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A constructive approach to affine and projective planes 6 1 0 2 n a J 9 1 ] T C . h t a m [ Achilleas Kryftis 1 v 8 9 9 Trinity College 4 and 0 . Department of Pure Mathematics and Mathematical Statistics 1 0 University of Cambridge 6 1 : v i This dissertation is submitted for the degree of X Doctor of Philosophy r a May 19, 2015 Thisdissertationistheresultofmyownworkandincludesnothing that is the outcome of work done in collaboration except where specifically indicated in the text. No part of this dissertation has been submitted for any other qualification. Achilleas Kryftis May 19, 2015 i A constructive approach to affine and projective planes Achilleas Kryftis Abstract In classical geometric algebra, there have been several treatments of affine and projective planes based on fields. In this thesis we approach affine and projective planes from a constructive point of view and we base our geometry on local rings instead of fields. Westartbyconstructingprojectiveandaffineplanesoverlocal rings and establishing forms of Desargues’ Theorem and Pappus’ Theorem which hold for these. From this analysis we derive co- herent theories of projective and affine planes. The great Greek mathematicians of the classical period used geometry as the basis for their theory of quantities. The modern version of this idea is the reconstruction of algebra from geometry. We show how we can construct a local ring whenever we are given an affine or a projective plane. This enables us to describe the classifying toposes of our theories of affine and projective planes as extensions of the Zariski topos by certain group actions. Through these descriptions of the classifying toposes, the links betweenthetheoriesoflocalrings, affineandprojectiveplanesbe- come clear. In particular, the geometric morphisms between these classifying toposes are all induced by group homomorphisms even though they demonstrate complicated constructions in geometry. In this thesis, we also prove results in topos theory which are applied to these geometric morphisms to give Morita equivalences between some further theories. iii Acknowledgements I would like thank my supervisor, Professor Martin Hyland, for his constant support and encouragement throughout my PhD and for sharing his wisdom with me. The category theory group of Cambridge has been a very important part of my life in Cambridge. My mathematical interactions with the group have been very fruitful and we have also shared fun moments in Cambridge and at conferences. I would like to thank Christina, Tamara, Guilherme, Zhen Lin, Sori, Enrico, Paige and all the members of the category theory group. I would like to thank Trinity College and DPMMS for financially supporting me during my PhD. I have been very fortune during my PhD to be surrounded by great friends. I would like to thank John, Gabriele, Ugo, Evangelia, Moses, Juhan, Giulio, Eleni, Dimitris, Anastasia, Richard and Anna for being such a great company. I would also like to thank my parents Georgios and Theano, and my siblings Maria and Yiannos for their constant love and support. v Contents Chapter 1. Introduction 1 Part 1. Constructive geometry 5 Chapter 2. Projective planes 7 2.1. Points and lines 7 2.2. Incidence 8 2.3. Duality 9 2.4. A few propositions and remarks 9 2.5. The theory of preprojective planes 18 2.6. Non-collinear points, non-concurrent lines 20 2.7. Morphisms of preprojective planes 23 2.8. Morphisms between projective planes over rings 24 2.9. Desargues’ theorem on the projective plane 31 2.10. Pappus’ theorem on the projective plane 37 2.11. The theory of projective planes 39 Chapter 3. Affine planes 43 3.1. Points 43 3.2. Lines 44 3.3. Incidence 45 3.4. Preaffine planes from preprojective planes with a line 45 3.5. The theory of preaffine planes 49 3.6. Morphisms of preaffine planes 51 3.7. Morphisms of projective planes from morphisms of preaffine planes 53 3.8. Morphisms between affine planes over rings 57 3.9. Desargues’ axioms on the affine plane 60 3.10. Further versions of Desargues’ theorem 63 3.11. Pappus’ axiom on the affine plane 74 Chapter 4. Constructing the local ring from an affine plane 77 4.1. Dilatations 77 4.2. Translations 79 4.3. The local ring of trace preserving homomorphisms 89 4.4. Introducing coordinates to an affine plane 96 4.5. Trace preserving homomorphisms and geometric morphisms 101 vii 4.6. Alternative construction of the local ring 106 4.7. Revisiting Desargues’ theorem 109 Chapter 5. Introducing coordinates to a Projective Plane 113 5.1. The local ring of a projective plane 113 5.2. The uniqueness of the local ring 116 5.3. The H-torsor 117 Part 2. Classifying toposes 119 Chapter 6. Discussion on Diaconescu’s theorem 121 6.1. Background and notation 121 6.2. Diaconescu’s theorem 124 6.3. [C,S] as a classifying topos 128 Chapter 7. Results about E[G] 131 7.1. Group(E) → Top 131 7.2. The H-endomorphisms of L 135 Chapter 8. The classifying topos for affine planes 139 8.1. The internal automorphism group of the generic local ring 139 8.2. Affine planes in Z and Z[G] 140 8.3. Explicit construction of an affine plane from a G-torsor 141 8.4. Geometric morphisms over Z 146 8.5. Z[G] −f→A Aff −f→G Z[G] is isomorphic to the identity 148 8.6. Aff −f→G Z[G] −f→A Aff is isomorphic to the identity 149 8.7. Aff (cid:39) Z[G ] 149 Pt 3 Chapter 9. The classifying topos for projective planes 151 9.1. Projective planes in Z and Z[H] 151 9.2. Explicit construction of a projective plane from an H-torsor 152 9.3. Geometric morphisms over Z 156 9.4. Z[H] −f→P Proj −f−H→ Z[H] is isomorphic to the identity 158 9.5. Proj −f−H→ Z[H] −f→P Proj is isomorphic to the identity 158 Chapter 10. Geometric morphisms between Z, Aff and Proj 161 10.1. Overview 161 10.2. Z as a slice of the topos Aff 161 10.3. Z as a slice of the topos Proj 162 10.4. Aff → Proj 163 10.5. Aff → Aff 164 Pt 10.6. Duality of the projective plane 165 References 169 viii