End (category theory)

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

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

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

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

Ends can also be described using limits. If ${\displaystyle \mathbf {X} }$ is complete and ${\displaystyle \mathbf {C} }$ 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 ${\displaystyle S(c',c')\to S(c,c')}$.