symmetric monoidal (∞,1)-category of spectra
An idempotent monad is a monad that “squares to the identity” in the evident category-theoretic sense. Idempotent monads hence serve as categorified projection operators, in that they encode reflective subcategories and the reflection/localization onto these.
In terms of type theory idempotent monads interpret (co-)modal operators.
An idempotent monad is a monad $(T,\mu,\eta)$ on a category $C$ such that one (hence all) of the following equivalent statements are true:
$\mu\colon T T \to T$ is a natural isomorphism.
All components of $\mu\colon T T \to T$ are monomorphisms.
The maps $T\eta, \eta T\colon T \to T T$ are equal.
For every $T$-algebra ($T$-module) $(M,u)$, the corresponding $T$-action $u\colon T M \to M$ is an isomorphism.
The forgetful functor $C^T \to C$ (where $C^T$ is the Eilenberg-Moore category of $T$-algebras) is a full and faithful functor.
There exists a pair of adjoint functors $F\dashv U$ such that the induced monad $(UF, U\epsilon F)$ is isomorphic to $(T,\mu)$ and $U$ is a full and faithful functor.
e.g. (Borceux, prop. 4.3.2).
$1\Rightarrow 2$ is trivial.
$2\Rightarrow 3$ Compositions $\mu\circ T\eta$ and $\mu\circ\eta T$ are always the identity (unit axioms for the monad), and in particular agree; if $\mu$ has all components monic, this implies $T\eta = \eta T$.
$3\Rightarrow 4$ Compatibility of action and unit is $u \circ \eta_M = id_M$, hence also $T(u)\circ T(\eta_M) = id_{T M}$. If $T\eta = \eta T$ then this implies $id_{T M} = T(u)\circ \eta_{T M} = \eta_M\circ u$, where the naturality of $\eta$ is used in the second equality. Therefore we exhibited $\eta_M$ both as a left and a right inverse of $u$.
$4\Rightarrow 1$ If every action is iso, then the components of multiplication $\mu_M\colon T T M\to T M$ are isos as a special case, namely of the free action on $T M$.
$4\Rightarrow 5$ For any monad $T$, the forgetful functor from Eilenberg-Moore category $C^T$ to $C$ is faithful: a morphism of $T$-algebras is always a morphism of underlying objects in $C$. To show that it is also full, we consider any pair $(M,u)$, $(M',u')$ in $C^T$ and must show that any $f\colon M\to M'$ is actually a map $f\colon (M,u)\to (M',u')$; i.e. $u'\circ T f = f\circ u$. But we know that $\eta_M, \eta_{M'}$ are inverses of $u,u'$ respectively and the naturality for $\eta$ says $\eta_{M'}\circ f = T f \circ \eta_M$. Compose that equation with $u$ on the right and $u'$ on the left with the result (notice that we used just the invertibility of $u$).
$5\Rightarrow 6$ Trivial, because the Eilenberg-Moore construction induces the original monad by the standard recipe.
$6\Rightarrow 3$ By $6$ the counit $\epsilon$ is iso, hence $U\epsilon F$ has a unique 2-sided inverse; by triangle identities, $T\eta$ and $\eta T$ are both right inverses of $U\epsilon F$, hence 2-sided inverses, hence they are equal.
$6\Rightarrow 1$ If $F\dashv U$ is an adjunction with $U$ fully faithful, then the counit $\epsilon$ is iso. Since $D(FU X,Y)\simeq C(UX,UY)\simeq D(X,Y)$ where the last equivalence holds since $U$ is full and faithful; hence by essential unicity of the representing object there is an iso $FUX\stackrel{\sim}{\to} X$ ; let $X=Y$ then the adjoint of this identity is the counit of the adjunction; since the hom objects correspond bijectively, the counit is an isomorphism. Hence the multiplication of the induced monad $\mu = U\epsilon F$ is also an iso.
