F W Lawvere | 29 Mar 04:30 2006




	The question is not whether mathematics should be applied. Most of
us agree that it should. The concern is rather that our subject is
sometimes being used as a mystifying smoke screen to protect
pseudo-applications against the scrutiny of the general public and of the
scientific colleagues in adjacent disciplines. We need to ensure that
applications themselves be maximally effective, not clouded by

	Some of the most important applications of our unifying efforts as
categorists have been to the
	teaching of algebraic topology
	teaching of algebraic geometry
	teaching of logic and set theory
	teaching of differential geometry
These subjects all arose from the efforts to clarify and apply calculus;
thus some of us have applied category theory to the teaching of calculus.

	But it seems that we have not taught category theory itself well
enough. Several recent writings reveal that basic misunderstandings about
category theory are still prevalent, even among people who use it. Some of
these concern the myth that category theory is the "insubstantial part" of
mathematics and that it heralds an era when precise axioms are no longer
needed. (Other myths revolve around the false belief that there are "size
problems" if one tries to do category theory in a way harmonious with the
standard practice of professional set theorists; see next posting.) The
first of these misunderstandings is connected with taking seriously the
jest "sets without elements". The traditions of algebraic geometry and of
category theory are completely compatible about elements, as I now show.

	Contrary to Fregean rigidity, in mathematics we never use
"properties" that are defined on the universe of "everything". There is
the "universe of discourse" principle which is very important: for
example, any given group, (or any given topological space, etc.) acts as a
universe of discourse. As these examples suggest, a universe of discourse
typically carries a structure which permits interesting properties and
constructions on it. As the examples also show, there are typically many
objects of a given mathematical category and also many categories, so
transformation is an essential part of the content. As quantity includes
zero, so structure includes the case of no structure, which Cantor
considered one of his most profound and exciting discoveries. (His
conjecture that the continuum hypothesis holds in that realm is probably
true. [Bulletin of Symbolic Logic 9 (2003) 213-223].) Dedekind, Hausdorff,
and most of 20th century mathematics followed the paradigm whereby
structures have two aspects, a theory and an interpretation of it in such
a featureless background. Because the background thus contributes minimal
distortion to the assumptions of the theory, the completeness theorems of
first-order logic, the Nullstellensatz, and related results are available.
The more geometric background categories which receive models are also
viewed as structures (of an opposite kind) in abstract sets, for example
the classifying topos for local rings as a background for algebraic
groups. Such is "set theory" in the practice of mathematics; it is part
of the essence from which organization emerges.

	By contrast, the "set theory" studied by 20th century set
theorists has a different aim and architecture. The aim is "justification"
of mathematics, and the architecture is that of the cumulative hierarchy.
The alleged need for justification arose in connection with the re-naming
of Cantor's theorem as "Russell's paradox"; Cantor's theorem had shown
that the system proposed by Frege was inconsistent, but there were those
who dreamed nonetheless of restoring that rigidity. There was a bitter
controversy between Cantor and Frege, and Zermelo swore allegiance to
Frege [Cantor G.: Abhandlungen mathematischen und philosophischen Inhalts,
1966, page 441, remarks of Zermelo on Cantor's 1884 review of Frege]. Von
Neumann based himself on Zermelo and made explicit the cumulative
hierarchy, which Bernays and Goedel used and which many subsequent set
theorists presumed was the only architecture to be studied. The
justificational aspect stems from the supposed construction of the
hierarchy by a bizarre parody of ordinary iteration, parameterized by
infinite ordinal numbers (Cantor's third discovery), entities which from
the point of view of ordinary mathematics are even more in need of
justification than the analysis that supposedly needed it. (Indeed, in
attempting to describe what these alleged infinite ordinals are and do,
people often resort to stories about gods and demons.) Little or no
progress has been made on this "justification" problem in a century, but
work with the hierarchy has produced some knowledge about the
possibilities for categories of sets. By adopting a standard definition of
map and discarding the mock iteration (with its concomitant complicated
structure), each model of the cumulative hierarchy yields a category of
abstract nearly featureless sets; most of the usual set-theoretical issues
depend only on the mere category: measurable cardinals,
Goedel-constructibility, the continuum hypothesis, etc.

