nLab
lax-idempotent 2-monad

Contents

Idea

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

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.

Definition

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:

(1)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):

(2)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 }

Lax-idempotent monads are also called Kock–Zöberlein or KZ monads.

Equivalent conditions

Theorem

A 2-monad TT as above 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:1ηAa\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.

Proof

(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.

Algebras

Theorem 1 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.

Theorem

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.

Examples

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 monad is the monad on Cat/BCat/B that takes p:EBp \colon E \to B to the projection B/ppB/p \to p 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.

References

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

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

  • Power–Cattani–Winskel, A representation result for free cocompletions, JPAA 151:273–286, 2000.

  • Marmolejo–Wood, Kan extensions and lax idempotent pseudomonads, TAC 26, p. 1–29 (2011)

Revised on January 14, 2014 10:28:12 by Urs Schreiber (89.204.153.51)