In Topoi Goldblatt uses category theory to explore the logical foundations of mathematics, while using logic as the motivation for learning category theory. (The alternative approach from algebraic topology is largely ignored. Sheaves get a brief mention around page 100, but are only used in the last third of the book, while functors and natural transformations are only touched on. It is possible to read the larger part of Topoi without knowing what a topological space is!)

Topoi begins with an introduction to category theory and a steady build up to explaining how sets — or a generalisation thereof, what is known as a topos — can be defined without the concept of membership. This is well-motivated, with Goldblatt using analogies to set-theoretical ideas which might have been disdained by a category theory purist.

Goldblatt proceeds with more or less independent chapters taking a categorial approach to different facets of mathematical logic: classical logic and semantics, the algebra of subobjects (implication and extensionality), intuitionist logic (Heyting algebras, Kripke semantics, and a five page history of Intuitionism), functors, set concepts and validity, elementary truth, the category of sets, and arithmetic.

The last third covers local truth (Grothendieck topoi, geometric modality, Kripke-Joyal semantics), adjunctions and quantifiers, and logical geometry. Some of this is considerably more difficult — I confess to skipping parts of it — but it remains well-motivated and Goldblatt is willing "to take an approach that will be more descriptive than rigorous".

Originally published in 1984 but understandably a classic, Topoi has fortunately been reprinted by Dover as a cheap paperback.

February 2014

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