End (category theory)

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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) 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 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) or \int_{}^\mathbf{C} S.
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