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. C 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 2 C to T is a family of morphisms (γA : C ! T (A; A))A2A 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 Instead of a diagram, we now consider a functor op T : A × A ! C of mixed variance. A wedge from an object C 2 C to T is a family of morphisms (γA : C ! T (A; A))A2A 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 Ends as universal wedges The end of T is the terminal wedge to T . γA C !A end(T ) T (A; A) ! 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 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 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 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 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 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 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 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 jX j wedges from 1 to T . fxg T (A; A) T (A; f ) T (B; B) T (A; B) T (f ; B) ∼ Q ∼ Thus Wedge(X ; T ) = x2X Wedge(fxg; 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 jX j wedges from 1 to T . fxg T (A; A) T (A; f ) T (B; B) T (A; B) T (f ; B) ∼ Q ∼ Thus Wedge(X ; T ) = x2X Wedge(fxg; 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 jX j wedges from 1 to T . fxg T (A; A) T (A; f ) T (B; B) T (A; B) T (f ; B) ∼ Q ∼ Thus Wedge(X ; T ) = x2X Wedge(fxg; 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 jX j wedges from 1 to T . fxg T (A; A) T (A; f ) T (B; B) T (A; B) T (f ; B) ∼ Q ∼ Thus Wedge(X ; T ) = x2X Wedge(fxg; 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 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)A2A 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)A2A 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)A2A 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.