Re: Small is beautiful

[Thomas Streicher <streicher@mathematik.tu-darmstadt.de>]

A few little comments on "small is beautiful"

1) The nice thing about small cats is that externalizing them gives rise to
a split fibration and that allows one to speak about equality of objects.
But if we have got a split fibration P : XX -> BB it may be considered as
a small cat over \widehat{BB} = Set^{BB^op} (for Set big enough). To my
knowledge this observation is due to Jean B'enabou and also found its way
into Bart Jacob's book.

2) Identifying small with representable seems to be an idea going back to
Grothendieck already (as told to me by Jean B'enabou and taken up by him).
Namely Grothendieck's notion of representable morphism in \widehat{BB} captures
the notion of a family of small things indexed by a possibly large index object
(arbitrary presheaf). The identification of small as representable lies at the
heart of B'enabou's definitions of the properties locally small and well powered
for fibrations.
In this sense the definition of n elementary topos also amounts to a smallness
condition: namely as a category EE with finite limits such that its fundamental
fibration EE^2 -> E is well powered.

3) Use of the idea "small is representable" has been made in
Algebraic Set Theory (AST) in the formulation of Awodey, Simpson
and collaborators (see www.phil.cmu.edu/projects/ast/ for more information).
There starting from a topos EE or (when working "predicatively") from a locally
cartesian closed pretopos EE one considers the topos Sh(EE) of sheaves over EE
w.r.t. the coherent, i.e. finite cover topology. Sh(EE) is thought of a
category of classes and the full subcat of representables as the full subcat
of sets. But Sh(EE) is a bit too large because for objects X in Sh(EE) the
diagonal \delta_X (equality on X) need not be a representable morphism (and
well behaved predicates should be since otherwise separation would lead out
of sets). Thus instead of Sh(EE) one considers the full subcategory Idl(EE)
of Sh(EE) on those separated objects, i.e. those where the diagonal is a
representable mono). Notice that separated for a presheaf over EE (a split
discrete fibration over EE) means that equality is definable in the sense
of B'enabou. It was suggested to Awodey et.al. by Joyal that the separated
objects in Sh(EE) can be characterized as those presheaves over EE which can
be obtained as an "ideal colimit" of representable objects ("ideal" meaning
directed diagram of monos). A further nice characterization of X being in
Idl(EE) is that the image of a map y(A) -> X (taken in Sh(EE)) is again
representable.
Now working in Idl(EE) one can define for X in Sh(EE) its "class of subsets"
P(X) as follows: P(X)(Y) is the collection of subobjects of y(I) x X whose
source is representable, i.e. monos of the form y(J) >--> y(I) x X.
By iterating P one obtains fixpoints (not representable) of P which serve as
universes for interpreting appropriately weak set theories.

As already mentioned by Bob the set theorist Vopenka wrote a lot about
set theories where subclasses of sets needn't be sets again. He called
"semiset" a subclass of a set which is not a set itself. Although Vopenka
doesn't emphasize this point he is working in an ultra power extension of
V_\omega (because he wants the negation of the Infinity axiom to hold) and
there subclasses of a set need not be in the ultra power extension. As I have
heard (and seen some notes of talks by him) B'enabou quite some time ago worked
on a Nonstandard Theory of Classes which relates to Nelson's Internal Set Theory
like GBN to ZFC. I am vaguely aware of extensive work by NSA people on
nonstandard class and set theories motivated by similar ideas (but this was
much later) but they have a somewhat richer ontology. There are 2 books to
mention in this context

Anatoly G. Kusraev, E. I. Gordon, S. S. Kutateladze
"Infinitesimal Analysis"
Kluwer Academic Pub (2002)

V. Kanovei, M. Reeken
"Nonstandard Analysis, Axiomatically"
Springer 2004

Both views have in common that "set" has nothing to do with size but rather
with "being definable in a reasonable sense" (the collection of standard
elements of a set is typically not a set because "standard" is not a clear
cut notion). This was concealed by early axiomatizations of class theory.

I wonder now whether these two notions of "smallness" (better called "sethood")
can be reconciled more precisely.

-- Thomas


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