ satisfying ${\int }^{\eta }\circ {\rho }_c={\rho }_c^{\mathrm{\prime }}\circ {\eta }_{c,c}$ for every $c$.

Moreover $\int 1_P=1_e$, and $\int ({\eta }^{\mathrm{\prime }}\circ \eta )=\int {\eta }^{\mathrm{\prime }}\circ \int \eta$ whenever $\eta :P\Rightarrow P^{\mathrm{\prime }}$ and ${\eta }^{\mathrm{\prime }}:P^{\mathrm{\prime }}\Rightarrow P^{\mathrm{\prime }\mathrm{\prime }}$ are natural and all three ends exist.

The two functor laws in steps 3.1 and 3.2 are proved from uniqueness alone and use nothing about $\eta$ beyond its defining equation. So on any full subcategory of functors all of whose objects have a chosen end, the assignment $P\mapsto e$ with $\eta \mapsto \int \eta$ is a functor; making that statement precise for a family of parameters, where the choice has to be made for every parameter value at once, is A chosen family of ends is the object part of exactly one functor making the counit natural in the parameters.

Let $\mathcal{C}$ be a category (Category, object, morphism, domain, codomain, identity, composition, and hom-collection). Its twisted arrow category $\mathrm{Tw}(\mathcal{C})$ has the following data. Its objects are the morphisms of $\mathcal{C}$. For objects $f:c\to c^{\mathrm{\prime }}$ and $g:d\to d^{\mathrm{\prime }}$, a morphism $f\to g$ is a pair $(a,b)$ with $bfa=g$, where $a:d\to c$ and $b:c^{\mathrm{\prime }}\to d^{\mathrm{\prime }}$ are morphisms of $\mathcal{C}$; the identity of $f$ is $(1_c,1_{c^{\mathrm{\prime }}})$, and the composite of $(a,b):f\to g$ with $(a^{\mathrm{\prime }},b^{\mathrm{\prime }}):g\to h$ is

$(a^{\mathrm{\prime }},b^{\mathrm{\prime }})\circ (a,b):=(a\circ a^{\mathrm{\prime }},b^{\mathrm{\prime }}\circ b):f\to h\mathrm{.}$

That composite is a morphism $f\to h$ because $(b^{\mathrm{\prime }}b)f(a{a}^{\mathrm{\prime }})=b^{\mathrm{\prime }}(bfa)a^{\mathrm{\prime }}=b^{\mathrm{\prime }}g{a}^{\mathrm{\prime }}=h$, associativity and the identity laws are inherited from $\mathcal{C}$, and the composite of the two identities is the identity, so these data satisfy Category, object, morphism, domain, codomain, identity, composition, and hom-collection.

The twisted arrow projection is the assignment

$\pi :\mathrm{Tw}(\mathcal{C})⟶{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C},(f:c\to c^{\mathrm{\prime }})⟼(c,c^{\mathrm{\prime }}),(a,b)⟼(a,b),$

where in the target the first coordinate $a$ is read as the morphism $c\to d$ of ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$ corresponding to $a:d\to c$ (Opposite category ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$, Product category and its projection functors). It preserves identities by construction, and it preserves composites because composition in ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$ is componentwise with the first coordinate reversed, which is the order written in the display above; so $\pi$ is a functor (Covariant functor, identity functor, composite functor, and contravariant functor).

The name records the twist: a morphism of $\mathrm{Tw}(\mathcal{C})$ acts by precomposition in one coordinate and by postcomposition in the other, so the first coordinate runs backwards. That is exactly what makes $\pi$ land in ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$ rather than in $\mathcal{C}\times \mathcal{C}$, and it is why a diagram indexed by $\mathrm{Tw}(\mathcal{C})$ can see a functor of two variables of opposite variance.

The opposite orientation is also in use in the literature, with $\mathrm{Tw}(\mathcal{C})$ naming what is here $\mathrm{Tw}(\mathcal{C}{)}^{\mathrm{o}\mathrm{p}}$. The orientation fixed above is in force everywhere on this page, and Orientation and notation conventions in force on this page states it alongside the integral conventions; every statement below that names $\mathrm{Tw}(\mathcal{C})$ is to be read with the definition given here and is false under the other one.

Let $\mathcal{C}$ be a locally small category (Small, locally small, and large categories) and let $\mathcal{C}(-,-):{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}\to \mathbf{S}\mathbf{e}\mathbf{t}$ be its hom-bifunctor (The covariant and contravariant hom-assignments and the hom-bifunctor of a locally small category, The hom-assignment $\mathcal{C}(-,-):{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}\to \mathbf{S}\mathbf{e}\mathbf{t}$ is a bifunctor). Write $\int \mathcal{C}(-,-)$ for its category of elements (The category of elements of a covariant functor or a presheaf) and ${\pi }_{\int }:\int \mathcal{C}(-,-)\to {\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$ for the projection sending $((a,b),x)$ to $(a,b)$.

The assignment

$\mathrm{\Phi }:\mathrm{Tw}(\mathcal{C})⟶\int \mathcal{C}(-,-),(f:c\to c^{\mathrm{\prime }})⟼((c,c^{\mathrm{\prime }}),f),(a,b)⟼(a,b)$

is an isomorphism of categories (The twisted arrow category and its projection to ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$): it is a bijection on objects and on morphisms, it preserves identities and composites, and it satisfies ${\pi }_{\int }\circ \mathrm{\Phi }=\pi$.