Hello every one, I'm reading "Programming with Categories Brendan Fong Bartosz Milewski David I. Spivak" and here there's an example of monoid in Haskell:
Consider the monoids Z× � (Z, 1, ×) and BAND � (B, true, AND). Let is_odd : Z → B be the function that sends odd numbers to true and even numbers to false. This is a monoid homomorphism. It preserves identities because 1 is odd, and it preserves composition because the product of any two odd numbers is odd, but the product of anything with an even number is even.
I'd like representing it graphically I mean the object Zx and arrows about is_odd function, but I do not know how can do it.
This is a valid example of monoid homomorphism and I'd like to know the Haskell implementation and the corresponding categorically view.
Thanks Co
Extending Li-yao's excellent answer, you could draw a single diagram using the free monoid monad. Disregarding non-termination and infinite lists, the free monoid is the list monad [*]. Every monoid has a structure map, namely mconcat :: Monoid m => [m] -> m satisfying equations (2) and (3) below. A function h :: z -> b between monoids is a monoid homomorphism iff it commutes with the structure map: h . mconcat = mconcat . fmap f (1) mconcat [Z] ---------> Z | | | fmap h | h | | v v [B] ---------> B mconcat This commutative square subsumes Li-yao's two squares. Instantiate the equation (1) with empty and two-element lists to recover Li-yao's squares, where mconcat [] = mempty, mconcat [x,y] = x `mappend` y. In "Programming with Categories" terminology: Monoid homomorphisms are []-Algebra homomorphisms (Definition 4.9). What is not contained in "Programming with Categories" are the Eilenberg-Moore algebra laws. Namely, mconcat . pure = id (2) mconcat . concat = mconcat . fmap mconcat (3) See also chapters in 6 and 7 in [1]. There, the example is linear functions between vector spaces, which are algebra homomorphisms of the monad of formal linear combinations. Olaf [*] Mathematically, the free monoid over a set A is the set of finite words in the alphabet A, with the empty word as 'mempty' and concatenation of words as 'mappend'. E.g. the free monoid over Char is Text. [1] https://hub.darcs.net/olf/haskell_for_mathematicians