From: James Borger <james.borger@anu.edu.au>
To: categories@mta.ca
Subject: filtered monads? [resent]
Date: Thu, 27 May 2010 13:06:14 +1000 [thread overview]
Message-ID: <E1OIPAn-0007Gc-Fc@mailserv.mta.ca> (raw)
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.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
next reply other threads:[~2010-05-27 3:06 UTC|newest]
Thread overview: 6+ messages / expand[flat|nested] mbox.gz Atom feed top
2010-05-27 3:06 James Borger [this message]
2010-05-29 20:55 ` Richard Garner
2010-05-30 4:05 ` filtered monads? Patrik Eklund
2010-05-30 11:37 ` John Baez
2010-05-30 15:40 ` Peter May
[not found] ` <AANLkTimKrlok4-5_VtORytawuPBh6v7sBu3dAB9Htj1n@mail.gmail.com>
2010-06-01 1:11 ` James Borger
Reply instructions:
You may reply publicly to this message via plain-text email
using any one of the following methods:
* Save the following mbox file, import it into your mail client,
and reply-to-all from there: mbox
Avoid top-posting and favor interleaved quoting:
https://en.wikipedia.org/wiki/Posting_style#Interleaved_style
* Reply using the --to, --cc, and --in-reply-to
switches of git-send-email(1):
git send-email \
--in-reply-to=E1OIPAn-0007Gc-Fc@mailserv.mta.ca \
--to=james.borger@anu.edu.au \
--cc=categories@mta.ca \
/path/to/YOUR_REPLY
https://kernel.org/pub/software/scm/git/docs/git-send-email.html
* If your mail client supports setting the In-Reply-To header
via mailto: links, try the mailto: link
Be sure your reply has a Subject: header at the top and a blank line
before the message body.
This is a public inbox, see mirroring instructions
for how to clone and mirror all data and code used for this inbox;
as well as URLs for NNTP newsgroup(s).