nLab lax-idempotent 2-monad




A lax-idempotent 2-monad, also called a Kock–Zöberlein or KZ 2-monad, encodes a certain kind of property-like structure that a category, or more generally an object of a 2-category, can carry.

Lax-idempotent 2-monads have occasionally also been called KZ monads in the literature, but this terminology may be confusing, as it is inconsistent with terminology of lax-idempotent nn-monads: a 1-monad may be viewed as a 2-monad on a locally-discrete 2-category, in which case lax-idempotence is equivalent to idempotence.

The archetypal examples are given by 2-monads TT on Cat that take a category CC to the free cocompletion TCT C of CC under a given class of colimits – then an algebra TCCT C \to C is a category CC with all such colimits, which are of course essentially unique. Moreover, given thus-cocomplete categories CC and DD, a functor F:CDF \colon C \to D, and a diagram SS in CC, there is a unique arrow colimTFSF(colimS)colim T F S \to F(colim S) given by the universal property of the colimit. It is this property that lax-idempotence generalizes.


A 2-monad TT on a 2-category KK is called lax-idempotent if given any two (strict) TT-algebras a:TAAa \colon T A \to A, b:TBBb \colon T B \to B and a morphism f:ABf \colon A \to B, there exists a unique 2-cell f¯:bTffa\bar f \colon b \circ T f \Rightarrow f \circ a making (f,f¯)(f,\bar f) a lax morphism of TT-algebras:

TA Tf TB a f¯ b A f B \array{ T A & \overset{T f}{\to} & T B \\ a \downarrow & \swArrow \bar f & \downarrow b \\ A & \underset{f}{\to} & B }

Dually, a 2-monad TT is called colax-idempotent if f:ABf \colon A \to B gives rise to a colax TT-morphism (f,f˜)(f,\tilde f):

TA Tf TB a f˜ b A f B \array{ T A & \overset{T f}{\to} & T B \\ a \downarrow & \neArrow \tilde f & \downarrow b \\ A & \underset{f}{\to} & B }

Equivalent conditions

A 2-monad (T,μ,η)(T, \mu, \eta) is lax idempotent if and only if, for every object AA, there is an adjunction Tη Aμ AT \eta_A \dashv \mu_A with invertible counit or an adjunction μ Aη TA\mu_A \dashv \eta_{T A} with invertible counit (with either adjunction implying the other).

An extensive list of equivalent conditions is given on the page of lax-idempotent 2-adjunctions.


A 2-monad TT as above, with unit η:1T\eta: 1 \to T, is lax-idempotent if and only if for any TT-algebra a:TAAa \colon T A \to A there is a 2-cell θ a:1η Aa\theta_a \colon 1 \Rightarrow \eta_A \circ a such that (θ a,1 1 A)(\theta_a ,1_{1_A}) are the unit and counit of an adjunction aη Aa \dashv \eta_A.


(Adapted from Kelly–Lack). The multiplication μ A:T 2ATA\mu_A \colon T^2 A \to T A is a TT-algebra on TAT A, and η A:ATA\eta_A \colon A \to T A is a morphism from the underlying object of aa to that of μ A\mu_A. So there is a unique η¯ A:μ ATη A=1 TAη Aa\bar\eta_A \colon \mu_A \circ T \eta_A = 1_{T A} \Rightarrow \eta_A \circ a making η A\eta_A into a lax TT-morphism. Set θ a=η¯ A\theta_a = \bar\eta_A. The triangle equalities then require that:

  1. aη¯ A:aaη Aa=aa \bar\eta_A \colon a \Rightarrow a \circ \eta_A \circ a = a is equal to 1 a1_a. The composite aη¯ Aa \circ \bar\eta_A makes aη Aa \circ \eta_A a lax TT-morphism from aa to aa (paste η¯ A\bar\eta_A with the identity square aμ A=aTaa \circ \mu_A = a \circ T a). But aη A=1 Aa \circ \eta_A = 1_A, and 1 a1_a also makes this into a lax TT-morphism, so by uniqueness aη¯ A=1 aa \bar\eta_A = 1_a.

