Sheaves in geometry and logic: a first introduction to topos theory
type: Item
instance of: scholarly article
title: Sheaves in geometry and logic: a first introduction to topos theory
zbMATH Open document ID: 0822.18001
author: Ieke Moerdijk
author: Saunders Mac Lane
published in: Universitext
publication date: 1992-09-18T00:00:00Z
review text: The summer of 1995 marked two anniversaries. The first is that of the 1945 paper of Eilenberg and MacLane which formally introduced the terms ``category'', ``functor'', and ``natural transformation'' to the world of mathematics [cf. \textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231-294 (1945; Zbl 0061.092)]. From the perspective of fifty years later, it seems quite safe to say that the 1945 paper, along with the ideas and seemingly endless supply of examples which flowed from it, changed forever the working vocabulary of mathematics. For some, the change was, and still is, viewed much as classically trained physicists viewed the ``gruppenpest'': ``Ignore it and perhaps it will go away''. For many, at least those who still see it as merely giving a formal meaning to Emmy Noether's 1930's then lonely and revolutionary, ``Don't just study the Betti numbers, but rather the groups and the boundary homomorphisms which give rise to them!'', it has only somewhat trivially changed the way that they have long studied and communicated ``modern'' mathematics. But for a smaller and younger group, that 1945 paper provoked a change not just in terminology or even methodology, but in their entire philosophical approach to mathematics. To see how profound that change could be after only twenty-five years, one needs only to look at the second of these 1995 anniversaries, that of the 1970 International Congress of Mathematicians in Nice where Lawvere introduced his joint work with Tierney and the concept of an ``elementary topos'' [cf. \textit{F. W. Lawvere}, Actes congr. internat. Math. 1970, 1, 329-334 (1971; Zbl 0261.18010)]. Still in active development, the ideas embodied in the non set-theoretic, categorical language of ``elementary topos theory'' not only incorporate the already extraordinary set-based categorical view of geometry (as abstract sheaf theory) pioneered by the Grothendieck school, but even lead to a totally new way of looking at the logic…
Mathematics Subject Classification ID: 18-01
Mathematics Subject Classification ID: 03-01
Mathematics Subject Classification ID: 18B25
Mathematics Subject Classification ID: 03G30
Mathematics Subject Classification ID: 18-02
Mathematics Subject Classification ID: 03-02
Mathematics Subject Classification ID: 03E99
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 18F20
Mathematics Subject Classification ID: 54B40
zbMATH DE Number: 53911
zbMATH Keywords: elementary topos
zbMATH Keywords: sheaf theory
zbMATH Keywords: model theory
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 53911