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A monad is an endofunctor $T:C\rightarrow C$ with natural transformations $\eta:id_C\rightarrow T$ and $\mu:T^2\rightarrow T$.

Being natural transformations mean that $$T(f)\circ \eta_A = \eta_B\circ id_C(f)$$ should hold and $$T(f)\circ\mu_A = \mu_B\circ T^2(f)$$ also. These are the algebraic expressions that the defining diagrams of natural transformations commute.

Translating this to the language of functional programming, $\eta$ is unit, and a polymorphic one, so $\eta_{nat}$ is the monomorphisation of unit to the nat type. $\mu$ is join in the same sense. The functor $T$ consists of two things: it takes object and morphisms. The latter part is called map in functional programming. So the above equations translate to

map f o unit = unit o f

where, if f is A -> B then this is

map f o unit(A) = unit(B) o f

but that is usually handled with a polymorphic type system and the second equation is

map f o join = join o map (map f)

So, map is the morphism-translating part of the endofunctor $T$.

But what is the object-translating part of $T$ in functional programming? Is it the type operator M when we write unit: 'a -> 'a M ?

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    $\begingroup$ Yes, the action on objects of a monad M is precisely M itself, mapping A to M A. $\endgroup$
    – varkor
    Sep 14, 2023 at 18:42

1 Answer 1

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Objects are isomorphic to identity morphisms. So, the object-translating portion of the endofunctor is a subportion of the morphism-translating portion, also carried by map.

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