End (category theory) – Wikipedia

In category theory, an end of a functor

is a universal extranatural transformation from an object e of X to S.

More explicitly, this is a pair

, where e is an object of X and

is an extranatural transformation such that for every extranatural transformation

there exists a unique morphism

of X with

for every object a of C.

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

) and is written

${displaystyle e=int _{c}^{}S(c,c){text{ or just }}int _{mathbf {C} }^{}S.}$

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

${displaystyle int _{c}S(c,c)to prod _{cin C}S(c,c)rightrightarrows prod _{cto c’}S(c,c’),}$

where the first morphism being equalized is induced by

and the second is induced by

.

The definition of the coend of a functor

is the dual of the definition of an end.

Thus, a coend of S consists of a pair

, where d is an object of X and

is an extranatural transformation, such that for every extranatural transformation

there exists a unique morphism

of X with

for every object a of C.

The coend d of the functor S is written

${displaystyle d=int _{}^{c}S(c,c){text{ or }}int _{}^{mathbf {C} }S.}$

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

${displaystyle int ^{c}S(c,c)leftarrow coprod _{cin C}S(c,c)leftleftarrows coprod _{cto c’}S(c’,c).}$

Examples[edit]

• Natural transformations:

  Suppose we have functors

  ${displaystyle F,G:mathbf {C} to mathbf {X} }$

  then

  ${displaystyle mathrm {Hom} _{mathbf {X} }(F(-),G(-)):mathbf {C} ^{op}times mathbf {C} to mathbf {Set} }$

  .

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

  ${displaystyle int _{c}mathrm {Hom} _{mathbf {X} }(F(c),G(c))=mathrm {Nat} (F,G)}$

  

  the natural transformations from

  ${displaystyle F}$

  to

  ${displaystyle G}$

  . Intuitively, a natural transformation from

  ${displaystyle F}$

  to

  ${displaystyle G}$

  is a morphism from

  ${displaystyle F(c)}$

  to

  ${displaystyle G(c)}$

  for every

  ${displaystyle c}$

  in the category with compatibility conditions. Looking at the equalizer diagram defining the end makes the equivalence clear.

• Geometric realizations:

  Let

  ${displaystyle T}$

  be a simplicial set. That is,

  ${displaystyle T}$

  is a functor

  ${displaystyle Delta ^{mathrm {op} }to mathbf {Set} }$

  . The discrete topology gives a functor

  ${displaystyle mathbf {Set} to mathbf {Top} }$

  , where

  ${displaystyle mathbf {Top} }$

  is the category of topological spaces. Moreover, there is a map

  ${displaystyle gamma :Delta to mathbf {Top} }$

  sending the object

  ${displaystyle [n]}$

  of

  ${displaystyle Delta }$

  to the standard

  ${displaystyle n}$

  -simplex inside

  ${displaystyle mathbb {R} ^{n+1}}$

  . Finally there is a functor

  ${displaystyle mathbf {Top} times mathbf {Top} to mathbf {Top} }$

  that takes the product of two topological spaces.

  Define

  ${displaystyle S}$

  to be the composition of this product functor with

  ${displaystyle Ttimes gamma }$

  . The coend of

  ${displaystyle S}$

  is the geometric realization of

  ${displaystyle T}$

  .

References[edit]

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

External links[edit]