idempotent monad




2-Category theory

Modalities, Closure and Reflection

Idempotent monads



An idempotent monad is a monad (T,μ,η)(T,\mu,\eta) on a category CC such that one (hence all) of the following equivalent statements are true:

  1. μ:TTT\mu\colon T T \to T is a natural isomorphism.

  2. All components of μ:TTT\mu\colon T T \to T are monomorphisms.

  3. The maps Tη,ηT:TTTT\eta, \eta T\colon T \to T T are equal.

  4. For every TT-algebra (TT-module) (M,u)(M,u), the corresponding TT-action u:TMMu\colon T M \to M is an isomorphism.

  5. The forgetful functor C TCC^T \to C (where C TC^T is the Eilenberg-Moore category of TT-algebras) is a full and faithful functor.

  6. There exists a pair of adjoint functors FUF\dashv U such that the induced monad (UF,UϵF)(UF, U\epsilon F) is isomorphic to (T,μ)(T,\mu) and UU is a full and faithful functor.

Proof of equivalence (in more than one way).

121\Rightarrow 2 is trivial.

232\Rightarrow 3 Compositions μTη\mu\circ T\eta and μηT\mu\circ\eta T are always the identity (unit axioms for the monad), and in particular agree; if μ\mu has all components monic, this implies Tη=ηTT\eta = \eta T.

343\Rightarrow 4 Compatibility of action and unit is uη M=id Mu \circ \eta_M = id_M, hence also T(u)T(η M)=id TMT(u)\circ T(\eta_M) = id_{T M}. If Tη=ηTT\eta = \eta T then this implies id M=T(u)η TM=η Muid_M = T(u)\circ \eta_{T M} = \eta_M\circ u, where the naturality of η\eta is used in the second equality. Therefore we exhibited η M\eta_M both as a left and a right inverse of uu.

414\Rightarrow 1 If every action is iso, then the components of multiplication μ M:TTMTM\mu_M\colon T T M\to T M are isos as a special case, namely of the free action on TMT M.

454\Rightarrow 5 For any monad TT, the forgetful functor from Eilenberg-Moore category C TC^T to CC is faithful: a morphism of TT-algebras is always a morphism of underlying objects in CC. To show that it is also full, we consider any pair (M,u)(M,u), (M,u)(M',u') in C TC^T and must show that any f:MMf\colon M\to M' is actually a map f:(M,u)(M,u)f\colon (M,u)\to (M',u'); i.e. uTf=fuu'\circ T f = f\circ u. But we know that η M,η M\eta_M, \eta_{M'} are inverses of u,uu,u' respectively and the naturality for η\eta says η Mf=Tfη M\eta_{M'}\circ f = T f \circ \eta_M. Compose that equation with uu on the right and uu' on the left with the result (notice that we used just the invertibility of uu).

565\Rightarrow 6 Trivial, because the Eilenberg-Moore construction induces the original monad by the standard recipe.

636\Rightarrow 3 By 66 the counit ϵ\epsilon is iso, hence UϵFU\epsilon F has a unique 2-sided inverse; by triangle identities, TηT\eta and ηT\eta T are both right inverses of UϵFU\epsilon F, hence 2-sided inverses, hence they are equal.

616\Rightarrow 1 If FUF\dashv U is an adjunction with UU fully faithful, then the counit ϵ\epsilon is iso. since D(FUX,Y)C(UX,UY)D(X,Y)D(FU X,Y)\simeq C(UX,UY)\simeq D(X,Y) where the last equivalence is since UU is full and faithful; hence by essential unicity of the representing object there is an iso FUXXFUX\stackrel{\sim}{\to} X.; let X=YX=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 μ=UϵF\mu = U\epsilon F is also an iso.

Part 5 means that in such a case C TC^T is, up to equivalence a full reflective subcategory of CC. Conversely, the monad induced by any reflective subcategory is idempotent, so giving an idempotent monad on CC is equivalent to giving a reflective subcategory of CC.

