Iterated distributive laws

We give a framework for combining $n$ monads on the same category via distributive laws satisfying Yang-Baxter equations, extending the classical result of Barr and Wells which combines two monads via one distributive law. We show that this corresponds to iterating $n$-times the process of taking the 2-category of monads in a 2-category, extending the result of Street characterising distributive laws. We show that this framework can be used to construct the free strict $n$-category monad on $n$-dimensional globular sets; we first construct for each $i$ a monad for composition along bounding $i$-cells, and then we show that the interchange laws define distributive laws between these monads, satisfying the necessary Yang-Baxter equations.

💡 Research Summary

The paper develops a systematic theory for combining an arbitrary finite collection of monads on a single category, extending the classical Barr‑Wells result that handles only two monads via a single distributive law. The authors introduce the notion of a “distributive series” of monads: given monads (T_{1},\dots ,T_{n}) on a category (\mathcal{C}), for every pair (i>j) there must be a natural transformation (\lambda_{ij}:T_{i}T_{j}\Rightarrow T_{j}T_{i}) satisfying the usual compatibility axioms with the monad structures of (T_{i}) and (T_{j}). In addition, for every triple (i>j>k) the three possible ways of “braiding’’ the three monads must coincide; this is expressed by a Yang‑Baxter equation (Equation 2.1). The main theorem (Theorem 2.1) proves that these two conditions are sufficient to guarantee that the ordered composite (T_{1}T_{2}\cdots T_{n}) carries a canonical monad structure, and that any intermediate split of the sequence into two blocks also yields a distributive law of the right block over the left block. Consequently all possible parenthesisations give the same monad.

The authors place this construction in a 2‑categorical context using Street’s theory of monads in a 2‑category. For any 2‑category (\mathcal{B}) one can form the 2‑category (\mathrm{Mnd}(\mathcal{B})) of monads, monad morphisms and monad transformations. Iterating this construction yields (\mathrm{Mnd}^{n}(\mathcal{B})). By induction they show that a 0‑cell of (\mathrm{Mnd}^{n}(\mathcal{B})) is precisely a distributive series of (n) monads together with the required Yang‑Baxter equations. Thus the abstract 2‑categorical picture exactly captures the concrete algebraic conditions introduced earlier.

The theory is applied to the construction of the free strict (n)-category monad on the category of (n)-globular sets. For each dimension (i) (with (0\le i<n)) a monad (T_{i}) is defined that freely adds composition along bounding (i)-cells while leaving lower‑dimensional structure untouched. The classical interchange law for strict (n)-categories provides, for each pair (i<j), a natural transformation (\lambda_{ji}:T_{j}T_{i}\Rightarrow T_{i}T_{j}). These interchange transformations satisfy the Yang‑Baxter equations for all triples (i<j<k). Hence, by the main theorem, the composite (T_{0}T_{1}\cdots T_{n-1}) is the free strict (n)-category monad. This construction mirrors the intuition that a strict (n)-category is a globular set in which every 2‑dimensional sub‑globular set is a strict 2‑category, i.e. all higher‑dimensional interchange laws hold strictly.

Beyond this primary example, the paper discusses the relevance of the framework to “interchange‑strict” higher categories, where all structure is strict except possibly the unit laws. It points to recent work by Joyal‑Kock and Simpson, suggesting that such semi‑strict models may capture homotopy (n)-types. The authors also indicate future work on comparing Trimble’s definition of (n)-categories (which has strict interchange but weak units) with Batanin’s operadic approach, using the iterated distributive law machinery developed here.

In summary, the paper provides:

1. A minimal set of axioms (pairwise distributive laws + Yang‑Baxter equations) that guarantee the existence of a well‑defined composite monad for any finite family of monads.
2. A 2‑categorical characterisation of these structures via iterated monad 2‑categories, establishing a deep conceptual link between distributive series and higher‑dimensional monad theory.
3. A concrete application to the free strict (n)-category monad, showing how the interchange laws of higher categories naturally give rise to the required distributive laws.
4. Perspectives on how this framework can be leveraged in the study of semi‑strict higher categories and in comparisons between different models of higher categorical structures.