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. Of course (4) => (3) => (2) => (1). There are a number of in-between concepts, which are generally less natural and of interest primarily for technical reasons. Please see my recent survey "A survey of graphical languages for monoidal categories" for a far more detailed discussion (http://arxiv.org/abs/0908.3347). Particularly the table on p.60 shows the whole taxonomy on one page. I will briefly mention two of the "less natural" notions: * A "braided autonomous" category is a monoidal category that is both braided and autonomous (with no axioms relating the two structures). This notion is entirely uninteresting, except to note that a braided left autonomous category is automatically right autonomous, due to the existence of isomorphisms A -> A**, and to note that it is NOT automatically pivotal, because said isomorphism is not monoidal. * A "braided pivotal category" is a monoidal category that is both braided and autonomous (again with no axioms relating the two structures). This notion is also completely uninteresting, except to note that a braided pivotal category is exactly the same thing as a balanced autonomous category (because on a braided autonomous category, giving a pivotal structure is precisely equivalent to giving a balanced structure). Such categories were studied by Freyd and Yetter, but arguably they were superseded by the better notion of tortile categories. These categories have a graphical language up to "regular isotopy", which means that one of the three Reidemeister moves fails. I have come to the opinion that it is a very good thing that notions (1)-(3) above have distinct names, and are not just distinguished by adjectives. It would be tempting to call a pivotal category a "[something] autonomous category", and to call a tortile category for example "[something else] braided pivotal" or "[something else] balanced autonomous". But the most natural adjective for [something] would be "pivotal", and the most natural adjective for [something else] would be "tortile", which would only make the names longer without adding any information. I do believe that the term (4) "compact closed" is something of an oddity, since "symmetric autonomous" would be similarly succinct, more systematic, and much more descriptive - in fact, it requires no additional definition if "symmetric" and "autonomous" have already been defined. Also, as Michael has pointed out, the name "compact" here has little to do with its usual meaning in mathematics. If this concept were invented today, one should certainly call it "symmetric autonomous". But in light of the fact that "compact closed" was historially the first of notions (1)-(4) defined, and that the term "compact closed" is already extremely well-known and wide-spread, this is one case where I believe it is better to stick with the existing terminology rather than trying to force it into a taxonomy. That doesn't mean that slow incremental change is not possible. For example, it seems reasonable to write "note that a compact closed category is the same as a symmetric autonomous category" whenever giving the definition. Perhaps after a few years, people will write "note that a symmetric autonomous category is also known as a compact closed category", and maybe after many more years, the term "symmetric autonomous" will even become standard. But such changes should come about through repeated and incremental use by a community, and not by unilateral choices. As a general rule, I think it is good manners when changing terminology (or inventing new unsystematic terminology) to give the old (or systematic) terminology in parentheses at least once per paper. In Oxford, compact closed categories are nowadays called "compact categories". I try not to follow this convention because it replaces one bad term with a shorter, but equally bad one. It would also clash with the standard meaning of "compact" in cases where the category was actually a topological space. But it seems like a benign enough change and is catching on rapidly. My last comment is that, unlike what Jeff Egger claimed, "autonomous category" is not a special case of "*-autonomous category", because no symmetry is assumed in autonomous categories. Unless of course one first drops symmetry from the definition of *-autonomous categories, as Jeff has also suggested. As it stands, neither of "autonomous" and "*-autonomous" implies the other, which is perfectly fine in my opinion, since they are two different words. -- Peter Michael Shulman wrote:
On Mon, May 10, 2010 at 2:28 PM, Jeff Egger <jeffegger@yahoo.ca> wrote:
the fact that "autonomous category" is a special case (and, from one point of view, a rather uninteresting special case) of "star-autonomous category", whereas it sounds like "star-autonomous category" should mean an "autonomous category" with some extra structure.
I agree, it does sound like that, but there is at least a long tradition of such names in mathematics (not that that makes them a good thing). (http://ncatlab.org/nlab/show/red+herring+principle)
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, but "compact" and "rigid" are words with definite and inapplicable intuitive meanings for me. 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?
Mike
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