This article is about ends (and coends) in category theory. For ends in topology, see at end compactification.


Category theory

Enriched category theory

Limits and colimits



An end is a special kind of limit over a functor of the form F:C op×CDF : 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

cCF(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. Dually, the coend of FF is the universal quotient of cCF(c,c)\coprod_{c \in C} F(c,c) that forces the two actions of FF on that object to be equal.

A classical example of an end is the VV-object of natural transformations between VV-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

Definition via extranatural transformations

In ordinary category theory, given a functor F:C op×CXF: C^{op} \times C \to X, an end of FF in XX is an object ee of XX equipped with a universal extranatural transformation from ee to FF. This means that given any extranatural transformation from an object xx of XX to FF, there exists a unique map xex \to e which respects the extranatural transformations.

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

π c: c:CF(c,c)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

θ c:xF(c,c),\theta_c: x \to F(c, c),

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

θ c=π cf\theta_c = \pi_c f

for every object cc of CC.

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

ι c:F(c,c) c:CF(c,c)\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.


Let F:C op×CXF: C^{op} \times C \to X be a functor. A wedge e:wFe: w \to F is an object ww and maps e c:wF(c,c)e_c: w\to F(c, c) for each cc, such that given any morphism f:ccf: c \to c', the following diagram commutes:

w e c F(c,c) e c F(f,c) F(c,c) F(c,f) F(c,c) \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:wFe: w \to F and a map f:vwf: v \to w, we obtain a wedge ef:vFe f: v \to F by composition. Then we define the end as follows:


Let F:C op×CXF: C^{op} \times C \to X be a functor. An end of FF is a universal wedge, ie. a wedge e:wFe: w \to F such that any other wedge e:wFe': w' \to F factors through ee via a map www' \to w.

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

In enriched category theory

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

End of VV-valued functors

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

Then in particular there is a covariant action of CC on FF, with components

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

(where C(d,e)C(d, e) is customary notation for the hom-object hom C(d,e)\hom_C(d, e) of CC in VV), and a contravariant action of CC on FF, with components

ρ c,d,e:F(d,e)C(c,d)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)[F(c,d),F(c,e)])Hom V(C(d,e),[F(c,d),F(c,e)]) (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)])

under the Hom-adjunction

Hom V(C(d,e),[F(c,d),F(c,e)])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 VV. Similarly for the contravariant action.


Even if VV is not closed monoidal, we can still define a notion of covariant CC-action, sometimes called a “left” CC-module, as consisting of a function F:Ob(C)Ob(V)F \colon Ob(C) \to Ob(V) together with an Ob(V)×Ob(V)Ob(V) \times Ob(V)-indexed collection of morphisms

F(c)×C(c,d)F(d)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 VV-functor CVC \to V. Similarly we have an evident notion of contravariant CC-action as a stand-in for a VV-functor C opVC^{op} \to V; notice that we don’t even the symmetry to make sense of this. Finally, we can combine these notions into one of CC-bimodule, where we have a function F:Ob(C)×Ob(C)Ob(V)F \colon Ob(C) \times Ob(C) \to Ob(V) together with a collection of morphisms

C(a,b)F(b,c)C(c,d)F(a,d)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 VV-functor of the form C opCVC^{op} \otimes C \to V.

A VV- extranatural transformation

θ:vF\theta: v \stackrel{\bullet}{\to} F

from vv to FF consists of a family of arrows in VV,

θ c:vF(c,c),\theta_c: v \to F(c, c),

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

vC(c,d)θ c1F(c,c)C(c,d)λ c,c,dF(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)
vC(c,d)θ d1F(d,d)C(c,d)ρ c,d,dF(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 VV-enriched end of FF is an object c:CF(c,c)\int_{c: C} F(c, c) of VV equipped with a VV-extranatural transformation

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

such any VV-extranatural transformation θ\theta from vv to FF is obtained by pulling back the components of π\pi along f:v c:CF(c,c)f: v \to \int_{c: C} F(c, c), for some unique map ff. That is,

θ c=π cf\theta_c = \pi_c f

for all objects cc of CC.

End of CC-valued functors for CVCatC \in V\Cat

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

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

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

π c:IX( c:CF(c,c),F(c,c)) \pi_c: I \to X(\int_{c: C} F(c, c), F(c, c))

in VV, such that for every object xx of XX, the family of maps

X(x,π c):X(x, c:CF(c,c))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, c:CF(c,c))X(x, \int_{c: C} F(c, c)) as the corresponding end c:CX(x,F(c,c))\int_{c: C} X(x, F(c, c)) in VV.

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 opSet F : C^{op} \to Set

is given by the equalizer of

cObj(C)F(c) fMor(c)(F(f)p t(f)) fMor(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))


cObj(C)F(c) fMor(c)(p s(f)) fMor(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)

(c 1fc 2)Mor(C)F(c 1) c 1,c 2Obj(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 VV-enriched category.

Still equivalently but suggestively rewriting the above, we now obtain the limit over FF as the equalizer of

cObj(C)F(c)λρ c 1,c 2Obj(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

ρ c 1,c 2:F(c 1)[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)*[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 Id F(c 1)Id_{F(c_1)}) and where

λ c 1,c 2:F(c 2)[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)[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

ρ:= c 1,c 2Cρ c 1,c 2pr F(c 1) \rho := \prod_{c_1, c_2 \in C} \rho_{c_1,c_2}\circ pr_{F(c_1)}


λ:= c 1,c 2Cλ c 1,c 2pr 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 λ\lambda and ρ\rho on equal footing. That leads to the following definition.

