1. A Tale of Two Monads: Category-theoretic and
Computational viewpoints
Liang-Ting Chen

2. What is … a monad?
Functional Programmer:

3. What is … a monad?
Functional Programmer:
• a warm, fuzzy, little thing
Monica Monad, by FalconNL

4. 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 a
m >>= return = m
m >>= (x-> k x >>= h) =
(m >>= k) >>= h

5. •
•
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
}

6. Kleisli Triple
T : Obj(C) ! Obj(C)
⌘A : A ! T A
⇤
( ) : hom(A, T B) ! hom(T A, T B)
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
`M :
(return)
` [M ]T : T
` M : T⌧
x:⌧ `N :T
(bind)
` letT (x ( M ) in N : T
monad laws
⇤
⌘A
⇤
= id T A
⌘A ; f = f
⇤ ⇤
⇤
f ; g = (f ; g)

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

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

9. “A monad in X is just a monoid in the category
of endofunctors of X, what’s the problem?”
–James Iry, A Brief, Incomplete and Mostly Wrong History of
Programming Languages

10. “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.”
–Saunders Mac Lane, Categories for the Working Mathematician, p.138

11. monad on a category
What is … a monad?
Mathematician:
• a monoid in the category of
endofunctors
T: C !C
⌘ : I !T
˙
µ : T 2 !T
˙
monad laws
T
3
Tµ
/ T2
µ
µT
✏
T2
T
T⌘
µ
✏
/T
2
/T o
⌘T
µ
id
✏ ~
T
id
T

12. monad in a bicategory
• 0-cell a;
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
• 1-cell t : a ! a;
• 2-cell ⌘ : 1a ! t, and µ : tt ! t
monad laws
ttt
tµ
/ tt t
µ
µt
✏
tt
µ
✏
/t
t⌘
/ tt o
⌘t
µ
id
✏ 
t
id
t

13. 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

14. 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
/T
2
µ
µT
✏
T2
µ
✏
/T
T
T⌘
2
/T o
⌘T
µ
id
✏ ~
T
id
T

15. 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

16. 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)

17. Monads are derivable from algebraic
operations and equations if and only if they
have finite rank.
–G. M. Kelly and A. J. Power, Adjunctions whose counits are
coequalizers, and presentations of finitary enriched monads, 1993.

18. 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

19. 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

20. 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

21. 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

22. 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))

23. A Model of an Algebraic Theory (an instance)
• a set M with
• an n-ary function M for each operation
satisfying each equation
with ar( ) = n

24. A Monad with Finite Rank
MX =
[
{ M i[M S] | i : S ✓f X }
(M i : M S ! M X)

25. Examples of Algebraic Effects
•
maybe X 7! X + 1
•
exceptions X 7! X + E
•
nondeterminism X 7! Pfin (X)
•
side-effects X 7! (X ⇥ State)
State
but continuations is not algebraic X 7! R
(RX )

26. Algebraic Theory of Exception
• nullary operations raisee for each e 2 E
• no equations
A monadic program
f :: A -> B + E
corresponds to a homomorphism between free algebras

27. 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.

28. 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

29. 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