Examples of (co)lax idempotent pseudocomonads on Cat

Question

A lax idempotent pseudomonad, also called a KZ doctrine or KZ monad, is a pseudomonad $(T, \mu, \eta)$ with the property that $T \eta \dashv \mu \dashv \eta T$. Lax idempotent pseudomonads were introduced to axiomatise colimit structure on categories, and it turns out that there is a strong sense in which lax idempotent pseudomonads exactly characterise colimit structure: Power–Cattani–Winskel prove in Theorem 16 of A representation result for free cocompletions that a 2-monad on $\mathrm{Cat}$ is lax idempotent and dense if and only if there is a class of weights $\Phi$ for which the 2-monad is the free $\Phi$-cocompletion. (Density is a technical condition that serves to exclude pathological counterexamples, like the terminal 2-monad.)

Dually, colax idempotent pseudomonads axiomatise limit structure on categories.

I would like to know whether (co)lax idempotent pseudocomonads (i.e. KZ codoctrines, or KZ comonads) may be characterised analogously. Since I do not expect a general classification result exists, like that for lax idempotent pseudomonads, I am really looking for a few (nontrivial) examples of (co)lax idempotent pseudocomonads on $\mathrm{Cat}$, to get an intuitive for what their coalgebras look like.

Answer 1

This isn't quite an example of what you're asking for, since the domain isn't $\rm Cat$. However, the complete dearth of answers suggests that there aren't a lot of (co)lax idempotent pseudocomonads around, so maybe an example on a different domain will still be interesting.

Let $T$ be a strict 2-monad on a strict 2-category $K$. If $K$ and $T$ are sufficiently nice, the inclusion $T\text{-}{\rm Alg_s} \to T\text{-}\rm Alg_l$ of algebras-and-strict-morphisms into algebras-and-lax-morphisms has a left 2-adjoint, called the "lax morphism classifier". In Enhanced 2-categories and limits for lax morphisms (Lemma 2.5), Steve Lack and I proved that the 2-comonad on $T\text{-}\rm Alg_s$ induced by this adjunction is lax idempotent. Similarly, if you use colax morphisms, you get a colax idempotent 2-comonad, and if you use pseudo morphisms you get a pseudo-idempotent 2-comonad.

We didn't give a general description of the coalgebras for these 2-comonads, but in the particular case where $T$-algebras are $\rm Cat$-valued functors on a small 2-category we found that the coalgebras for the lax morphism classifier are precisely the weights $W$ such that $W$-weighted limits lift to the category of colax morphisms for an 2-monad, and dually. Moreover, in the pseudo case these weights are precisely the PIE-weights. (This is in section 6 of our paper. The rest of the paper is about $\mathcal F$-enriched limits, which are similar but carry more information about strictness.