Enriched ends over VV-valued functors as equalizers

For VV a symmetric monoidal category, CC a VV-enriched category and F:C op×CVF \colon C^{op} \times C \to V a VV-enriched functor, the end of FF is the equalizer

(1) cCF(c,c) cObj(C)F(c,c)λρ c 1,c 2Obj(C)[C(c 1,c 2),F(c 1,c 2)] \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 ρ\rho in components given by

ρ c 1,c 2:F(c 1,c 1)[C(c 1,c 2),F(c 1,c 2)] \rho_{c_1, c_2} \colon F(c_1,c_1) \longrightarrow [C(c_1,c_2), F(c_1,c_2)]

being the adjunct of

F(c 1,):C(c 1,c 2)[F(c 1,c 1),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)]


λ c 1,c 2:F(c 2,c 2)[C(c 1,c 2),F(c 1,c 2)] \lambda_{c_1, c_2} \colon F(c_2,c_2) \longrightarrow [C(c_1,c_2), F(c_1,c_2)]

being the adjunct of

F(,c 2):C(c 1,c 2)[F(c 2,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 FF.

Dually, the coend of FF is the coequalizer

(2) c 1,c 2C(c 2,c 1)F(c 1,c 2) cF(c,c) cF(c,c) \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 FF.

End as a weighted limit

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

For CC and DD both VV-categories and F:C op×CDF : C^\op \times C \to D an VV-enriched functor, the end of FF is the weighted limit of FF

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

with weight Hom C:C op×CVHom_C : C^{op} \times C \to V. The coend of FF is the colimit

cCF(c,c)Hom C op*F=colim Hom C opF \int^{c \in C} F(c,c) \coloneqq Hom_{C^{op}} \ast F = \colim^{Hom_{C^{op}}} F

of FF weighted by the hom functor of C opC^{op}.

Connecting the two definitions

If CC is a VV-category, then the hom functor C(,):C op×CVC(-,-) \colon C^{op} \times C \to V is the coequalizer in

c,cC(,c)×C(c,c)×C(c,) cC(,c)×C(c,)C(,) \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*V,F}{W,{V,F}} \{W \ast V, F\} \cong \{W, \{V-, F\}\}


(W*V)*GW*(V*G) (W \ast V) \ast G \cong W \ast (V \ast G)

This implies that {,F}\{-,F\} takes the coequalizer above to an equalizer, which, after some fiddling with the Yoneda lemma, turns out to be isomorphic to (1). Similarly, (*F)(- \ast F) takes the analogous coequalizer presentation of C op(,)C^{op}(-,-) to (2).


SetSet-enriched coends as ordinary colimits

Let the enriching category be 𝒱=\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.



We have a natural isomorphism in CC

dDW(d)F(d)lim ((elW) opDFC) \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 WW.

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×DCF: D^{op} \times D \to C and cCc \in C, we have the isomorphisms

C( xF(x,x),c) xC(F(x,x),c) C(c, xF(x,x)) xC(c,F(x,x)) \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}

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

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

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

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

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

F() kDD(k,)F(k). 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).


(Fubini theorem for ends)

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


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

be a 𝒱\mathcal{V}-enriched functor. Then:

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

(A,B)𝒜T(A,B,A,B) A𝒜 BT(A,B,A,B) \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 𝒜𝒜\mathcal{A} \otimes \mathcal{B} \simeq \mathcal{B} \otimes \mathcal{A} this implies that

B A𝒜T(A,B,A,B) A𝒜 BT(A,B,A,B) \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.


Natural transformations


Let F,G:CDF, G: C \to D be functors between two categories, and let Nat(F,G)Nat (F, G) be the set of natural transformations between them. Then we have

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

An element of cCD(F(c),G(c))\int_{c \in C} D(F(c), G(c)) is by definition a collection τ c:F(c)G(c)\tau_c: F(c) \to G(c) of morphisms in DD such that for any morphism f:cdf: c \to d in CC, the following square commutes:

F(c) F(f) F(d) τ c τ d G(c) G(f) G(d) \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 FGF \to G.

Enriched functor categories

In light of Proposition 3, we can define the natural transformations object for enriched functors as an end:

For CC and DD both VV-enriched categories, the VV-enriched functor category [C,D][C,D] is the VV-enriched category whose

  • objects are VV-enriched functors F:CDF : C \to D;

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

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

Kan extension

If the VV-enriched category DD is tensored over VV, then the (left) Kan extension of a functor F:CDF : C \to D along a functor p:CBp : C \to B is given by the coend

LanF:b cChom(p(c),b)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:CDF : C \to D along the Yoneda embedding Y:C[C op,V]Y : C \to [C^{op},V].

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

|X|:= cCF(c)hom(Y(c),X) cCF(c)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=ΔC = \Delta is the simplex category and where F:ΔTopF : \Delta \to Top regards the abstract nn-simplex as the standard simplex as a topological space.

Tensor product of functors

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

S CT= cS(c)D(c), S \otimes_C T = \int^c S(c) \otimes D(c),

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

(Co)end calculus

The formal properties of (co)ends in Propositions 1, 2 and 3 allow us to prove certain results by abstract nonsense.


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

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

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

Set( cCC(c,c)×F(c),y) cCSet(C(c,c)×F(c),y) cCSet(C(c,c),Set(F(c),y)) [C,Set](C(c,),Set(F(),y)) Set(F(c),y). \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) cCC(c,c)×F(c). F(c) \simeq \int^{c' \in C} C(c,c')\times F(c').

More examples can be found in Fosco.

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


The standard reference is

  • Max Kelly, Basic concepts in enriched category theory (pdf)

    • ends of VV-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 VV-enriched functors in terms of weighted limits is in section 3.10 .

  • Ends, nn-Category Café discussion.

  • Fosco Loregian, This is the (co)end, my only (co)friend (arXiv).

Revised on August 20, 2016 09:19:06 by Dexter Chua (