FOM: foundations of category theory

[Andrej.Bauer@cs.cmu.edu]

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

> The answer to this is that category theory *cannot* get around the
> Russell paradox.  When category theorists speak of ``the category of
> all categories'' (as they often do), they are whistling in the dark.

Clarification:

When a category theorist speaks of ``the category of all categories''
(or more technically ``the 2-category of categories'') what is meant
precisely is ``the large category of all small categories''. Rarely
they mean something larger than that, say, the super-large category of
large categories.

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.

Claims about existence of "categories of categories" of various sorts
can easily be analyzed in terms of set-theoretic concepts, such as
large cardinals and classes. This analysis is useful because it lets a
category theorist know exactly how shaky his or her assumptions are.
The vast majority of category theory happens within the 2-category of
small categories, which is a class, so that's not shaky at all.

Let my try to explain my view of why there is a misunderstanding that
causes comments like the above one about ``whistling in the dark.''

Category theorists typically do not pay much attention to questions
about existence of "category of all categories". They talk about
"category of all categories" in a sloppy way, since they do not have
to worry about these matters because set theorists can easily provide
answers about what existence of such categories amounts to. The
questions about existence and consistency of category theory are NOT
category-theoretic questions, they are set-theoretic questions.

Having said that, you might wonder what other foundational questions
there are that category theory might be considering and that are not
present in set theory. The point is that category theory studies
*different* foundational concepts than set theory does. Let me just
mention adjoint functors here. That alone is enough to establish
category theory as a foundational field, since it exposes a concept
that is not treated by set theory and is without a doubt of
a general foundational interest.

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 believe that such opinions can only
be caused by lack of acquaintance 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.

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