James Borger wrote: Does the concept of "filtered monad" exist in the literature? I don't know, but your concept does indeed seem to come up a lot... I'm tempted to formalize it a bit. Let's look at an example:
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.
It seems that *one* aspect of what you've got here is a lax monoidal functor from the multiplicative monoid of natural numbers to End(C). Such a thing consists of a functor F_n: C -> C for each natural number n, together with natural transformations F_m o F_n => F_{mn} and 1_C => F_1 satisfying appropriate coherence laws. (For example, you can build two natural transformations from F_m o F_n o F_k to F_{mnk}, but they're equal.) But there's more: you also have natural transformations F_m => F_n whenever m is less than or equal to n. And this seems to be an important aspect of the intuition that's making you use the word "filtered". So, instead of treating the natural numbers as a mere monoid, I think you are treating them as a monoidal poset. In other words: there's a monoidal category M with natural numbers as objects, a single morphism m -> n whenever m is less than or equal to n, and multiplication as tensor product. And, I think you've got a lax monoidal functor F: M -> End(C) It's possible that whenever C has colimits, you can take a pointwise colimit of the functors F_n in this situation and get an actual monad. But it seems your lax monoidal functor is a bit better than average: you have F_1 = 1_C, not just a natural transformation from 1_C to F_1. 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.
Here it seems you're using a different monoidal poset, coming from the additive monoid of natural numbers with its usual ordering. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]