The hom-set of morphisms in the Kleisli category of from to is canonically equivalent (using excluded middle) to the set of partial functions from to ; its Eilenberg–Moore category is equivalent to the category of pointed sets and pointed function?s, i.e. functions of the form and commuting squares between such functions.
The successor monad as defined here is also interesting in that it stabilizes the finite von Neumann ordinals and monotone maps between them, and that and are also monotone. Thus the successor monad restricts as a monad to the augmented simplex category. Furthermore, every monotone map of finite ordinals can be written as a composite of arrows of the form and . Indeed, the monoidal category is generated by the monoid object .
JCMcKeown: I want to say something like generates the (skeletal augmented) simplex category; there is surely a right way to say that, but what is it?
Toby: I don't think that it's quite true that generates the simplex category, because you need an object to start applying it to. But I'd agree that generates it starting from . I don't know any slick way to say that.
I've put in a suggestion by Finn above. How's that?