FOM: small category theory
butz at math.mcgill.ca
Fri Apr 30 14:27:29 EDT 1999
(sorry for the long quotes here at the beginning, but I didn't want to
take them out of the context)
> > but there are what are called _essentially algebraic structures_,
> > which roughly means that there might be some operations (here:
> > composition) which are not total, but defined on a definable
> > sub-type. This class of structures behaves very much like algebraic
> > ones (forgetful functors have adjoints, i.e., "free" structures
> > exist, etc). For (small) elementary toposes it was Peter Freyd who
> > first mentioned that they form an essentially algebraic structure.
> But fields aren't ``essentially algebraic'' in this sense either, are
> they? But small categories are, right?
You are right.
> Anyway, I apologize for stating my point somewhat carelessly. But I
> still think my point is correct. When I said that a small category is
> an algebraic structure, I meant this in a naive sense, i.e. it is a
> set (the set of arrows) together with a partially defined operation
> (composition of arrows), and maybe a distinguished subset (the
> identity arrows) and is required to satisfy certain simple laws
> (associativity etc). My point is that you could write an algebra book
> consisting of chapter 1 on groups, chapter 2 on rings, chapter 3 on
> fields, chapter 4 on small categories, etc. It is all algebra.
> Here is another way to make my point. Logically, the issue of small
> (i.e. set size) versus large (i.e. proper class size) is not really
> peculiar to category theory. It applies to any kind of algebraic
> structure whatsoever. If you insist on talking about large
> categories, you might as well also insist on talking about large
> groups, large rings, large fields, etc. [ I seem to remember somebody
> saying that John H. Conway's surreal numbers form a large field .... ]
I disagree a little (just a little) the way you argue here. As said, one
aspect is the theory of small categories, which is much like universal
algebra. It could fit, indeed, in any book on standard algebra. But:
(1) Category theory _does_ want to speak about large categories, people do
want to study the category of all (small) groups, of all (small) vector
spaces over some fixed field, of all (small) topological spaces, etc.
(2) There is fortunately not much difference between small and large
categories (after all, the large ones are small relative to some
universe/large cardinal...). And, as well, by accepting a large cardinal
axiom we _do_ speak about large rings, fields etc (after all, what else is
the monster model used in stability theory).
By the way, there is currently a lot of research going on in (large)
higher-dimensional category theory. This stuff _is_ to a large extend
large (universal) algebra, and people know this.
Maybe it is interesting to mention here that recently F. Borceux wrote a
book (in fact, three volumes) of what "ideally someone should know
before starting research in category theory" (this is not verbatim, but he
says something like this in the introduction). The book is called
Handbook of categorical algebra (!), which not (just) mean that you do
algebra using category theory.
Yesterday I said that when restricting to small categories you can loose
something. I owe you some positive and negative examples: Of course, the
forgetful functor from countable rings to countable groups has a
left-adjont (simply, the free ring generated by a countable group is
countable), the same is true for many other "forgetful" functors, like
the ones from countable groups to countable semi-groups, countable
boolean algebras to countable lattices, etc. (You can use your favourite
cardinals in these examples.)
But, look at the "contra-variant" powerset functor
P: Set^op --> Set
from the opposite of the category of sets, to sets. It sends a set X to
P(X), the set of all its subsets, and sends a map to its inverse image.
This functor has a left-adjoint, which is simply the same functor, now
viewed as a functor Set --> Set^op.
If we restrict P to some subcategory like Set(k), sets of cardinality at
most k, the above adjointness is clearly lost.
> My feeling is that the *best* place to make the small/large
> (i.e. set/class) distinction is set theory, because sets and classes
> have the least amount of extra structure, namely none. Historically I
> The bottom line is that the small/large (i.e. set/class) distinction
> belongs to f.o.m.
Again, yes and no. The distinction certainly belongs there (and comes from
there). But it does arise in "naive" category theory as well. As for sets
you can form (gramatically) the "category of all categories" (this is
different to other structures: the collection of all groups does not have
a group structure, ...). After Russell one _has_ to reflect on this and
think about whether this gives problems or not. Since such a step from
naive to non-naive is the study of foundations, at the end I probably
agree with Steve:
The distinction small/large does belong to f.
(I ommitted the o.m. on purpose. Steve will probably say, ... but
category theory is then part of mathematics, so that at the end the
distinction does belong to f.o.m. ... Let's please not start a discussion
on that (minor) issue. Thanks.)
Dept. of Mathematics and Statistics
McGill University, Montreal
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