A recognition criterion for lax-idempotent pseudomonads

We describe a simple criterion which makes it easy to recognise when a pseudomonad is lax-idempotent. The criterion concerns the behaviour of colax bilimits of arrows - certain comma objects - and is easy to verify in examples. Building on this, we obtain a new characterisation of lax-idempotent pseudomonads on 2-categories with colax bilimits of arrows.

💡 Research Summary

The paper presents a practical recognition criterion for lax‑idempotent pseudomonads in the setting of 2‑categories. A lax‑idempotent pseudomonad (also known as a KZ‑pseudomonad) is characterized by the multiplication µ being left adjoint to the unit η, giving rise to a chain of adjunctions η T ⊣ µ ⊣ T η. While this definition is abstract, an equivalent “left Kan” description shows that such a pseudomonad is completely determined by the objects T X and the unit components η_X, reflecting its role as a co‑completion process.

The central result, Theorem 3.1, states that for a biadjunction F ⊣ U : A → B, if the right adjoint U is locally full (i.e., it reflects 2‑cells) and every arrow f : U A → U B admits a colax bilimit of the form

 U A ←p– U C –q→ U B with a 2‑cell λ : q ⇒ f p,

then the biadjunction is lax‑idempotent and the induced pseudomonad T = U F is lax‑idempotent. A colax bilimit is precisely a comma‑type object (the “colax” version of a comma object) satisfying a universal property: for any X, the functor

 C(X, C) → C(X, B) / C(X, f)

is an equivalence. The proof constructs the required adjunction U ε ⊣ η U by exploiting the universal property of the colax bilimit and the local fullness of U to guarantee the invertibility of the necessary 2‑cells.

The paper then illustrates the criterion in several familiar contexts:

1. Categories with binary coproducts (BCop) – The forgetful 2‑functor U : BCop → Cat preserves the colax bilimit B/f (the comma category) and its projections, because coproducts are computed pointwise in B/f. Hence U satisfies the criterion and yields a lax‑idempotent pseudomonad.

2. Categories with limits of a class Φ (Φ‑Lim) – While limits in B/f exist pointwise when f preserves the relevant limits, the comparison maps need not be invertible, so the colax bilimit property fails. This shows that not every locally full functor gives a lax‑idempotent pseudomonad.

3. Regular categories (Reg) → Lex (finite‑limit‑preserving categories) – Regular categories have finite limits and stable co‑equalisers of kernel pairs. For a finite‑limit‑preserving functor f, the comma category B/f inherits these structures pointwise, and the projections reflect them. Thus the inclusion Reg → Lex satisfies the colax bilimit property, producing a lax‑idempotent pseudomonad.

4. 2‑dimensional monad theory – For a 2‑monad T, the inclusion of strict algebras into pseudo‑algebras (or lax‑algebras) has left 2‑adjoints when T‑Alg_s is sufficiently cocomplete. The left adjoints are precisely the pseudomorphism and lax‑morphism classifiers. Theorem 3.1 generalizes the classical result that these inclusions give rise to lax‑idempotent pseudomonads.

Building on Theorem 3.1, the author proves Theorem 4.5, which characterizes lax‑idempotent pseudomonads on any 2‑category that possesses all colax bilimits of arrows. Theorem 4.8 refines Beck’s monadicity theorem for pseudomonads: if a pseudofunctor U is locally full and preserves colax bilimits of arrows, then the induced pseudomonad is lax‑idempotent and U is monadic in the bicategorical sense. This version of Beck’s theorem is easier to verify because it replaces the usual preservation of all limits/colimits with the single colax‑bilimit condition.

In summary, the paper reduces the verification of lax‑idempotency—a condition that traditionally requires checking a whole family of adjunctions—to the existence and preservation of a specific type of comma‑object (colax bilimit) together with a modest local fullness assumption. This yields a clear, verifiable criterion applicable to many standard 2‑categorical constructions, thereby simplifying the analysis of co‑completion processes, monadicity, and related structures in higher‑category theory.