End (category theory)

This article is about the type of transformation. For the category of morphisms denoted as End, see Endomorphism.

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 ${\displaystyle S}$.^[1]

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

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

Characterization as limit: 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 ${\displaystyle S(c',c')\to S(c,c')}$.

Coend

The definition of the coend of a functor ${\displaystyle S\colon \mathbf {C} ^{\mathrm {op} }\times \mathbf {C} \to \mathbf {X} }$ is the dual of the definition of an end.

Thus, a coend of ${\displaystyle S}$ consists of a pair ${\displaystyle (d,\zeta )}$, where ${\displaystyle d}$ is an object of ${\displaystyle \mathbf {X} }$ and ${\displaystyle \zeta \colon S{\ddot {\to }}d}$ is an extranatural transformation, such that for every extranatural transformation ${\displaystyle \gamma \colon S{\ddot {\to }}x}$ there exists a unique morphism ${\displaystyle g\colon d\to x}$ of ${\displaystyle \mathbf {X} }$ with ${\displaystyle \gamma _{a}=g\circ \zeta _{a}}$ for every object ${\displaystyle a}$ of ${\displaystyle \mathbf {C} }$.

The coend ${\displaystyle d}$ of the functor ${\displaystyle S}$ is written

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

Characterization as colimit: Dually, if ${\displaystyle \mathbf {X} }$ is cocomplete and ${\displaystyle \mathbf {C} }$ is small, then the coend can be described as the coequalizer in the diagram

${\displaystyle \int ^{c}S(c,c)\leftarrow \coprod _{c\in C}S(c,c)\leftleftarrows \coprod _{c\to c'}S(c',c).}$

Examples

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 ${\displaystyle d:\mathbf {Set} \to \mathbf {Top} }$, 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 ${\displaystyle dT\times \gamma }$. The coend of ${\displaystyle S}$ is the geometric realization of ${\displaystyle T}$.