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^}\times\mathbf\to \mathbf] 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 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 [\omega]) and is written

[e=\int_c^ S(c,c)] or just [\int_\mathbf^ S].

The definition of the coend of a functor [S:\mathbf^}\times\mathbf\to\mathbf] is the dual of 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 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 S].

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