>> is a monad really just a monoid in the category of endofunctors?
> In category theory, a monad is a structure that is similar to a monoid in the category of endofunctors. A monoid is a mathematical object that consists of a set of elements, along with a binary operation that combines the elements in a specific way. For example, the set of integers, with the operation of addition, forms a monoid.
> In the category of endofunctors, a monad is a similar structure, but it is defined in terms of functions instead of elements. An endofunctor is a function that takes a mathematical object, called the "domain," and maps it to another mathematical object, called the "codomain." For example, the function that maps a set to its power set (the set of all subsets of the set) is an endofunctor.
> A monad in the category of endofunctors is a structure that consists of an endofunctor, along with two natural transformations that satisfy certain conditions. A natural transformation is a function that maps one endofunctor to another, in a way that is consistent with the structure of the domain and codomain of the endofunctors. The two natural transformations that are part of a monad are called the "unit" and the "multiplication" natural transformations.
> The unit natural transformation is a function that maps an object in the domain of the endofunctor to the endofunctor itself. The multiplication natural transformation is a function that maps the endofunctor to itself, in a way that is consistent with the structure of the domain and codomain of the endofunctor. The unit and multiplication natural transformations must satisfy certain conditions in order for the structure to be a monad.
> Overall, a monad in the category of endofunctors is a structure that is similar to a monoid in the category of sets. It consists of an endofunctor, along with two natural transformations that satisfy certain conditions. This structure is useful in category theory and related fields, because it allows us to abstract away from the specific details of the endofunctors and focus on the general properties of the structure.
> In category theory, a monad is a structure that is similar to a monoid in the category of endofunctors. A monoid is a mathematical object that consists of a set of elements, along with a binary operation that combines the elements in a specific way. For example, the set of integers, with the operation of addition, forms a monoid.
> In the category of endofunctors, a monad is a similar structure, but it is defined in terms of functions instead of elements. An endofunctor is a function that takes a mathematical object, called the "domain," and maps it to another mathematical object, called the "codomain." For example, the function that maps a set to its power set (the set of all subsets of the set) is an endofunctor.
> A monad in the category of endofunctors is a structure that consists of an endofunctor, along with two natural transformations that satisfy certain conditions. A natural transformation is a function that maps one endofunctor to another, in a way that is consistent with the structure of the domain and codomain of the endofunctors. The two natural transformations that are part of a monad are called the "unit" and the "multiplication" natural transformations.
> The unit natural transformation is a function that maps an object in the domain of the endofunctor to the endofunctor itself. The multiplication natural transformation is a function that maps the endofunctor to itself, in a way that is consistent with the structure of the domain and codomain of the endofunctor. The unit and multiplication natural transformations must satisfy certain conditions in order for the structure to be a monad.
> Overall, a monad in the category of endofunctors is a structure that is similar to a monoid in the category of sets. It consists of an endofunctor, along with two natural transformations that satisfy certain conditions. This structure is useful in category theory and related fields, because it allows us to abstract away from the specific details of the endofunctors and focus on the general properties of the structure.
pretty good explanation, actually