(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.