The first time I came across the phrase topos theory was in the book The Trouble with Physics by Lee Smolin in which he describes topos theory as an area of logic that “a few mathematicians have used to explore radical ideas about the nature of time.” While that certainly piqued my curiosity, I still have no idea what the quote from Smolin’s book is all about. After reading through Robert Goldblatt’s Topoi: The Categorial Analysis of Logic, however, I did finally learn something about topos theory as an area of logic.
In his prospectus to the book Goldblatt states
The purpose of this book is to introduce the reader to the notion of a topos, and to explain what its implications are for logic and the foundations of mathematics.
The author aims to make the book accessible to a wide and mixed audience of logicians, mathematicians and philosophers without prior background in much of the mathematics required to define the notion of a topos. Specifically, Goldblatt does not assume that the reader has experience in category theory. Indeed, as a novice myself in category theory, mathematical logic, and topos theory, I appreciate the author’s exposition, which involves carefully motivating the concepts, presenting simple examples, and gradually proceeding from the specific to the general.
I can imagine that readers with more experience may find the presentation overly simple. For example, much of the category theory covered in the text could likely be skipped by a reader with a knowledge of categories at the level presented in many graduate level algebra texts. Additionally, the discussion of some topics in the book (such as bundles and sheaves) are a little hand-wavy. Nevertheless, I feel that there is considerable value in Goldblatt’s approach. There exist, after all, many other references which provide greater rigor that the reader may use to supplement the exposition in Topoi: The Categorial Analysis of Logic. In fact, the reader of the Dover edition of Goldblatt’s book can find an updated list of references on topos theory in the new preface (see pages xv–xvii).
Before going into additional detail on Topoi: The Categorial Analysis of Logic, I will briefly describe the notion of a topos and some of the motivation for their study. Most mathematicians grow up with the idea that set theory provides the foundation for our subject. There are a number of important constructions in set theory that have proved extremely useful in mathematics. Examples are the product of two sets, the power set of a set, and the set of all functions between two sets. A topos is a category which allows for constructions analogous to those. Examples of topoi are the category of sets and the category of sheaves of sets on a topological space. “A startling aspect of topos theory is that it unifies two seemingly wholly distinct mathematical subjects: on the one hand, topology and algebraic geometry, and on the other hand, logic and set theory.” (From Sheaves in Geometry and Logic, by MacLane and Moerdijk.)
In this book Goldblatt explains that there are essentially two separate historical sources for topos theory. One was in algebraic geometry, where Alexander Grothendieck and others sought to define a notion of cohomology inspired by conjectures of Andr&eqcute; Weil that would be suitable for the solution of important problems in that field. Another came from attempts by F. William Lawvere to develop a categorical (or categorial) foundation for set theory. Note that there is a Wikipedia article on the history of topos theory and the article “What Is… a Topos?” by Luc Illusie in the October 2004 Notices of the AMS elegantly explains the genesis of topoi in algebraic geometry.
One of the key steps toward the definition of a topos is to define common set theoretical notions without referring to elements. Instead, one uses commutative diagrams and universal properties. Some constructions that are particularly important are the pullback and its dual notion the pushout. A careful presentation of this is given in Topoi: The Categorial Analysis of Logic. The author always shows how the usual construction in sets can be recast without elements, using functions and arrows, and then gives the general categorical definition. I find this to be one of the major strengths of Goldblatt’s book, making categorical constructions seem natural. Plus, the reader gets a lot of experience in diagrammatic reasoning, which is a valuable tool. On the other hand, Goldblatt splits up his treatment of category theory. The first few chapters discuss categories and morphisms, but the reader does not see functors until chapter 9. Adjunctions appear only in chapter 15. In each case, these are preceded by several chapters dealing with topics from logic.
After “elementless set theory” we come to the definition of topos on page 84. To paraphrase, a topos is a category that has initial and final objects, pushouts and pullbacks, exponentiation, and a sub-object classifier. Next, examples of toposes are given and Goldblatt proceeds to consider their relation to logic. Some other topics that receive interesting coverage are intuitionist logic, model theory and the continuum hypothesis.
The text is very readable and the author does a nice job motivating the theory and describing the historical development of the subject. While the book may not provide insight into radical ideas about the nature of time, I still think that Topoi: The Categorial Analysis of Logic is worth reading, especially if you have an interest in the philosophy or foundations of mathematics and logic but are not necessarily interested in becoming an expert in the field.
Jason M. Graham is an assistant professor in the department of mathematics at the University of Scranton, Scranton, Pennsylvania. His current professional interests are in teaching applied mathematics and mathematical biology, and collaborating with biologists specializing in the collective behavior of groups of organisms.