Dear category theorists, Does the concept of "filtered monad" exist in the literature? Here are two basic models of what I have in mind. 1. Let C be the category of sets, let F:C->C be the set underlying the free monoid on S, and let F_n(S) be the subset of F(S) consisting of words of length at most n. Then the monad structure map F o F-->F restricts to maps F_m o F_n-->F_{mn}, and F_1 is the identity functor. 2. Let C be the category of R-modules (R a given ring), F(M) is the tensor product R[x] \otimes_R M, and F_n(M) is the sub-R-module M + xM + ... + x^n M of F(M). Then the monad structure map F o F --> F restricts to a map F_m o F_n --> F_{m+n}, and F_0 is the identity functor. So, in the first example, I'd say that the monad F is filtered by the ordered monoid of non-negiative integers under multiplication, and in the second example, it's filtered by that under addition. There seems to be a pretty obvious way of formalizing this, and since many monads in practice come with such a structure, I'd guess that this concept is in the literature, but I didn't find anything on the internet or in the textbooks on my shelf. But perhaps that's because it's not called a "filtered monad" or because it's a special case of a general concept with a completely different name. So, does this concept exist in the literature? I'm writing something about a particular monad with a a filtered structure, and after I define it, I'd like to have the sentence "In the language of [5], the functors F_n endow F with a filtered monad structure." Yours, James Borger ps I'm not at the moment a subscriber to the mailing list, so please cc to me any responses to the list. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]