# Reference request: the category of adjunctions between posets as categories that induce a partiuclar monad

I am interested in the category $A$ of adjunctions that induce a monad $c : C \to C$ where $C$ is a poset. (The description of $A$ is in a previous math.se post.) For a general $C$, of course, $A$ could be very complicated, but I would imagine that for a poset $C$ it is more feasible to give an explicit description of $A$.

Is there a good source from which I can learn about such a description of this category? Even if this is not possible for general posets, is it feasible for special posets (lattices, locales, etc)?

• Can you please give a precise definition of $A$? What are the objects, what are the morphisms? – HeinrichD Mar 30 '17 at 16:32
• I believe the intention is that $A$ is the category in which the Kleisli and Eilenberg-Moore splittings are initial and terminal, respectively? I actually can't think of any reference that deals with these categories in any detail. – Malice Vidrine Mar 30 '17 at 23:50
• @MaliceVidrine I meant that category. (Is the category that I described not that category?) – Pteromys Mar 31 '17 at 1:42
• @pteromys - It's the only such category I know, but I didn't want to put words in your mouth in case there was another I wasn't aware of :) – Malice Vidrine Mar 31 '17 at 4:02
• If $C$ and $D$ are posets, an adjunction between them is just a Galois connection between $C$ and $D^{op}$, and the monads are exactly the closure operators: ncatlab.org/nlab/show/Galois+connection, ncatlab.org/nlab/show/closure+operator – Berci Apr 1 '17 at 20:03