End (category theory)

In category theory, an end of a functor $S:{\mathbf{𝐂}}^{\mathrm{o}\mathrm{p}}\times \mathbf{𝐂}\to \mathbf{𝐗}$ 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

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

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

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

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

Coend

The definition of the coend of a functor $S:{\mathbf{𝐂}}^{\mathrm{o}\mathrm{p}}\times \mathbf{𝐂}\to \mathbf{𝐗}$ 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

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

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

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

Examples

• Natural transformations:

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

  ${\mathrm{H}\mathrm{o}\mathrm{m}}_{\mathbf{𝐗}}(F(-),G(-)):{\mathbf{𝐂}}^{op}\times \mathbf{𝐂}\to \mathbf{𝐒}\mathbf{𝐞}\mathbf{𝐭}$.

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

  $\underset{c}{\int }{\mathrm{H}\mathrm{o}\mathrm{m}}_{\mathbf{𝐗}}(F(c),G(c))=\mathrm{N}\mathrm{a}\mathrm{t}(F,G)$

  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 ${\mathrm{\Delta }}^{\mathrm{o}\mathrm{p}}\to \mathbf{𝐒}\mathbf{𝐞}\mathbf{𝐭}$. The discrete topology gives a functor $d:\mathbf{𝐒}\mathbf{𝐞}\mathbf{𝐭}\to \mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$, where $\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$ is the category of topological spaces. Moreover, there is a map $\gamma :\mathrm{\Delta }\to \mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$ sending the object $[n]$ of $\mathrm{\Delta }$ to the standard $n$-simplex inside ${\mathbb{ℝ}}^{n+1}$. Finally there is a functor $\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}\times \mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}\to \mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$ 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 in nLab