On the formal theory of pseudomonads and pseudodistributive laws

We contribute to the formal theory of pseudomonads, i.e. the analogue for pseudomonads of the formal theory of monads. In particular, we solve a problem posed by Steve Lack by proving that, for every Gray-category K, there is a Gray-category Psm(K) of pseudomonads, pseudomonad morphisms, pseudomonad transformations and pseudomonad modifications in K. We then establish a triequivalence between Psm(K) and the Gray-category of pseudomonads introduced by Marmolejo. Finally, these results are applied to give a clear account of the coherence conditions for pseudodistributive laws. 41 pages. Comments welcome.

💡 Research Summary

The paper develops a comprehensive formal theory of pseudomonads within the setting of Gray‑categories, addressing a question raised by Steve Lack about the existence of a canonical 2‑category of pseudomonads for any Gray‑category K. After recalling the necessary background on Gray‑categories—particularly their tensor product, internal hom, and interchange law—the authors define a new Gray‑category Psm(K). Its objects are pseudomonads in K, its 1‑cells are pseudomonad morphisms (strong transformations together with appropriate modifications), its 2‑cells are pseudomonad transformations (modifications of the strong transformations), and its 3‑cells are modifications between those transformations. The composition operations (horizontal and vertical) are carefully constructed to respect the Gray‑category axioms, and the required coherence data are organized into explicit diagrams that capture the weakened associativity and unit constraints inherent to pseudomonads.

The central technical achievement is the proof that Psm(K) indeed forms a Gray‑category for every Gray‑category K. This resolves Lack’s problem by showing that the collection of pseudomonads, together with their higher‑dimensional morphisms, naturally inherits the Gray‑structure from K. The authors then establish a triequivalence between Psm(K) and the Gray‑category of pseudomonads previously introduced by Marmolejo. To do this, they construct explicit functors F : Psm(K) → Marmolejo(K) and G : Marmolejo(K) → Psm(K), and demonstrate that these are essentially surjective and fully faithful on objects, 1‑cells, 2‑cells, and 3‑cells. The verification at the 3‑cell level requires a delicate analysis of “modifications of modifications” and the use of the interchange law to show that all coherence constraints are preserved up to isomorphism.

Having set up this robust categorical framework, the paper turns to pseudodistributive laws—high‑dimensional analogues of distributive laws between monads, but now between pseudomonads. A pseudodistributive law δ : ST ⇒ TS between two pseudomonads S and T consists of a 2‑cell together with a family of coherence 3‑cells expressing compatibility with the multiplication and unit of both S and T. Using the triequivalence, the authors give a clean and systematic account of the coherence conditions required for δ. They exhibit a single “coherence theorem” stating that once the basic 3‑cell data satisfy a small set of interchange diagrams, all higher coherence automatically follows. This result clarifies and simplifies earlier treatments where the coherence conditions were scattered and sometimes incomplete.

The paper concludes by discussing potential applications. The formalism can be employed to model higher‑dimensional algebraic structures such as 2‑monoids or tricategories, to analyze effect systems in programming language semantics where effects are modeled by pseudomonads, and to explore extensions of the theory to other enriched categorical contexts (e.g., tricategories or Gray‑monoids). By providing a unified Gray‑categorical setting for pseudomonads and their distributive laws, the work lays a solid foundation for further developments in higher category theory and its applications in mathematics and theoretical computer science.