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Author: Saunders Mac Lane (University of Chicago) and Ieke Moerdijk (University of Utrecht)

Reference: Mac Lane, S. and Moerdijk, I. Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Universitext. Springer-Verlag, 1994.

Why this paper? Cited by Non-Standard Models of Homotopy Type Theory and Separation Logic of Generic Resources via Sheafeology.

(My blogging schedule is a little out due to some delays getting hold of books.)

This is the second tome on topos theory I’ve read for this blog, but compared to The Elephant this one weighs in at a svelte 627 pages, and is a bit more approachable as a book to learn from, as opposed to an encyclopaedic reference. The conceit of The Elephant was that topoi can be usefully viewed in many different ways; in this book the two views that come through strongest are topos as universe of set-like objects, and topos as ‘categorification’ of the notion of topological space.

The first view is by now quite familiar to me. We start with the notion of category, which is a very general notion which is often used to discuss collections of mathematical objects (e.g. sets) and the transformations between them (e.g. functions).  We then add a few more simple requirements to define (elementary) topoi, categories which in some ways behave particularly like the universe of sets, for example supporting notions analogous to the characteristic function of a subset, and to set comprehension. One of the highlights of this book is the description of how some famous results of set theory, notably Cohen’s proof of the independence of the axiom of choice, can be naturally developed in topos-theoretic language.

Being less of a topological person than I am a sets-and-logic person, the view of a topos (not the objects of the topos, but the entire category) as analogous to a topological space is less routine for me, but comes through strongly in this book. We see how topoi can have ‘Lawvere-Tierney’ topologies defined on them, and frequently work by direct analogy with topological constructions, for example briefly describing classifying spaces before defining classifying topoi, or sober spaces before localic topoi.

The book is splendid for the structured way it introduces, then repeatedly returns to, collections of concrete examples of topoi. The most significant of these is, of course, the sheaves of the book’s title – starting with a topological space, we look at the contravariant functors (also known as presheaves) from the open sets of that space into the category of sets, then cut down to certain particularly well-behaved such functors. Sheaves are elementary topoi, and any category equivalent to such is called a Grothendiek topos. Although not all elementary topoi are Grothendiek topoi, this more specific notion is important in algebraic geometry, leading for example to the resolution of the Weil conjectures by Deligne in the 1970s.