Subject: categories: WHY ...CONCERNED? III

From: F W Lawvere <[email-masked]>
Date: Fri, 31 Mar 2006 09:15:28 -0500 (EST)

WHY ARE WE CONCERNED? III

The second main misconception about category theory

Part of the perception that category theory is "foundations" (in the
pejorative sense of being remote from applications and development) is due
to a preoccupation with huge size. Since such perceptions hold back the
learning of category theory, and hence facilitate its misuse as a
mystifying shield, they are among our concerns. We need to deal with the
size preoccupation head on.

	Experience has shown that we cannot build up or construct
mathematical concepts from nothing. On the contrary, centuries of
experience become concentrated in concepts such as "there must be a group
of all rotations" and we then place ourselves conceptually within that
creation; we state succinctly the properties which that creation as a
structure seems to have, and then develop rigorously the consequences of
those properties taken as axioms. The notion of category arose in that
way, and in turn serves as a powerful instrument for guiding further such
developments. Placing ourselves conceptually within the metacategory of
categories, we routinely make use of the leap which idealizes the category
of all finite sets as an object. The question is, what more? Of course we
make use of the experience of those who have labored to justify
mathematics, and it is fortunate that ultimately our results are
compatible with theirs. (Mac Lane's use of the term metacategory is not
mysterious; it simply refers to the universe of discourse of any model, in
the special context where the elements of such a model are themselves
called categories and functors. In the spirit of algebra, we do not
concentrate on the cumulative hierarchy which might have been used to
present the metacategory, but rather on the mathematical category itself.)

	The supposed size problems of category theory are often
concentrated in functor category formation. For any two categories that
are objects of the metacategory, the category of functors from one to the
other exists in the sense that it also is an object in the metacategory
(it is unique by exponential adjointness). That existence statement is
compatible with standard set theory, although it is often presumed to be
incompatible.

	In the original 1945 exposition of category theory, it was the
Goedel-Bernays account of the cumulative hierarchy (see posting II) that
was cited as probably relevant (in case the problem of justifying category
theory should come up). As a result, category theorists have been worried
about supposed "illegitimacies" that might arise from violating the
Goedel-Bernays rules (which in essence stemmed from von Neumann). These
rules expressed an expediency which was a very effective trick at the
time, identifying two kinds of membership relation and truncating the
content at a plausible level. The Goedel-Bernays theory is well known to
have the same logical strength as the Zermelo-Fraenkel system. An
important advantage is that the greater expressive power of Goedel-Bernays
permits it to be finitely axiomatizable, whereas Zermelo-Fraenkel is not;
the greater expressive power concerns an element V of any model in which
all small sets of the model can be embedded (just as another smaller
element captures all finite sets). But the greater expressive power still
allows mutual relative consistency: To every model of Goedel-Bernays, a
model of Zermelo-Fraenkel can be constructed in a fairly straightforward
manner: just take the small elements; in the converse direction there are
two procedures (left and right adjoint?): given a model of
Zermelo-Fraenkel, one can take all definable subsets of it, or just all
subsets, and in either case a model of Goedel-Bernays apparently results.
Because these mutual interpretations are hypothetical, relatively weak
assumptions are required on the background category of sets taken as the
recipient of models. In fact, with only slightly stronger assumptions on
the background category one can construct, for any model of
Zermelo-Fraenkel, a model of what set theorists use daily as BG+, which
contains as elements not only V but W = V^V, V^W etc.

	Our practice is consistent with the minimal assumptions of
professional set theorists: For any model of BG+ the presented
metacategory of categories is both cartesian closed (in the usual
elementary sense) and also has an object S of small sets. (Those facts
strongly augment well-known properties, such as the existence of the first
four finite ordinals and their adequacy in the metacategory relative to
the sub-metacategory of discrete categories; of course these same ordinals
also co-represent one of the "2-category" structures on the metacategory).

	The category S is itself cartesian closed, and the categories of
structures of geometry and analysis are enriched in it. Of course functor
categories may no longer enjoy the same enrichment, just as functor
categories starting from finite sets may not have finite hom-sets; but
that is no reason to avoid considering them, and functionals on them, etc.
when such considerations serve mathematics.

	It is of special interest to note that the restrictive "law"
(under which categorists have been chafing) was already repealed forty
years ago by Goedel and Bernays themselves. In their correspondence of
1963, it appears that they had been informed that a student of Eilenberg
was working on a project to base set theory and mathematics on category
theory; their immediate response was that mathematics will have to
consider finite types over the class of small sets. (The relative
consistency was presumably obvious to them.)

	Even though most set-theorists have themselves maintained clarity
on the distinction, the identification of two kinds of membership in a
formalized theory may have fostered in the minds of others a confusion
between smallness (of a class or set) and existence as an element of the
(meta)universe. Certainly, the specific meaning of smallness needs to be
clarified (although for some purposes it can be taken as a parameter).
There is a way of specifying smallness that is directly related to
fundamental space/quantity dualities (rather than to imagined "building
up" by stronger and stronger closure properties).

	Just as Dedekind finite sets X are characterized by the condition
that a natural map
			X --->Hom(Q^X, Q)

is an isomorphism, so indications from the study of rings of continuous
functions and other branches of analysis strongly suggest that all small
sets X should satisfy the same sort of isomorphism, with the truth-value
space Q being replaced by the real line (in both cases, Hom refers to the
binary algebraic operations on the object Q). There is the possibility to
assume that conversely all sets X satisfying that isomorphism are small
i.e. that, like the Dedekind-finite sets, they belong to a single
uniquely-determined category S. That possibility in itself would imply no
commitment concerning the existence or non-existence of super-huge objects
in the metacategory "beyond" S, S^S, etc.  Such an axiom would be somewhat
stronger than ZF, but much weaker than the standard discussions of
contemporary set theorists.

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F. William Lawvere
Mathematics Department, State University of New York
244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA
Tel. 716-645-6284
HOMEPAGE:  http://www.acsu.buffalo.edu/~wlawvere
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