• Source: End (category theory)

In category theory, an end of a functor



S
:


C



o
p



×

C

→

X



{\displaystyle 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
)


{\displaystyle (e,\omega )}

, where e is an object of X and



ω
:
e



→
¨



S


{\displaystyle \omega :e{\ddot {\to }}S}

is an extranatural transformation such that for every extranatural transformation



β
:
x



→
¨



S


{\displaystyle \beta :x{\ddot {\to }}S}

there exists a unique morphism



h
:
x
→
e


{\displaystyle h:x\to e}


of X with




β

a


=

ω

a


∘
h


{\displaystyle \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


{\displaystyle \omega }

) and is written




e
=

∫

c





S
(
c
,
c
)

or just


∫


C






S
.


{\displaystyle e=\int _{c}^{}S(c,c){\text{ or just }}\int _{\mathbf {C} }^{}S.}


Characterization as limit: If X is complete and C is small, the end can be described as the equalizer in the diagram





∫

c


S
(
c
,
c
)
→

∏

c
∈
C


S
(
c
,
c
)
⇉

∏

c
→

c
′



S
(
c
,

c
′

)
,


{\displaystyle \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 being equalized is induced by



S
(
c
,
c
)
→
S
(
c
,

c
′

)


{\displaystyle S(c,c)\to S(c,c')}

and the second is induced by



S
(

c
′

,

c
′

)
→
S
(
c
,

c
′

)


{\displaystyle S(c',c')\to S(c,c')}

.




Coend


The definition of the coend of a functor



S
:


C



o
p



×

C

→

X



{\displaystyle 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
)


{\displaystyle (d,\zeta )}

, where d is an object of X and



ζ
:
S



→
¨



d


{\displaystyle \zeta :S{\ddot {\to }}d}


is an extranatural transformation, such that for every extranatural transformation



γ
:
S



→
¨



x


{\displaystyle \gamma :S{\ddot {\to }}x}

there exists a unique morphism




g
:
d
→
x


{\displaystyle g:d\to x}

of X with




γ

a


=
g
∘

ζ

a




{\displaystyle \gamma _{a}=g\circ \zeta _{a}}

for every object a of C.
The coend d of the functor S is written




d
=

∫




c


S
(
c
,
c
)

or


∫





C



S
.


{\displaystyle d=\int _{}^{c}S(c,c){\text{ or }}\int _{}^{\mathbf {C} }S.}


Characterization as colimit: Dually, if X is cocomplete and C is small, then the coend can be described as the coequalizer in the diagram





∫

c


S
(
c
,
c
)
←

∐

c
∈
C


S
(
c
,
c
)
⇇

∐

c
→

c
′



S
(

c
′

,
c
)
.


{\displaystyle \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
:

C

→

X



{\displaystyle F,G:\mathbf {C} \to \mathbf {X} }

then






H
o
m



X



(
F
(
−
)
,
G
(
−
)
)
:


C


o
p


×

C

→

S
e
t



{\displaystyle \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





∫

c




H
o
m



X



(
F
(
c
)
,
G
(
c
)
)
=

N
a
t

(
F
,
G
)


{\displaystyle \int _{c}\mathrm {Hom} _{\mathbf {X} }(F(c),G(c))=\mathrm {Nat} (F,G)}


the natural transformations from



F


{\displaystyle F}

to



G


{\displaystyle G}

. Intuitively, a natural transformation from



F


{\displaystyle F}

to



G


{\displaystyle G}

is a morphism from



F
(
c
)


{\displaystyle F(c)}

to



G
(
c
)


{\displaystyle G(c)}

for every



c


{\displaystyle c}

in the category with compatibility conditions. Looking at the equalizer diagram defining the end makes the equivalence clear.

Geometric realizations:

Let



T


{\displaystyle T}

be a simplicial set. That is,



T


{\displaystyle T}

is a functor




Δ


o
p



→

S
e
t



{\displaystyle \Delta ^{\mathrm {op} }\to \mathbf {Set} }

. The discrete topology gives a functor



d
:

S
e
t

→

T
o
p



{\displaystyle d:\mathbf {Set} \to \mathbf {Top} }

, where




T
o
p



{\displaystyle \mathbf {Top} }

is the category of topological spaces. Moreover, there is a map



γ
:
Δ
→

T
o
p



{\displaystyle \gamma :\Delta \to \mathbf {Top} }

sending the object



[
n
]


{\displaystyle [n]}

of



Δ


{\displaystyle \Delta }

to the standard



n


{\displaystyle n}

-simplex inside





R


n
+
1




{\displaystyle \mathbb {R} ^{n+1}}

. Finally there is a functor




T
o
p

×

T
o
p

→

T
o
p



{\displaystyle \mathbf {Top} \times \mathbf {Top} \to \mathbf {Top} }

that takes the product of two topological spaces.

Define



S


{\displaystyle S}

to be the composition of this product functor with



d
T
×
γ


{\displaystyle dT\times \gamma }

. The coend of



S


{\displaystyle S}

is the geometric realization of



T


{\displaystyle T}

.




Notes






References






External links


end at the nLab

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