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/ ]