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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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/ ]