In the language of stuff, structure, property, an idempotent monad may be said to equip its algebras with properties only (since C TCC^T\to C is fully faithful), unlike an arbitrary monad, which equips its algebras with at most structure (since C TCC^T\to C is, in general, faithful but not full).

If TT is idempotent, then it follows in particular that an object of CC admits at most one structure of TT-algebra, that this happens precisely when the unit η X:XTX\eta_X\colon X\to T X is an isomorphism, and in this case the TT-algebra structure map is η X 1:TXX\eta_X^{-1}\colon T X \to X. However, it is possible to have a non-idempotent monad for which any object of CC admits at most one structure of TT-algebra, in which case TT can be said to equip objects of CC with property-like structure; an easy example is the monad on semigroups whose algebras are monoids.


Let us be in a 22-category KK. Part of the structure of an idempotent monad (C,T,η,μ)(C,T,\eta,\mu) in KK is of course an idempotent morphism T:CCT:C\to C. More precisely (Definition 1.1.9) considers μ\mu as part of the structure such that an idempotent 1-cell has a 2-isomorphism μ:TTT\mu:TT\to T such that μT=Tμ\mu T=T\mu. Equivalently an idempotent morphism is a normalized pseudofunctor from the two object monoid {*,e}\{*,e\} with e 2=ee^2=e to KK.

Recall that a splitting of an idempotent (T,μ)(T,\mu) consists of a pair of 1-cells I:DCI:D\to C and R:CDR:C\to D and a pair of 2-isomorphisms a:RIid Da:RI\to id_D and b:TIRb:T\to IR such that μ=b 1(IAR)(bb)\mu=b^{-1}(I\circ A\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 KK has equalizers or coequalizers, then all its idempotents split.

Now if (I,R,a,b)(I,R,a,b) is a splitting of an idempotemt monad, then RIR\dashv I are adjoint. And in this case the splitting of an idempotent is equivalently an Eilenberg-Moore object for the monad (C,T,η,μ)(C,T,\eta,\mu). In this case DD is called an adjoint retract of CC.

(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.

(Johnstone, B 1.1.9, p.248-249)


Algebras for an idempotent monad and Localization


Let (T,η,μ)(T, \eta, \mu) be an idempotent monad on a category EE. The following conditions on an object ee of EE are equivalent:

  1. The object ee carries an TT-algebra structure.

  2. The unit ηe:eTe\eta e\colon e \to T e is a split monomorphism.

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

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


The implication 1. \Rightarrow 2. is immediate. Next, if ξ:Mee\xi\colon M e \to e is any retraction of ηe\eta e, we have both ξηe=1 e\xi \circ \eta e = 1_e and

ηeξ = (Mξ)(ηMe) naturality ofη = (Mξ)(Mηe) see definitions above = M(ξηe) functoriality = 1 Me \array{ \eta e \circ \xi & = & (M \xi)(\eta M e) & & \text{naturality of}\, \eta \\ & = & (M \xi)(M \eta e) & & \text{see definitions above} \\ & = & M(\xi \circ \eta e) & & \text{functoriality} \\ & = & 1_{M e} & & }

so 2. implies 3. Finally, if ηe\eta e is an isomorphism, put ξ=(ηe) 1\xi = (\eta e)^{-1}. Then ξηe=1 e\xi \circ \eta e = 1_e (unit condition), and the associativity condition for ξ\xi,

ξμe=ξMξ,\xi \circ \mu e = \xi \circ M \xi,

follows by inverting the naturality equation ηMeηe=Mηeηe\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.

The associated idempotent monad of a monad

Theorem (Fakir)

Let CC be a complete, well-powered category, and let M:CCM\colon C \to C be a monad with unit u:1Mu\colon 1 \to M and multiplication m:MMMm\colon M M \to M. Then there is a universal idempotent monad, giving a right adjoint to the inclusion

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

Given a monad MM, define a functor MM' as the equalizer of MuM u and uMu M:

MMMuuMMM.M' \hookrightarrow M \stackrel{\overset{u M}{\to}}{\underset{M u}{\to}} M M.

