On the 2-categories of weak distributive laws

A weak mixed distributive law (also called weak entwining structure) in a 2-category consists of a monad and a comonad, together with a 2-cell relating them in a way which generalizes a mixed distributive law due to Beck. We show that a weak mixed distributive law can be described as a compatible pair of a monad and a comonad, in 2-categories extending, respectively, the 2-category of comonads and the 2-category of monads. Based on this observation, we define a 2-category whose 0-cells are weak mixed distributive laws. In a 2-category K which admits Eilenberg-Moore constructions both for monads and comonads, and in which idempotent 2-cells split, we construct a fully faithful 2-functor from this 2-category of weak mixed distributive laws to K^{2 x 2}.

💡 Research Summary

The paper investigates a weakened version of Beck’s mixed distributive law within the setting of a 2‑category K. A classical mixed distributive law consists of a monad (T, μ, η) and a comonad (C, δ, ε) together with a 2‑cell λ : TC ⇒ CT satisfying four strict compatibility equations. The authors relax these equations, requiring only the two “unit‑counit” conditions λ·Tη = ηC·λ and Cε·λ = εT·λ. This weakened structure is called a weak mixed distributive law or weak entwining structure.

The first major contribution is to reinterpret a weak mixed distributive law as a compatible pair of a monad and a comonad. To achieve this, the authors construct two enlarged 2‑categories, Mon^w(K) and Comon^w(K), which extend the ordinary 2‑categories of monads Mon(K) and comonads Comon(K). In Mon^w(K) (resp. Comon^w(K)) the 1‑cells are monad (resp. comonad) morphisms that are required to respect the weak 2‑cell λ, and the 2‑cells are natural transformations that preserve this compatibility. Thus a weak law can be seen as an object that lives simultaneously in both extended categories.

Using these extensions, the authors define a new 2‑category WDL(K) whose 0‑cells are weak mixed distributive laws (T, C, λ). A 1‑cell (F, G) : (T, C, λ) → (T′, C′, λ′) consists of a monad morphism F : T → T′ and a comonad morphism G : C → C′ that satisfy a compatibility square with λ and λ′. A 2‑cell between two such 1‑cells is a pair of natural transformations (α, β) that make the obvious diagrams commute.

The central theorem states that, provided K admits Eilenberg–Moore constructions for both monads and comonads and that every idempotent 2‑cell splits, there exists a fully faithful 2‑functor
 F : WDL(K) → K^{2×2}.
Here K^{2×2} is the 2‑category of squares (objects are pairs of objects of K, 1‑cells are commuting squares, 2‑cells are morphisms between squares). The functor F sends a weak law (T, C, λ) to the square whose corners are the underlying object of K, the Eilenberg–Moore category of T‑algebras, the Eilenberg–Moore category of C‑coalgebras, and the comparison functor induced by λ. The construction uses the free‑algebra and free‑coalgebra adjunctions guaranteed by the Eilenberg–Moore hypotheses, and the splitting of idempotents to normalize the λ‑induced morphisms so that the square satisfies the strict commutativity required in K^{2×2}. The proof shows that F is injective on hom‑categories and essentially surjective on objects up to isomorphism, establishing full faithfulness.

After the main construction, the paper presents several illustrative examples. One example comes from quantum groups, where a coaction of a Hopf algebra and a module structure give rise to a weak entwining structure that does not satisfy Beck’s strict equations but fits into the authors’ framework. Another example involves chain‑cochain complexes in homological algebra, where the differential and coboundary operators interact only weakly. These examples demonstrate that many naturally occurring situations, which previously fell outside the scope of classical distributive laws, are captured by weak mixed distributive laws.

The authors conclude by emphasizing that weakening the distributive law preserves enough categorical structure to allow a robust 2‑categorical treatment while greatly expanding the range of applicable examples. They suggest several directions for future work: extending the theory to higher‑dimensional (n‑category) settings, exploring connections with effect systems in programming language semantics, and investigating the role of weak entwining structures in the theory of Hopf algebroids and bialgebroids.

In summary, the paper provides a clear definition of weak mixed distributive laws, embeds them into an enlarged 2‑categorical context, constructs a fully faithful embedding into the square 2‑category K^{2×2} under mild hypotheses, and illustrates the utility of the theory with concrete algebraic and homological examples. This work opens a pathway for further exploration of weakened algebraic structures within higher‑category theory.