Natural transformation

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Informally, the notion of a natural transformation states that a particular map between functors can be done consistently over an entire category.

Indeed, this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications.

Definition

If

F

{\displaystyle F}

and

G

{\displaystyle G}

are functors between the categories

C

{\displaystyle {\mathcal {C}}}

and

D

{\displaystyle {\mathcal {D}}}

(both from

C

{\displaystyle {\mathcal {C}}}

to

D

{\displaystyle {\mathcal {D}}}

), then a natural transformation

η

{\displaystyle \eta }

from

F

{\displaystyle F}

to

G

{\displaystyle G}

is a family of morphisms

(

η

X

)

X

∈

ob

(

C

)

{\displaystyle (\eta _{X})_{X\in {\text{ob}}({\mathcal {C}})}}

, where

η

X

:

F

(

X

)

→

G

(

X

)

{\displaystyle \eta _{X}:F(X)\to G(X)}

. The morphism

η

X

{\displaystyle \eta _{X}}

is called "the component of

η

{\displaystyle \eta }

at

X

{\displaystyle X}

" or "the

X

{\displaystyle X}

component of

η

{\displaystyle \eta }

." The components of

η

{\displaystyle \eta }

are such that for every morphism

f

:

X

→

Y

{\displaystyle f:X\to Y}

in

C

{\displaystyle {\mathcal {C}}}

we have

η

Y

∘

F

(

f

)

=

G

(

f

)

∘

η

X

.

{\displaystyle \eta _{Y}\circ F(f)=G(f)\circ \eta _{X}.}

This equation can be conveniently expressed by the following commutative diagram:

If both

F

{\displaystyle F}

and

G

{\displaystyle G}

are instead contravariant functors, the vertical arrows in the right diagram are reversed. If

η

{\displaystyle \eta }

is a natural transformation from

F

{\displaystyle F}

to

G

{\displaystyle G}

, we write

η

:

F

→

G

{\displaystyle \eta :F\to G}

or

η

:

F

⇒

G

{\displaystyle \eta :F\Rightarrow G}

. This is also expressed by saying the family of morphisms

η

X

:

F

(

X

)

→

G

(

X

)

{\displaystyle \eta _{X}:F(X)\to G(X)}

is natural in

X

{\displaystyle X}

.