Re: terminology (was: bilax_monoidal_functors)

[Michael Shulman <shulman@uchicago.edu>]

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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