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Category theory/Natural transformation

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1 Example:

 map even $ maybeToList $ Just 5

yields the same as

 maybeToList $ fmap even $ Just 5

yields: both yield


In the followings, this example will be used to illustrate the notion of natural transformation.

2 Definition

  • 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 ηA the component of η at A.
  • \eta_Y \cdot \Phi(f) = \Psi(f) \cdot \eta_X

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

Natural transformation.png

2.1 Vertical arrows: sides of objects

… showing how the natural transformation works.

\eta : \Phi \to \Psi
maybeToList :: Maybe a -> [a]

2.1.1 Left: side of X object

\eta_X : \Phi(X) \to \Psi(X)
maybeToList :: Maybe Int -> [Int]
Just 0
Just 1

2.1.2 Right: side of Y object

\eta_Y : \Phi(Y) \to \Psi(Y)
maybeToList :: Maybe Bool -> [Bool]
Just True
Just False

2.2 Horizontal arrows: sides of functors

f : X \to Y
 even :: Int -> Bool

2.2.1 Side of Φ functor

\Phi(f) : \Phi(X) \to \Phi(Y)
fmap even:: Maybe Int -> Maybe Bool
Just 0
Just True
Just 1
Just False

2.2.2 Side of Ψ functor

\Psi(f) : \Psi(X) \to \Psi(Y)
map even:: [Int] -> [Bool]

2.3 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
Just 0
Just 1

2.4 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).

3 Operations

3.1 Functor and natural transformation

Let us imagine a parser library, which contains functions for parsing a form. There are two kinds of cells:

  • containing data which are optional (e.g. name of spouse)
  • containing data which consist of an enumaration of items (e.g. names of acquired languages)
 spouse :: Parser (Maybe String)
 languages :: Parser [String]

Let us imagine we have any processing (storing, archiving etc.) function which processes lists (or any other reason which forces us to convert our results tu list fomrat instead of lists). (Maybe all this example is unparactical and exaggerated!)

We can convert
to list with
But if we want to do something similar with a parser on Maybe's to achieve a parser on list, then
is not enough alone, we must
it. E.g. if we want to convert a parser like
to be of the same type as
 fmap maybeToList spouse

Let us see the types: We start with

 spouse :: Parser (Maybe String)

or using notion of composing functors


We want to achieve

 fmap maybeToList spouse :: Parser [String]

thus we can infer

 fmap maybeToList :: Parser (Maybe [String]) -> Parser [String]

In fact, we have a new “datatype converter”: converting not Maybe's to lists, but parser on Maybe to Parser on list. Let us notate the corresponding natural transformation with Λη:

To each X \in \mathbf{Ob}(\mathcal C) we associate (\Lambda\eta)_X \in \mathrm{Hom}_{\mathcal D}(\Lambda\Phi)(X),\;(\Lambda\Psi)(X))
\Lambda\eta : \Lambda\Phi \to \Lambda\Psi
(Λη)X = Λ(ηX)

3.2 Natural transformation and functor

\eta\Delta : \Phi\Delta \to \Psi\Delta
(ηΔ)X = ηΔ(X)

It can be illustrated by Haskell examples, too. Understanding it is made harder (easier?) by the fact that Haskell's type inference “dissolves” the main point, thus there is no “materialized” manifestation of it.

4 External links