## Definition

We give the definitions in Cat and leave it to future readers and writers to generalise.

Let $\left(C,D,\ell ,r,\iota ,ϵ\right)$ be an adjunction in $\mathrm{Cat}$; that is, $\ell :C\to D$ and $r:D\to C$ are adjoint functors with $\ell ⊣r$, where $\iota$ and $ϵ$ are the unit and counit. Let $T$ be $r\circ \ell$; $T$ has the structure of a monad on $C$, so consider the Eilenberg–Moore category ${C}^{T}$ of modules (algebras) for $T$. Then $r\circ ϵ:T\circ r\to r$ endows $r:D\to C$ with a $T$-algebra structure, hence defines a functor $k:D\to {C}^{T}$.

The adjunction $\ell ⊣r$ is monadic if this functor $k$ is an equivalence of categories.

Beck’s Monadicity Theorem gives a necessary and sufficient condition for an adjunction to be monadic. Namely, the adjunction $\left(C,D,\ell ,r,\iota ,ϵ\right)$ is monadic iff:

• $r$ reflects isomorphisms; and

• $D$ has coequalizers of $r$-split coequalizer pairs, and $r$ preserves those coequalizers.

See monadicity theorem for more details and variants.

## Algebraic categories

The typical categories studied in algebra, such as Grp, Ring, etc, all come equipped with monadic adjunctions from Set. Specifically, the right adjoint is the forgetful functor from algebras to sets, and the left adjoint maps each set to the free algebra on that set. Their composite (a monad on $\mathrm{Set}$) may be thought of as mapping a set $A$ to the set of words with alphabet taken from $A$ and the connections between letters taken from the appropriate algebraic operations, with two words identified if they can be proved equal by the appropriate algebraic axioms.

Abstractly, one may define an algebraic category to be a category equipped with a monadic adjunction from $\mathrm{Set}$. However, there are now more examples than the ones from algebra; the best known of these is the category of compact Hausdorff spaces, which corresponds to the ultrafilter monad. (This result relies on the ultrafilter principle, regardless of whether one interprets ‘space’ here as referring to topological spaces or locales.)

## Discussion

I imagine something like this: there’s a functor (or 2-functor)

$\left[\mathrm{adjunctions}\right]\to \left[\mathrm{monads}\right]$[adjunctions] \to [monads]

$\mathrm{Kleisli}:\left[\mathrm{monads}\right]\to \left[\mathrm{adjunctions}\right]$Kleisli : [monads] \to [adjunctions]
$\mathrm{Eilenberg}\mathrm{Moore}:\left[\mathrm{monads}\right]\to \left[\mathrm{adjunctions}\right]$Eilenberg Moore: [monads] \to [adjunctions]

As evidence, note that the Kleisli category gives the initial object among adjunctions that give rise to a specified monad, while the Eilenberg-Moore category gives the terminal one. See the Wikipedia article.

Mike Shulman: This adjunction does exist (although you sometimes have to be careful about size issues) and is called the semantics-structure adjunction. The EM-category functor $\mathrm{Mnd}\left(C\right)\to \mathrm{RAdj}/C$ is called the “semantics” functor (here $\mathrm{RAdj}/C$ is the subcategory of $\mathrm{Cat}/C$ consisting of the right adjoints) and its left adjoint (the monad underlying an adjunction) is called the “structure” functor. In fact, the structure functor is defined on a larger subcategory of $\mathrm{Cat}/C$, namely those functors $g:A\to C$ such that ${\mathrm{Ran}}_{g}g$ exists (if $g$ has a left adjoint $f$ then ${\mathrm{Ran}}_{g}g$ always exists and is equal to $gf$). In this case ${\mathrm{Ran}}_{g}g$ is the image of $g$ under “structure”, also called its codensity monad. Presumably by duality, “structure” also has a left adjoint “cosemantics” given by the Kleisli construction. The semantics-structure adjunction can be found in Chapter II of Dubuc’s “Kan Extensions in Enriched Category Theory” and also in section 2 of “The Formal Theory of Monads”.