Re: filtered monads?

From my perspective, the original definition of operad was designed to produce a (nonnegatively) filtered monad with the same algbras. That was central to all of the early applications, although I don't think I spelled it out categorically. I did spell it out when using it for calculations. The filtration on the monad C was given by subfunctors F_jC such that the unit Id >--> C factored through F_1C and the product CC >--> C was given by restricted functors F_kCF_jC >--> F_{kj}C. (I also preferred F_0C to be the constant functor at the unit object of the underlying symmmetric monoidal category, but that was negotiable). [For admin and other information see: http://www.mta.ca/~cat-dist/ ]