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A recognition criterion for lax-idempotent pseudomonads

John Bourke

TL;DR

The paper addresses when a pseudomonad arising from a biadjunction is lax-idempotent by introducing a colax bilimit property that is straightforward to verify in examples. It proves a core recognition theorem: if $U$ is locally full and each arrow has a colax bilimit of a specified form, then the induced pseudomonad is lax-idempotent, with dual and pseudo variants. It then provides a characterisation: a pseudomonad $T$ on a suitably complete 2-category is lax-idempotent exactly when the associated forgetful 2-functor is locally full and satisfies the colax bilimit property, using Kan injectives to lift bilimits to pseudo-algebras. Finally, it connects these ideas to a refined Beck-type monadicity framework through a 2-equivalence with Kan injectives, clarifying when monadic descriptions arise in 2-dimensional settings.

Abstract

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.

A recognition criterion for lax-idempotent pseudomonads

TL;DR

The paper addresses when a pseudomonad arising from a biadjunction is lax-idempotent by introducing a colax bilimit property that is straightforward to verify in examples. It proves a core recognition theorem: if is locally full and each arrow has a colax bilimit of a specified form, then the induced pseudomonad is lax-idempotent, with dual and pseudo variants. It then provides a characterisation: a pseudomonad on a suitably complete 2-category is lax-idempotent exactly when the associated forgetful 2-functor is locally full and satisfies the colax bilimit property, using Kan injectives to lift bilimits to pseudo-algebras. Finally, it connects these ideas to a refined Beck-type monadicity framework through a 2-equivalence with Kan injectives, clarifying when monadic descriptions arise in 2-dimensional settings.

Abstract

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.

Paper Structure

This paper contains 9 sections, 7 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Basic $2$-categorical concepts and notation
  5. "Adjoint adjunctions" and adjunctions with invertible counit
  6. Colax bilimits of arrows
  7. Recognising lax-idempotent biadjunctions and pseudomonads
  8. A characterisation of lax-idempotent pseudomonads
  9. Kan injectives and lax-idempotent pseudomonads

Key Result

Lemma 2.1

Let $f \colon A \to B$ and $u\colon B \to A$ in a $2$-category $\mathcal{C}\xspace$. Then $f \dashv u$ if and only if there exists 2-cells $\eta \colon 1 \to uf$ and $\varepsilon \colon fu \to 1$ such that that the composites $u\varepsilon \circ \eta_u \colon u \to ufu \to u$ and $\varepsilon_f \ci

Theorems & Definitions (17)

  • Lemma 2.1: Lemma 2.1.11 of RV
  • Corollary 2.2
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Example 3.5
  • Theorem 4.1: Theorem 2.7 of DLS
  • Remark 4.2
  • ...and 7 more