Ends and Coends

Home , End (category theory), Yoneda lemma

Deﬁnitions Basic properties Applications

Ends and Coends

Graham Manuell

June 2016

1 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Outline

1 Deﬁnitions

2 Basic properties

3 Applications

2 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Limits

Recall that the limit of a diagram D : A → C is the terminal cone over D.

f f X lim D Y

πX πY

DX DY Df

3 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Wedges

Instead of a diagram, we now consider a functor op T : A × A → C of mixed variance. A wedge from an object C ∈ C to T is a family of morphisms (γA : C → T (A, A))A∈A such that the following diagram commutes for all f : A → B in A.

γA C T (A, A)

γB T (A, f )

T (B, B) T (A, B) T (f , B)

4 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Wedges

γA C T (A, A)

γB T (A, f )

T (B, B) T (A, B) T (f , B)

4 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Ends as universal wedges

The end of T is the terminal wedge to T .

γA C

ωA end(T ) T (A, A)

ω γB B T (A, f )

T (B, B) T (A, B) T (f , B)

5 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Deﬁnition as a representable functor

op We may deﬁne a functor Wedge(−, T ): C → Set which sends an element C to the set of wedges from C to T . For f : C → D, Wedge(f , T ): Wedge(D, T ) → Wedge(C, T ) acts like precomposition by f .

δAf C

δA f D T (A, A)

δB δB f T (A, f )

T (B, B) T (A, B) T (f , B)

6 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Deﬁnition as a representable functor

δAf C

δA f D T (A, A)

δB δB f T (A, f )

T (B, B) T (A, B) T (f , B)

6 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Deﬁnition as a representable functor

δAf C

δA f D T (A, A)

δB δB f T (A, f )

T (B, B) T (A, B) T (f , B)

6 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Deﬁnition as a representable functor

This is directly analogous to the definition of the functor op Cone(−, D) = Nat(∆(−), D): C → Set which sends an object to its set of cones. Recall that D has a limit iff Cone(−, D) is representable and then lim D is the representing object. Similarly, Wedge(−, T ) is representable iff T has an ending wedge and then end(T ) is the representing object.

op op× (Wedge extends to a functor Wedge: C × CA A → Set, but we won’t consider this just yet.)

7 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Deﬁnition as a representable functor

op op× (Wedge extends to a functor Wedge: C × CA A → Set, but we won’t consider this just yet.)

7 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Deﬁnition as a representable functor

op op× (Wedge extends to a functor Wedge: C × CA A → Set, but we won’t consider this just yet.)

7 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Deﬁnition as a representable functor

op op× (Wedge extends to a functor Wedge: C × CA A → Set, but we won’t consider this just yet.)

7 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Alternative deﬁnition when C = Set

op Suppose T : A × A → Set. Since X =∼ 1 + 1 + ··· + 1, a wedge from X to T is uniquely deﬁned by |X | wedges from 1 to T . {x} T (A, A)

T (A, f )

T (B, B) T (A, B) T (f , B) ∼ Q ∼ Thus Wedge(X , T ) = x∈X Wedge({x}, T ) = Hom(X , Wedge({∗}, T )). And so end(T ) =∼ Wedge({∗}, T ). Again this is analogous to how lim D =∼ Cone({∗}, D).

8 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Alternative deﬁnition when C = Set

op Suppose T : A × A → Set. Since X =∼ 1 + 1 + ··· + 1, a wedge from X to T is uniquely deﬁned by |X | wedges from 1 to T . {x} T (A, A)

T (A, f )

8 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Alternative deﬁnition when C = Set

op Suppose T : A × A → Set. Since X =∼ 1 + 1 + ··· + 1, a wedge from X to T is uniquely deﬁned by |X | wedges from 1 to T . {x} T (A, A)

T (A, f )

8 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Alternative deﬁnition when C = Set

op Suppose T : A × A → Set. Since X =∼ 1 + 1 + ··· + 1, a wedge from X to T is uniquely deﬁned by |X | wedges from 1 to T . {x} T (A, A)

T (A, f )

8 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications An important example

Consider the example T (X , Y ) = Hom(FX , GY ) for some functors F , G : A → B. A wedge from {∗} to T is has the following form.

