End (category theory) explained

In category theory, an end of a functor

S: C ^{op} x C \to X is a universal dinatural transformation from an object e of X to S.

More explicitly, this is a pair

(e,\omega) , where e is an object of X and \omega:e\ddot\to S is an extranatural transformation such that for every extranatural transformation \beta : x\ddot\to S there exists a unique morphism h:x\to e of X with \beta _{a=\omega} _{a\circ} h for every object a of C.

By abuse of language the object e is often called the end of the functor S (forgetting

\omega ) and is writtene = \int _{c} S(c,c) orjust \int _{C} S.

Characterization as limit: If X is complete and C is small, the end can be described as the equalizer in the diagram

\int _{c} S(c, c) \to \prod _{c} S(c, c) \rightrightarrows \prod _{c} S(c, c'),

where the first morphism being equalized is induced by

S(c, c) \to S(c, c') and the second is induced by S(c', c') \to S(c, c') .

Coend

The definition of the coend of a functor

S: C ^{op} x C \to X is the dual of the definition of an end.

Thus, a coend of S consists of a pair

(d,\zeta) , where d is an object of X and \zeta:S\ddot\to d is an extranatural transformation, such that for every extranatural transformation \gamma:S\ddot\to x there exists a unique morphismg:d\to x of X with \gamma _{a=g\circ\zeta} _{a} for every object a of C.

The coend d of the functor S is written

d = \int ^{c} S(c,c) or \int ^{C} S.

Characterization as colimit: Dually, if X is cocomplete and C is small, then the coend can be described as the coequalizer in the diagram

\int ^{c} S(c, c) \leftarrow \coprod _{c} S(c, c) \leftleftarrows \coprod _{c} S(c', c).

Examples

• Natural transformations:Suppose we have functors
  F, G : C \to X thenHom _{X} (F( - ), G( - )) : C ^{op} x C \to Set .

  In this case, the category of sets is complete, so we need only form the equalizer and in this case

  \int _{c} Hom _{X} (F(c), G(c)) = Nat (F, G)

  the natural transformations from

  F to G . Intuitively, a natural transformation from F to G is a morphism from F(c) to G(c) for every c in the category with compatibility conditions. Looking at the equalizer diagram defining the end makes the equivalence clear.
• Geometric realizations:Let T be a simplicial set. That is, T is a functor \Delta ^{op} \to Set . The discrete topology gives a functor d: Set \to Top , where Top is the category of topological spaces. Moreover, there is a map \gamma:\Delta \to Top sending the object [n] of \Delta to the standard n -simplex inside R ^{n+1} . Finally there is a functor Top x Top \to Top that takes the product of two topological spaces.Define S to be the composition of this product functor with dT x \gamma . The coend of S is the geometric realization of T .

References

• Book: Mac Lane . Saunders . Saunders Mac Lane. Categories For the Working Mathematician . 2013 . Springer Science & Business Media . 222–226.
• Loregian . Fosco . This is the (co)end, my only (co)friend . 1501.02503. math.CT . 2015 .