I apologize for making a "mess" by including symmetry where it was not supposed to be; I thought I was just repeating what I had heard elsewhere. Perhaps my memory was faulty, or perhaps someone else made the same error (or mess). Jeff's comment probably contributed to my confusion too. (I have read your very nice paper, but since I am only interested in a few of the notions, I didn't take the trouble to memorize all the different shadings of meaning.) Thanks for setting me straight. I am happy with "symmetric autonomous," although I guess it loses out to "compact" on the score of brevity. Mike On Fri, May 14, 2010 at 9:43 AM, Peter Selinger <selinger@mathstat.dal.ca> wrote:
Argh, Michael, you have managed to make a mess of the existing terminology. The terminology is confusing, but it is actually settled. While many concepts have more than one name, thankfully no name refers to more than one concept so far (and I am working hard to keep it that way - for example by discouraging redefinitions of "autonomous"). Here are, for reference, the four most common notions of (1-)categories with duals:
(1) An "autonomous category" is a monoidal category where every object has a left dual and a right dual. Note that it is not assumed to be symmetric. There is also the notion of a "left autonomous category", where only left duals are assumed, and analogously "right autonomous category". Note that duals, where they exist, are unique up to isomorphism, so being autonomous is a property of monoidal categories, not an additional structure.
"Rigid category" is a synonym of "autonomous category", preferred by certain communities of authors.
(2) A "pivotal category" is an autonomous category equipped with a monoidal natural isomorphism A -> A**. (A right autonomous category with such an isomorphism is automatically left autonomous too, so the right/left distinction does not apply to pivotal categories).
"Sovereign category" is a synonym of "pivotal category" used by Freyd and Yetter in one paper, but it does not seem to have caught on. It was a word play suggesting something that is even more than autonomous.
(3) A "tortile category" is a braided pivotal (equivalently balanced autonomous) category satisfying theta* = theta (where theta is the twist).
"Ribbon category" is a synonym of "tortile category", preferred by certain communities of authors.
(4) A "compact closed category" is a tortile category that is symmetric (as a balanced monoidal category), or equivalently, an autonomous symmetric monoidal category.
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