Category theory/Natural transformation

Example: maybeToList

 map even $ maybeToList $ Just 5

yields the same as

 maybeToList $ fmap even $ Just 5

yields: both yield

 [False]

Commutative diagram

• Let ${\mathcal {C}}$, ${\mathcal {D}}$ denote categories.
• Let $\Phi ,\Psi :{\mathcal {C}}\to {\mathcal {D}}$ be functors.
• Let $X,Y\in \mathbf {Ob} ({\mathcal {C}})$. Let $f\in \mathrm {Hom} _{\mathcal {C}}(X,Y)$.

Let us define the $\eta :\Phi \to \Psi$ natural transformation. It associates to each object of ${\mathcal {C}}$ a morphism of ${\mathcal {D}}$ in the following way (usually, not sets are discussed here, but proper classes, so I do not use term “function” for this $\mathbf {Ob} ({\mathcal {C}})\to \mathbf {Mor} ({\mathcal {D}})$ mapping):

• $\forall A\in \mathbf {Ob} ({\mathcal {C}})\longmapsto \eta _{A}\in \mathrm {Hom} _{\mathcal {D}}(\Phi (A),\Psi (A))$. We call $\eta _{A}$ the component of $\eta$ at A.
• $\eta _{Y}\cdot \Phi (f)=\Psi (f)\cdot \eta _{X}$

Thus, the following diagram commutes (in ${\mathcal {D}}$):

Vertical arrows: sides of objects

… showing how the natural transformation works.

$\eta :\Phi \to \Psi$

maybeToList :: Maybe a -> [a]

Left: side of X object

$\eta _{X}:\Phi (X)\to \Psi (X)$

	maybeToList :: Maybe Int -> [Int]
Nothing	[]
Just 0	[0]
Just 1	[1]

Right: side of Y object

$\eta _{Y}:\Phi (Y)\to \Psi (Y)$

	maybeToList :: Maybe Bool -> [Bool]
Nothing	[]
Just True	[True]
Just False	[False]

Horizontal arrows: sides of functors

$f:X\to Y$

 even :: Int -> Bool

Side of $\Phi$ functor

$\Phi (f):\Phi (X)\to \Phi (Y)$

	fmap even:: Maybe Int -> Maybe Bool
Nothing	Nothing
Just 0	Just True
Just 1	Just False

Side of $\Psi$ functor

$\Psi (f):\Psi (X)\to \Psi (Y)$

	map even:: [Int] -> [Bool]
[]	[]
[0]	[True]
[1]	[False]

Commutativity of the diagram

$\Psi (f)\cdot \eta _{X}=\eta _{Y}\cdot \Phi (f)$

both paths span between

$\Phi (X)\to \Psi (Y)$

	Maybe Int -> [Bool]
	map even . maybeToList	maybeToList . fmap even
Nothing	[]	[]
Just 0	[True]	[True]
Just 1	[False]	[False]

Remarks

• even has a more general type (Integral a => a -> Bool) than described here
• Words “side”, “horizontal”, “vertical”, “left”, “right” serve here only to point to the discussed parts of a diagram, thus, they are not part of the scientific terminology.
• If You want to modifiy the #Commutative diagram, see its source code (in LaTeX using amscd).

External links

• The corresponding HaWiki article is not migrated here yet, so You can see it for more information.
• Wikipedia's Natural transformation article