Re: Teaching Category Theory

[Vaughan Pratt <pratt@cs.stanford.edu>]

> A rough and ready solution would be to tell them that there is a
> distinction between "small" and "large" collections, otherwise known as
> "sets" and "proper classes".  This would necessitate giving them an
> example of a large collection, and I guess the obvious choice is the class
> in Russell's Paradox. [...]
>
> There's probably a better solution involving an axiomatization of the
> category of sets (along the lines of the Lawvere-Rosebrugh book), or at
> least a listing of some its properties.

The question is more when than whether to bring up this distinction.
Although the Lawvere-Rosebrugh book doesn't define "large" it does
define "small relative to Set" in the antepenultimate paragraph of the
book (p.250), namely as "can be parameterized by an object of Set".
Near the midpoint of the book (p.130) is the statement of Cantor's
theorem X < 2^X with the consequence that Set cannot be parameterized by
any of its objects (and hence by the definition on p.250 cannot be small
relative to itself).

In contrast Borceux in his 3-volume series gets the size issue out of
the way on pages 1-4 of Volume 1.  CTWM is in between, addressing it on
p.22 after covering categories, natural transformations, monics, epis,
and zeros.

One benefit of getting the distinction out of the way near the beginning
is that the students won't feel so mystified when they run across it
while reading other category-relevant material (as the better students
will).   The combinatorics of sets (n^m functions from a set of m
elements to a set of n elements, etc., which they definitely should
know) in no way prepares one for the possibility of an object larger
than any set, for which Cantor's theorem is very helpful.

Of the above three positionings, I like CTWM's best: early on, yet not
so early as to exaggerate its importance relative to the fundamental
concepts of CT.

Not to say that CTWM starts out ideally.  Spending three pages defining
"metacategory" and then defining "category" as "any interpretation of
the category axioms within set theory" is impenetrably idiosyncratic for
students used to more conventional introductions in their other pure
maths courses.  Once past the size issue Borceux is much more
conventional and direct, except for the relatively mild criticism that
his definition of "category" is actually the definition of "locally
small category."  But that's not at all the stumbling block to
understanding that "metacategory" presents, in fact if anything it is
helpful not to be distracted at the outset by the prospect of large
homobjects.

While on the topic of size of homobjects, what drawbacks are there to
regarding both the objects and homobjects of any n-category as all lying
within the n-th Grothendieck universe for a suitable hierarchy U_1 < U_2
< ... of such (with U_0 = 1)?  Although admittedly idiosyncratic, it
seems very natural to account for large homobjects in a category C with
the explanation that C is really a 2-category, whether or not one is
making use of the 2-cells.  I can see a methodological objection, namely
that there is no logical connection between size and existence of
n-cells for a given n.  Does it create any actual difficulty somewhere?

Vaughan