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 $S:\mathbf {C} ^{\mathrm {op} }\times \mathbf {C} \to \mathbf {X}$ 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 written

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

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

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

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

Coend

The definition of the coend of a functor $S:\mathbf {C} ^{\mathrm {op} }\times \mathbf {C} \to \mathbf {X}$ 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 morphism $g:d\to x$ of X with $\gamma _{a}=g\circ \zeta _{a}$ for every object a of C.

The coend d of the functor S is written

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

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

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

Examples

• Natural transformations:

  Suppose we have functors $F,G:\mathbf {C} \to \mathbf {X}$ then

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

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

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

  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 $\Delta ^{\mathrm {op} }\to \mathbf {Set}$. The discrete topology gives a functor $d:\mathbf {Set} \to \mathbf {Top}$, where $\mathbf {Top}$ is the category of topological spaces. Moreover, there is a map $\gamma :\Delta \to \mathbf {Top}$ sending the object $[n]$ of $\Delta$ to the standard $n$-simplex inside $\mathbb {R} ^{n+1}$. Finally there is a functor $\mathbf {Top} \times \mathbf {Top} \to \mathbf {Top}$ that takes the product of two topological spaces.

  Define $S$ to be the composition of this product functor with $dT\times \gamma$. The coend of $S$ is the geometric realization of $T$.

References

• 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

• end at the nLab