Sheaves in Geometry and Logic: A First Introduction to Topos Theory

by Saunders MacLane, Ieke Moerdijk

Sheaves arose in geometry as coefficients for cohomology and as descriptions of the functions appropriate to various kinds of manifolds. Sheaves also appear in logic as carriers for models of set theory. This text presents topos theory as it has developed from the study of sheaves. Beginning with several examples, it explains the underlying ideas of topology and sheaf theory as well as the general theory of elementary toposes and geometric morphisms and their relation to logic.

Genres: Mathematics, Logic, Algebra, Textbooks, Theory, Geometry

642 pages, Paperback

First published May 14, 1992

Reviews

[Gary · Rating 4 out of 5]

Several folks at Amazon have already commented on how this volume provides an excellent formal introduction to category theory and topoi. I have nothing to add to those excellent reviews and encourage others to read them. What I wish to do here is comment on a somewhat tangential subject of my own interest, and that is the philosophy of logic.

This book makes clear the very significant connections between logic, what I would call "general spatial reasoning," and category theory. For anyone interested in the underlying core and structure of formal reasoning beyond the questionable dogmas of Russell-Frege proof-theoretic approaches, this book is an absolute must have. The proof-theoretic methods that have swamped the thinking about logic in most philosophical circles has seriously undermined our understanding of the relevant issues by blinding scholars to the genuine wealth of ideas that exists within mathematics. I would argue that this book, in conjunction with such works as Corry's "Modern Algebra and the Rise of Mathematical Structures" is a fundamental step away from the shackles of the Russell-Frege vision of formal logic that dominates so much thought in the philosophy of logic. (Corry's work places algebraic thinking within an historical context that the mere formal study of the subject tramples right over. Such historical context is an essential element in the philosophical -- as opposed to purely formal -- study of such topics.)

The materials in _Sheaves_ are presented in an accessible way for the non-mathematician, *provided* that person still has a reasonably solid background in some such topic(s) as formal logic, model theory, abstract algebra, etc. The focus of the text on those relational structures known as "sheaves" provides an especially illuminating approach to the connections between algebraic logic, category theory, and such "purely" logical topics as proofs and models.

Also, let me add that I am writing this review of the *Kindle* edition. Obviously the wood-pulp version is wonderful, but the eBook version is well formatted, with none of the crazy symbol and fornatting issues that dogged earlier attempts to migrate mathematical texts into an electronic format.

So I recommend this book to anyone with even a passing interest in philosophical logic. The time has come to move beyond Russell-Frege, and this text is an excellent instrument for taking such a step!

[E. · Rating 5 out of 5]

standard reference. some definitions disagree with the mainstream approach today, but overall still relevant.