Part 5 means that in such a case $C^T$ is, up to equivalence a full reflective subcategory of $C$. Conversely, the monad induced by any reflective subcategory is idempotent, so giving an idempotent monad on $C$ is equivalent to giving a reflective subcategory of $C$.
In the language of stuff, structure, property, an idempotent monad may be said to equip its algebras with properties only (since $C^T\to C$ is fully faithful), unlike an arbitrary monad, which equips its algebras with at most structure (since $C^T\to C$ is, in general, faithful but not full).
If $T$ is idempotent, then it follows in particular that an object of $C$ admits at most one structure of $T$-algebra, that this happens precisely when the unit $\eta_X\colon X\to T X$ is an isomorphism, and in this case the $T$-algebra structure map is $\eta_X^{-1}\colon T X \to X$. However, it is possible to have a non-idempotent monad for which any object of $C$ admits at most one structure of $T$-algebra, in which case $T$ can be said to equip objects of $C$ with property-like structure; an easy example is the monad on semigroups whose algebras are monoids.
Let us be in a $2$-category $K$. Part of the structure of an idempotent monad $(C,T,\eta,\mu)$ in $K$ is of course an idempotent morphism $T:C\to C$. More precisely (Definition 1.1.9) considers $\mu$ as part of the structure such that an idempotent 1-cell has a 2-isomorphism $\mu:TT\to T$ such that $\mu T=T\mu$. Equivalently an idempotent morphism is a normalized pseudofunctor from the two object monoid $\{*,e\}$ with $e^2=e$ to $K$.
Recall that a splitting of an idempotent $(T,\mu)$ consists of a pair of 1-cells $I:D\to C$ and $R:C\to D$ and a pair of 2-isomorphisms $a:R I\to id_D$ and $b:T\to I R$ such that $\mu=b^{-1}(I\circ s\circ R)(b\circ b)$ where $\circ$ denotes horizontal composition of 2-cells. Equivalently an splitting of an idempotent is a limit or a colimit of the defining pseudofunctor. If $K$ has equalizers or coequalizers, then all its idempotents split.
Now if $(I,R,a,b)$ is a splitting of an idempotemt monad, then $R\dashv I$ are adjoint. And in this case the splitting of an idempotent is equivalently an Eilenberg-Moore object for the monad $(C,T,\eta,\mu)$. In this case $D$ is called an adjoint retract of $C$.
(Johnstone, B 1.1.9, p.248-249)
Equivalences (resp. cores) in an allegory are precisely those symmetric idempotents which are idempotent monads (resp. comonads). In an allegory the following statements are equivalent: all symmetric idempotents split, idempotent monads split, idempotent comonads split. A similar statement holds at least for some 2-categories.
Given an algebra $(X, u:TX\to X)$ , by (1) and (4) the action $u$ yields an isomorphism in $C^T$ between the free algebra $(TX, \mu_X)$ and $(X,u)$ i.e. for an idempotent monad the Eilenberg-Moore and the Kleisli categories coincide.
Let $(T, \eta, \mu)$ be an idempotent monad on a category $E$. The following conditions on an object $e$ of $E$ are equivalent:
The object $e$ carries an $T$-algebra structure.
The unit $\eta e\colon e \to T e$ is a split monomorphism.
The unit $\eta e$ is an isomorphism.
(It follows from 3. that there is at most one algebra structure on $e$, given by $\xi = (\eta e)^{-1}\colon T e \to e$.)
The implication 1. $\Rightarrow$ 2. is immediate. Next, if $\xi\colon M e \to e$ is any retraction of $\eta e$, we have both $\xi \circ \eta e = 1_e$ and
so 2. implies 3. Finally, if $\eta e$ is an isomorphism, put $\xi = (\eta e)^{-1}$. Then $\xi \circ \eta e = 1_e$ (unit condition), and the associativity condition for $\xi$,
follows by inverting the naturality equation $\eta M e \circ \eta e = M \eta e \circ \eta e$. Thus 3. implies 1.
