A well kept secret

[Paul Taylor <pt09@PaulTaylor.EU>]

I'm not too sure what the context was, but Andre' Joyal said on 7
December,

 > Category theory is a powerful mathematical language.  It is extremely
good
 > for organising, unifying and suggesting new directions of research.

I completely agree.

 > It is probably the most important mathematical developpement of
 > the 20th century.

It is too early to tell.

[Comment attributed to Zhou Enlai (Chinese Communist leader 1949-76)
when asked his opinion of the French Revolution.]

 > But we cant say that publically.

I think we should be wary of slapping ourselves on the back too much.

The fact is that category theory alienated the rest of the mathematical
world.   Since the damage had been done in the 1970s, well before my
time,
I have never managed to work out how this happenned, or who was
responsible.

Probably it was the result of haughty claims about being the "most
important mathematical development",  and about being the foundations
of mathematics before any serious technical work was done to justify
this.
Of course the ignorance and arrogance of mathematicians outside our
subject
had a lot to do with it too.

Indeed, I believe that there is nothing wrong with pre-1980 category
theory that cannot be attributed to the fact that it was done by pure
mathematicians,  and nor is there anything wrong with the post-1980
subject that is not the result of its having been done by computer
scientists.

However,  discussion on that is not going to get us very far.  What is
more relevant and able to be fixed is the point in Andre's title, that
category theory is a
       WELL KEPT SECRET.

Secrecy, like charity, begins at home.   For example, the notion of
       ARITHMETIC UNIVERSE
was one of the most insightful developments of 1970s categorical logic.

It captures exactly what is taught as "discrete mathematics" to
computer science students (and is relevant to combinatorial
mathematics),
namely products, equalisers, stable disjoint sums, stable effective
quotients of equivalence relations and FINITE powersets.  It is the
least structure that is capable of constructing the free internal gadget
of the same kind,  so the original idea was to prove Godel's
incompleteness
theorem categorically.

Recently I was looking though the archives of the "Foundations of
Mathematics" (FOM) mailing list at   cs.nyu.edu/pipermail/fom/
and, amongst all of the personal abuse directed at Colin McLarty and
Steve Awodey, came across an interesting argument against category
theory, namely that the notion of elementary topos was merely an
aping of the axioms of set theory.   Arithmetic universes answer that
objection extremely well.

The work on arithmetic universes was done THIRTY SIX YEARS AGO, and
many people since then have been nagging the author to write it up,
indeed I myself have been doing so for half of that time now.

I don't want anybody to read this as a personal attack -- it is
simply an example of a general phenomenon, albeit an important example
because of the importance of the material.   Anybody in my generation
or younger can cite lots of examples of "well known"  "folklore"
results that were supposedly discovered in the 1970s but have never
been written up.   The worst thing is that any younger person who
is so impertinent as to write out a proof of one of these results
has their paper rejected.

To give another example, the theory of continuous lattices is crucial
as background for my work on Abstract Stone Duality.   I asked exactly
the people who should have written it whether there was an introduction
to continuous lattices suitable for analysts.   There isn't, so
I had to write my own.   In this, I stated without proof that the
evaluation map   Sigma^X x X --> Sigma   is continuous (when the
topology Sigma^X is itself given the Scott topology)   iff  X is locally
compact,  and in this case Sigma^X is itself locally compact and
obeys the adjunction   Yx(-) -| Sigma^(-).    The referee quite
reasonably asked for a reference to a proof, but, so far as I can
gather, no such proof exists in the literature.

Two more examples: when is some Australian going to write
"2-categories for the working categorist"?
Where is the textbook on universal algebra based on monads?

So, to answer Andre's question about why category theory is such a well
kept secret -- it is because category theorists KEEP it as a secret.

Each of us can help to leak this secret by doing two things:

PUBLISH (= make freely available on the Web) all of the papers that
you PRIVATISED by handing them over to commercial journals.

WRITE textbook or encyclopedia accounts of your work for resources
like the "n-cat lab",   ncatlab.org/nlab/show/HomePage

Paul Taylor



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