Subject: categories: question about symmetric monoidal categories

From: Gaunce Lewis <[email-masked]>
Date: Wed, 14 Mar 2001 23:55:00 -0500

I'm looking for a reference for two notions related to symmetric monoidal 
categories and symmetric monoidal functors.

Assume that A and B are symmetric monoidal categories, and denote the 
product operation on both by \tensor.  Assume that L is a lax (symmetric) 
monoidal functor from A to B.

Here are the two closely related notions:

1.  Let G be a functor from A to B for which there is a natural transformation

L(a_1) \tensor G(a_2) -> G(a_1 tensor a_2)

for all a_1, a_2 in A making the obvious diagrams commute (one an 
associativity diagram and the other a unit diagram).  One can think of L as 
acting on G, or of G as "modular" over L  (I'm working in the abelian 
category context, so it is natural to think in terms of rings and modules).

2.  Now suppose that L has a right adjoint R and G has a left adjoint 
F.  Also assume that L is strictly (or is it strongly) monoidal, and that R 
has the "dual" monoidal structure.  There seems to be a natural common 
source for an action of L on G and of R on F.  This source is a natural 
isomorphism of the form

F(L(a) \tensor b) \iso a \tensor F(b)

for all a in A and b in B.  If the appropriate diagrams commute, then it is 
easy to derive the actions of L on G and R on F  from this 
isomorphism.  What intrigues me most about this situation is the way in 
which the adjunctions go in the opposite direction.

These two situations appear throughout some stuff in algebraic topology 
that I am working on right now.  Is there a reference in the literature 
where these notions are discussed?

Here is a standard (somewhat degenerate) example of this situation.  Let h: 
S -> T be a homomorphism of commutative rings.  There is an associated 
pullback functor from T-Mod to S-mod which plays the role of the functors R 
and F above.  This functor has both left adjoint (playing the role of L) 
and right adjoint (playing the role of G).  The pullback functor R is lax 
monoidal, and its left adjoint L is strict  monoidal.  The functor L acts 
on G, and R = F acts on F via the lax monoidal structure on R.  In the 
examples that show up in my topological research, R and F are distinct, and 
neither F nor G have monoidal structures.

Thanks,
Gaunce Lewis