Let $F : C \to D, U : D \to C$ be an adjunction. We get a monad $T = FU : C \to C.$ This is something like a map $\{\text{adjunctions}\} \to \{\text{monads}\}$ (actually, this can be encoded by a certain $2$-functor). Conversely,  suppose we have an abstract monad $T : C \to C.$ One can construct an adjunction between $C$ and its so-called ‘Eilenberg-Moore category of algebras’ $C^T.$

Definition (Eilenberg-Moore Category).  Let $(T,\eta)$ be a monad on $C.$ Then, the category of T-algebras (or Eilenberg-Moore category$C^T$ has

• objects: an object $x \in C$ together with a morphism $h:Tx \to x$ in $C$ such that the diagramscommute.
• morphisms: a morphism $\varphi : (x,h) \to (x',h')$ of $T$-algebras is a $C$-morphism $f: x \to x'$ such that TFDC

Example. Consider the monad $(-)_\ast : Set \to Set$ which adjoins an element $\ast.$ Then, $C^T$ is the category of pointed sets.

The Eilenberg-Moore construction gives an adjunction. There is forgetful functor $U C^T \to C$ 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 $F : C \to C^T$ where

• $F(x) = \mu_x : T^2x \to Tx$
• $F(f : x \to x') = Tf : Tx \to Tx'$

Now, suppose our monad came from an adjunction between categories $C,D.$ There is a canonical comparison functor $k : D \to C^T.$ If $k$ is an equivalence of categories, the adjunction is said to be monadic. Beck’s monadicity theorem characterizes monadic adjunctions. In general, $k$ is not an equivalence of categories.

Counterexample ([1]). Consider the forgetful functor $U : Top \to Set.$ Its left adjoint is the discrete space functor. So, $Top$ cannot possibly be the Eilenberg-Moore category. Ultimately, this is related to the fact that $U$ does not reflect isomorphisms (a hypothesis of Beck’s monadicity theorem).

References