This means that the Eilenberg-Moore category of an idempotent monad is equivalently the reflective subcategory (a “localization” of the ambient category) whose embedding-reflection adjunction gives the idempotent monad.
See also (Borceux, volume 2, corollary 4.2.4).
Hence dually the co-algebras over an idempotent comonad form a coreflective subcategory, hence a “co-localization” of the ambient category.
In modal type theory one thinks of a (idempotent) (co-)monad as a (co-)modal operator and of its algebras as (co-)modal types. In this terminology the above says that categories of (co-)modal types are precisely the (co-)reflective localizations of the ambient type system.
We discuss here how under suitable conditions, for every monad $T$ there is a “completion” to an idempotent monad $\tilde T$ in that the completion construction is right adjoint to the inclusion of idempotent monads into all monads, exhibiting idempotent monads as a coreflective subcategory. Here $\tilde T$ inverts the same morphisms that $T$ does and hence exhibits the localization(reflective subcategory) at the $T$-equivalences, and in fact the factorization of any adjunction inducing $T$ through that localization (Fakir 70, Applegate-Tierney 70, Day 74 Casacuberta-Frei 99. Lucyshyn-Wright 14).
Let $C$ be a complete, well-powered category, and let $M\colon C \to C$ be a monad with unit $u\colon 1 \to M$ and multiplication $m\colon M M \to M$. Then there is a universal idempotent monad, giving a right adjoint to the inclusion
Given a monad $M$, define a functor $M'$ as the equalizer of $M u$ and $u M$:
This $M'$ acquires a monad structure (see this MathOverflow thread for some detailed discussion). It might not be an idempotent monad (although it will be if $M$ is left exact). However we can apply the process again, and continue transfinitely. Define $M_0 = M$, and if $M_\alpha$ has been defined, put $M_{\alpha+1} = M_{\alpha}'$; at limit ordinals $\beta$, define $M_\beta$ to be the inverse limit of the chain
where $\alpha$ ranges over ordinals less than $\beta$. This defines the monad $M_\alpha$ inductively; below, we let $u_\alpha$ denote the unit of this monad.
Since $C$ is well-powered (i.e., since each object has only a small number of subobjects), the large limit
exists for each $c$. Hence the large limit $E(M) = \underset{\alpha \in Ord}{\lim} M_\alpha$ exists as an endofunctor. The underlying functor
reflects limits (irrespective of size), so $E = E(M)$ acquires a monad structure defined by the limit. Let $\eta\colon 1 \to E$ be the unit and $\mu\colon E E \to E$ the multiplication of $E$. For each $\alpha$, there is a monad map $\pi_\alpha\colon E \to M_\alpha$ defined by the limit projection.
$E$ is idempotent.
For this it suffices to check that $\eta E = E \eta\colon E \to E E$. This may be checked objectwise. So fix an object $c$, and for that particular $c$, choose $\alpha$ so large that $\pi_\alpha (c)\colon E(c) \to M_\alpha(c)$ and $\pi_\alpha E(c)\colon E E(c) \to M_{\alpha} E(c)$ are isomorphisms. In particular, $\pi_\alpha \pi_\alpha(c)\colon E E (c) \to M_\alpha M_\alpha(c)$ is invertible.
Now $u_\alpha M_\alpha(c) = M_{\alpha} u_{\alpha} c$, since $\pi_\alpha\colon E \to M_\alpha$ factors through the equalizer $M_{\alpha + 1} \hookrightarrow M_\alpha$. Because $\pi_\alpha$ is a monad morphism, we have
as required.
