In mathematics, specifically in category theory, an extranatural transformation is a generalization of the notion of natural transformation.
Let
F
:
A
×
B
o
p
×
B
→
D
and
G
:
A
×
C
o
p
×
C
→
D
two functors of categories. A family
η
(
a
,
b
,
c
)
:
F
(
a
,
b
,
b
)
→
G
(
a
,
c
,
c
)
is said to be natural in a and extranatural in b and c if the following holds:
η
(
−
,
b
,
c
)
is a natural transformation (in the usual sense).
(extranaturality in b)
∀
(
g
:
b
→
b
′
)
∈
M
o
r
B
,
∀
a
∈
A
,
∀
c
∈
C
the following diagram commutes
(extranaturality in c)
∀
(
h
:
c
→
c
′
)
∈
M
o
r
C
,
∀
a
∈
A
,
∀
b
∈
B
the following diagram commutes
Extranatural transformations can be used to define wedges and thereby ends (dually co-wedges and co-ends), by setting
F
(dually
G
) constant.
Extranatural transformations can be defined in terms of Dinatural transformations.
