Mike wrote: One reason I like "autonomous" to mean a symmetric monoidal category in

which all objects have duals is that the only alternative names I have heard for such a thing convey misleading intuition to me. They are sometimes called "compact closed" or (I think) "rigid" monoidal categories...

Yes, I think "rigid" is traditional in algebraic geometry. Perhaps some wiser head could explain how it originated! Personally I often use "compact', since "compact closed" seems redudant. And if I were king of the world, I'd use "with duals for objects". Regarding the relation of this use of "compact" to the use in topology: 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 don't know if this is what people were thinking when they first applied "compact" to categories, or just my own rationalization, but: A compact subset is closed, but it has a very nice property: its image under any continuous map is again closed. Similarly a compact category is a closed, but it has a very nice property: its essential image under any symmetric monoidal functor is again compact. I don't claim this justifies the terminology, but it helped me learn to live with it. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]