FOM: Yoneda's Lemma

[Todd Wilson]

Steve Simpson 5 May 1999 17:24:51 writes:
> Either of these formulations of the Yoneda lemma is perfectly good for
> applications.  Nothing important is lost.
> 
> But apparently category theorists don't like these formulations.  Why
> not?  If set theory is a good enough foundational framework for most
> mathematicians, why isn't it good enough for category theorists?

I can't speak for all (or even a significant proportion) of category
theorists, but anyway here is my answer to this question.  Category
theory owes its extistence to a certain frame of mind or point of
view.  Technically, the whole field of category theory is eliminable
in the sense that the application of any one of its theorems to a
particular category can always be replaced by reasoning "within" that
category; results involving several categories and functors between
them can similarly be expressed using more elementary notions
involving the particular categories concerned; and so on.  (I'm not
quoting a metatheorem here, but the process is clear enough, I trust.)

What good is category theory, then, if it can be eliminated in this
way?  Answer:  It is the apropriate carrier for many "structural"
results, when these are expressed in their most general form.  Without
category theory, we would not be able to express in a sufficiently
final way, for example, that limits for all diagrams can be
constructed from products and equalizers (or many other similar
results).  What attracts many researchers to category theory, I
imagine, is this aspect of finality, of capturing the essence of a
structural insight and making it available in the widest range of
situations.

Therefore, when a category theorist sees a structural result like the
Yoneda Lemma, which basically proves the existence of a natural
isomorphism between hom sets in different categories, and which only
involves, at its core, the notions of composition of morphisms and
functor application, he or she wants to highlight those simple
structural properties without having to clutter the picture with
not-so-simple notions from set theory -- especially notions involving
certain classes of (large) cardinals.  This may not always be
possible, but that is what one is looking for.  If current
set-theoretic foundations do not allow these simple structural results 
to be expressed in their most elegant form, the one is led to look for 
alternative foundations for which this is possible.

I don't think category theorists are out to denegrate f.o.m.  I think
their (apparent) hostility towards the subject is more the result of a 
reaction against a misunderstanding of their own purposes and
intuitions. 

-- 
Todd Wilson
Computer Science Department
California State University, Fresno