## 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 ([1]). **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).

**References**

[1] MathStackExchange: https://math.stackexchange.com/questions/1752842/non-monadic-adjunction/1752856 April, 2016.

[2.] Awodey, S. *Category Theory. *Oxford University Press, 2006.