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copied!<p>It seems to me that you omitted important things:</p> <ul> <li>There is the word “monoid” in the definition. You did not consider it in your post.</li> <li>It is better to replace it with “monoid object” in order to be precise. Monoids live in abstract algebra, monoid objects live in category theory. Different species. <ul> <li>Monoids are monoid objects in some category, but this is not relevant here.</li> </ul></li> <li>A monoid object may be defined in a monoidal category only.</li> </ul> <p>So, a path to understanding the definition is as follows.</p> <ul> <li>You consider a category of endofunctors (lets call it <strong>E</strong>) on the category of Haskell types and functions <strong>Hask</strong>. Its objects are functors on <strong>Hask</strong>, a morphism in it from F to G is a polymorphic function of type F a -> G a with some property.</li> <li>You consider a structure on <strong>E</strong> which turns it into a monoidal category, i.e. composition of functors and the identity functors.</li> <li>You consider definition of a monoid object, e.g. from Wikipedia.</li> <li>You consider what it means in your specific category <strong>E</strong>. It means that M is an endofunctor, and μ has the same type as “join”, and η has the same type as “return”.</li> <li>“(>>=)” is defined via “join”.</li> </ul> <p>An advice. Do not try to express everything in Haskell. It is designed for writing programs, not mathematics. Mathematical notation is more convenient here because you can write composition of functors as “∘” without going into trouble with type checker.</p>

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