Summary: Clear, concise, and well motivated.
Rating: 5
I have only read the first seventy pages of this book, but so far it is excellent. The character of the book can be likened to Kleene's "Introduction to Metamathematics" or "Mathematical Logic." The subject is well motivated by brief philosophical discussions, but the philosophy is not allowed to interrupt the flow of mathematical development. There are few exercises, but enough for the casual reader to check from time to time that he or she is understanding the material. The exposition itself is perfectly clear and concise, by my lights. Nothing important is skipped--one does not need to fill in the blanks in proofs--yet the book moves at a good pace. The focus of the book is on the development of Category theory in relation to its applications in logic. For this reason it is probably of more interest to a logician or a philosopher than to a computer scientist. Indeed, for the logician who is interested in Category theory this book cannot be missed. To put it bluntly, Goldblatt strikes me as a superior mind, even his brief remarks on set theory were enlightening. The book has been (to this point, at least) a very enjoyable read.
Summary: Great introduction to the theory of toposes.
Rating: 5
When Goldblatt's book first hit the stands, it was blasted by reviewers who had a geometric predilection. They claimed that Goldblatt had trivialized this essentially rich geometric subject by giving a set-theoretic treatment.I became fascinated with topos theory in the summer of 1989. My only experience with category theory at that time was some dabbling that every graduate student gets in studying modules and in functional analysis. I didn't have much background in algebraic topology where the subject is usually developed considerably further. Frankly, I found Goldblatt's "gentle" approach to categorical concepts and the concept of a topos very rewarding. I worked through most of the book in a summer, and was fully prepared to take on the more advanced texts at that point. Later, I had the opportunity to teach topos theory at the graduate level to people with backgrounds similar to mine (i.e., without a strong background in category theory). We worked through most of Goldblatt's book in the first semester and I guarantee that all the students were very grateful, as I had been, for Goldblatt's approach to the subject.The book is well written, accessible to graduate students, filled with fun and often challenging exercises, and packed with references. In my opinion, it is the right place to start if you want to become proficient in topos theory (and you don't already have significant proficiency in category theory).

 

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