Re: tensorial strength

[rjwood@mathstat.dal.ca (RJ Wood)]

(I wrote this to Jeff and Anders a few minutes ago. Since Anders has replied
to all I am circulating it more widely.)

Dear Jeff and Anders
In my thesis (supervised by Bob Pare at Dalhousie in 1976) I considered
(for reasons I won't labour here) (V^op,set)-categories, for monoidal V.
The monoidal structure I took on (V^op,set) was Brian Day's convolution.
Necessarily, a (V^op,set)-category A gives rise to a functor
P:A^op x V^op x A ---> set and this reveals that there are three special
kinds of (V^op,set)-categories:

1) those for which P(-,v,b) is representable, for all v and b, by {v,b} say

2) those for which P(a,-,b) is representable, for all a and b, by [a,b] say

3) those for which P(a,v,-) is representable, for all a and v, by v@a say.

Ordinary V-categories are given by 2). The others have been known
by various names but they are best understood in terms of actions.

Now suppose that F:A--->B is a (V^op,set)-functor where A is of type i)
and B is of type j) as above. Each of the nine possibilities admits a
simple encoding of the enrichment as displayed in the following table:

i)\j)	      1)		      2)		      3)
  1)	F{v,b}--->{v,Fb}	v--->[F{v,b},Fb]	v@F{v,b}--->Fb
  2)	Fa--->{[a,b],Fb}	[a,b]--->[Fa,Fb]	[a,b]@Fa--->Fb
  3)	Fa--->{v,F(v@a)}	v--->[Fa,F(v@a)]	v@Fa--->F(v@a)

Susan Niefield, Robin Cockett, and I are writing a paper whose sequel
will deal with later developments of this topic.
Best to all, Richard

> 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.