Actions of monoidal functors [was Re: Arens product]

Actions of monoidal functors [was Re: Arens product]

[Jeff Egger]

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.

Re: Actions of monoidal functors [was Re: Arens product]

[Ross Street]

Dear Jeff

"Monoid" and "object on which a monoid acts" make sense in any
multicategory. A monoidal functor is a monoid in the convolution
multicategory [V,W] of functors from V to W. The T of which you speak
is an object on which M acts in [V,W].

Regards,
Ross

On 18/07/2007, at 4:11 AM, Jeff Egger wrote:

> 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.
> --------------------
> 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?