Idiom

Functional pearl 15
Use the MFunctor laws and the above definition of ι to prove the equation
fmap swap (ι x ⋆ u) = u ⋆ ι x
for the function
swap :: (a, b) → (b, a)
swap (a, b) = (b, a)
The proof makes essential use of higher-order functions.
9.1 Strong lax monoidal functors
In the first-order language of category theory, such data flow must be explicitly
plumbed using strong functors, i.e. functors F equipped with a tensorial strength
tAB : A × F B −→ F (A × B)
that makes the following diagrams commute.
1 × F A ∼ ⏐ = FA ⏐
t⏐
F (1 × A) ∼ = F A
(A × B) × F C ∼ = A × (B ⏐×
F C)
⏐
⏐
A × t
t
A × F (B × C)
⏐
⏐
⏐
⏐
⏐
⏐
t F ((A × B) × C) ∼ = F (A × (B × C))
The naturality axiom above then becomes strong naturality: the natural transformation
m corresponding to ‘⋆’ must also respect the strength:
(A × B) × (F C × F D) ∼ ⏐
= (A × F C) × ⏐ (B × F D)
⏐
⏐
(A × B) × m⏐
⏐
t × t
(A × B) × ⏐F
(C × D) F (A × C) × ⏐ F (B × D)
⏐
⏐
t⏐
⏐
m F ((A × B) × (C × D)) ∼ = F ((A × C) × (B × D))
Note that B and F C swap places in the above diagram: strong naturality implies
commutativity with pure computations.
Thus in categorical terms idioms are strong lax monoidal functors. Every strong
monad determines two such functors, as the definition is symmetrical.
10 Conclusions and further work
References
Baars, A.I., Löh, A., & Swierstra, S.D. (2004). Parsing permutation phrases. Journal of
functional programming, 14(6), 635–646.
Barr, Michael, & Wells, Charles. (1984). Toposes, triples and theories. Grundlehren der
Mathematischen Wissenschaften, no. 278. New York: Springer. Chap. 9.