A Non-Monadic Adjunction

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 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 categoryC^T has

  • objects: an object x \in C together with a morphism h:Tx \to x in C such that the diagramsScreen Shot 2017-05-20 at 12.31.17 PMcommute.
  • morphisms: a morphism \varphi : (x,h) \to (x',h') of T-algebras is a C-morphism f: x \to x' such that TFDC Screen Shot 2017-05-20 at 12.34.40 PM

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).


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

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



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