Saunders MacLane

Duality for groups

If F is known to be free, with generators gi, /3 may be constructed by setting j3g; = &», with bi so chosen that pbi = agi. Conversely, let F have the cited property and represent F as a quotient group Fo/Ro> where F0 is a free abelian group. Choose A—F and B = Fo in (1.1), let a be the identity, and p the given homomorphism of .Fo onto F with kernel Ro. Then, by (1.2), a=p/3 is an isomorphism, hence /3 has kernel 0 and thus is an isomorphism of F into F0. Therefore F is isomorphic to a subgroup of a free group F0, so is itself free. The analogous theorem is true for free nonabelian groups, when A and B are interpreted as arbitrary (not necessarily abelian) groups; the proof uses the Schreier theorem [l4] that a subgroup of a free group is free. An abelian group D is said to be infinitely divisible if for each dÇ:D and each integer m there exists in D an element x such that mx = d. Such groups may be characterized by a similar diagram

Sets, Topoi, and Internal Logic in Categories

Publisher Summary A recent development in category theory has been the description and study of a useful class of categories called “elementary topoi.” This class includes the ordinary category of sets: diagrams of sets and sheaves of sets on a topological space. The axiomatic description of these categories provides a formulation of axiomatic set theory wholly different from the usual set-theoretic axioms on the membership relation and the further study casts considerable light on a number of problems of foundations. Systematic developments of the properties of a topos from axioms are discussed in this chapter. Other categories with internal logic and quantifiers and their relations to standard logical systems are also described in the chapter. Given the basic geometric character of sheaf theory, this common development of ideas from geometry and concepts from set theory embodies exciting prospects.

The Influence of M. H. Stone on the Origins of Category Theory

This talk is a small piece of historical investigation, intended to be an example of history in the retrospective sense: Starting with some currently active ideas in category theory, it will examine their origins in particular in certain work of Marshall Stone. Hence this talk will not even mention many of Stone’s contributions (his theorem on one-parameter unitary groups, the Stone-Weierstrass theorem, or his results on spectra, on integration, or on convexity); instead we will examine the connection of just a few of his ideas with the subsequent development of category theory.

A conjecture of Ore on chains in partially ordered sets

In a recent investigation, Ore has given a form of the JordanHolder theorem valid for an arbitrary partially ordered set P . This theorem involves essentially the deformation of one chain into another by successive steps, each step being like that used in the conventional Jordan-Holder theorem. Ore observes that his first theorem would be slightly easier to apply if it were proved under a weaker hypothesis. The modified theorem runs as follows :

E. the Impact of Modern Mathematics

of the work in high school, and too often the college courses have merely ir.tensified many of the fallacious judgments in content and approach that have been discussed above. E. A. Cameron, writing in the American Matbematical Monthly for March, 1953, makes the following observations about the introductory courses in college mathematics: The type of courses which offers most promise of substantial contribution to a general education is not adequately described by merely listing the topics covered. The spirit in which the subject is treated is of the greatest importance. An understanding of the nature and significance of mathematics is sought through an emphasis on basic concepts, the logical processes used in developing the subject, and the relation of the discipline to other fields through a consideration of its origins and its applications. Techniques, of course, are necessary, but there is plenty of evidence that many students pass their freshman courses by memorizing techniques without obtaining the slightest insight into the true nature of mathematics.... Increasing dissatisfaction with the inadequate contributions of traditional courses to a general education is impelling some institutions to undertake serious investigation in this area. The number of good textbooks suitable for a course of this character is extremely small. This fact has undoubtedly discouraged some institutions from instituting such a course.