Michael Barr Charles Wells Toposes, Triples and Theories Version 1.1 10 September 2000 Copyright 2000 by Michael Barr and Charles Frederick Wells. This version may be downloaded and printed in unmodified form for private use only. It is available at http://www.cwru.edu/artsci/math/wells/pub/ttt.html and ftp.math.mcgill.ca/pub/barr as any of the files ttt.dvi, ttt.ps, ttt.ps.zip, ttt.pdf, ttt.pdf.zip. Michael Barr Peter Redpath Professor Emeritus of Mathematics, McGill University
[email protected] Charles Wells Professor Emeritus of Mathematics, Case Western Reserve University Affiliate Scholar, Oberlin College
[email protected] To Marcia and Jane Contents Preface vi 1. Categories 1.1 Definition of category 1.2 Functors 1.3 Natural transformations 1.4 Elements and Subobjects 1.5 The Yoneda Lemma 1.6 Pullbacks 1.7 Limits 1.8 Colimits 1.9 Adjoint functors 1.10 Filtered colimits 1.11 Notes to Chapter I 1 1 11 16 20 26 29 35 48 54 67 71 2. Toposes 2.1 Basic Ideas about Toposes 2.2 Sheaves on a Space 2.3 Properties of Toposes 2.4 The Beck Conditions 2.5 Notes to Chapter 2 74 74 78 86 92 95