FOM: Conway's book; small category formulation of the Yoneda lemma

Stephen G Simpson simpson at
Wed May 5 17:24:51 EDT 1999

Joe Shipman 05 May 1999 12:32:09 writes:

 > You are being unfair to Conway here.  Please read "On Numbers and
 > Games" before accusing him of anti-foundationalism.  

OK Joe, I will get ``On Numbers and Games'' out of the library and
have a look at it.  I already did have a look at it way back in the
1970's when it was published, but I wasn't very impressed and I don't
remember much of what was in it.  Judging from Johnstone's remarks, I
conjecture that it suffers from an anti-f.o.m. tone.  This would not
necessarily be inconsistent with what you have said about it.

By the way, Mossokowski 05 May 1999 00:04:05 touts the (trivial)
Yoneda lemma as an example of something that goes beyond VNBG, but
actually the following formulation of it makes perfectly good sense in

  For any small (i.e. set size) category A, the small (i.e. set size)
  functor hom: A -> [A^op,Set] is full and faithful.

Furthermore, if you want to totally avoid large (i.e. proper class
size) categories, you can replace the large category Set by any small
(i.e. set size) full subcategory S of Set such that all of the hom
sets of A belong to S.  In this way you get a purely small category
formulation of the Yoneda lemma.  This makes perfect sense in ZFC.

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?

-- Steve

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