Hi - Given a monad T: C -> C, any algebra A of this monad gives rise to a simplicial algebra of this monad, say BA, via the bar construction. It's finally occurred to me to wonder: how do various extra structures on the monad give extra structures on the bar construction? For example, suppose C is monoidal. Then we say T is a "strong" monad if it's a monad not just in Cat but in the 2-category of C-actions. This amounts to having natural transformations A tensor TB -> T(A tensor B) obeying various laws. If T is strong, what does this do for its bar construction? Similarly, we say T is a "monoidal" monad if it's a monad not just in Cat but in the 2-category of monoidal categories, lax monoidal functors and monoidal natural transformations. This amounts to having a natural transformation TA tensor TB -> T(A tensor B) and a morphism I -> TI obeying various laws. If T is monoidal, what does this do for its bar construction? I could ask the same question for commutative monads, or for a pair of monads related by a distributive law. I would enjoy figuring these out, but someone must have done it already, and I'd rather enjoy doing something new. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]