nLab idempotent monad

Idea

An idempotent monad is a monad that “squares to itself” 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; see modal type theory.

Definition

Definition

An idempotent monad is a monad $(T,\mu,\eta)$ on a category $\mathcal{C}$ such that one (hence all) of the following equivalent statements are true:

1. $\mu\colon T T \to T$ is a natural isomorphism.

2. All components of $\mu \,\colon\, T T \to T$ are monomorphisms.

3. The maps $T\eta, \eta T \,\colon\, T \to T T$ are equal (that is, $(T, \eta)$ is a well-pointed endofunctor).

4. For every $T$-algebra (aka: $T$-module) $(M,u)$, the corresponding $T$-action $u\colon T M \to M$ is an isomorphism (i.e. every algebra is a fixed point).

5. The forgetful functor $\mathcal{C}^T \to \mathcal{C}$ (where $C^T$ is the Eilenberg-Moore category of $T$-algebras) is a full and faithful functor.

6. There exists a pair of adjoint functors $F \dashv U$ such that the induced monad $(T \colon U F, \mu \coloneqq U \epsilon F, \eta)$ is isomorphic to $(T,\mu, \eta)$ and $U$ is a full and faithful functor.

   6’.

   There is a full subcategory $U \colon C^T \hookrightarrow C$ and a natural bijection of the form

   $$\array{ Hom_{\mathcal{C}}\big( T(c) ,\, U(a) \big) & \overset{\sim}{\longrightarrow} & Hom_{\mathcal{C}}\big( c ,\, U(a) \big) \\ \big( T(c) \overset{f}{\longrightarrow} U(a) \big) &\mapsto& \big( c \overset{\eta}{\longrightarrow} T(c) \overset{f}{\longrightarrow} U(a) \big) }$$

   (This is the form in which modal operators tend to be axiomatized in modal type theory, e.g. UP13, Def. 7.7.1.)

7. The identity natural transformation $T T \Rightarrow T T$ is a distributive law.

e.g. (Borceux, prop. 4.2.3). For (7) see Lemma 3.4 of RW95.

Proposition

The conditons in Def. are indeed equivalent.

Proof (in more than one way).

• $1\Rightarrow 2$

  This 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 $u$).

• $5\Rightarrow 6$

  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.

• $6 \Leftrightarrow 6'$

  This is the special case of the characterization (here) of (adjunction units of) adjoint functors $F \dashv U$ as systems of universal arrows when $U$ is fully faithful.

• $6\Rightarrow 1$

  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 isomorphism $FUX\stackrel{\sim}{\to} X$ ; 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.

Remark

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 idempotent 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)

Remark

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.

Remark

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.

Properties

Algebras for an idempotent monad and Localization

Proposition

Let $(T, \eta, \mu)$ be an idempotent monad on a category $E$. The following conditions on an object $e$ of $E$ are equivalent:

1. The object $e$ carries an $T$-algebra structure.

2. The unit $\eta e\colon e \to T e$ is a split monomorphism.

3. The unit $\eta e$ is an isomorphism.

(It follows from 3. that there is at most one algebra structure on $e$, given by $\xi = (\eta e)^{-1}\colon T e \to e$.)

Proof

The implication 1. $\Rightarrow$ 2. is immediate. Next, if $\xi\colon T e \to e$ is any retraction of $\eta e$, we have both $\xi \circ \eta e = 1_e$ and

$$\array{ \eta e \circ \xi & = & (T \xi)(\eta T e) & & \text{naturality of}\, \eta \\ & = & (T \xi)(T \eta e) & & \text{see definitions above} \\ & = & T(\xi \circ \eta e) & & \text{functoriality} \\ & = & 1_{T e} & & }$$

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 $\xi$,

$$\xi \circ \mu e = \xi \circ T \xi,$$

follows by inverting the naturality equation $\eta T e \circ \eta e = T \eta e \circ \eta e$. Thus 3. implies 1.

Remark

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).

Remark

Hence dually the co-algebras over an idempotent comonad form a coreflective subcategory, hence a “co-localization” of the ambient category.

Remark

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.

The associated idempotent monad of a monad

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 the category of all monads on a given category, exhibiting the subcategory of 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).

Theorem

(Fakir 70)
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

$$IdempotentMonad(C) \hookrightarrow Monad(C)$$

Proof

Given a monad $M$, define a functor $M'$ as the equalizer of $M u$ and $u M$:

$$M' \hookrightarrow M \stackrel{\overset{u M}{\longrightarrow}}{\underset{M u}{\longrightarrow}} M M.$$

This $M'$ acquires a unique monad structure such that $M' \hookrightarrow M$ is a morphism of monads (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

$$\ldots \hookrightarrow M_{\alpha} \hookrightarrow \ldots \hookrightarrow M$$

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

$$E(M)(c) = \underset{\alpha \in Ord}{\lim} M_\alpha(c)$$

exists for each $c$. Hence the large limit $E(M) = \underset{\alpha \in Ord}{\lim} M_\alpha$ exists as an endofunctor. The underlying functor

$$Monad(C) \to Endo(C)$$

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.

Lemma

$E$ is idempotent.

