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

[Stephen G Simpson]

This is a reply to two postings on category theory by Andrej Bauer
yesterday (Feb 20, 2000).

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.

 > In general, categories are proper classes (the category of all groups,
 > the category of all topological spaces, the category of all sets).
 > Many important examples of categories are actually just sets (the
 > "small" categories). But this is not a reason to consider category
 > theory as something in the reach of ZFC.

Why not?  

For example, group theory and ring theory are generally viewed as
being ``in the reach of set theory'' (i.e., amenable to a
set-theoretical foundation), precisely because groups and rings can be
viewed, and usually if not always are viewed, as sets or classes.  And
the same goes for categories -- they are usually if not always viewed
as sets or classes.

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.

 > 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?  

The truth of the matter is that category theorists tend to have a
certain viewpoint about foundational questions, which is very
different from everybody else's viewpoint.  I don't think it is a good
idea to try to blur or gloss over very significant differences among
various positions on foundational issues, or to pretend that such
differences don't exist.

 > 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.  

Bauer's statement reminds me of ``Cherlin's Thesis'': that
f.o.m. embraces all conceptual analysis in mathematics.  (See my FOM
posting of Sun Dec 05 20:02:08 1999.)  This thesis strikes me as
absurd, because conceptual analysis is an integral, indispensable part
of mathematics as a whole and of every branch of mathematics.  If we
accept Cherlin's Thesis, f.o.m. would subsume all of mathematics.

 > To a degree, what is and isn't a fundamental concept is a matter of
 > opinion. For example, is "cardinal number" a fundamental concept?

To a degree, the question ``which concepts are fundamental'' is a
matter of opinion.  But there is also a way of attacking such
questions objectively.

A concept C is said to be fundamental with respect to a subject X if C
``lies at the base of'' X, i.e., X is built on C and cannot be
explained or reduced to concepts that are more fundamental than C and
that also belong to subject X.

For example, the concept ``group'' is fundamental with respect to
group theory, but it is not fundamental with respect to mathematics as
a whole, because in the context of mathematics as a whole, the concept
``group'' can be (and always is) explained in terms of more
fundamental concepts such as ``set'' and ``operation''.

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.

See also my short essay attempting to answer the question, ``What is
foundations of mathematics?'', at,

  http://www.math.psu.edu/simpson/hierarchy.html

Bauer continues:

 > There are concepts that are fundamental but are not at all exposed,
 > clarified, or easily studied within set theory. These are the basic
 > category-theoretic concepts, such as "natural transformation",
 > "adjoint functor", "limit", and "colimit". 

I disagree.

Why do you think that these concepts are not amenable to study within
set theory?  For example, the Herrlich/Strecker textbook of category
theory defines and develops all of these concepts on the basis of an
explicitly set-theoretical foundation.  And the same goes for other
category theory textbooks.  Of course some of these books (not
Herrlich/Strecker) are deliberately sloppy about foundational issues,
and some of them even take an anti-set-theory tone, but nevertheless,
the foundational setup in them is set-theoretical.

 > 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.

 > and it brings to light deep connections between these
 > branches of mathematics.

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

 > It is without a doubt of interest to foundations of mathematics to
 > study such concepts, in order to see what else is out there apart
 > from sets, sets, and still more sets.

``Without a doubt?''  I for one doubt it.  What has the study of
adjoint functors contributed that is of interest for f.o.m.?

 > Category theory is a foundation of mathematics in its own right,

A highly dubious claim.  See the earlier FOM discussion on
``categorical dys-foundations''.

 > because it makes it possible to study fundamental concepts that are
 > invisible from the set-theoretic point of view.

I find this formulation very questionable.

Would you also say that group theory ``is a foundation of mathematics
in its own right, because it makes it possible to study fundamental
concepts that are invisible from the set-theoretic point of view''?
Is the concept ``group'' such a concept?

Could you please give an example of a concept in category theory which
you think is ``invisible from the set-theoretic point of view''?  Note
that the concept ``adjoint functor'' is not such a concept.  Adjoint
functors are always defined in the context of the usual
set-theoretical foundation for category theory.

 > category theory is not part of set theory, and vice versa.

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.

 > Category theorists DO NOT presume that there is indeed a category of
 > all categories that includes itself as a category. They know better
 > than that, of course. They did not forget the lessons learned in set
 > theory.

Yes, category theorists must take seriously the lessons they learned
from set theory.  They cannot do otherwise, because category theory is
based on set theory.

Nevertheless, I find it interesting that category theorists seem to
derive pleasure from talking about ``the category of all categories'',
as if they were free of the Russell paradox.

 > questions about existence and consistency of category theory are NOT
 > category-theoretic questions, they are set-theoretic questions.

Yes.  Category theory is based on set theory.  I am glad you recognize
that fact.  I hope all category theorists recognize that fact.

 > So, the misunderstanding comes from the fact that some people do
 > not know, see, or accept the fundamental category-theoretic
 > concepts as having foundational importance.

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 believe that such opinions can only be caused by lack of
 > acquaintance with category theory,

Perhaps I myself am a counterexample to this belief, since I am not
unacquainted with category theory.

 > and set-theoretic and logic biases which cause one to overlook the
 > importance of structure that is not directly captured by set
 > theory.

Well, maybe I have some ``bias'' toward set theory and logic, because
I am a researcher in related areas of f.o.m.   But, could you please
explain what you mean by ``structure that is not directly captured by
set theory''?  Do you think the concept ``group'' is an example of
this?

-- Steve