Categories Mailing List
Tue, 28 Mar 2006 21:30:59 -0500 (EST)
WHY ARE WE CONCERNED? II
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 misunderstanding.
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 E.]
F. William Lawvere
Mathematics Department, State University of New York
244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA