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

[Andrej.Bauer@cs.cmu.edu]

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

> The issue of so-called ``categorical foundations'' has already been
> discussed in great detail on FOM.  I suggest you have a look at the
> earlier discussion, so that we can perhaps avoid going over old
> ground.

The FOM archives seem to be rather extensive. I'd appreciate more
specific pointers (approximately what year and month should I be
looking at)? Thanks.
 
> Why then do you think category theory is different from group theory
> or ring theory in this respect?  After all, a category is just another
> kind of algebraic structure, and category theory is just another
> branch of algebra.

I am not saying that category theory is somehow different and cannot
be expressed in terms of sets and classes. I am saying that category
theory brings to light concepts which are or ought to be of general
foundational interest, *and* just by looking at set theory we would
never discover those concepts (just like we would never discover
groups by studying cardinal arithmetic).

>  > Let my try to explain why, in a sense, category theory is not
>  > "within the reach of set theory", from the foundational point of
>  > view.
> 
> Your statement implies that there is a unique ``foundational point of
> view''.  Do you really believe this?  

I don't even want to discuss this :-) I'd prefer to waste my time on
talking about specific things. I propose that I take the position that
"adjoint functor" is a concept of general foundational interest. I am
sure there will be no lack of people who will disagree with this, so
it should make a reasonable discussion. If this is already in FOM
archives, then I can just go back to writing my dissertation, which
wouldn't be such a bad thing.

>  > A goal of foundations of mathematics is to expose, clarify, and
>  > study those mathematical concepts and ideas that are fundamental
>  > for all of mathematics, or at least very large portions of it.
> 
> I question this loaded statement.
> 
> For example, cohomology is a mathematical concept that is fundamental
> for very large portions of core mathematics (parts of analysis,
> algebra, topology, geometry, number theory, ...).  Is it part of the
> purpose of f.o.m. (= foundations of mathematics) to expose, clarify,
> and study cohomology theory?  In my view, it is not.  In my view, the
> study of cohomology is a matter for specialists in those core
> mathematical areas which make use of cohomology.

Good point. I definitely would not count cohomology to be part of
f.o.m. Clearly, something more is needed then for a concept to be part 
of f.o.m. Let's see if I can convince you that adjoint functors have
the "something more".

> Similarly, the concept ``category'' is fundamental for category
> theory, but it is not fundamental for mathematics as a whole, because
> it can be (and always is) explained in terms of sets, classes, and
> operations.

You can play this game the other way around: sets, classes and
operations can be explained in terms of objects and morphisms.
But as I say, that's NOT THE POINT.

>  > the notion of an adjoint functor is fundamental to algebra,
>  > geometry and logic at the same time,
> 
> I disagree.  A great many algebraists, geometers, and logicians never
> use the concept ``adjoint functor''.  Many of them can't even define
> it, and this in no way hinders their research in algebra, geometry,
> and logic.

Examples of adjoint functors:

1. The free group (ring, module, etc.) construction is the left
   adjoint of the forgetful functor.

2. Implication is the right adjoint of conjunction.

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

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

5. Exponentiation (of sets, for example) is right adjoint to cartesian
   product.

6. Existential quantification is the left adjoint to the inverse image 
   functor.

7. Universal quantification is the right adjoint to the inverse image
   functor.

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

And yes, people's ignorance of adjoint functors greatly hinders
research in algebra, logic, and geometry. You sound like a physicist
of a few decades ago who would advise against teaching Lie groups to
nuclear physicists.

> 
> This I agree with.  Adjoint functors sometimes help to expose and make
> precise certain otherwise vague analogies in disparate branches of
> mathematics.
>

This is the crux of the matter.

Cauchy and Weirestrass exposed and made precise certain vague notions
in analysis. Their epsilons and deltas sometimes helped expose and
make precise shaky arguments in analysis. So their work lies at the
foundation of analysis.

The great mathematicians of late 19th and early 20th century exposed
and made precise certain otherwise unclear concepts of "function" and
"collection". Their work reshaped how algebra, analysis, geometry,
topology, and practically ever other branch of mathematics was done.
So their work lies at the foundation of mathematics.

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

> In this respect the concept `` adjoint functor''rivals the concept
> ``group'', which also brings to light deep connections between various
> branches of mathematics and makes precise what would otherwise be
> vague analogies.

> However, this does not mean that a study of adjoint functors is the
> particular concern of f.o.m., any more than group theory is.

However, as should be clear from the examples above, adjoint functors
are much, much, more wide-spread than groups. In fact, they are so
wide-spread throughout mathematics that an f.o.m. researchers should
be asking what's up with that.

> This is true only in the same sense that group theory is not part of
> set theory and vice versa.  They are two different branches of
> mathematics, and neither is part of the other.  However, set theory
> has a valid claim to being a foundation for all of mathematics, while
> group theory and category theory do not.

I do not wish to defend the position that category theory can be a
foundation for all of mathematics (that would lead us to discussions
that have already taken part on FOM). I just want to defend the
position that "adjoint functors" are of a general interest to f.o.m.

> I guess I am one who has this misunderstanding.  For example, I do not
> see the concept ``adjoint functor'' as having foundational importance.
> I see it as something like the concept ``group''.  It is a useful
> concept that can serve as a framework or language for expressing
> various interesting things, and it can be studied in its own right as
> a subject within mathematics, but that's all.  (And, so far as I can
> tell, the concept ``group'' is actually much more useful and pervasive
> in core mathematics than the concept ``adjoint functor''.)

I disagree that groups are more pervasive in core mathematics than
adjoint functors. Every time you use an implication or form a
cartesian product you have an adjoint functor at hand. Is that
pervasive enough?

If you would like more examples of adjoint functors, a basic list can
be found on page 85 of MacLane's ``Categories for the Working
Mathematician'', but that list is slanted towards algebra. A list
coming from categorical logic might be more relevant to this
discussion.

P.S. Maybe we should rename the instance to "Are Adjoint Functors part 
of f.o.m.?"

--
Andrej Bauer
Graduate Students in Pure and Applied Logic
Carnegie Mellon University
http://andrej.com