symmetric monoidal (∞,1)-category of spectra
One can turn monads into adjunctions and adjunctions into monads, but one doesn't always return where one started. Every monad comes from an adjunction, but only a monadic adjunction comes from a monad via a monadic functor. (To be fair, there are two ways to turn a monad into an adjunction, given by the Kleisli category and the Eilenberg–Moore category; we are talking about the latter here.)
We give the definitions in Cat and leave it to future readers and writers to generalise. See for instance (Riehl-Verity 13).
Let $(C,D,\ell,r,\iota,\epsilon)$ be an adjunction in $Cat$; that is, $\ell: C \to D$ and $r: D \to C$ are adjoint functors with $\ell \dashv r$, where $\iota$ and $\epsilon$ 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 $r \circ \epsilon: T \circ r \to r$ endows $r: D \to C$ with a $T$-algebra structure, hence defines a functor $k: D \to C^T$.
The adjunction $\ell \dashv r$ 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 $(C,D,\ell,r,\iota,\epsilon)$ 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.
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 $Set$) 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 $Set$. 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.)
Michael Barr and Charles Wells, Toposes, Triples and Theories (online)
Emily Riehl and Dominic Verity, Homotopy coherent adjunctions and the formal theory of monads ( arxiv’d )
John Baez: We read above: “One can turn monads into adjunctions and adjunctions into monads, but one doesn't always return where one started.” This suggests that there is something like an adjunction between monads and adjunctions! What’s the precise story?
I imagine something like this: there’s a functor (or 2-functor)
and this has left and right adjoints (or 2-adjoints)
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.
Zoran Škoda: the best-well-known article on monads, R. Street, Formal theory of monads, JPAA 2, 149–168, 1972, has the basic fact that Eilenberg-Moore construction as a correspondence from monads to categories, extends to a 2-functor which is right 2-adjoint to the trivial monad 2-functor and the left adjoint is the underlying 2-functor (forgetting actions). Now you are trying to view EM construction as an adjoint to adjunctions to monads correspondence. First of all, I could imagine several different 2-categories of adjunctions, hence several different questions in the game (and monads make 2-category in more than one way, but with lessvariety than adjunctions). But there are better people to ask about this…
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 $Mnd(C) \to RAdj/C$ is called the “semantics” functor (here $RAdj/C$ is the subcategory of $Cat/C$ 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 $Cat/C$, namely those functors $g:A\to C$ such that $Ran_g g$ exists (if $g$ has a left adjoint $f$ then $Ran_g g$ always exists and is equal to $g f$). In this case $Ran_g g$ 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”.
Emily Riehl: I’m interrupting Mike to comment on duality. The Kleisli construction is really a functor $Mnd(C)^{op} \to C/LAdj$. A monad map $(C,S) \to (C,T)$ consists of a 2-cell $T \Rightarrow S$ satisfying conditions. This defines a map of Kleisli categories $C_T \to C_S$ that commutes with the left adjoints but not with the right adjoints (that apply the monads to objects to get back to $C$).
I didn’t check all the details but it does seem to be the case that “structure” and “cosemantics” are mutual left adjoints, with adjunct maps defined in exactly the way you’d expect.
Mike Shulman: If anyone can give a nice conceptual explanation of the terms “semantics” and “structure” in this context, Daniel Schaeppi is currently writing a paper which could benefit from such an explanation. I’ve never found anyone who really explains the words (nor is it entirely clear who pioneered their use in this context—Dubuc perhaps?) It makes sense to me that the E-M category can be called the “semantics” of a monad, but my intuition for “structure” is fuzzier, except that of course any monad can be regarded as a notion of structure with which one can equip objects. But why is that “the” structure associated to an adjunction?
Mike Shulman: In case anyone is following this, Daniel has come up with what seems to be the right answer to this question; I’m hoping that eventually he will write about it here.
Discussion for quasi-categories is around definition 6.1.15 and definition 7.1.6 in