Finally we must check that $M \mapsto E(M)$ satisfies the appropriate universal property. Suppose $T$ is an idempotent monad with unit $v$, and let $\phi\colon T \to M$ be a monad map. We define $T \to M_\alpha$ by induction: given $\phi_\alpha\colon T \to M_\alpha$, we have
so that $\phi_{\alpha}$ factors uniquely through the inclusion $M_{\alpha + 1} \hookrightarrow M_\alpha$. This defines $\phi_{\alpha + 1}\colon T \to M_{\alpha + 1}$; this is a monad map. The definition of $\phi_\alpha$ at limit ordinals, where $M_\alpha$ is a limit monad, is clear. Hence $T \to M$ factors (uniquely) through the inclusion $E(M) \hookrightarrow M$, as was to be shown.
For $(L \dashv R) \;\colon\; \mathcal{C} \stackrel{\overset{L}{\longrightarrow}}{\underset{R}{\longleftarrow}} \mathcal{D}$ a pair of adjoint functors with induced monad $T = R\circ L$ on the complete and well-powered category $\mathcal{C}$, then the idempotent monad $\tilde T$ of theorem 1 corresponds via remark 4 to a reflective subcategory inclusion $\mathcal{C}_T \stackrel{i}{\hookrightarrow} \mathcal{C}$ which factors the original adjunction
such that $L'$ is a conservative functor.
(Lucyshyn-Wright 14, theorem 4.15)
The factorization in theorem 2 has its analog in homotopy theory in the concept of Bousfield localization of model categories: given a Quillen adjunction
then (if it exists) the Bousfield localized model category structure $\mathcal{C}_W$ obtained from $\mathcal{C}$ by adding the $L$-weak equivalences factors this into two consecutive Quillen adjunctions of the form
On the (∞,1)-categories presented by these model categories this gives a factorization of the derived (∞,1)-adjunction through localization onto a reflective sub-(∞,1)-category followed by a conservative (∞,1)-functor.
Let $A$ be a commutative ring, and let $f\colon A \to B$ be a flat (commutative) $A$-algebra. Then the forgetful functor
from $B$-modules to $A$-modules has a left exact left adjoint $f_! = B \otimes_A -$. The induced monad $f^\ast f_!$ on the category of $B$-modules preserves equalizers, and so its associated idempotent monad $T$ may be formed by taking the equalizer
(To be continued. This example is based on how Joyal and Tierney introduce effective descent for commutative ring homomorphisms, in An Extension of the Galois Theory of Grothendieck. I would like to consult that before going further – Todd.)
Mike Shulman: How about some examples of monads and their associated idempotent monads?
Do 2-monads have associated lax-, colax-, or pseudo-idempotent 2-monads?
The analog in model category theory of the localization at idempotent monad is the content of the Bousfield-Friedlander theorem.
General discussion includes
Francis Borceux, Handbook of Categorical Algebra, vol.2, p. 196.
Pierre Gabriel and Michel Zisman, Calculus of Fractions and Homotopy Theory
Peter Johnstone, Sketches of an Elephant, A.4.3.11, p.194, B1.1.9, p.249
The idempotent monad which exhibits the localization at the $T$-equivalences for a given monad $T$ is discussed in
H. Applegate and Myles Tierney, Iterated cotriples, Lecture Notes in Math. 137 (1970), 56-99
S. Fakir, Monade idempotente associée à une monade, C. R. Acad. Sci. Paris Ser. A-B 270 (1970), A99-A101. (gallica)
Brian Day, On adjoint-functor factorisation, Lecture Notes in Math. 420 (1974), 1-19.
Carles Casacuberta, Armin Frei, Localizations as idempotent approximations to completions, Journal of Pure and Applied Algebra 142 (1999), 25-33 (pdf)
and for enriched category theory in
Extension of idempotent monads along subcategory inclusions is discussed in
Carles Casacuberta, Armin Frei, Tan Geok Choo, Extending localization functors , Journal of Pure and Applied Algebra 103 (1995), 149-165. (pdf)
A. Deleanu, A. Frei, P. Hilton, Idempotent triples and completion , Math. Z. 143 (1975) pp.91-104. (pdf)