  2. η¯ Aη A:η Aη Aaη A=η A\bar\eta_A \eta_A \colon \eta_A \Rightarrow \eta_A \circ a \circ \eta_A = \eta_A is equal to 1 η A1_{\eta_A}. But this follows directly from the unit coherence condition for the lax TT-morphism η¯ A\bar\eta_A.

Conversely, suppose θ a\theta_a, algebras a,ba,b on A,BA,B and f:ABf \colon A \to B are given. Take f¯\bar f to be the mate of 1 f:bTfηA=ff1_f \colon b \circ T f \circ \eta A = f \Rightarrow f with respect to the adjunctions aη Aa \dashv \eta_A and 111 \dashv 1, which is given in this case by pasting with θ a\theta_a, so we have that f¯=bTfθ a\bar f = b \circ T f \circ \theta_a. The mate of f¯\bar f in turn is given by f¯η A\bar f \circ \eta_A, which because mates correspond bijectively is equal to 1 f1_f. So f¯\bar f satisfies the unit condition.

Consider the diagrams expressing the multiplication condition: because aμ A=aTaa \circ \mu_A = a \circ T a (and the same for bb), their boundaries are equal, so we have 2-cells α,β:bTbT 2ffaTa\alpha, \beta \colon b \circ T b \circ T^2 f \Rightarrow f \circ a \circ T a. Their mates under the adjunction (Tθ a,1):TaTη A(T\theta_a, 1) \colon T a \dashv T\eta_A are given by pasting with Tη AT \eta_A. One is f¯\bar f pasted with Tf¯Tη A=T(fη A)=T1 f=1 TfT \bar f \circ T \eta_A = T(f \circ \eta_A) = T 1_f = 1_{T f}, and the other is given by composing Tη AT \eta_A with the identity μ BT 2f=Tfμ A\mu_B \circ T^2 f = T f \circ \mu_A (and then pasting with f¯\bar f), but because μ ATη A=1 TA\mu_A \circ T \eta_A = 1_{T A} this is also equal to 1 Tf1_{T f}. The two original 2-cells are hence equal, because their mates are equal, and so f¯\bar f is indeed a lax TT-morphism.

Since TT‘s multiplication μ\mu makes TT itself into a (generalized) TT-algebra, the above implies (and in fact is implied by) the requirement that there exist a modification :1 T 2ηTμ\ell \colon 1_{T^2} \to \eta T \circ \mu making (,1):μηT(\ell,1) \colon \mu \dashv \eta T. Conversely, given an algebra a:TAAa \colon T A \to A, the 2-cell θ a\theta_a is given by Ta ATη AT a \circ \ell_A \circ T \eta_A.

A different but equivalent condition is that there be a modification d:TηηTd \colon T \eta \to \eta T such that dη=1d \eta = 1 and μd=1\mu d = 1; and given \ell as above, dd is given by Tη\ell \circ T \eta.

These various conditions can also be regarded as ways to say that the Eilenberg-Moore adjunction for TT is a lax-idempotent 2-adjunction. Thus, TT is a lax-idempotent 2-monad exactly when this 2-adjunction is lax-idempotent, and therefore also just when it is the 2-monad induced by some lax-idempotent 2-adjunction.

Dually, for TT to be colax-idempotent, it is necessary and sufficient that any of the following hold.

  • For any TT-algebra a:TAAa \colon T A \to A there is a 2-cell ζ a:η Aa1\zeta_a \colon \eta_A \circ a \Rightarrow 1 such that (1,ζ a):η Aa(1,\zeta_a) \colon \eta_A \dashv a.

  • There is a modification m:μηT1m \colon \mu \circ \eta T \to 1 making (1,m):ηTμ(1,m) \colon \eta T \dashv \mu.

  • There is a modification e:ηTTηe \colon \eta T \to T\eta such that eη=1e\eta = 1 and μe=1\mu e = 1.


Theorem gives a necessary condition for an object AA to admit a TT-algebra structure, namely that η A:ATA\eta_A : A \to T A admit a left adjoint with identity counit. In the case of pseudo algebras, this necessary condition is also sufficient.


