End (category theory)

In category theory, an end of a functor $S:{𝐂}^{\mathrm{o}\mathrm{p}}\times 𝐂\to 𝐗$ is a universal dinatural transformation from an object $e$ of $𝐗$ to $S$.Template:Sfnp

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

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

Ends can also be described using limits. If $𝐗$ is complete and $𝐂$ is small, the end can be described as the equalizer in the diagram

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

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 $S:{𝐂}^{\mathrm{o}\mathrm{p}}\times 𝐂\to 𝐗$ 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

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

Coends have a characterization using limits dual to the characterization of ends. 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 $F,G:𝐂\to 𝐗$ then

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

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

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

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

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External links

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