Re: Evil in bicategories

[Vaughan Pratt <pratt@cs.stanford.edu>]

On 9/13/2010 1:04 PM, Steve Vickers wrote (in response to Jean Benabou's
question as to whether CT is a religion):
> I'm not so sure of the answer for general category theory, but Topos
> Theory is undoubtedly a religion. At a monastery in Meteora, the monks
> have erected a sign saying "O topos einai ieros", i.e. "The topos is
> holy". And as Psalm 126 says, "He shall doubtless come again with
> rejoicing, bringing his sheaves with him."

For my part I'm not so sure of the answer for topos theory, but for
general category theory I would reply in the same vein as I did when
Paul Taylor opined to this list back in October of 1992:

    The strings, braids, etc., are a completely different matter.
    It is not belittling the work done on these topics, in
    Australia in particular, to say that they are of minority
    interest, compared to the use of the simpler forms of
    diagrams.

It seemed to me that Paul was making a religious issue out of category
theory that might have been sound English orthodoxy in those days but
that came across as heresy to Australian ears.  I therefore replied at
the time as follows.

=================================

Paul's representation of strings and braids as a flash in the category
theory pan is not borne out by the literature.  Strings have been an
integral part of the arrow business for a long time, witness Psalms
21:12, "Therefore shalt thou make them turn their back, when thou shalt
make ready thine arrows upon thy strings against the face of them."

Besides their supporting role with arrows, strings are also known to
every linguist and computer scientist to be an integral part of
language, as Mark 7:35 attests: "And straightway his ears were opened,
and the string of his tongue was loosed, and he spake plain."

Commutative algebra has been synonymous with mainstream algebra for a
long time.  But noncommutative algebra has been more than just a cottage
industry for many years, and moreover has found a warmer welcome in
linguistics and CS than the commutative variety, providing us with an
algebraic basis for Turing machine computations and formal grammar
derivations, both of which can be conveniently laid out in the
(oriented) plane, where they lend themselves to a 2-categorical formulation.

But as Zadeh reminds us, we live in a fuzzy world, neither clearly
commutative nor clearly orientedly planar, but somewhere in between.
The laws of this in-between world are braidal, with planarity
corresponding to the initial or discrete braids and commutativity to the
final braids, which can pass through themselves without losing their
identity altogether (commutativity without idempotence).  Under these
circumstances we can only keep our 2-categorical cool with braids,
suggesting the following slogan:

      Braids are the rule, of which commutativity and noncommutativity
are its two extremes.

This appears to be a natural idea in both senses of "natural."	It is a
natural mathematical idea to suggest and pursue; and it appears to be
one that can be found in nature, witness the Yang-Baxter equations
arising early on in physics, whose braidal character is now clear and
about which several mathematical physicists have started writing, e.g.
John Baez's recent MIT lecture notes on "Braids and Quantization."

Nor would I be accusing Ross of riding a bandwagon if I suggest that in
five years' time he'll probably be interested in something else and
drawing a completely different kind of diagram.

Nor would I be calling Paul shortsighted if I suggest that in five years
time many of us in both mathematics and physics, and conceivably even
philosophy, will be drawing braids.  (This is not to suggest that Jon
Barwise's reaction to my explanation of linear logic last February would
have been any different had I omitted the section on braids, which
included five braid diagrams I had to do in ASCII that I am looking
forward to being able to render in Taylorese.)

As for Ross, I rather expect that in five years time he will be drawing
whatever it is that those of us in the trenches will be drawing in ten
years time, and one might hope that these too would appear in some
diagram package, preferably in 1997 rather than 2002.

Meanwhile others on the fringe of the expanding categorical cosmos will
only just be learning to use commutative \square's.

This brings me back to my first theme.	Categories have been a pons
asinorum for "the rest of us" for a very long time, ever since the exam
in category theory given to the young lad who appeared briefly in the
story of David and Jonathan [I Samuel 20:21-22]:

      And, behold, I will send a lad, saying, "Go, find out the arrows."
If I expressly say unto the lad, "Behold, the arrows are on this side of
thee, take them;" then come thou: for there is peace to thee, and no
hurt; as the Lord liveth.  But if I say thus unto the young man,
"Behold, the arrows are beyond thee;" go thy way: for the LORD hath sent
thee away.

As it turned out the arrows were indeed beyond the lad, who "knew not
anything, only David and Jonathan knew the matter," and the lad was sent
off to the city [20:37-40], a drop-out who for all we know may have
later become the Bill Gates of his day.

Another biblical character who struggled mightily with the subject was
Job.  "For the arrows of the Almighty are within me, the poison whereof
drinketh up my spirit: the terrors of God do set themselves in array
against me." [Job 6:4]  One imagines him tackling either metacategories
or coherence on that occasion.

Job was thus afflicted for a long time, during which he complained
bitterly of his plight to his three friends and the Lord in three major
jam-sessions.  In the last of these, the Lord showed up in a Whirlwind
evidently hoping to be able undo Satan's mischief and set things
straight at last.  After spending the better part of three chapters
extolling the virtues of His nobler creatures and getting Job into the
proper frame of mind, the Lord came to the whale, of which He said "The
arrow cannot make him flee." [Job 41:28]  That apparently did the
trick:	Job immediately apologized for his ignorance of the subject: "I
have heard of thee by the hearing of the ear, but now mine eye seeth
thee.  Wherefore I abhor myself, and repent in dust and ashes." [Job
42:5-6].

The Lord then in unexpectedly firm tones told Job's friends that Job now
understood the subject better even than they did and to treat him
properly henceforth.  And He gave Job twice what he had before.  This
came to 14,000 sheep, 6,000 camels, 1,000 oxen, 1,000 she-asses, 7 sons,
and 3 daughters, so you can figure out what he had before, at least for
the livestock.  If Job's arrow anxiety lasted 25-35 years, that's around
2-3% p.a., probably a good rate for those days.

But I digress.	Anyway you can read it for yourself, you'll see it's
exactly as I said.

Coming from the electrical engineering side of the business myself, the
advice to "Cast forth lightning, and scatter them: shoot out thine
arrows, and destroy them." [Psalms 144:6] speaks more directly to me.

 	Vaughan Pratt



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