Now we have encountered comparison functors, we can introduce another classical topic in monad theory, that of monadicity. A functor $U : \mathcal{D} \rightarrow \mathcal{C}$ is monadic if it has a left adjoint, and the Eilenberg-Moore comparison functor for the induced monad is an equivalence of categories. Monadicity is a big topic, with a lot of new concepts and results to digest. For now, we shall restrict ourselves to some introductory examples. Along the way, we shall introduce a new class of monads, the idempotent monads.

Example: For any equational presentation $(\Sigma,E)$, the obvious forgetful functor $\mathsf{Alg(\Sigma,E)} \rightarrow \mathsf{Set}$ is monadic. This example motivates the term Eilenberg-Moore algebra. For example the category $\mathsf{Ab}$ of Abelian groups is monadic over $\mathsf{Set}$.

The previous example is important for intuitions about monadic functors. Informally, it is common to think of monadicity as evidence that a category is “algebraic” in nature. For monadic functors $\mathcal{D} \rightarrow \mathsf{Set}$, this idea can be made mathematically precise, which we may pursue in detail in a later post.

Example: A reflective subcategory is a full subcategory such that the inclusion functor has a left adjoint. It is useful to know a subcategory is reflective, for example completeness and cocompleteness properties can then be established from standard results.

Reflective subcategories correspond to a special class of the monads. Every reflective subcategory induces an idempotent monad, that is one who’s multiplication is an isomorphism. Furthermore, the Eilenberg-Moore category of an idempotent monad is equivalent to a reflective subcategory of the base category, so these concepts are tightly connected.

For example:

• A group is said to be torsion free if $x^n = 1$ implies $x = 1$. The category of torsion free Abelian groups is a reflective subcategory of the category of Abelian groups. The category $\mathsf{TFAb}$ of torsion free Abelian groups is a reflective subcategory of the category of Abelian groups $\mathsf{Ab}$.
• The category of Abelian groups is a reflective subcategory of the category of groups.
• The category of symmetric graphs is a reflective subcategory of the category of graphs.

The corresponding monads are all idempotent.

The notion of idempotent monad has many equivalent characterisations, which we may discuss in a later post.

Of course, not every functor with a left adjoint is monadic, so it would be useful to see some counterexamples.

Counterexample: Let $\mathsf{Pre}$ be the category of preorders and monotone maps. There is an obvious forgetful functor $U: \mathsf{Pre} \rightarrow \mathsf{Set}$, and this has a left adjoint such that $F(X)$ has underlying set $X$, and for all $x_1, x_2 \in X$ $x_1 \leq x_2 \quad\Leftrightarrow\quad x_1 = x_2$

This is sometimes referred to as the discrete preorder. The composite $U \circ F$ is the identity monad. $\mathsf{Set}^{\mathsf{Id}}$ is equivalence to $\mathsf{Set}$, which is certainly not equivalent to $\mathsf{Pre}$.

The following is a well-known counterexample, refuting a natural conjecture.

Counterexample: Monadic functors are not closed under composition. As we have seen, the there are monadic functors $\mathsf{TFAb} \rightarrow \mathsf{Ab}$ and $\mathsf{Ab} \rightarrow \mathsf{Set}$. The composite functor $U : \mathsf{TFAb} \rightarrow \mathsf{Ab} \rightarrow \mathsf{Set}$

has a left adjoint given by composing the two component left adjoints, but is not monadic. To see this, consider the action of the left adjoint. The groups in the image of the adjoint $\mathsf{Set} \rightarrow \mathsf{Ab}$ are all torsion free. Therefore the functor $\mathsf{Ab} \rightarrow \mathsf{TFGrp}$ leaves them unchanged up to isomorphism. Therefore the composite functor $F : \mathsf{Set} \rightarrow \mathsf{Ab} \rightarrow \mathsf{TFAb}$

maps a set to the free Abelian group over that set. The algebras of the induced monad are simply the Abelian groups, and so the composite forgetful functor is not monadic.

## Why would we care if a functor is monadic?

There are a lot of standard theorems allowing is to derive nice properties of Eilenberg-Moore categories from properties of the monad and its base category. For example, we might be able to establish the presence of certain limits or colimits, or that the category is regular or locally finite presentable. In some concrete situations, it might be easier to establish these properties via direct calculation, but monadicity results allow us to work axiomatically, simultaneously deriving results that apply in many situations.

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