The forgetful functor $U : \mathsf{Ab} \to \mathsf{Set}$ is monadic, this follows from Beck’s monadicity theorem or some other general result. Anyway, I would like to prove this directly, thereby solving an exercise in Mac Lane’s Categories for the working mathematician. This amounts to prove an equivalent definition of an abelian group, using $\mathbb{Z}$-linear combinations […]

After answering this question here about Kleisli triples, I realized that this whole Kleisli triple construction: $T:{\rm Ob}\mathcal C\to{\rm Ob}\mathcal C$, $\ \eta_A:A\to TA$ for all $A\in {\rm Ob}\mathcal C$ and $f\mapsto f^* $ for all $f:A\to B^T$ such that $\eta_A^*=1_{TA}$ $\eta_Af^*=f\ $ for all $\ f:A\to TB\ $ (writing composition from left to right) […]

I’ve been trying to figure out what having a monad in a monoid (i.e. a category with one object) would mean. As far as I can tell it would be a homomorphism (functor) $T : M → M$, with two elements (natural transformation components) $\eta, \mu : M$, such that $\forall x. \eta x = […]

I have been learning some functional programming recently and I so I have come across monads. I understand what they are in programming terms, but I would like to understand what they are mathematically. Can anyone explain what a monad is using as little category theory as possible?

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