Iterated distributive laws

Abstract

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.