End (category theory)

Not to be confused with the use of End to represent (categories of) endomorphisms.

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 a 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 a 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