Core Method: map(Function<A,B> f, Kind<F,A> fa) -> Kind<F,B>

Purpose: Transform values inside a context without changing the context structure

Use When:

• You have a simple transformation function A -> B
• The context/container should remain unchanged
• No dependency between input and output contexts

Laws:

• Identity: map(identity) == identity
• Composition: map(g ∘ f) == map(g) ∘ map(f)

Common Instances: List, Optional, Maybe, Either, IO, CompletableFuture

Example:

// Transform string to length, preserving Optional context
Kind<OptionalKind.Witness, Integer> lengths = 
    optionalFunctor.map(String::length, optionalString);

Think Of It As: Applying a function "inside the box" without opening it

Core Methods:

• of(A value) -> Kind<F,A> (lift pure value)
• ap(Kind<F,Function<A,B>> ff, Kind<F,A> fa) -> Kind<F,B> (apply wrapped function)

Purpose: Combine independent computations within a context

Use When:

• You need to combine multiple wrapped values
• Operations are independent (don't depend on each other's results)
• You want to accumulate errors from multiple validations

Key Insight: map2, map3, etc. are built on ap for combining 2, 3, or more values

Laws: Identity, Composition, Homomorphism, Interchange

Common Patterns:

• Form validation (collect all errors)
• Combining configuration values
• Parallel computations

Example:

// Combine two independent validations
Kind<ValidatedKind.Witness<List<String>>, User> user = 
    applicative.map2(
        validateUsername(input.username()),
        validatePassword(input.password()),
        User::new
    );

Think Of It As: Combining multiple "boxes" when contents are independent

Core Method: flatMap(Function<A,Kind<F,B>> f, Kind<F,A> fa) -> Kind<F,B>

Purpose: Sequence dependent computations within a context

Use When:

• Each step depends on the result of the previous step
• You need to chain operations that return wrapped values
• You want short-circuiting behaviour on failure

Key Difference from Applicative: Operations are sequential and dependent

Laws:

• Left Identity: flatMap(f, of(a)) == f(a)
• Right Identity: flatMap(of, m) == m
• Associativity: flatMap(g, flatMap(f, m)) == flatMap(x -> flatMap(g, f(x)), m)

Utility Methods:

• as(B value, Kind<F,A> fa) - replace value, keep effect
• peek(Consumer<A> action, Kind<F,A> fa) - side effect without changing value

Example:

// Chain database operations where each depends on the previous
Kind<OptionalKind.Witness, Account> account = 
    monad.flatMap(user -> 
        monad.flatMap(profile -> 
            findAccount(profile.accountId()),
            findProfile(user.id())),
        findUser(userId));

Think Of It As: Chaining operations where each "opens the box" and "puts result in new box"

Core Methods:

• raiseError(E error) -> Kind<F,A> (create error state)
• handleErrorWith(Kind<F,A> fa, Function<E,Kind<F,A>> handler) -> Kind<F,A> (recover from error)

Purpose: Add explicit error handling to monadic computations

Use When:

• You need to handle specific error types
• You want to recover from failures in a workflow
• You need to distinguish between different kinds of failures

Key Insight: Error type E is fixed for each MonadError instance

Common Error Types:

• Throwable for CompletableFuture
• Unit for Optional/Maybe (absence as error)
• Custom domain error types for Either/Validated

Recovery Methods:

• handleError(fa, Function<E,A> handler) - recover to pure value
• recover(fa, A defaultValue) - provide default value

Example:

// Handle division by zero gracefully
Kind<EitherKind.Witness<String>, Double> result = 
    monadError.handleErrorWith(
        divideOperation,
        error -> monadError.of(0.0) // recover with default
    );

Think Of It As: try-catch for functional programming

Core Method: combine(A a1, A a2) -> A

Purpose: Types that can be combined associatively

Key Property: Associativity - combine(a, combine(b, c)) == combine(combine(a, b), c)

Use When:

• You need to combine/merge two values of the same type
• Order of combination doesn't matter (due to associativity)
• Building blocks for parallel processing

Common Instances:

• String concatenation: "a" + "b" + "c"
• Integer addition: 1 + 2 + 3
• List concatenation: [1,2] + [3,4] + [5,6]
• Set union: {1,2} ∪ {2,3} ∪ {3,4}

Example:

// Combine error messages
Semigroup<String> stringConcat = Semigroups.string("; ");
String combined = stringConcat.combine("Error 1", "Error 2");
// Result: "Error 1; Error 2"

Think Of It As: The + operator generalised to any type

Core Methods:

• combine(A a1, A a2) -> A (from Semigroup)
• empty() -> A (identity element)