This MM' acquires a monad structure. It might not be an idempotent monad (although it will be if MM is left exact). However we can apply the process again, and continue transfinitely. Define M 0=MM_0 = M, and if M αM_\alpha has been defined, put M α+1=M αM_{\alpha+1} = M_{\alpha}'; at limit ordinals β\beta, define M βM_\beta to be the inverse limit of the chain

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

where α\alpha ranges over ordinals less than β\beta. This defines the monad M αM_\alpha inductively; below, we let u αu_\alpha denote the unit of this monad.

Since CC is well-powered (i.e., since each object has only a small number of subobjects), the large limit

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

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

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

reflects limits (irrespective of size), so E=E(M)E = E(M) acquires a monad structure defined by the limit. Let η:1E\eta\colon 1 \to E be the unit and μ:EEE\mu\colon E E \to E the multiplication of EE. For each α\alpha, there is a monad map π α:EM α\pi_\alpha\colon E \to M_\alpha defined by the limit projection.


EE is idempotent.


For this it suffices to check that ηE=Eη:EEE\eta E = E \eta\colon E \to E E. This may be checked objectwise. So fix an object cc, and for that particular cc, choose α\alpha so large that π α(c):E(c)M α(c)\pi_\alpha (c)\colon E(c) \to M_\alpha(c) and π αE(c):EE(c)M αE(c)\pi_\alpha E(c)\colon E E(c) \to M_{\alpha} E(c) are isomorphisms. In particular, π απ α(c):EE(c)M αM α(c)\pi_\alpha \pi_\alpha(c)\colon E E (c) \to M_\alpha M_\alpha(c) is invertible.

Now u αM α(c)=M αu αcu_\alpha M_\alpha(c) = M_{\alpha} u_{\alpha} c, since π α:EM α\pi_\alpha\colon E \to M_\alpha factors through the equalizer M α+1M αM_{\alpha + 1} \hookrightarrow M_\alpha. Because π α\pi_\alpha is a monad morphism, we have

ηE(c) = (π απ α(c)) 1(u αM α(c))π α(c) = (π απ α(c)) 1(M αu α(c))π α(c) = Eη(c)\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 ME(M)M \mapsto E(M) satisfies the appropriate universal property. Suppose TT is an idempotent monad with unit vv, and let ϕ:TM\phi\colon T \to M be a monad map. We define TM αT \to M_\alpha by induction: given ϕ α:TM α\phi_\alpha\colon T \to M_\alpha, we have

(u αM α)ϕ α=ϕ αϕ α(vT)=ϕ αϕ α(Tv)=(M αu α)ϕ α(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 ϕ α\phi_{\alpha} factors uniquely through the inclusion M α+1M αM_{\alpha + 1} \hookrightarrow M_\alpha. This defines ϕ α+1:TM α+1\phi_{\alpha + 1}\colon T \to M_{\alpha + 1}; this is a monad map. The definition of ϕ α\phi_\alpha at limit ordinals, where M αM_\alpha is a limit monad, is clear. Hence TMT \to M factors (uniquely) through the inclusion E(M)ME(M) \hookrightarrow M, as was to be shown.


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

f *=Ab f:Ab BAb Af^\ast = Ab^f\colon Ab^B \to Ab^A

from BB-modules to AA-modules has a left exact left adjoint f !=B Af_! = B \otimes_A -. The induced monad f *f !f^\ast f_! on the category of BB-modules preserves equalizers, and so its associated idempotent monad TT may be formed by taking the equalizer

T(M)B AMηf *f !Mf *f !ηMB AB AMT(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?


Revised on March 28, 2014 21:07:33 by Tim Porter (