symmetric monoidal (∞,1)-category of spectra
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One can turn monads into adjunctions and adjunctions into monads (see there), 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.)
The relationship between monads and adjunctions itself constitutes an adjunction called the semantics-structure adjunction. Explicitly, for a category $C$ there exist contravariant functors $Str:Cat_{/C}^*\to Mon(C):Sem$ with $Str\dashv Sem$ where $Cat_{/C}^*$ denotes the full subcategory of $Cat_{/C}$ consisting of functors admitting a codensity monad; $Str$ sends a functor to its corresponding codensity monad and $Sem$ sends a monad to the forgetful functor from its E-M category to $C$. Intuitively speaking we may think of a monad as a kind of structure with which the objects of $\mathcal{C}$ can be equipped presented in a syntax-independent way, and we may think of the E-M category of a monad (viewed as a syntax independent presentation of an equational theory) as the category of models of this theory, which is often referred to by logicians as the semantics of the theory. For more on this, see for instance section 5 of (Schäppi 2009).
Discussion for quasi-categories is around definition 6.1.15 and definition 7.1.6 in
Emily Riehl, Dominic Verity, Homotopy coherent adjunctions and the formal theory of monads (arXiv:1310.8279)
Daniel Schäppi, Tannaka duality for comonoids in cosmoi (arXiv:0911.0977)
Alec Rhea (MO user page), Semantics-structure adjunction, URL (version: 2019-01-13): https://mathoverflow.net/q/320698
Last revised on August 10, 2023 at 15:15:38. See the history of this page for a list of all contributions to it.