End (category theory)

In category theory, an end of a functor $S:{\mathbf {C}}^{{{\mathrm {op}}}}\times {\mathbf {C}}\to {\mathbf {X}}$ is a universal extranatural 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, the end can be described as the equaliser 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 is induced by $S(c,c)\to S(c,c')$ and the second morphism 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, 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 ${\mathbf {Set}}\to {\mathbf {Top}}$, where ${\mathbf {Top}}$ is the category of topological spaces. Moreover, there is a map $\gamma :\Delta \to {\mathbf {Top}}$ which sends the object $[n]$ of $\Delta$ to the standard $n$ simplex inside $\mathbb {R} ^{n+1}$. Finally there is a functor ${\mathbf {Top}}\times {\mathbf {Top}}\to {\mathbf {Top}}$ which takes the product of two topological spaces.

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

References

• end in nLab