End (category theory)

In category theory, an end of a functor $S\colon \mathbf {C} ^{\mathrm {op} }\times \mathbf {C} \to \mathbf {X}$ is a universal dinatural transformation from an object $e$ of $\mathbf {X}$ to $S$ .^[1]

More explicitly, this is a pair $(e,\omega )$ , where $e$ is an object of $\mathbf {X}$ and $\omega \colon e{\ddot {\to }}S$ is an extranatural transformation such that for every extranatural transformation $\beta \colon x{\ddot {\to }}S$ there exists a unique morphism $h\colon x\to e$ of $\mathbf {X}$ with $\beta _{a}=\omega _{a}\circ h$ for every object $a$ of $\mathbf {C}$ .

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.

Ends can also be described using limits. If $\mathbf {X}$ is complete and $\mathbf {C}$ 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'}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 $S\colon \mathbf {C} ^{\mathrm {op} }\times \mathbf {C} \to \mathbf {X}$ 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 $\mathbf {X}$ and $\zeta \colon S{\ddot {\to }}d$ is an extranatural transformation, such that for every extranatural transformation $\gamma \colon S{\ddot {\to }}x$ there exists a unique morphism $g\colon d\to x$ of $\mathbf {X}$ with $\gamma _{a}=g\circ \zeta _{a}$ for every object $a$ of $\mathbf {C}$ .

The coend $d$ of the functor $S$ is written

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

Coends have a characterization using limits dual to the characterization of ends. If $\mathbf {X}$ is cocomplete and $\mathbf {C}$ 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'}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 $d:\mathbf {Set} \to \mathbf {Top}$ , where $\mathbf {Top}$ is the category of topological spaces. Moreover, there is a map $\gamma :\Delta \to \mathbf {Top}$ sending the object $[n]$ of $\Delta$ to the standard $n$ -simplex inside $\mathbb {R} ^{n+1}$ . Finally there is a functor $\mathbf {Top} \times \mathbf {Top} \to \mathbf {Top}$ that takes the product of two topological spaces.