From: Jeff Egger <jeffegger@yahoo.ca>
To: categories@mta.ca
Subject: Actions of monoidal functors [was Re: Arens product]
Date: Tue, 17 Jul 2007 14:11:46 -0400 (EDT) [thread overview]
Message-ID: <E1IAxKz-0005Hf-Qb@mailserv.mta.ca> (raw)
Dear all,
Anders Kock's reply to Yemon Choi gives me a good opportunity to pose a
question which I have been meaning to ask the list for a while:
> The V-enrichment ("strength") of an endofunctor T on V can be encoded
> without reference to the closed structure of V as a transformation
> T(A)@B-->T(A@B) ("tensorial strength", introduced in [4]).
This notion of "tensorial strength" is just a special case of what
I would call "an action of a monoidal functor on a (mere) functor".
Specifically, it is a right-action of the identity monoidal functor
on the functor T.
In general, given a monoidal functor M:V-->W and a functor T:V-->W, a
right-action of M on T should be a n.t. of the form T(A)@M(B)-->T(A@B)
satisfying the obvious associativity and unitality axioms.
For instance, if we regard a G-graded algebra as a monoidal functor G-->Vec,
then a right-action of this on a mere functor G-->Vec is precisely the same
thing as a G-graded right-module. [Here the monoid G (G can also stand for
grading-object!) is considered as a discrete monoidal category.]
I have always assumed that this concept is well-known, but I haven't
succeeded in finding a reference in the literature for it... perhaps
some of the more well-read readers of this list could help me out?
Cheers,
Jeff.
P.S. Upon reviewing [4], I see that there is a more general notion of
tensorial strength which can be applied to a functor A-->B whenever A
and B are tensored over some monoidal category V. There is a similar
adaptation of the notion of action of a monoidal functor V-->W to
functors A-->B whenever A is tensored over (or I would say, acted on by)
V, and B over (by) W.
> [4] Strong functors and monoidal monads, Archiv der Math. 23 (1972),
> 113-120.
next reply other threads:[~2007-07-17 18:11 UTC|newest]
Thread overview: 2+ messages / expand[flat|nested] mbox.gz Atom feed top
2007-07-17 18:11 Jeff Egger [this message]
2007-07-19 4:46 Ross Street
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=E1IAxKz-0005Hf-Qb@mailserv.mta.ca \
--to=jeffegger@yahoo.ca \
--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).