End (category theory)

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(Redirected from Coend)

This page is not about 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 dinatural transformation from an object e of X to S.

More explicitly, this is a pair $(e,ω)$ , where e is an object of X and

$\omega:e\ddot\to S$

is a dinatural transformation, such that for every dinatural 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 $ω$ ) and is written

$e=\int_c^{} S(c,c)\text{ or just }\int_\mathbf{C}^{} S.$

[edit] 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 $(d,ζ)$ , where d is an object of X and

$\zeta:S\ddot\to d$

is a dinatural transformation, such that for every dinatural 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

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

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End (category theory)

From Wikipedia, the free encyclopedia

[edit] Coend

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