{∗} Hom(FA, GA)

∗ tA

Gf ◦ −

tB Gf ◦ tA = tB ◦ Ff

Hom(FB, GB) Hom(FA, GB) − ◦ Ff

9 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications An important example

Consider the example T (X , Y ) = Hom(FX , GY ) for some functors F , G : A → B. A wedge from {∗} to T is has the following form.

{∗} Hom(FA, GA)

∗ tA

Gf ◦ −

tB Gf ◦ tA = tB ◦ Ff

Hom(FB, GB) Hom(FA, GB) − ◦ Ff

9 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications An important example

So such a wedge deﬁnes a family of maps (tA : FA → GA)A∈A satisfying Gf ◦ tA = tB ◦ Ff . This just means t is a natural transformation from F to G.

Hence Wedge({∗}, T ) =∼ Nat(F , G). And thus end(Hom(F (−), G(=))) =∼ Nat(F , G).

10 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications An important example

So such a wedge deﬁnes a family of maps (tA : FA → GA)A∈A satisfying Gf ◦ tA = tB ◦ Ff . This just means t is a natural transformation from F to G.

Hence Wedge({∗}, T ) =∼ Nat(F , G). And thus end(Hom(F (−), G(=))) =∼ Nat(F , G).

10 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications An important example

So such a wedge deﬁnes a family of maps (tA : FA → GA)A∈A satisfying Gf ◦ tA = tB ◦ Ff . This just means t is a natural transformation from F to G.

Hence Wedge({∗}, T ) =∼ Nat(F , G). And thus end(Hom(F (−), G(=))) =∼ Nat(F , G).

10 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Limits as ends

We may consider any diagram D : A → C as a functor op D : A × A → C that is constant in the ﬁrst component. Then a wedge into D is the same as a cone over D. γA C D(A)

γB D(f )

D(B) D(B)

Thus the end of D and the limit of D clearly coincide.

11 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Limits as ends

We may consider any diagram D : A → C as a functor op D : A × A → C that is constant in the ﬁrst component. Then a wedge into D is the same as a cone over D. γA C D(A)

γB D(f )

D(B) D(B)

Thus the end of D and the limit of D clearly coincide.

11 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Coends

The dual of the notion of an end is that of a coend. op The coend of T : A × A → C is just the end of op op op T : A × A → C . coend(T ) is the initial cowedge from T . coend(T ) is representing object of Cowedge(T , −): C → Set.

T (f , A) T (B, A) T (A, A)

T (B, f ) γA

T (B, B) C γB

12 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Coends

T (f , A) T (B, A) T (A, A)

T (B, f ) γA

T (B, B) C γB

12 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Notation

It is traditional to use integral notation for ends and coends. We write R T (a, a) for end(T ). a∈A And R a∈A T (a, a) for coend(T ).

This notation is bold and fairly nice to manipulate, but the analogy between ends and integrals in calculus is tenuous. We will use it when convenient.

13 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Notation

It is traditional to use integral notation for ends and coends. We write R T (a, a) for end(T ). a∈A And R a∈A T (a, a) for coend(T ).

This notation is bold and fairly nice to manipulate, but the analogy between ends and integrals in calculus is tenuous. We will use it when convenient.

13 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Outline

1 Deﬁnitions

2 Basic properties

3 Applications

14 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Ends are functorial

op× We can turn end into a functor end: CA A → C. 0 ωA end(T 0) T 0(A, A)

end(τ) τA,A 0 ωB T 0(A, f ) end(T ) T (A, A) ωA

T 0(f , B) T 0(B, B) T 0(A, B) ωB τB,B T (A, f ) τA,B

T (B, B) T (A, B) T (f , B)

15 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Ends are functorial

op× We can turn end into a functor end: CA A → C. 0 ωA end(T 0) T 0(A, A)

end(τ) τA,A 0 ωB T 0(A, f ) end(T ) T (A, A) ωA

T 0(f , B) T 0(B, B) T 0(A, B) ωB τB,B T (A, f ) τA,B

T (B, B) T (A, B) T (f , B)

15 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Ends are functorial

op× We can turn end into a functor end: CA A → C. 0 ωA end(T 0) T 0(A, A)

end(τ) τA,A 0 ωB T 0(A, f ) end(T ) T (A, A) ωA

T 0(f , B) T 0(B, B) T 0(A, B) ωB τB,B T (A, f ) τA,B

T (B, B) T (A, B) T (f , B)

