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.^[1]

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 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'}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 $S:\mathbf {C} ^{\mathrm {op} }\times \mathbf {C} \to \mathbf {X}$ 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 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'}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 $\gamma :\Delta \to \mathbf {Top}$ 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.