fyi: "coalgebras"
Jun. 5th, 2020 08:46 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
juan_gandhiSay, you have a monad M, there's a category CM of M-algebras, M[X] => X, with proper properties, and a category of M-coalbegras, X => M[X], with proper properties.
What are these coalgebras, X => M[X]? They are just objects of Kleisli category CM for the monad M. Definition? Same objects, but arrows of the form X => M[X].
What are these coalgebras, X => M[X]? They are just objects of Kleisli category CM for the monad M. Definition? Same objects, but arrows of the form X => M[X].
That's it. So, a terminal ("final", as Kiselyov says) coalgebra is just a terminal object in CM.
E.g. for the functor 1+_ (which is a monad) in Sets, the Kleisli category consists of the same objects and partial functions; and, btw, the terminal object ("final coalgebra") is (see https://en.wikipedia.org/wiki/Initial_algebra#Final_coalgebra) ℕ∪{ω} with "pred" as the partial endomorphism.
Do you need details?
