Dear Categorists - I made some mistakes in my account of Cheng's work, saying "monad on a category" at some points where I should have said "monad in a 2-category". Here's a fixed version: Street noted that we can talk about monads, not just in the 2-category of categories, but in any 2-category. I actually explained monads at this level of generality back in "week89". Indeed, for any 2-category C, there's a 2-category Mnd(C) of monads in C. And, he noted that a monad in Mnd(C) is a pair of monads in C related by a distributive law! That's already mindbogglingly beautiful. According to Eugenia, it's practically the last sentence of Street's paper. But in her new work: 18) Eugenia Cheng, Iterated distributive laws, available as arXiv:0710.1120. she goes a bit further: she considers monads in Mnd(Mnd(C)), and so on. Here's the punchline, at least for today: she shows that a monad in Mnd(Mnd(C)) is a triple of monads F, G, H related by distributive laws satisfying the Yang-Baxter equation: \F G/ |H F| G\ /H \ / | | \ / / | | / / \ | | / \ / \ | \ / \ | \ / \ / | | / = / | | / \ / \ | | / \ / \ | \ / | | \ / \ / | | \ / / | | / / \ | | / \ /H \G |F H| G/ \F This is also just what you need to make the composite FGH into a monad! By the way, the pathetic piece of ASCII art above is lifted from "week1", where I first explained the Yang-Baxter equation. That was back in 1993. So, it's only taken me 14 years to learn that you can derive this equation from considering monads in the category of monads in the category of monads in a 2-category. Also, I should have given a reference to earlier work on Gelfand duality in a topos: Bernhard Banachewski and Christopher J. Mulvey, A globalisation of the Gelfand duality theorem, Ann. Pure Appl. Logic 137 (2006), 62-103. Also available at http://www.maths.sussex.ac.uk/Staff/CJM/research/pdf/globgelf.pdf They show that any commutative C*-algebra A in a Grothendieck topos is canonically isomorphic to the C*-algebra of continuous complex functions on the compact, completely regular locale that is its maximal spectrum (that is, the space of homomorphisms f: A -> C). Conversely, they show any compact completely regular locale X gives a commutative C*-algebra consisting of continuous complex functions on X.