[FP & LINQ via C# series]#

[Category Theory via C# series]#

Natural transformation and naturality#

If F: C → D and G: C → D are both functors from categories C to category D, the mapping from F to G is called natural transformation and denoted α: F ⇒ G. α: F ⇒ G is actually family of morphisms from F to G, For each object X in category C, there is a specific morphism αX: F(X) → G(X) in category D, called the component of α at X. For each morphism m: X → Y in category C and 2 functors F: C → D, G: C → D, there is a naturality square in D:

In another word, for m: X → Y in category C, there must be αY ∘ F(m) ≡ G(m) ∘ αX , or equivalently αY ∘ SelectF(m) ≡ SelectG(m) ∘ αX in category D.

In DotNet category, the following ToLazy<> generic method transforms Func<> functor to Lazy<> functor:

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public static partial class NaturalTransformations
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{
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    // ToLazy: Func<> -> Lazy<>
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    public static Lazy<T> ToLazy<T>(this Func<T> function) => new Lazy<T>(function);
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}

Apparently, for above natural transformation: ToLazy<>: Func<> ⇒ Lazy<>:

• for each specific object T, there is an object Func, an object Lazy, and a morphism ToFunc: Func → Lazy.

• For each specific morphism selector: TSource → TResult, there is a naturality square, which consists of 4 morphisms:

• ToLazy: Func → Lazy, which is the component of ToLazy<> at TResult

• FuncExtensions.Select(selector): Func → Func

• LazyExtensions.Select(selector): Lazy → Lazy

• ToLazy: Func → Lazy, which is the component of ToLazy<> at TSource

The following example is a simple naturality square that commutes for ToLazy<>:

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internal static void Naturality()
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{
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    Func<int, string> selector = int32 => Math.Sqrt(int32).ToString("0.00");
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    // Naturality square:
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    // ToFunc<string>.o(LazyExtensions.Select(selector)) == FuncExtensions.Select(selector).o(ToFunc<int>)
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    Func<Func<string>, Lazy<string>> funcStringToLazyString = ToLazy<string>;
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    Func<Func<int>, Func<string>> funcInt32ToFuncString = FuncExtensions.Select(selector);
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    Func<Func<int>, Lazy<string>> leftComposition = funcStringToLazyString.o(funcInt32ToFuncString);
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    Func<Lazy<int>, Lazy<string>> lazyInt32ToLazyString = LazyExtensions.Select(selector);
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    Func<Func<int>, Lazy<int>> funcInt32ToLazyInt32 = ToLazy<int>;
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    Func<Func<int>, Lazy<string>> rightComposition = lazyInt32ToLazyString.o(funcInt32ToLazyInt32);
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    Func<int> funcInt32 = () => 2;
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    Lazy<string> lazyString = leftComposition(funcInt32);
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    lazyString.Value.WriteLine(); // 1.41
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    lazyString = rightComposition(funcInt32);
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    lazyString.Value.WriteLine(); // 1.41
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}

And the following are a few more examples of natural transformations:

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// ToFunc: Lazy<T> -> Func<T>
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public static Func<T> ToFunc<T>(this Lazy<T> lazy) => () => lazy.Value;
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// ToEnumerable: Func<T> -> IEnumerable<T>
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public static IEnumerable<T> ToEnumerable<T>(this Func<T> function)
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{
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    yield return function();
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}
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// ToEnumerable: Lazy<T> -> IEnumerable<T>
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public static IEnumerable<T> ToEnumerable<T>(this Lazy<T> lazy)
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{
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    yield return lazy.Value;
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}

Functor Category#

Now there are functors, and mappings between functors, which are natural transformations. Naturally, they lead to category of functors. Given 2 categories C and D, there is a functor category, denoted DC:

• Its objects ob(DC) are the functors from category C to D .
• Its morphisms hom(DC) are the natural transformations between those functors.
• The composition of natural transformations α: F ⇒ G and β: G ⇒ H, is natural transformations (β ∘ α): F ⇒ H.
• The identity natural transformation idF: F ⇒ F maps each functor to itself

Regarding the category laws:

• Associativity law: As fore mentioned, natural transformation’s components are morphisms in D, so natural transformation composition in DC can be viewed as morphism composition in D: (β ∘ α)X: F(X) → H(X) = (βX: G(X) → H(X)) ∘ (αX: F(X) → G(X)). Natural transformations’ composition in DC is associative, since all component morphisms’ composition in D is associative
• Identity law: similarly, identity natural transform’s components are the id morphisms idF(X): F(X) → F(X) in D. Identity natural transform satisfy identity law, since all its components satisfy identity law.

Here is an example of natural transformations composition:

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// ToFunc: Lazy<T> -> Func<T>
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public static Func<T> ToFunc<T>(this Lazy<T> lazy) => () => lazy.Value;
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#endif
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// ToOptional: Func<T> -> Optional<T>
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public static Optional<T> ToOptional<T>(this Func<T> function) =>
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    new Optional<T>(() => (true, function()));
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// ToOptional: Lazy<T> -> Optional<T>
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public static Optional<T> ToOptional<T>(this Lazy<T> lazy) =>
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    // new Func<Func<T>, Optional<T>>(ToOptional).o(new Func<Lazy<T>, Func<T>>(ToFunc))(lazy);
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    lazy.ToFunc().ToOptional();
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}