To give a pseudo TT-algebra structure on an object AA is equivalently to give a left adjoint to η A:ATA\eta_A : A\to T A with invertible counit.

In particular, an object admits at most one pseudo TT-algebra structure, up to unique isomorphism. Thus, TT-algebra structure is property-like structure.

In many cases it is interesting to consider the pseudo TT-algebras for which the algebra structure TAAT A \to A has a further left adjoint, forming an adjoint triple. Algebras of this sort are sometimes called continuous algebras.


As mentioned above, the standard examples of lax-idempotent 2-monads are those on CatCat whose algebras are categories with all colimits of a specified class. In this case, the 2-monad is a free cocompletion operation. Dually, there are colax-idempotent 2-monads which adjoin limits of a specified class. A converse is given by (PowerCattaniWinskel), who show that a 2-monad is a monad for free cocompletions if and only if it is lax-idempotent and the unit η\eta is dense (plus a coherence condition).

Another important example of a colax-idempotent 2-monad is the monad on Cat/BCat/B that takes p:EBp \colon E \to B to the projection B/pBB/p \to B out of the comma category. The algebras for this monad are Grothendieck fibrations over BB; see also fibration in a 2-category. The monad pp/Bp \mapsto p/B is lax-idempotent, and its algebras are opfibrations.

This latter is actually a special case of a general situation. If TT is a (2-)monad relative to which one can define generalized multicategories, then often it induces a lax-idempotent 2-monad T˜\tilde{T} on the 2-category of such generalized multicategories (aka “virtual TT-algebras”), such that (pseudo) T˜\tilde{T}-algebras are equivalent to (pseudo) TT-algebras. When TT is the 2-monad whose algebras are strict 2-functors BCatB\to Cat and whose pseudo algebras are pseudofunctors BCatB\to Cat, then a virtual TT-algebra is a category over BB, and it is a pseudo T˜\tilde{T}-algebra just when it is an opfibration. Similarly, there is a lax-idempotent 2-monad on the 2-category of multicategories whose pseudo algebras are monoidal categories, and so on.


  • pseudo-distributive laws involving lax-idempotent 2-monads have an especially nice form; see (Marmolejo) and (Walker).

  • For ordinary 1-monads there exists a presentation due to Manes as “Kleisli triples” with primary data a family of unit morphisms and lifts avoiding the iteration of the endofunctor. A similar presentation exists for lax-idempotent 2-monads as shown in Marmolejo-Wood (2012). It is shown then in Walker (2017) that provided the units of this presentation are fully faithful (a reflection of the fully-faithfulness of the Yoneda embedding) (almost) all the axioms of a Yoneda structure are satisfied. In cases where size plays no role like e.g. the ideal completion of posets the two concepts coincide. For further details see at Yoneda structure or Walker (2017).


Classical references are

  • Max Kelly, Steve Lack, On property-like structures, TAC 3(9), 1997. (abstract)

  • Anders Kock, Monads for which structures are adjoint to units , Aarhus Preprint 1972/73 No. 35. (pdf)

  • Anders Kock, Monads for which structures are adjoint to units, JPAA 104:41–59, 1995.

  • Ross Street, Fibrations and Yoneda’s lemma in a 2-category, Lecture Notes in Mathematics, Vol. 420, 1974, pp. 104–133. [doi:10.1007/BFb0063102]

  • Ross Street, Fibrations in Bicategories , Cah. Top. Géom. Diff. XXI no.2 (1980). (numdam)

  • Volker Zöberlein, Doctrines on 2-categories , Math. Zeitschrift 148 (1976) pp.267-279. (gdz)

Textbook accounts:

See also:

Their distributive laws come into focus in

The relation to Yoneda structures:

  • Charles Walker, Yoneda Structures and KZ Doctrines, arxiv

The logical-syntactical side:

  • Jiri Adamek, Lurdes Sousa, KZ-monadic categories and their logic, tac

Discussion of the adjoint functor theorem:

Last revised on January 28, 2024 at 16:27:11. See the history of this page for a list of all contributions to it.