The archetypal examples are given by 2-monads on Cat that take a category to the free cocompletion of under a given class of colimits – then an algebra is a category with all such colimits, which are of course essentially unique. Moreover, given thus-cocomplete categories and , a functor , and a diagram in , there is a unique arrow given by the universal property of the colimit. It is this property that lax-idempotence generalizes.
Dually, a 2-monad is called colax-idempotent if gives rise to a colax -morphism :
Lax-idempotent monads are also called Kock–Zöberlein or KZ monads.
A 2-monad as above is lax-idempotent if and only if for any -algebra there is a 2-cell such that are the unit and counit of an adjunction .
(Adapted from Kelly–Lack). The multiplication is a -algebra on , and is a morphism from the underlying object of to that of . So there is a unique making into a lax -morphism. Set . The triangle equalities then require that:
is equal to . The composite makes a lax -morphism from to (paste with the identity square ). But , and also makes this into a lax -morphism, so by uniqueness .
is equal to . But this follows directly from the unit coherence condition for the lax -morphism .
Conversely, suppose , algebras on and are given. Take to be the mate of with respect to the adjunctions and , which is given in this case by pasting with , so we have that . The mate of in turn is given by , which because mates correspond bijectively is equal to . So satisfies the unit condition.
Consider the diagrams expressing the multiplication condition: because (and the same for ), their boundaries are equal, so we have 2-cells . Their mates under the adjunction are given by pasting with . One is pasted with , and the other is given by composing with the identity (and then pasting with ), but because this is also equal to . The two original 2-cells are hence equal, because their mates are equal, and so is indeed a lax -morphism.
Since ‘s multiplication makes itself into a (generalized) -algebra, the above implies (and in fact is implied by) the requirement that there exist a modification making . Conversely, given an algebra , the 2-cell is given by .
A different but equivalent condition is that there be a modification such that and ; and given as above, is given by .
These various conditions can also be regarded as ways to say that the Eilenberg-Moore adjunction for is a lax-idempotent 2-adjunction. Thus, 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 to be colax-idempotent, it is necessary and sufficient that any of the following hold.
For any -algebra there is a 2-cell such that .
There is a modification making .
There is a modification such that and .
Theorem 1 gives a necessary condition for an object to admit a -algebra structure, namely that admit a left adjoint with identity counit. In the case of pseudo algebras, this necessary condition is also sufficient.
To give a pseudo -algebra structure on an object is equivalently to give a left adjoint to with invertible counit.
In particular, an object admits at most one pseudo -algebra structure, up to unique isomorphism. Thus, -algebra structure is property-like structure.
In many cases it is interesting to consider the pseudo -algebras for which the algebra structure 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 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 is dense (plus a coherence condition).
Another important example of a colax-idempotent monad is the monad on that takes to the projection out of the comma category. The algebras for this monad are Grothendieck fibrations over ; see also fibration in a 2-category. The monad is lax-idempotent, and its algebras are opfibrations.
This latter is actually a special case of a general situation. If is a (2-)monad relative to which one can define generalized multicategories, then often it induces a lax-idempotent 2-monad on the 2-category of such generalized multicategories (aka “virtual -algebras”), such that (pseudo) -algebras are equivalent to (pseudo) -algebras. When is the 2-monad whose algebras are strict 2-functors and whose pseudo algebras are pseudofunctors , then a virtual -algebra is a category over , and it is a pseudo -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.
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.