FOM: category theory, cohomology, group theory, and f.o.m.

[Stephen G Simpson]

Reply to Andrej Bauer's posting of Feb 21, 2000.

You asked me to help you find the earlier FOM discussions of category
theory.  I think November 1997 to February 1998 was the most
extensive.  The conclusion was that category theory is no good as
a global foundational setup, because (i) it is more complicated than
set theory, (ii) it depends on set theory, (iii) it has no underlying
or motivating foundational picture.  There was also a lot of category
theory discussion on FOM around April-May 1999, about small vs large
categories, the set-theoretic basis of category theory, etc.  Let's
not go over all this ground again, unless you have some new point to
make.

 > sets, classes and operations can be explained in terms of objects
 > and morphisms.

No, they can't.  This was well covered in the earlier FOM discussion.

 > 3. Disjoint sum is left adjoint of the diagonal functor.

But the disjoint sum of two sets can be explained to 5-year old
children much better and more easily without category theory, in terms
of sets of marbles, etc.  This is part of why sets are fundamental to
math but adjoint functors are not.

 > 4. Cartesian product is the right adjoint of the diagonal functor.

Same remark as above for disjoint sum, except you might need a 12-year
old child, to explain Cartesian product in terms of a rectangular
array or table.

 > 6. Existential quantification is the left adjoint to the inverse image 
 >    functor.
 > 
 > 7. Universal quantification is the right adjoint to the inverse image
 >    functor.

As a non-category-theorist and a human being, I of course find this
way of viewing quantifiers somewhat unnatural.  But putting that
aside, don't you agree with me that this alleged definition of
quantifiers in terms of adjoint functors is circular?  Quantification
has to be understood *before* you can even define what you mean be a
category, let alone a functor and a left adjoint.  After all, a
category is defined as a certain kind of algebraic structure where
*every* pair of morphisms of a certain kind has a composition, and
*for every* object *there exists* an identify morphism, etc etc etc.
This illustrates why logic is more fundamental than category theory.

 > You may not *know* that you are using adjoint functors all the time,
 > but that's hardly an argument for anything.

Instead of casually dismissing the fact that the vast majority of
mathematicians don't know and don't care about adjoint functors, why
not ponder this fact and try to learn something from it?  For
instance, you could contrast it dramatically to the situation
regarding truly fundamental mathematical concepts such as set,
function, number, etc.

 > And yes, people's ignorance of adjoint functors greatly hinders
 > research in algebra, logic, and geometry.

Probably you can give some examples illustrating this by showing that,
with hindsight, somebody could have perhaps discovered some theorem
more quickly using adjoint functors.  Isn't hindsight wonderful?  :-)

And of course there are lots of examples illustrating the same point
for lots of other specialized concepts, where knowledge of those
concepts might be thought to help in unexpected ways.

But none of this proves that these specialized concepts are part of
f.o.m.

 > You sound like a physicist of a few decades ago who would advise
 > against teaching Lie groups to nuclear physicists.

I don't advise against teaching category theory to mathematicians.
But I think category theorists tend to overrate its value, especially
for f.o.m.

 > Category theorists have exposed and made precise certain vague
 > analogies in disparate branches of mathematics.

But the mathematics in question was perfectly rigorous and valid and
had a pefectly good foundation already, without category-theoretic
language.  Contrast this to the truly foundational work of Cauchy and
Weierstrass, where they were systematically replacing non-rigorous
math by rigorous math.

 > It turned out that these concepts (specifically "adjoint functors")
 > permeate almost every branch of mathematics, including algebra,
 > geometry, topology, and logic. And so, I claim, this puts certain
 > aspects of category theory at the foundations of mathematics.

The concept of ``group'' also permeates almost every branch of
mathematics.  Do you think this puts group theory at the foundation of
mathematics?  If not, ask yourself why not.

 > However, as should be clear from the examples above, adjoint functors
 > are much, much, more wide-spread than groups.

Most core mathematicians could give a much longer list of places in
mathematics where groups come up in highly non-trivial ways.  Think of
Klein's Erlanger Programm, etc.  Yet surely you would deny that group
theory is part of f.o.m.

 > Every time you use an implication or form a cartesian product you
 > have an adjoint functor at hand. Is that pervasive enough?

Define ``at hand''.  I can assure you that adjoint functors are the
farthest thing from my hand and my mind in these situations.

 > I just want to defend the position that "adjoint functors" are of a
 > general interest to f.o.m.

The pervasive interest and fundamental nature of sets and classes in
f.o.m. is extremely well established.  The interest of adjoint
functors in f.o.m. is much, much less well established, or maybe not
established at all.

As a mathematician, I am somewhat interested and curious to understand
why adjoint functors arise in a variety of contexts and explain a
number of analogies.  Similarly, I am interested and curious to
understand why groups arise in a variety of contexts and explain a
number of things.  But this doesn't convince me that adjoint functors
or groups are part of f.o.m.

-- Steve