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In mathematics, specifically in (higher) category theory, an extranatural transformation is a generalization of the notion of natural transformation.
Definition
Let
F
:
A
×
B
o
p
×
B
→
D
{\displaystyle F:A\times B^{op}\times B\rightarrow D}
G
:
A
×
C
o
p
×
C
→
D
{\displaystyle G:A\times C^{op}\times C\rightarrow D}
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 hold:
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 MorB}
∀
a
∈
A
{\displaystyle \forall a\in A}
∀
c
∈
C
{\displaystyle \forall c\in C}
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)}|\qquad &&_{\eta (a,b,c)}|\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 MorC}
∀
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')}|\qquad &&_{G(1,h,1)}|\qquad \\G(a,c',c')&{\xrightarrow {G(1,1,h)}}&G(a,c',c)\end{matrix}}}
