In the mathematical branch of category theory , an end is a special limit .
definition
Let there be categories , the category that is too dual and finally a functor .
C.
;
D.
{\ displaystyle {\ mathcal {C}}; {\ mathcal {D}}}
C.
O
p
{\ displaystyle {\ mathcal {C}} ^ {\ mathrm {op}}}
C.
{\ displaystyle {\ mathcal {C}}}
F.
:
C.
op
×
C.
→
D.
{\ displaystyle F \ colon {\ mathcal {C}} ^ {\ text {op}} \ times {\ mathcal {C}} \ to {\ mathcal {D}}}
One end of is a pair consisting of an object and an indexed family of arrows , called projections , such that for all objects and morphisms the diagram
F.
{\ displaystyle F}
(
E.
;
π
)
{\ displaystyle (E; \ pi)}
E.
∈
If
(
D.
)
{\ displaystyle E \ in \ operatorname {Ob} ({\ mathcal {D}})}
If
(
C.
)
{\ displaystyle \ operatorname {Ob} ({\ mathcal {C}})}
π
X
:
E.
→
F.
(
X
,
X
)
{\ displaystyle \ pi _ {X} \ colon E \ to F (X, X)}
X
,
Y
∈
If
(
C.
)
{\ displaystyle X, Y \ in \ operatorname {Ob} ({\ mathcal {C}})}
H
∈
C.
(
X
,
Y
)
{\ displaystyle h \ in {\ mathcal {C}} (X, Y)}
E.
→
π
X
F.
(
X
,
X
)
π
Y
↓
↓
F.
(
X
,
H
)
F.
(
Y
,
Y
)
→
F.
(
H
,
Y
)
F.
(
X
,
Y
)
{\ displaystyle {\ begin {array} {ccc} \ quad E & {\ xrightarrow {\ quad \ pi _ {X} \ quad}} & \! F (X, X) \\\ scriptstyle \ pi _ {Y} {\ Big \ downarrow} && \ quad {\ Big \ downarrow} \ scriptstyle F (X, h) \\ F (Y, Y) & {\ xrightarrow [{F (h, Y)}] {}} & F ( X, Y) \ end {array}}}
commutes. (In short: is a dinatural transformation .)
π
{\ displaystyle \ pi}
Δ
E.
→
F.
{\ displaystyle \ Delta E \ to F}
An end is also universal , that is, for each alternative
with corresponding projections there is a clearly defined arrow , so that applies to all .
E.
′
{\ displaystyle E '}
π
X
′
:
E.
′
→
F.
(
X
,
X
)
{\ displaystyle \ pi '_ {X} \ colon E' \ to F (X, X)}
k
:
E.
′
→
E.
{\ displaystyle k \ colon E '\ to E}
π
X
∘
k
=
π
X
′
{\ displaystyle \ pi _ {X} \ circ k = \ pi '_ {X}}
X
∈
If
(
C.
)
{\ displaystyle X \ in \ operatorname {Ob} ({\ mathcal {C}})}
notation
A common notation for the end of is is
F.
:
C.
op
×
C.
→
D.
{\ displaystyle F \ colon {\ mathcal {C}} ^ {\ text {op}} \ times {\ mathcal {C}} \ to {\ mathcal {D}}}
E.
≅
∫
X
∈
C.
F.
(
X
,
X
)
{\ displaystyle E \ cong \ int _ {X \ in {\ mathcal {C}}} F (X, X)}
.
example
Let functors be given for locally small categories . The amount of natural transformations of to is precisely one end of the functor
, by explained, wherein the hom functor of call.
C.
,
D.
{\ displaystyle {\ mathcal {C}}, {\ mathcal {D}}}
F.
,
G
:
C.
→
D.
{\ displaystyle F, G \ colon {\ mathcal {C}} \ to {\ mathcal {D}}}
F.
{\ displaystyle F}
G
{\ displaystyle G}
T
:
C.
op
×
C.
→
S.
e
t
{\ displaystyle T \ colon {\ mathcal {C}} ^ {\ text {op}} \ times {\ mathcal {C}} \ to \ mathbf {Set}}
T
(
X
,
Y
)
=
D.
(
F.
X
,
G
Y
)
{\ displaystyle T (X, Y) = {\ mathcal {D}} (FX, GY)}
D.
(
-
,
-
)
{\ displaystyle {\ mathcal {D}} (-, -)}
D.
{\ displaystyle {\ mathcal {D}}}
The above diagram is here
E.
→
π
X
D.
(
F.
X
,
G
X
)
π
Y
↓
↓
D.
(
F.
X
,
G
H
)
D.
(
F.
Y
,
G
Y
)
→
D.
(
F.
H
,
G
Y
)
D.
(
F.
X
,
G
Y
)
.
{\ displaystyle {\ begin {array} {ccc} \ quad E & {\ xrightarrow {\ quad \ pi _ {X} \ quad}} & \! {\ mathcal {D}} (FX, GX) \\\ scriptstyle \ pi _ {Y} {\ Big \ downarrow} && \ quad {\ Big \ downarrow} \ scriptstyle {\ mathcal {D}} (FX, Gh) \\ {\ mathcal {D}} (FY, GY) & {\ xrightarrow [{{\ mathcal {D}} (Fh, GY)}] {}} & {\ mathcal {D}} (FX, GY). \ end {array}}}
The projections of the end assign a component
to every natural transformation . So at the element level of , the diagram states that for components and
ψ
∈
E.
{\ displaystyle \ psi \ in E}
ψ
X
=
π
X
(
ψ
)
∈
D.
(
F.
X
,
G
X
)
{\ displaystyle \ psi _ {X} = \ pi _ {X} (\ psi) \ in {\ mathcal {D}} (FX, GX)}
E.
{\ displaystyle E}
ψ
X
{\ displaystyle \ psi _ {X}}
ψ
Y
{\ displaystyle \ psi _ {Y}}
G
H
∘
ψ
X
=
ψ
Y
∘
F.
H
{\ displaystyle Gh \ circ \ psi _ {X} = \ psi _ {Y} \ circ Fh}
applies. The universality ensures that it all contains natural transformations.
E.
{\ displaystyle E}
This example can also be interpreted as a definition of natural transformations. In this form the definition can easily be generalized to enriched categories and functors.
literature
GM Kelly: Basic Concepts of Enriched Category Theory . In: Lecture Notes in Mathematics 64 . Cambridge University Press, 1982 ( mta.ca [accessed March 8, 2014]).
Web links
end , entry in the nLab . (English)
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