A Tale of Two Monads: Category-theoretic and Computational viewpoints

Liang-Ting Chen

What is … a monad?

Functional Programmer:

What is … a monad?

Functional Programmer:• a warm, fuzzy, little thing

Monica Monad, by FalconNL

What is … a monad?

Functional Programmer:• a warm, fuzzy, little thing• return and bind with monad

laws

class Monad m where (>>=) ::m a->(a -> m b)->m b return::a ->m a!-- monad laws return a >>= k = k am >>= return = mm >>= (\x-> k x >>= h) = (m >>= k) >>= h

What is … a monad?

Functional Programmer:• a warm, fuzzy, little thing• return and bind with monad

laws• a programmable semicolon

• do { x' <- return x; f x’} ≡ do { f x }

• do { x <- m; return x } ≡ do { m }

• do { y <- do { x <- m; f x }

g y }

≡ do { x <- m; do { y <- f

x; g y }}

≡ do { x <- m; y <- f x; g y

}

What is … a monad?

Functional Programmer:• a warm, fuzzy, little thing• return and bind with monad

laws• a programmable semicolon• E. Moggi, “Notions of

computation and monads”, 1991

T : Obj(C) ! Obj(C)

⌘A : A ! TA

(�)⇤ : hom(A, TB) ! hom(TA, TB)

⌘⇤A = idTA

⌘A; f⇤ = f

f ⇤; g⇤ = (f ; g)⇤

Kleisli Triple

monad laws

� ` M : �

� ` [M ]T : T�(return)

� ` M : T ⌧ x : ⌧ ` N : T�

� ` letT (x ( M) inN : T�(bind)

“Hey, mathematician! What is a monad?”,you asked.

“A monad in X is just a monoid in the category of endofunctors of X, what’s the problem?”

–Philip Wadler

–James Iry, A Brief, Incomplete and Mostly Wrong History of Programming Languages

“A monad in X is just a monoid in the category of endofunctors of X, what’s the problem?”

–Saunders Mac Lane, Categories for the Working Mathematician, p.138

“A monad in X is just a monoid in the category of endofunctors of X, with product × replaced

by composition of endofunctors and unit set by the identity endofunctor.”

What is … a monad?

Mathematician:• a monoid in the category of

endofunctors T 3 Tµ//

µT✏✏

T 2

µ✏✏

T 2µ// T

TT⌘//

id

T 2

µ

✏✏

T⌘Too

id~~

T

monad laws

monad on a categoryT : C ! C

⌘ : I ˙�!T

µ : T 2 ˙�!T

What is … a monad?

Mathematician:• a monoid in the category of

endofunctors• a monoid in the endomorphism

category K(a,a) of a bicategory K

• 0-cell a;

• 1-cell t : a ! a;

• 2-cell ⌘ : 1a ! t, and µ : tt ! t

ttttµ//

µt✏✏

tt

µ✏✏

tt µ// t

tt⌘//

id��

tt

µ

✏✏

t⌘too

id��

t

monad in a bicategory

monad laws

What is … a monad?

Mathematician:• a monoid in the category of

endofunctors• a monoid in the endomorphism

category K(a,a) of a bicategory K• …

from Su Horng’s slide

Monads in Haskell, the Abstract Ones

• class Functor m => Monad m where unit :: a -> m a -- η join :: m (m a) -> m a -- μ

• --join . (fmap join) = join . join --join . (fmap unit) = join . unit = id

T 3 Tµ//

µT✏✏

T 2

µ✏✏

T 2µ// T

TT⌘//

id

T 2

µ

✏✏

T⌘Too

id~~

T

Kleisli Triples and Monads are Equivalent (Manes 1976)

• fmap :: Monad m => (a -> b) -> m a -> m b fmap f x = x >>= return . f join :: Monad m => m (m a) -> m a join x = x >>= id -- id :: m a -> m a

Kleisli Triples and Monads are Equivalent (Manes 1976)

• fmap :: Monad m => (a -> b) -> m a -> m b fmap f x = x >>= return . f join :: Monad m => m (m a) -> m a join x = x >>= id -- id :: m a -> m a

• (>>=) :: Monad m => m a -> (a -> m b) -> m b x >>= f = join (fmap f x) -- fmap f :: m a -> m (m b)

–G. M. Kelly and A. J. Power, Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads, 1993.

Monads are derivable from algebraic operations and equations if and only if they

have finite rank.

An Algebraic Theory: Monoid

• a set M with

• a nullary operation ✏ : 1 ! M

• a binary operation • : M ⇥M ! M

satisfying

• associativity: (a • b) • c = a • (b • c)

• identity: a • ✏ = ✏ • a = a

Monoids in Haskell:

class Monoid a where mempty :: a -- ^ Identity of 'mappend' mappend :: a -> a -> a -- ^ An associative operation!instance Monoid [a] where mempty = [] mappend = (++)!instance Monoid b => Monoid (a -> b) where mempty _ = mempty mappend f g x = f x `mappend` g x

An Algebraic Theory: Semi-lattice

• a set L with

• a binary operation _ : M ⇥M ! M

satisfying

• commutativity: a _ b = b _ a

• associativity: a _ (b _ c) = (a _ b) _ c

• idenpotency: a _ a = a

Semi-lattices in Haskell

class SemiLattice a where join :: a -> a -> a!instance SemiLattice Bool where join = (||)!instance SemiLattice v => SemiLattice (k -> v) where f `join` g = \x -> f x `join` g x!instance SemiLattice IntSet where join = union

An Algebraic Theory (defined as a type class in Haskell)

• a set of operations � 2 ⌃ and ar(�) 2 N

• a set of equations with variables, e.g. �1(�1(x, y), z) = �1(x, �1(y, z))

A Model of an Algebraic Theory (an instance)

• a set M with

• an n-ary function �M for each operation � with ar(�) = nsatisfying each equation

A Monad with Finite Rank

MX =[

{Mi[MS] | i : S ✓f X }

(Mi : MS ! MX)

• maybe

• exceptions

• nondeterminism

• side-effects

but continuations is not algebraic

Examples of Algebraic Effects

X 7! X + E

X 7! (X ⇥ State)State

X 7! Pfin(X)

X 7! R(RX)

X 7! X + 1

Algebraic Theory of Exception

A monadic program

f :: A -> B + E

corresponds to a homomorphism between free algebras

• nullary operations raisee for each e 2 E

• no equations

Why Algebraic Effects?

• Various ways of combination, e.g. sum, product, distribution, etc.

• Equational reasoning of monadic programming is simpler.

• A classification of effects: a deeper insight.

Conclusion

• Moggi’s formulation solves fundamental problems, e.g. a unified approach to I/O.

• Mathematicians bring new ideas to functional programming, e.g. algebraic effects, modular construction of effects

• Still an ongoing area

Conclusion

• Moggi’s formulation solves fundamental problems, e.g. a unified approach to I/O.

• Mathematicians bring new ideas to functional programming, e.g. algebraic effects, modular construction of effects

• Still an ongoing area