Re: bilax_monoidal_functors

[Andre Joyal <joyal.andre@uqam.ca>]

Dear Michael,

> Compact means small, finite,
> bounded, inaccessible by directed joins, etc. and "rigid" means "having few
> automorphisms," and I don't see that there is anything very compact or
> rigid about such categories.  The only relationship I can think of is that a
> compact subset of a Hausdorff space is closed, and a symmetric monoidal
> category with duals for objects is also automatically closed, but of course
> these two meanings of "closed" are totally different.  Perhaps someone
> can enlighten me?

 
I guess that in the category of R-modules over a commutative ring R, 
a module M has a (good) dual iff it is finitely generated projective
iff the endo-functor functor Hom(M,-) preserves all colimits
(M is *compact* in a strong sense). 

The rigidity terminology may have something to do with Tanaka duality.
If C is a rigid  monoidal category, then any monoidal natural 
transformation u:F-->G between (strong) monoidal functors C-->E
(where E is a monoidal category) is invertible.

I would prefer a different terminology for monoidal categories with duals.

What about "auto-dual monoidal category"?

It as a bit like "autonomous" category.

Best,
André





 



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]