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End (category theory) ：ウィキペディア英語版

End (category theory)

In category theory, an end of a functor

S:\mathbf^\to \mathbf

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 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'),

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^\to\mathbf

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_^\mathbf 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_ S(c, c) \leftleftarrows \coprod_ S(c', c).

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