End (category Theory)

In category theory, an end of a functor $S:\mathbf^\times\mathbf\to \mathbf$ 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

:$e=\int_c^ S(c,c)\text\int_\mathbf^ S.$

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

:$\int_c S(c, c) \to \prod_ S(c, c) \rightrightarrows \prod_ S(c, c'),$

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 $S:\mathbf^\times\mathbf\to\mathbf$ 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

:$d=\int_^c S(c,c)\text\int_^\mathbf S.$

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

:$\int^c S(c, c) \leftarrow \coprod_ S(c, c) \leftleftarrows \coprod_ S(c', c).$

Examples

• Natural transformations:
  
  
  Suppose we have functors $F, G : \mathbf \to \mathbf$ then

  :$\mathrm_(F(-), G(-)) : \mathbf^ \times \mathbf \to \mathbf$.

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

  :$\int_c \mathrm_(F(c), G(c)) = \mathrm(F, G)$

  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 $\Delta^ \to \mathbf$. The discrete topology gives a functor $\mathbf \to \mathbf$, where $\mathbf$ is the category of topological spaces. Moreover, there is a map $\gamma:\Delta \to \mathbf$ sending the object /math> of $\Delta$ to the standard $n$-simplex inside $\mathbb^$. Finally there is a functor $\mathbf \times \mathbf \to \mathbf$ that 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 $T$.
  

Notes

References

*
*

External links

*

{{Category theory

Functors