Proof

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

$$\array{ \eta E(c) & = & (\pi_\alpha \pi_\alpha (c))^{-1} (u_\alpha M_\alpha(c))\pi_\alpha(c) \\ & = & (\pi_\alpha \pi_\alpha (c))^{-1} (M_\alpha u_\alpha(c))\pi_\alpha(c) \\ & = & E \eta(c) }$$

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

$$(u_\alpha M_\alpha)\phi_\alpha = \phi_\alpha \phi_\alpha (v T) = \phi_\alpha \phi_\alpha (T v) = (M_\alpha u_{\alpha})\phi_\alpha$$

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.

Theorem

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 corresponds via remark to a reflective subcategory inclusion $\mathcal{C}_T \stackrel{i}{\hookrightarrow} \mathcal{C}$ which factors the original adjunction

$$(L\dashv R) \;\colon\; \mathcal{C} \stackrel{\overset{}{\longrightarrow}}{\underset{i}{\longleftarrow}} \mathcal{C}_T \stackrel{\overset{L'}{\longrightarrow}}{\underset{}{\longleftarrow}} \mathcal{D}$$

such that $L'$ is a conservative functor.

(Lucyshyn-Wright 14, theorem 4.15)

Remark

The factorization in theorem has its analog in homotopy theory in the concept of Bousfield localization of model categories: given a Quillen adjunction

$$(L \dashv R) \;\colon\; \mathcal{C} \stackrel{\longrightarrow}{\longleftarrow} \mathcal{D}$$

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

$$\mathcal{C} \stackrel{\overset{id}{\longrightarrow}}{\underset{id}{\longleftarrow}} \mathcal{C}_{W} \stackrel{\overset{L}{\longrightarrow}}{\underset{R}{\longleftarrow}} \mathcal{D} \,.$$

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.

Example

Let $A$ be a commutative ring, and let $f\colon A \to B$ be a flat (commutative) $A$-algebra. Then the forgetful functor

$$f^\ast = Ab^f\colon Ab^B \to Ab^A$$

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

$$T(M) \to B \otimes_A M \stackrel{\overset{f^\ast f_! \eta M}{\longrightarrow}}{\underset{\eta f^\ast f_! M}{\longrightarrow}} B \otimes_A B \otimes_A M$$

(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?

Idempotent strong monads are commutative

Theorem

Let $(\mathcal{C}, \otimes, I)$ be a monoidal category and let $(T, \mu, \eta)$ be an idempotent monad on $\mathcal{C}$ equipped with compatible left and right strengths $t_{A, B} : TA \otimes B \to T(A \otimes B)$ and $t'_{A, B} : A \otimes TB \to T(A \otimes B)$. Then $T$ is a commutative monad, or equivalently a monoidal monad.

Proof

The commutativity equation is obtained using the strength axioms, the monad axioms, and the fact that $\eta_T = T \eta$ for an idempotent monad:

Alternatively we can deduce this from the fact that that thunkable morphisms are central, since a monad is idempotent iff every Kleisli map is thunkable, and a monad is commutative iff every Kleisli map is central. (This fact is proved by Paul Levy there.)

Theorem

If furthermore $\mathcal{C}$ is a monoidal category with diagonals $\Delta_A : A \to A \otimes A$, then the induced monoidal monad structure on $T$ is idempotent in the sense that the following diagram commutes:

Proof

See 1Lab, Strong idempotent monads.

Examples

• The free/forgetful adjunction between groups and abelian groups generates an idempotent monad.

Related concepts

• idempotent monoid in a monoidal category

• idempotent adjunction

• reflective subcategory

• closure operator, modality

• lax-idempotent 2-monad

• idempotent (∞,1)-monad

• The analog in model category theory of the localization at idempotent monad is the content of the Bousfield-Friedlander theorem (“Quillen idempotent monad?”).

References

• Harry Appelgate, Myles Tierney, Section 6 in: Categories with models, p. 156-244 In: Beno Eckmann (ed.) Seminar on Triples and Categorical Homology Theory Lecture Notes in Mathematics, 80, Springer 1969 (doi:10.1007/BFb0083086, pdf)

Textbook accounts:

• Francis Borceux, Handbook of Categorical Algebra, vol.2, p. 196.

• Peter Johnstone, Sketches of an Elephant, A.4.3.11, p.194, B1.1.9, p.249

See also:

• Pierre Gabriel, Michel Zisman, Calculus of fractions and homotopy theory

The idempotent monad which exhibits the localization at the $T$-equivalences for a given monad $T$ is discussed in

• Harry Applegate, Myles Tierney, Iterated cotriples, Lecture Notes in Math. 137 (1970) 56-99 (doi:10.1007/BFb0060440)

• 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

• Rory Lucyshyn-Wright, Completion, closure, and density relative to a monad, with examples in functional analysis and sheaf theory, Theory Appl. Cat. 29 (2014) 31, 896–928 abs arXiv:1406.2361

Discussion in algebraic topology (localization, completion):

• John Frank Adams, Zbigniew Fiedorowicz, Localisation and Completion with an addendum on the use of Brown-Peterson homology in stable homotopy, based on 1973 lectures by Adams [arXiv:1012.5020]

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)

Discussion of idempotent comonads and their relation to adjoint strings is contained in

• Robert Rosebrugh, Richard Wood Distributive adjoint strings, Theory and applications of categories (1995) 119–145.

Formalisation in cubical Agda:

• 1lab, Idempotent monads