15 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Ends are functorial

op× We can turn end into a functor end: CA A → C. 0 ωA end(T 0) T 0(A, A)

end(τ) τA,A 0 ωB T 0(A, f ) end(T ) T (A, A) ωA

T 0(f , B) T 0(B, B) T 0(A, B) ωB τB,B T (A, f ) τA,B

T (B, B) T (A, B) T (f , B)

15 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Ends are functorial

op× We can turn end into a functor end: CA A → C. 0 ωA end(T 0) T 0(A, A)

end(τ) τA,A 0 ωB T 0(A, f ) end(T ) T (A, A) ωA

T 0(f , B) T 0(B, B) T 0(A, B) ωB τB,B T (A, f ) τA,B

T (B, B) T (A, B) T (f , B)

15 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Ends are functorial

op× We can turn end into a functor end: CA A → C. 0 ωA end(T 0) T 0(A, A)

end(τ) τA,A 0 ωB T 0(A, f ) end(T ) T (A, A) ωA

T 0(f , B) T 0(B, B) T 0(A, B) ωB τB,B T (A, f ) τA,B

T (B, B) T (A, B) T (f , B)

15 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Ends are functorial

op× We can turn end into a functor end: CA A → C. 0 ωA end(T 0) T 0(A, A)

end(τ) τA,A 0 ωB T 0(A, f ) end(T ) T (A, A) ωA

T 0(f , B) T 0(B, B) T 0(A, B) ωB τB,B T (A, f ) τA,B

T (B, B) T (A, B) T (f , B)

15 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Ends as limits

The universal property deﬁnition of an end makes it clear that it can be found as a certain limit. A straightforward encoding of the diagram works, but there is a more elegant solution.

1 We deﬁne the twisted arrow category TW(A). The objects are morphisms a: A → A0 in A. A morphism f : a → b (where b : B → B0 in A) is a pair of ∗ 0 0 morphisms f : B → A and f∗ : A → B such that the diagram commutes. a A A0 ∗ f f∗ B B0 b

1 op This is actually the category of elements of Hom:(A × A) → Set. 16 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Ends as limits

The universal property deﬁnition of an end makes it clear that it can be found as a certain limit. A straightforward encoding of the diagram works, but there is a more elegant solution.

1 op This is actually the category of elements of Hom:(A × A) → Set. 16 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Ends as limits

The universal property deﬁnition of an end makes it clear that it can be found as a certain limit. A straightforward encoding of the diagram works, but there is a more elegant solution.

1 op This is actually the category of elements of Hom:(A × A) → Set. 16 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Ends as limits

op There is an obvious forgetful functor TW(A) → A × A given by (a: A → A0) 7→ (A, A0). op× TW( ) This induces a functor E : CA A → C A . One can show that end =∼ lim ◦ E and coend =∼ colim ◦ E . Exercise: Prove this!

(In fact, certain ends and limits may fail to exist, so these functors may not be completely defined. But they will agree where they are both defined and the one is defined wherever the other is defined.)

17 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Ends as limits

17 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Ends in terms of products and equalisers

Recall that the limit of any diagram can be expressed in terms of products and an equaliser. Applying this to the diagrammatic deﬁnition of ends we ﬁnd

ϕ ! ∼ Y 1 Y end(T ) = eq T (A, A) ⇒ T (A, B) ϕ2 A∈A f :A→B

where πf ◦ ϕ1 = T (A, f ) ◦ πA and πf ◦ ϕ2 = T (f , B) ◦ πB and the π’s denote the appropriate product projections. This form is handy for computations.

18 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Ends in terms of products and equalisers

Recall that the limit of any diagram can be expressed in terms of products and an equaliser. Applying this to the diagrammatic deﬁnition of ends we ﬁnd

ϕ ! ∼ Y 1 Y end(T ) = eq T (A, A) ⇒ T (A, B) ϕ2 A∈A f :A→B

where πf ◦ ϕ1 = T (A, f ) ◦ πA and πf ◦ ϕ2 = T (f , B) ◦ πB and the π’s denote the appropriate product projections. This form is handy for computations.

