T Introduction v jj Chapter I - Sup-lattices , , 1 1. Definitions andduality 1 2. Limits and colimits 2 3. Free sup-lattices 3 4. Sub andquotient lattices 4 5. Tensor products 5 Chapter II - Rings, modules and descent 7 1. Rings andmodules 7 2. Tensor product of modules 8 3. Change of rings 9 4. Flatness, projectivity, andpurity 11 5. . Descent theory formodules . . 12 Chapter III - Locales 21 1. Locales andcommutative monoids 21 2. Limits and colimits . 21 3. Thefree locale 22 4. Local operators andquotients 22 5. The splitting locale 25 Chapter IV - Spaces 27 1. Subspaces 27 2. Points and discrete spaces 28 3. The Sierpinski space 29 4. Pullbacks and projective limits 30 5. The splitting space 31 Chapter V - Open maps of spaces 3 3 1. Open maps - definition 33 2. Open subspaces 34 3. Conditions for openness 35 4. Open surjections, pullbacks 38 5. A characterization of discrete spaces 40 Chapter VI - Change of base 42 1. Change of base for sup-lattices and locales 42 A °P A°P 2. Determination of si(S- ) and loc(S- ) 46 3. Determination of s£(sh(A)) and loc(sh(A)) for A e loc(S) . . . 49 4. The Beck-Che valleyv conditions 52 5. The spatial reflection 53 Chapter VII - Open morphisms of topoi 56 1. Open geometric morphisms 56 2. A site characterization of openness 58 3. The spatial cover 59 4. A characterization of atomic topoi 61 Chapter VIII - Descent morphisms of topoi 64 1. Effective descent morpnisms 64 2. Open surjections 65 3. Applications to the structure of topoi 68 References 71 v
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