Indeed, this is a bit confusing. In the lecture, we talked about the list functor as ListF a x = NilF | ConsF a x. It is actually a bifunctor, so its fixed point is a functor in a. The starting point of the exercise can be written as List a = Nil | …
The answer is that, in general, we don't know. If there is a unique fixed point then it must be both the initial algebra and the terminal coalgebra. Infinite streams are an example of a terminal coalgebra for the product functor Pair a x, but an ini…
This is what David Spivak told me: "In Set, an algebra for the list monad is a set X together with a function mult : List X -> X satisfying unitality and associativity. There is an equivalence between the category of monoids and the category of L…
Puzzle 168: For a programmer the most important adjunction is the one between product and function type. We call it currying. It's an adjunction between two endofunctors: \(F\) takes takes the set \(a\) and "multiplies" it by some fixed set \(c\) (t…
Keith: I presume \(f\) and \(g\) are morphisms, but what is \(x\)? The notation \(f(g(x))\) suggests functions, where \(x\) is an element of a set; and if you define composition of functions this way, it's automatically associative.
@Reuben : I wrote a blog post about Eilenberg-Moore categories and T-algebras, which might be more approachable. https://bartoszmilewski.com/2017/03/14/algebras-for-monads/