Non-monadic adjunction

Question

Could someone give some examples of a non-monadic adjunctions please? Possibly explaining why they are not monadic and how they contradict the monadicity theorem? Thanks!

Answer 1

A classical example : the forgetful functor from topological spaces to sets.

The left adjoint is the "discrete space" functor (sending a set $X$ to the discrete space with underlying space $X$), and the composition just gives the identity on Sets, so clearly Top is not the Eilenberg-Moore category of the monad.

You can see that the forgetful functor does not reflect isomorphisms (a homeomorphism is more than a bijective continuous function), so the monadicity theorem indeed cannot be applied.

Answer 2

This might not be the answer you are looking for, but it could give you a little insight on what monadic really means.

Given a monad $(T,\eta,\mu)$ on a category $\mathcal C$, you can form the category $\operatorname{Adj}(\mathcal C,T)$ of adjunctions above $T$:

• its objects are the ajdunction $F: \mathcal C \rightleftarrows \mathcal D :U$ ($F$ on the left) such that the induced monad $(UF,{\rm unit},U\,{\rm counit}\,F)$ equals $(T,\eta,\mu)$ ;
• a morphism from $F: \mathcal C \rightleftarrows \mathcal D :U$ to $F': \mathcal C \rightleftarrows \mathcal D' :U'$ is a functor $K \colon \mathcal D \to \mathcal D'$ such that $KF = F'$, $U'K = U$ and $K$ commutes with counits.

This category $\operatorname{Adj}(\mathcal C,T)$ has a terminal object: the Eilenberg-Moore category of $T$. The unique morphism from any other object is the so called comparison functor. Hence, by definition, a adjunction $F: \mathcal C \rightleftarrows \mathcal D :U$ is monadic if and only if it is terminal in $\operatorname{Adj}(\mathcal C,UF)$.

It should make you realize that, in a sense, most adjunctions are non-monadic. For example, the previous category $\operatorname{Adj}(\mathcal C,T)$ always have an initial object: the Kleisli category of $T$. And this category is terminal only if every algebra over $T$ is free! Which is quite uncommon. It produces a lot of examples of non-monadic adjunctions $$Set \rightleftarrows \text{Free algrebas over T}$$ for $T$ the monad of groups, monoids, modules over a non-field ring, etc.

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Note that the initial and terminal object can be the same (as in the example of Captain Lama), but still leaving some "room" for non monadic adjunctions in between.