idempotent monad




2-Category theory

Modalities, Closure and Reflection

Idempotent monads


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.



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.

e.g. (Borceux, prop. 4.3.2).


The conditons in Def. are indeed equivalent.

Proof (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 TM=T(u)η TM=η Muid_{T 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 holds 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:R I\to id_D and b:TIRb:T\to I R such that μ=b 1(IsR)(bb)\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 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.


Given an algebra (X,u:TXX)(X, u:TX\to X) , by (1) and (4) the action uu yields an isomorphism in C TC^T between the free algebra (TX,μ X)(TX, \mu_X) and (X,u)(X,u) i.e. for an idempotent monad the Eilenberg-Moore and the Kleisli categories coincide.


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

We discuss here how under suitable conditions, for every monad TT there is a “completion” to an idempotent monad T˜\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 T˜\tilde T inverts the same morphisms that TT does and hence exhibits the localization(reflective subcategory) at the TT-equivalences, and in fact the factorization of any adjunction inducing TT through that localization (Fakir 70, Applegate-Tierney 70, Day 74 Casacuberta-Frei 99. Lucyshyn-Wright 14).

Theorem (Fakir 70)

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}{\longrightarrow}}{\underset{M u}{\longrightarrow}} M M.

This MM' acquires a monad structure (see this MathOverflow thread for some detailed discussion). 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.


For (LR):𝒞RL𝒟 (L \dashv R) \;\colon\; \mathcal{C} \stackrel{\overset{L}{\longrightarrow}}{\underset{R}{\longleftarrow}} \mathcal{D} a pair of adjoint functors with induced monad T=RLT = R\circ L on the complete and well-powered category 𝒞\mathcal{C}, then the idempotent monad T˜\tilde T of theorem corresponds via remark to a reflective subcategory inclusion 𝒞 Ti𝒞\mathcal{C}_T \stackrel{i}{\hookrightarrow} \mathcal{C} which factors the original adjunction

(LR):𝒞i𝒞 TL𝒟 (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 LL' is a conservative functor.

(Lucyshyn-Wright 14, theorem 4.15)


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

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

then (if it exists) the Bousfield localized model category structure 𝒞 W\mathcal{C}_W obtained from 𝒞\mathcal{C} by adding the LL-weak equivalences factors this into two consecutive Quillen adjunctions of the form

𝒞idid𝒞 WRL𝒟. \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.


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?


General discussion includes

The idempotent monad which exhibits the localization at the TT-equivalences for a given monad TT 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)

Last revised on June 29, 2018 at 07:05:05. See the history of this page for a list of all contributions to it.