Dear category theorists, Many thanks for the responses to my question. I now have no doubt that the "right" way of defining M-filtered monad, where M is an ordered monoid, is as a lax monoidal functor as described by John Baez below. I admit that it doesn't capture every aspect of the examples I gave, but that's OK -- general definitions never do. So then the precise form of my original question would then be whether there is a reference, suitable for a non-category-theoretic paper, where this definition is written down purely in the language of functors and natural transformations. Since the monoidal category language is so useful here, such a reference probably wouldn't have been written by a category theorist. But since filtered monads are so common even in contexts where people don't talk much about monoidal categories, it *should* exist. So if anyone happens to know of one, please let me know. Yours, James Borger On 2010/05/30, at 9:37 PM, John Baez wrote:

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.

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