FOM: RE: Foundations of category theory

[Lawrence N. Stout]

Stephen G Simpson wrote:
  >
  > As McLarty has noted, Corry speculates on page 332 that Bourbaki
  > stayed away from categories because it's difficult to reconcile them
  > with structures.  In actual fact, such a reconciliation is difficult
  > only if you arbitrarily decide that, while structures must be sets,
  > categories may be proper classes.  There is no serious mathematical
  > reason forcing anyone to make such a decision, so Corry's
speculation
  > seems incorrect.
  >
 
A serious reason forcing such a decision is that the most basic examples
of categories, namely the category of Sets (with functions as
morphisma), the category of topological spaces (with continuous
functions), and the category of abelian groups (with group
homomorphisms), are all proper classes.  Steve seems to be arguing that
categorists _shouldn't_ want to be talking about the categories of _all_
sets, topological spaces, or abelian groups, but they _do_ want to.
Many of the interesting examples of topoi are built from the category of
Sets.  They could perhaps be built on small categories of sets of
restricted size, but that seems an unnecessary complication.
 
Actually there is a problem, but it isn't with category theory.
Independence and consistency results tell us that there isn't a single
_category of Sets_ since we have choices about what axioms sets will
satisfy and thus, presumably, about what will count as a set.  Large
cardinals ae relevant here--I can either assume that there is a
Grothendieck universe or not.  If I want certainty in my foundations, I
have to look at something more restricted than Sets.
 
Let me ask a foundational question:  is there an axiomatization which
completely specifies the category of _finite_ sets?  Characterization of
what finite means becomes quite complicated in the absence of the axiom
of infinity (so that finite cardinals are not available for comparison)
and in the absence of choice.  (This was pointed out by Tarski in the
20's.)  Working in an intuitionist setting makes the matter even more
complicated (a paper by Troelstra gives several infinite families of
defintions of finite).
 

-- 
Lawrence Neff Stout
Professor of Mathematics
Illinois Wesleyan University

http://www.iwu.edu/~lstout

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