	Having thus briefly understood the two visions which are called
set theory

(1) a category of Cantorian featureless sets which serves as the
background recipient for the structures of algebra, geometry and analysis;

(2) the cumulative hierarchy with its rigid Fregean structure aiming to
justify mathematics,

it is not surprising that the precise nature of the elementhood relations
appropriate to each are quite different. While the Fregean image involves
rigid inclusion and elementhood relations imagined to be given once and
for all for mathematics as a whole, the usual mathematical practice
instead considers inclusion and membership relations for subsets of a
given universe of discourse (such as R^3). Thanks to Grothendieck's Tohoku
observation, these mathematical local belonging relations are well
globalized within the notion of category, whose primitives are domain,
codomain, identity, and composition.

[The notion of category is a simple first-order theory of a semi-algebraic
kind. It has myriads of interpretations, some in "classes", some "locally
small" etc., but such undefined restrictions on interpretations have
nothing to do with the notion of category per se. Many properties are best
expressed within the first-order theory itself.]

	Composition is a kind of non-commutative multiplication, hence
there are two kinds of division problems. In any category, given any two
morphisms a and b we can ask whether there exists a morphism p such that
a = bp; if so, we may say that a belongs to b. This forces a and b to have
as codomain the same object, which serves as their common universe of
discourse. (The dual relation, f determines g, defined by "there exists m
with mf = g", is probably equally important in mathematics.) There are two
special cases of this belonging relation which are of special interest.
First we say that b is a part (or subset in the case of a category of
sets) of its codomain, if for all a belonging to b, the proof p of that
belonging is unique; this is immediately seen to be equivalent to the
usual notion of monomorphism. Then, if a and b are parts of the same
object, we say a included in b iff a belongs to b. Any arbitrary morphism
x with codomain X may be considered an element of X in the sense of
Volterra (also known as a figure in X); we say that x is a member of b iff
x belongs to b. Then clearly

 a is included in b iff for all x, if x is a member of a, then x is a
member of b.

The usual relationship between these two relations is thus maintained.
Because in category theory the domain relation is as important as the
codomain relation, we can be more precise about elements: very often it is
appropriate to consider a special property of objects, and restrict the
term element (or figure) to elements whose domain has that property, that
is, to figures whose shape has the property. For example, in algebraic
geometry the connected separable objects are appropriate domains for the
figures known as "points"; in the algebraically closed case it suffices to
consider elements with domain a terminal object 1 as points. On the other
hand, frequently it is of interest to choose a small class of figure
shapes which generates in the sense of Grothendieck, i.e. so that the
above equivalence between inclusion and universal implication of
memberships holds even when the figures x are restricted to those of the
prescribed shapes. A basic property of categories of Cantorian sets is
that this holds with x restricted to those with terminal domain 1. In
algebraic geometry, the figures whose domains have trivial cohomology are
adequate. Note that if f is a morphism from A to B and if x is an element
of A, then fx is an element of B of the same shape (of course in general
figures are singular in that they distort their shape, for example, fx may
be more singular than the figure x). Properties of x in A may be quite
different from the properies of fx in B.

	The mysterious distinction between x and singleton(x) in the
hierarchical Frege architecture takes quite a different form in the
categorical architecture where there is a natural transformation from the
identity functor to the covariant power set functor; this natural
transformation can be called singleton: singleton(x) is simply x
considered as a special element of PX, rather than of the original X.

	Professors may not consider the possibility of learning from
undergraduate text books, and some may feel bored that I have once again
repeated the above basic definitions and observations. But if these basics
were widely understood among algebraic geometers, perhaps misconceptions
like "category theory is the insubstantial part of mathematics" would not
have arisen. (As we know from experience, all of the substance of
mathematics can be fully expressed in categories.) Perhaps the general
term "A-points" for arbitrary rings A was confusing. "Spec(A)-shaped
figures" is a more accurate rendering of Volterra's "elements"; that could
be abbreviated to "A-figures", but points are in some sense special among
figures. On the other hand, we often vary the background category, so that
alternative terminology might involve passing from a category E to
E/spec(A), and restricting the notion of "point" in any category to mean
figure of terminal shape; then the A-figures become, on pulling back to
the new category, literally "moving points".

	Whatever the particular chosen terminology, the important
conclusion is to actively eliminate the mythology that spaces in
categories have no elements, because as we see, this mythology obscures
the simplicity of certain matters and thus provides a bogus basis for
insulating one field of mathematics from another.

	[The belonging relation is just the poset collapse of
the categories E/X, whose actual maps serve as incidence relations,
especially between figures in X. Thus every category E supports a certain
geometrical imagery wherein all maps are geometrically continuous, in that
they map figures to figures without tearing the incidence relations.
Precise axioms about E are a key to further progress because they
explicitly sum up and guide our experience with the objects and maps in

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