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