Applied Categorical Structures

, Volume 27, Issue 6, pp 567–617 | Cite as

Distributive Laws via Admissibility

  • Charles WalkerEmail author


This paper concerns the problem of lifting a KZ doctrine P to the 2-category of pseudo T-algebras for some pseudomonad T. Here we show that this problem is equivalent to giving a pseudo-distributive law (meaning that the lifted pseudomonad is automatically KZ), and that such distributive laws may be simply described algebraically and are essentially unique [as known to be the case in the (co)KZ over KZ setting]. Moreover, we give a simple description of these distributive laws using Bunge and Funk’s notion of admissible morphisms for a KZ doctrine (the principal goal of this paper). We then go on to show that the 2-category of KZ doctrines on a 2-category is biequivalent to a poset. We will also discuss here the problem of lifting a locally fully faithful KZ doctrine, which we noted earlier enjoys most of the axioms of a Yoneda structure, and show that a bijection between oplax and lax structures is exhibited on the lifted “Yoneda structure” similar to Kelly’s doctrinal adjunction. We also briefly discuss how this bijection may be viewed as a coherence result for oplax functors out of the bicategories of spans and polynomials, but leave the details for a future paper.


KZ-doctrines Lax-idempotent pseudomonads Pseudo-distributive laws 

Mathematics Subject Classification

18A35 18C15 18D05 


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsMacquarie UniversitySydneyAustralia

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