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Generalization of natural transformations
In mathematics , specifically in category theory , an extranatural transformation [ 1] natural transformation .
Definition
Let
F
:
A
×
B
o
p
×
B
→
D
{\displaystyle F:A\times B^{\mathrm {op} }\times B\rightarrow D}
G
:
A
×
C
o
p
×
C
→
D
{\displaystyle G:A\times C^{\mathrm {op} }\times C\rightarrow D}
functors of categories.
A family
η
(
a
,
b
,
c
)
:
F
(
a
,
b
,
b
)
→
G
(
a
,
c
,
c
)
{\displaystyle \eta (a,b,c):F(a,b,b)\rightarrow G(a,c,c)}
natural in a and extranatural in b and c if the following holds:
η
(
−
,
b
,
c
)
{\displaystyle \eta (-,b,c)}
(extranaturality in b )
∀
(
g
:
b
→
b
′
)
∈
M
o
r
B
{\displaystyle \forall (g:b\rightarrow b^{\prime })\in \mathrm {Mor} \,B}
∀
a
∈
A
{\displaystyle \forall a\in A}
∀
c
∈
C
{\displaystyle \forall c\in C}
diagram commutes
F
(
a
,
b
′
,
b
)
→
F
(
1
,
1
,
g
)
F
(
a
,
b
′
,
b
′
)
F
(
1
,
g
,
1
)
↓
η
(
a
,
b
′
,
c
)
↓
F
(
a
,
b
,
b
)
→
η
(
a
,
b
,
c
)
G
(
a
,
c
,
c
)
{\displaystyle {\begin{matrix}F(a,b',b)&\xrightarrow {F(1,1,g)} &F(a,b',b')\\_{F(1,g,1)}\downarrow \qquad &&_{\eta (a,b',c)}\downarrow \qquad \\F(a,b,b)&\xrightarrow {\eta (a,b,c)} &G(a,c,c)\end{matrix}}}
(extranaturality in c )
∀
(
h
:
c
→
c
′
)
∈
M
o
r
C
{\displaystyle \forall (h:c\rightarrow c^{\prime })\in \mathrm {Mor} \,C}
∀
a
∈
A
{\displaystyle \forall a\in A}
∀
b
∈
B
{\displaystyle \forall b\in B}
F
(
a
,
b
,
b
)
→
η
(
a
,
b
,
c
′
)
G
(
a
,
c
′
,
c
′
)
η
(
a
,
b
,
c
)
↓
G
(
1
,
h
,
1
)
↓
G
(
a
,
c
,
c
)
→
G
(
1
,
1
,
h
)
G
(
a
,
c
,
c
′
)
{\displaystyle {\begin{matrix}F(a,b,b)&\xrightarrow {\eta (a,b,c')} &G(a,c',c')\\_{\eta (a,b,c)}\downarrow \qquad &&_{G(1,h,1)}\downarrow \qquad \\G(a,c,c)&\xrightarrow {G(1,1,h)} &G(a,c,c')\end{matrix}}}
Properties
Extranatural transformations can be used to define wedges and thereby ends [ 2]
F
{\displaystyle F}
G
{\displaystyle G}
Extranatural transformations can be defined in terms of dinatural transformations , of which they are a special case.[ 2]
See also
References
^ Eilenberg and Kelly , A generalization of the functorial calculus, J. Algebra 3 366–375 (1966)^ a b Fosco Loregian, This is the (co)end, my only (co)friend , arXiv preprint [1]
External links
