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 dinatural 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 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 $$d:\mathbf{Set} \to \mathbf{Top}$$, 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. Define $$S$$ to be the composition of this product functor with $$dT \times \gamma$$. The coend of $$S$$ is the geometric realization of $$T$$.  