categories | 14 May 16:43 2010
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Re: terminology (was: bilax_monoidal_functors)

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 <at> 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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>

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