filtered monads?

filtered monads?

[categories]

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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Re: filtered monads?

[James Borger]

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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Re: filtered monads?

[Peter May]

  From my perspective, the original definition of
operad was designed to produce a (nonnegatively)
filtered monad with the same algbras. That was
central to all of the early applications, although
I don't think I spelled it out categorically.
I did spell it out when using it for calculations.

The filtration on the monad C was given by subfunctors
F_jC such that the unit Id >--> C factored through F_1C
and the product CC >--> C was given by restricted
functors F_kCF_jC >--> F_{kj}C.

(I also preferred F_0C to be the constant functor at
the unit object of the underlying symmmetric monoidal
category, but that was negotiable).


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Re: filtered monads?

[John Baez]

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


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Re: filtered monads?

[Patrik Eklund]

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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