Purpose: Semigroups with an identity/neutral element

Key Properties:

• Associativity (from Semigroup)
• Identity: combine(a, empty()) == combine(empty(), a) == a

Use When:

• You need a starting value for reductions/folds
• Implementing fold operations over data structures
• You might be combining zero elements

Common Instances:

• String: empty = "", combine = concatenation
• Integer addition: empty = 0, combine = +
• Integer multiplication: empty = 1, combine = *
• List: empty = [], combine = concatenation
• Boolean AND: empty = true, combine = &&

Example:

// Sum a list using integer addition monoid
Integer sum = listFoldable.foldMap(
    Monoids.integerAddition(),
    Function.identity(),
    numbersList
);

Think Of It As: Semigroup + a "starting point" for combinations

Core Method: foldMap(Monoid<M> monoid, Function<A,M> f, Kind<F,A> fa) -> M

Purpose: Reduce a data structure to a single summary value

Use When:

• You want to aggregate/summarise data in a structure
• You need different types of reductions (sum, concat, any/all, etc.)
• You want to count, find totals, or collapse collections

Key Insight: Different Monoids give different aggregations from same data

Common Operations:

• Sum numbers: use integer addition monoid
• Concatenate strings: use string monoid
• Check all conditions: use boolean AND monoid
• Count elements: map to 1, use integer addition monoid

Example:

// Multiple aggregations of the same list
List<Integer> numbers = List.of(1, 2, 3, 4, 5);

// Sum
Integer sum = foldable.foldMap(Monoids.integerAddition(), 
    Function.identity(), numbers); // 15

// Concatenate as strings  
String concat = foldable.foldMap(Monoids.string(), 
    String::valueOf, numbers); // "12345"

// Check all positive
Boolean allPositive = foldable.foldMap(Monoids.booleanAnd(), 
    n -> n > 0, numbers); // true

Think Of It As: Swiss Army knife for data aggregation

Core Method: traverse(Applicative<G> app, Function<A,Kind<G,B>> f, Kind<F,A> fa) -> Kind<G,Kind<F,B>>

Purpose: Apply an effectful function to each element and "flip" the contexts

Use When:

• You have a collection and want to apply an effect to each element
• You want to validate every item and collect all errors
• You need to "turn inside-out": F<G<A>> becomes G<F<A>>

Key Operations:

• traverse: apply function then flip
• sequence: just flip contexts (when you already have F<G<A>>)

Common Patterns:

• Validate every item in a list
• Make async calls for each element
• Parse/process each item, collecting all failures

Example:

// Validate every string in a list, collect all errors
List<String> inputs = List.of("123", "abc", "456");

Kind<ValidatedKind.Witness<List<String>>, Kind<ListKind.Witness, Integer>> result =
    listTraverse.traverse(
        validatedApplicative,
        this::parseInteger, // String -> Validated<List<String>, Integer>
        LIST.widen(inputs)
    );

// Result: either Valid(List[123, 456]) or Invalid(["abc is not a number"])

Think Of It As: Applying effects to collections while flipping the "nesting order"

Core Methods:

• lmap(Function<C,A> f, Kind2<P,A,B> pab) -> Kind2<P,C,B> (contravariant on input)
• rmap(Function<B,D> g, Kind2<P,A,B> pab) -> Kind2<P,A,D> (covariant on output)
• dimap(Function<C,A> f, Function<B,D> g, Kind2<P,A,B> pab) -> Kind2<P,C,D> (both)

Purpose: Adapt inputs and outputs of two-parameter types (especially functions)

Use When:

• Building flexible data transformation pipelines
• Creating API adapters that convert between different formats
• You need to preprocess inputs or postprocess outputs
• Building reusable validation or transformation logic

Key Insight:

• lmap = preprocess the input (contravariant)
• rmap = postprocess the output (covariant)
• dimap = do both transformations

Common Instance: Function<A,B> is the canonical Profunctor

Example:

// Adapt a string length function to work with integers and return formatted strings
Function<String, Integer> stringLength = String::length;

// Input adaptation: Integer -> String  
Kind2<FunctionKind.Witness, Integer, Integer> intToLength = 
    profunctor.lmap(Object::toString, FUNCTION.widen(stringLength));

// Output adaptation: Integer -> String
Kind2<FunctionKind.Witness, String, String> lengthToString = 
    profunctor.rmap(len -> "Length: " + len, FUNCTION.widen(stringLength));

// Both adaptations
Kind2<FunctionKind.Witness, Integer, String> fullAdaptation = 
    profunctor.dimap(Object::toString, len -> "Result: " + len, 
        FUNCTION.widen(stringLength));

Think Of It As: The adapter pattern for functional programming