18 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Preservation of ends

We say the end of T is preserved by a functor F : C → D if it sends the ending wedge of T to an ending wedge of F ◦ T . Since ends are limits, they are preserved by any continuous functor. R ∼ R Furthermore, A S(A, A, −) (B) = A S(A, A, B) naturally in op B for any S : A × A × B → C whenever the right hand side exists for all B (by the similar result for limits).

We obtain the following isomorphisms, natural in X . Z Z Hom(X , T (A, A)) =∼ Hom(X , T (A, A)) A A Z Z A Hom(T (A, A), X ) =∼ Hom( T (A, A), X ) A

19 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Preservation of ends

We obtain the following isomorphisms, natural in X . Z Z Hom(X , T (A, A)) =∼ Hom(X , T (A, A)) A A Z Z A Hom(T (A, A), X ) =∼ Hom( T (A, A), X ) A

19 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Preservation of ends

We obtain the following isomorphisms, natural in X . Z Z Hom(X , T (A, A)) =∼ Hom(X , T (A, A)) A A Z Z A Hom(T (A, A), X ) =∼ Hom( T (A, A), X ) A

19 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications The Fubini theorem for ends

op op Let T : A × A × B × B → C. We can ﬁnd the ‘end of the end’ of this as either R R T (A, A, B, B) or R R T (A, A, B, B). A∈A B∈B B∈B A∈A Or we could compute it at once as R T (A, A, B, B). (A,B)∈A×B The Fubini theorem for ends says that these all coincide (as long as the relevant ends exist).

We omit the proof since I do not know a nice one.

20 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications The Fubini theorem for ends

We omit the proof since I do not know a nice one.

20 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications The Fubini theorem for ends

We omit the proof since I do not know a nice one.

20 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications The Fubini theorem for ends

We omit the proof since I do not know a nice one.

20 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Summary of identities

In summary we have the following formulae.

Z Z Z T (A, A, B, B) =∼ T (A, A, B, B). A∈A B∈B (A,B)∈A×B Z Z Hom(X , T (A, A)) =∼ Hom(X , T (A, A)) A A Z A Z Hom( T (A, A), X ) =∼ Hom(T (A, A), X ) A Z Hom(F (A), G(A)) =∼ Nat(F , G) A

21 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Outline

1 Deﬁnitions

2 Basic properties

3 Applications

22 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Powers and copowers

Let Y be an object in C and let S be a set. ` The copower of Y by S is given by S · Y := s∈S Y . It forms part of the adjunction ∼ HomC(S · Y , Z) = HomSet(S, HomC(Y , Z)).

S Q The power of Y by S is given by Y := s∈S Y . It is part of the contravariant adjunction S ∼ HomC(X , Y ) = HomSet(S, HomC(X , Y )).

23 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Powers and copowers

Let Y be an object in C and let S be a set. ` The copower of Y by S is given by S · Y := s∈S Y . It forms part of the adjunction ∼ HomC(S · Y , Z) = HomSet(S, HomC(Y , Z)).

S Q The power of Y by S is given by Y := s∈S Y . It is part of the contravariant adjunction S ∼ HomC(X , Y ) = HomSet(S, HomC(X , Y )).

23 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications Powers and copowers

Let Y be an object in C and let S be a set. ` The copower of Y by S is given by S · Y := s∈S Y . It forms part of the adjunction ∼ HomC(S · Y , Z) = HomSet(S, HomC(Y , Z)).

S Q The power of Y by S is given by Y := s∈S Y . It is part of the contravariant adjunction S ∼ HomC(X , Y ) = HomSet(S, HomC(X , Y )).

23 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications The co-Yoneda lemma

Recall that if F : A → Set, the Yoneda lemma states that Nat(Hom(X , −), F ) =∼ FX (naturally in X ). R ∼ We may express this as A HomSet(HomA(X , A), FA) = FX .

op Now let G : A → Set and take F = Hom(G(−), Y ). Z ∼ HomSet(GX , Y ) = Hom(Hom(X , A), Hom(GA, Y )) A Z =∼ Hom(Hom(X , A) × GA, Y ) A Z A =∼ Hom( Hom(X , A) × GA, Y ).

All of the above isomorphisms are natural in X and Y . So by the Yoneda lemma we ﬁnd G(X ) =∼ R A Hom(X , A) × GA.

24 / 28 Graham Manuell Ends and Coends Deﬁnitions Basic properties Applications The co-Yoneda lemma