Identity Monad (Id)

The Identity Monad, often referred to as Id, is the simplest possible monad. It represents a computation that doesn't add any additional context or effect beyond simply holding a value. It's a direct wrapper around a value.

While it might seem trivial on its own, the Identity Monad plays a crucial role in a higher-kinded type library for several reasons:

  1. Base Case for Monad Transformers: Many monad transformers (like StateT, ReaderT, MaybeT, etc.) can be specialized to their simpler, non-transformed monad counterparts by using Id as the underlying monad. For example:

    • StateT<S, Id.Witness, A> is conceptually equivalent to State<S, A>.
    • MaybeT<Id.Witness, A> is conceptually equivalent to Maybe<A>. This allows for a unified way to define transformers and derive base monads.

  2. Generic Programming: When writing functions that are generic over any Monad<F>, Id can serve as the "no-effect" monad, allowing you to use these generic functions with pure values without introducing unnecessary complexity.

  3. Understanding Monads: It provides a clear example of the monadic structure (of, flatMap, map) without any distracting side effects or additional computational context.

What is Id?

An Id<A> is simply a container that holds a value of type A.

  • Id.of(value) creates an Id instance holding value.
  • idInstance.value() retrieves the value from the Id instance.

Key Classes and Concepts

id_monad.svg

  • Id<A>: The data type itself. It's a final class that wraps a value of type A. It implements Kind<Id.Witness, A>.
  • Id.Witness: A static nested class (or interface) used as the first type parameter to Kind (i.e., F in Kind<F, A>) to represent the Id type constructor at the type level. This is part of the HKT emulation pattern.
  • IdKindHelper: A utility class providing static helper methods:
    • narrow(Kind<Id.Witness, A> kind): Safely casts a Kind back to a concrete Id<A>.
    • widen(Id<A> id): widens an Id<A> to Kind<Id.Witness, A>. (Often an identity cast since Id implements Kind).
    • narrows(Kind<Id.Witness, A> kind): A convenience to narrow and then get the value.
  • IdentityMonad: The singleton class that implements Monad<Id.Witness>, providing the monadic operations for Id.

Using Id and IdentityMonad

Example 1: Creating Id Instances

public void createExample(){
  // Direct creation
  Id<String> idString = Id.of("Hello, Identity!");
  Id<Integer> idInt = Id.of(123);
  Id<String> idNull = Id.of(null); // Id can wrap null

  // Accessing the value
  String value = idString.value(); // "Hello, Identity!"
  Integer intValue = idInt.value();   // 123
  String nullValue = idNull.value(); // null
}

Example 2: Using with IdentityMonad

The IdentityMonad provides the standard monadic operations.

public void monadExample(){
  IdentityMonad idMonad = IdentityMonad.instance();

  // 1. 'of' (lifting a value)
  Kind<Id.Witness, Integer> kindInt = idMonad.of(42);
  Id<Integer> idFromOf = ID.narrow(kindInt);
  System.out.println("From of: " + idFromOf.value()); // Output: From of: 42

  // 2. 'map' (applying a function to the wrapped value)
  Kind<Id.Witness, String> kindStringMapped = idMonad.map(
      i -> "Value is " + i,
      kindInt
  );
  Id<String> idMapped = ID.narrow(kindStringMapped);
  System.out.println("Mapped: " + idMapped.value()); // Output: Mapped: Value is 42

  // 3. 'flatMap' (applying a function that returns an Id)
  Kind<Id.Witness, String> kindStringFlatMapped = idMonad.flatMap(
      i -> Id.of("FlatMapped: " + (i * 2)), // Function returns Id<String>
      kindInt
  );
  Id<String> idFlatMapped = ID.narrow(kindStringFlatMapped);
  System.out.println("FlatMapped: " + idFlatMapped.value()); // Output: FlatMapped: 84

  // flatMap can also be called directly on Id if the function returns Id
  Id<String> directFlatMap = idFromOf.flatMap(i -> Id.of("Direct FlatMap: " + i));
  System.out.println(directFlatMap.value()); // Output: Direct FlatMap: 42

  // 4. 'ap' (applicative apply)
  Kind<Id.Witness, Function<Integer, String>> kindFunction = idMonad.of(i -> "Applied: " + i);
  Kind<Id.Witness, String> kindApplied = idMonad.ap(kindFunction, kindInt);
  Id<String> idApplied = ID.narrow(kindApplied);
  System.out.println("Applied: " + idApplied.value()); // Output: Applied: 42
}

Example 3: Using Id with Monad Transformers

As mentioned in the StateT Monad Transformer documentation, State<S,A> can be thought of as StateT<S, Id.Witness, A>.

Let's illustrate how you might define a State monad type alias or use StateT with IdentityMonad:

  public void transformerExample(){
  // Conceptually, State<S, A> is StateT<S, Id.Witness, A>
  // We can create a StateTMonad instance using IdentityMonad as the underlying monad.
  StateTMonad<Integer, Id.Witness> stateMonadOverId =
      StateTMonad.instance(IdentityMonad.instance());

  // Example: A "State" computation that increments the state and returns the old state
  Function<Integer, Kind<Id.Witness, StateTuple<Integer, Integer>>> runStateFn =
      currentState -> Id.of(StateTuple.of(currentState + 1, currentState));

  // Create the StateT (acting as State)
  Kind<StateTKind.Witness<Integer, Id.Witness>, Integer> incrementAndGet =
      StateTKindHelper.stateT(runStateFn, IdentityMonad.instance());

  // Run it
  Integer initialState = 10;
  Kind<Id.Witness, StateTuple<Integer, Integer>> resultIdTuple =
      StateTKindHelper.runStateT(incrementAndGet, initialState);

  // Unwrap the Id and then the StateTuple
  Id<StateTuple<Integer, Integer>> idTuple = ID.narrow(resultIdTuple);
  StateTuple<Integer, Integer> tuple = idTuple.value();

  System.out.println("Initial State: " + initialState);       // Output: Initial State: 10
  System.out.println("Returned Value (Old State): " + tuple.value()); // Output: Returned Value (Old State): 10
  System.out.println("Final State: " + tuple.state());         // Output: Final State: 11
}

This example shows that StateT with Id behaves just like a standard State monad, where the "effect" of the underlying monad is simply identity (no additional effect).