A Non-Monadic Adjunction
“One can turn monads into adjunctions and adjunctions into monads, but one doesn’t always return where one started.” – John Baez
Let be an adjunction. We get a monad This is something like a map (actually, this can be encoded by a certain -functor). Conversely, suppose we have an abstract monad One can construct an adjunction between and its so-called ‘Eilenberg-Moore category of algebras’
Definition (Eilenberg-Moore Category). Let be a monad on Then, the category of T-algebras (or Eilenberg-Moore category) has
- objects: an object together with a morphism in such that the diagramscommute.
- morphisms: a morphism of -algebras is a -morphism such that TFDC
Example. Consider the monad which adjoins an element Then, is the category of pointed sets.
The Eilenberg-Moore construction gives an adjunction. There is forgetful functor given on objects by projection onto the first factor and on morphisms by identity. It admits a left adjoint. This is the free T-algebra functor where
Now, suppose our monad came from an adjunction between categories There is a canonical comparison functor If is an equivalence of categories, the adjunction is said to be monadic. Beck’s monadicity theorem characterizes monadic adjunctions. In general, is not an equivalence of categories.
Counterexample (). Consider the forgetful functor Its left adjoint is the discrete space functor. So, cannot possibly be the Eilenberg-Moore category. Ultimately, this is related to the fact that does not reflect isomorphisms (a hypothesis of Beck’s monadicity theorem).
 MathStackExchange: https://math.stackexchange.com/questions/1752842/non-monadic-adjunction/1752856 April, 2016.
[2.] Awodey, S. Category Theory. Oxford University Press, 2006.