Dear Jim, A suitable reference would be: Ross Street, "Two constructions on lax functors", Cahiers de topologie et géométrie différentielle catégoriques, tome 13, no 3 (1972) p. 217--264. The relevance of this paper to your notion is as follows. We learnt from Bénabou that a monad on a category may be expressed as a lax functor of 2-categories 1 --> Cat, with the unique object of 1 being sent to the category C on which the monad subsists; the unique morphism of 1 being sent to the underlying functor F of the monad; and the lax preservation of identities and composition yielding the natural transformations eta : 1 -> F and mu : FF -> F, together with the monad laws they are required to satisfy. Now given a monoid M, your notion of filtered monad with respect to M may similarly be expressed as a lax functor into Cat; but one whose domain is no longer the terminal category, but rather the monoid M, viewed as a one-object category. Indeed, such a lax functor picks out, firstly, a category C, this being the image of the unique object; then, for every element m of the monoid, an endofunctor F_m of C; and finally, by virtue of the lax functoriality, coherent natural transformations F_n o F_m -> F_{nm} and id => F_1. (Note that this last condition slightly generalises your situation, in in which you required that F_1 should actually be the identity functor.) The paper of Ross's which I reference above is concerned precisely with lax functors A -> Cat for some ordinary category A; and states one of its reasons for studying such as being "to provide a generalization of the theory of triples". It therefore seems an eminently suitable reference for your purposes. Best wishes, Richard --On 27 May 2010 13:06 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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