End (category theory)

In category theory, an end of a functor ${\displaystyle S:\mathbf {C} ^{\mathrm {op} }\times \mathbf {C} \to \mathbf {X} }$ is a universal dinatural transformation from an object e of X to S.^[1]

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

By abuse of language the object e is often called the end of the functor S (forgetting ${\displaystyle \omega }$) 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

${\displaystyle \int _{c}S(c,c)\to \prod _{c\in C}S(c,c)\rightrightarrows \prod _{c\to c'}S(c,c'),}$

where the first morphism being equalized is induced by ${\displaystyle S(c,c)\to S(c,c')}$ and the second is induced by ${\displaystyle S(c',c')\to S(c,c')}$.