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

[Andrej.Bauer@cs.cmu.edu]

Stephen G Simpson <simpson at math.psu.edu> writes:

> 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. .... [text deleted] ....Let's not go over all this ground
> again, unless you have some new point to make.

Thank you for the pointers. I think I've read most of those
discussions. I'll try to not go there. Let's stick to the discussion
about adjoint functors. And let's try to follow Todd Wilson's
suggestion of "good will".

DIGRESSION:

>  > 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.

These arguments about what a five year old can and cannot understand,
are you serious about them? I think they're completely irrelevant.
Maybe you can explain why a five year old's grasp of mathematical
concepts has anything to do with f.o.m. (on a separate instance, or at
least a separate posting, please. I don't want this to become a main
issue).

END OF DIGRESSION

> As a non-category-theorist and a human being, I of course find this
> way of viewing quantifiers somewhat unnatural.

Yes, well, I think everyone does when they first see it.

> 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.

I am not proposing that universal quantification be defined in terms
of adjoint functors. I am not proposing that adjoint functors are more 
fundamental than universal quantification. I am just saying that
universal quantification is an example of adjoint functors.

> This illustrates why logic is more fundamental than category theory.

This game is not about what is more fundamental. Let me reiterate: I
am not saying that category theory, or adjoint functors, are more
fundamental than X. I am just saying: adjoint functors are important
for f.o.m.

The reason for listing all those examples of adjoint functors was NOT
to claim that adjoint functors are more fundamental than all the other
concepts. I was doing that in order to convince you that adjoint
functors are as pervasive as "sets", and so they are more like the
concept of "set" than the concept of "group".

But it seems you don't quite believe yet that adjoint functors are
far, far more general and omnipresent than groups.

DIGRESSION:

> 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.

Well, I would put it like this. If we find out that a vast majority of
mathematicians are aware of and are using concept X, that indicates
that concept X might be of foundational interest. But if they are not
aware of it, then that does not mean much.

> 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.

And I never claimed that it does prove anything. It was a reaction to
your claim that mathematicians are not hindered by their ignorance of
adjoint functors.

> 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.

Is your work on reverse mathematics foundational, given that it's all
within the realm of perfectly rigorous and valid mathematics with a
perfectly good foundation? I don't think you can draw the conclusion
that a mathematical activity has no foundational interest just because
it has not exposed a foundational deficiency.

But we digress. Let us focus on adjoint functors.

END OF DIGRESSION

> 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.

Let me reiterate, and focus just on f.o.m.-style concepts that are
adjoint functors:

1. Logical connectives (and, or, imply) are adjoint functors.

2. Logical quantifiers (for-all, exists) are adjoint functors.

3. Products, disjoint sums, and power sets, are adjoint functors.

4. Application of a function to an argument is the counit of an adjunction.

Do you still think that adjoint functors can be compared to groups
fairly? Can you make statements like the above about groups?

In fact, can you think of any other mathematical concepts, other than
adjoint functors, for which you can make similar claims as the ones
above, i.e., a concept that draws connections between the most basic
f.o.m. concepts such as logical connectives, quantifiers, and the
basic set-theoretic constructs?

When you find such a concept, that's what adjoint functors should be
compared to. Not groups.

>  > 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.

Quite *trivially*, some of the most basic f.o.m. concepts ARE adjoint
functors, but they are not groups, cohomologies, or differential
operators. That is reason enough for adjoint functors to stand out as
having foundational importance, and for groups not to.

> 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.

I am establishing it as I speak :-)
 
> 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.

If you are interested in understanding why the logical connectives and
quantifiers are adjoint functors, that would be a foundational
interest, in my opinion.

I hope this posting makes it clear that adjoint functors are not like
groups.

--
Andrej Bauer
Graduate Student in Pure and Applied Logic
School of Computer Science
Carnegie Mellon University
http://andrej.com