Am 16.07.2018 um 21:48 schrieb Olaf Klinke:
Joachim Durchholz wrote:
Nope, monoid is a special case of monad (the case where all input and output types are the same). (BTW monoid is associativity + neutral element. Not 100% sure whether monad's "return" qualifies as a neutral element, and my monoid-equals-monotyped-monad claim above may fall down if it is not.
For every monad M, the type M () is a monoid,
That doesn't make M a monoid, just its type. Besides, type-level monoids aren't relevant to data-level properties.
I'm sorry, I misunderstood your question, then. Of course monads are not monoids. You could say that every monad harbors a monoid, and that every monoid arises this way. But of course the whole and the part are not the same.
Does every monoid arise this way? Yes: Since base-4.9 there is the monad instance Monoid a => Monad ((,) a) and of course a and (a,()) are isomorphic (disregarding bottoms).
An isomorphism does not make it the same, just easily mappable.
This is pretty near to hair-splitting, I know. However, from a software design perspective, it's important to keep things separate even if they are easily mappable: temperatures are just floats, it's an incredibly easy mapping - but the code becomes clearer and more maintainable if you keep the Temperature type separate from the Float type. You may even want to forbid multiplying Temperatures (but you want the ability to multiply with a number, and temperature quotients can be useful, too).
I totally agree. As a domain theorist I am aware that a and (a,()) are not isomorphic because neither is () the terminal object of Hask nor is (,) the categorical product. But in terms of total elements, there is a monoid isomorphism between the two.
Not 100% sure how much of this is really applicable.
Also, there is the famous tongue-in-cheek saying "monads are just monoids in the category of endofunctors"
Tongue-in-check does not make it real. And being a monoid in a category does not make it a monoid directly.
That depends on the definition of "monoid". If a monoid is a set with certain structure, then a monad is not a monoid. If a monoid is a monoid object in some monoidal category, then a monad is a monoid. Maybe we can agree on this distinction? monoid = monoid object in Set monoid object = any monoid object in monoidal closed category.
There's also a final argument: If monad and monoid are really the same, why do mathematicians still keep the separate terminology?
Because they are different, as we hopefully agree. Olaf