categories and disdain

categories and disdain

[edubuc]

this is about the recent thread  "categories and functors" and "disdain
for categories"

Ten points:

1) It seems clear that E-M arrived to categories and functors by
abstraction from the usual large categories of sets, groups, boolean
algebras, modules, etc etc

2) It seems (less clearly) that Ehresman arrived to categories and
functors by generalization from groupoids and morphisms of groupoids.

3) I agree with  "It's been said before that the real insight of
category theory
--as something more general than groupoids, monoids, and posets--
is the notion of adjoint functors (including limits, etc)."

Concerning this, I think that real breakthrough made by categories is
the simple fact that they furnish the appropriate abstract structure to
define the Bourbaki's  concept of universal property. The fact that the
singleton set is characterized by being a terminal object opens the way
to characterize thousands of objects and constructions by being the
terminal object in the appropriate category. Yoneda's lemma is the
milestone. Everything is due to it.

4) I think the small-large business played no role at all in the rise of
the concept of categories, neither in the rise of the disdain to them by
many mathematicians.

 5) "working" mathematicians were never afraid about paradoxes. In
consequence, I think that phrases as "dare regarding these (perceived)
monsters ...",   "fear of the "illegitimately large" size", etc etc,
are  misleading and out of place.

6) Cantor did not  "dare  to think that there could be different levels
of infinity", he discovered that they were different levels of infinity,
and proceed to study this phenomena. This was not a bold action, he was
just fascinated by the existence of different levels of infinity. He was
not afraid of paradoxes either, he was very well aware of Russell
paradox, but for him it  was just another theorem.

7) It is often repeated that axiomatic set theory arise in order to
eliminate paradoxes. False, axiomatic set theory arise in an attempt to
understand Von Neumann accumulation process:  Which were the axioms
satisfied by the output of that process ?  Answer: axiomatic set theory.

8) "In my experience, disdain for cat theory is due to papers with a very
high density of unfamiliar names", I agree with this in the sense that
this fact contributed to the rise of the disdain, but not as the single
reason. I agree also with "Complicated sentences like this can be found
in every fields. They are often the mark of a poor paper".

It follows there must be other reasons (besides  the abundance of poor
papers in category theory, a fact that I found true) to explain the disdain.

9) A profound reason could be an instinctive opposition  to real change
in many people.  The instinctive reaction against progress that may
disrupt their own comfortable position.

10) The so self proclaimed "problem solvers" who disdain abstract
theories often do not resolve any real problem. The just do "concrete
nonsense". People who really solve true problems usually have a great
respect for abstract theories.  Of course, they are also many who just
do "abstract nonsense" instead of contribute to the meaningful
development of theories.

eduardo dubuc