COMMENTS ON THE DEVELOPMENT OF TOPOS THEORY

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Reprints in Theory and Applications of Categories, No. 24, 2014, pp. 1–22.COMMENTS ON THE DEVELOPMENT OF TOPOS THEORYF. WILLIAM LAWVEREPresentationSummarizing several threads in the development of the Elementary Theory of Toposes inits first 30 years 1970-2000, this historical article prepares the reader for later publicationsuch as Johnstone’s Elephant (2002) and for the author’s own steps toward an improvedfoundation for algebraic geometry in the Grothendieck spirit, but using the tools of categorical logic and taking up the theme of axiomatic cohesion.Addendum:An important fact should be noted. It was inaccessible to me at the time of writing thishistorical paper. It concerns the origins of the function-space concept that now embodiesthe basic topological example of cartesian-closed category. I cited seven contributors tothat subject at the end of section 4. Later, when I telephoned David Gale to inquireabout his 1950 paper, he informed me that indeed it was in lectures at Princeton in thelate 1940’s that Witold Hurewicz defined and used the notion of k-spaces to present hissolution of the problem that he had posed to Fox (and which Fox had solved for the sequential case in the work cited here). It seems that (directly or indirectly) it was Hurewiczhimself who by that example inspired the other six works cited here.There are two corrections to note:On p. 719, the caption on the photo of the author should read F. William Lawvere.On p. 734, the author’s address should read:244 Mathematics BuildingUNIVERSITY AT BUFFALOThe State University of New YorkBuffalo, N.Y. 14260USAwlawvere@buffalo.eduOriginally published in DEVELOPMENT OF MATHEMATICS 1950 - 2000, edited by J-P Pier,pp 715-734, Birkhäuser, Basel, 2000.Received by the editors 2014-01-15.Transmitted by M. Barr, R. Rosebrugh, and R.J. Wood. Reprint published on 2014-05-22.2010 Mathematics Subject Classification: 03G30, 18B25.Key words and phrases: topos, category theory, sheaves.1

REPRINTS IN THEORY AND APPLICATIONS OF CATEGORIES will disseminate articles from thebody of important literature in Category Theory and closely related subjects which have never beenpublished in journal form, or which have been published in journals whose narrow circulation makesaccess very difficult. Publication in ‘Reprints in Theory and Applications of Categories’ will permitfree and full dissemination of such documents over the Internet. Articles appearing have been criticallyreviewed by the Editorial Board of Theory and Applications of Categories. Only articles of lastingsignificance are considered for publication. Distribution is via the Internet tools WWW/ftp.Subscription information. Individual subscribers receive (by e-mail) abstracts of articles as they arepublished. To subscribe, send e-mail to tac@mta.ca including a full name and postal address. For institutional subscription, send enquiries to the Managing Editor, Robert Rosebrugh, rrosebrugh@mta.ca.Selection of reprints. After obtaining written permission from any copyright holder, any threeTAC Editors may propose a work for TAC Reprints to the Managing Editor. The Managing Editormay either accept the proposal or require that the Editors vote on it. If a 2/3 majority of those TACEditors responding within one month agrees, the work will be accepted for TAC Reprints. After a workis accepted, the author or proposer must provide to TAC either a usable TeX source or a PDF documentacceptable to the Managing Editor that reproduces a typeset version. Up to five pages of corrections,commentary and forward pointers may be appended by the author. When submitting commentary,authors should read and follow the Format for submission of Theory and Applications of Categories athttp://www.tac.mta.ca/tac/.Editorial boardManaging editor Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.caTEXnical editor Michael Barr, McGill University: barr@math.mcgill.caAssistant TEX editor Gavin Seal, Ecole Polytechnique Fédérale de Lausanne:gavin seal@fastmail.fmTransmitting editorsClemens Berger, Université de Nice-Sophia Antipolis: cberger@math.unice.frRichard Blute, Université d’ Ottawa: rblute@uottawa.caLawrence Breen, Université de Paris 13: breen@math.univ-paris13.frRonald Brown, University of North Wales: ronnie.profbrown(at)btinternet.comValeria de Paiva: valeria.depaiva@gmail.comEzra Getzler, Northwestern University: getzler(at)northwestern(dot)eduKathryn Hess, Ecole Polytechnique Fédérale de Lausanne: kathryn.hess@epfl.chMartin Hyland, University of Cambridge: M.Hyland@dpmms.cam.ac.ukAnders Kock, University of Aarhus: kock@imf.au.dkStephen Lack, Macquarie University: steve.lack@mq.edu.auF. William Lawvere, State University of New York at Buffalo: wlawvere@buffalo.eduTom Leinster, University of Edinburgh: Tom.Leinster@ed.ac.ukIeke Moerdijk, Radboud University Nijmegen: i.moerdijk@math.ru.nlSusan Niefield, Union College: niefiels@union.eduRobert Paré, Dalhousie University: pare@mathstat.dal.caJiri Rosicky, Masaryk University: rosicky@math.muni.czGiuseppe Rosolini, Università di Genova: rosolini@disi.unige.itAlex Simpson, University of Edinburgh: Alex.Simpson@ed.ac.ukJames Stasheff, University of North Carolina: jds@math.upenn.eduRoss Street, Macquarie University: street@math.mq.edu.auWalter Tholen, York University: tholen@mathstat.yorku.caMyles Tierney, Rutgers University: tierney@math.rutgers.eduRobert F. C. Walters, University of Insubria: robert.walters@uninsubria.itR. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca

Clemens Berger, Universit e de Nice-Sophia Antipolis: cberger@math.unice.fr Richard Blute, Universit e d’ Ottawa: rblute@uottawa.ca Lawrence Breen, Universit e de Paris 13: breen@math.u

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