Dear James, There is a pretty obviuous way to formalize it, but very few does it. It's the existence/construction thing. "Free" is an angel and devil is in the detail. What I mean is just to construct each F_m and see what happens in this transfinite induction. At the end you pick up all F_m's to produce F = \cup_m F_m, and formulating F_m is the main formalizing part. When you do this over C=Set, it looks natural, and almost canonic, but when moving over to other C's it is not all that jazz. Many will say, yes, it's already in the literature, "it's been there for the past 50 years", "everything is pretty standard", and so on and so forth. This angelic self-confidence makes no good in your case, I believe(!). You need to work out the filtered steps and complete the transfinite induction. eta : id -> F often behaves, and is sometimes even unique, and obviously mu : F o F -> F requires detail. We are doing these things for the term monad. Let me know if you need some references. In your case your are a bit more abstract algebra, and terms mean you are more universal algebra, but that shouldn't make any difference, I guess. Cheers, Patrik On Thu, 27 May 2010, James Borger wrote:
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.
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