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Ultraproducts and Los's Theorem Ultraproducts andLo´s’sTheorem: A Category-Theoretic Analysis by Mark Jonathan Chimes Dissertation presented for the degree of Master of Science in Mathematics in the Faculty of Science at Stellenbosch University Department of Mathematical Sciences, University of Stellenbosch Supervisor: Dr Gareth Boxall March 2017 Stellenbosch University https://scholar.sun.ac.za Declaration By submitting this dissertation electronically, I declare that the entirety of the work contained therein is my own, original work, that I am the sole author thereof (save to the extent explicitly otherwise stated), that reproduction and publication thereof by Stellenbosch University will not infringe any third party rights and that I have not previously in its entirety or in part submitted it for obtaining any qualification. Signature: . Mark Jonathan Chimes Date: March 2017 Copyright c 2017 Stellenbosch University All rights reserved. i Stellenbosch University https://scholar.sun.ac.za Abstract Ultraproducts andLo´s’sTheorem: A Category-Theoretic Analysis Mark Jonathan Chimes Department of Mathematical Sciences, University of Stellenbosch, Private Bag X1, Matieland 7602, South Africa. Dissertation: MSc 2017 Ultraproducts are an important construction in model theory, especially as applied to algebra. Given some family of structures of a certain type, an ul- traproduct of this family is a single structure which, in some sense, captures the important aspects of the family, where “important” is defined relative to a set of sets called an ultrafilter, which encodes which subfamilies are considered “large”. This follows fromLo´s’sTheorem, namely, the Fundamental Theorem of Ultraproducts, which states that every first-order sentence is true of the ultraproduct if, and only if, there is some “large” subfamily of the family such that it is true of every structure in this subfamily. In this dissertation, ul- traproducts are examined both from the standard model-theoretic, as well as from the category-theoretic view. Some potential problems with the category- theoretic definition of ultraproducts are pointed out, and it is argued that these are not as great an issue as first perceived. A general version ofLo´s’s Theorem is shown to hold for category-theoretic ultraproducts in general. This makes use of the concept of injectivity of a (compact) tree, which is intended to generalize truth of first-order formulae (under given assignments of variables), and, in the category of relational structures, corresponds exactly to first-order formulae. This type of thinking leads to a means of characterizing fields in the category of rings, and a new proof that every ultraproduct of fields is a field, which takes place entirely in the category of rings (along with the inclu- sion of the category of fields). Finally, the family of all (category-theoretic) ultraproducts on a given family is shown to arise from the “codensity monad” ii Stellenbosch University https://scholar.sun.ac.za ABSTRACT iii of the functor which includes the category of finite families into the category of families. In this sense, it is shown that ultraproducts are a rather natural construction category-theoretically speaking. Ultraprodukte enLo´sse Stelling: ’n Category-Teoretiese Analiese Mark Jonathan Chimes Department Wiskunde Wetenskappe, Universiteit Stellenbosch, Privaatsak X1, Matieland 7602, Suid Afrika. Thesis: MSc 2016 Ultraprodukte is ’n belangrike konstruksie in modelteorie, veral in hul toepass- ings in algebra. Gegewe ’n reeks strukture van ’n sekere tipe, is ’n ultrapro- duk van hierdie reeks ’n enkele struktuur wat, op ’n manier, die belangrikste aspekte van die reeks bevat, waar “belangrik” hier gedefinie¨erword met be- trekking tot ’n versameling reekse wat ’n ultrafilter genoem word. Hierdie ultrafilter verteenwoordig watter subreekse deur die ultraproduk as “groot” beskou word. Dit is ’n gevolgtrekking vanLo´sse Stelling, dit wil sˆe,’n eerste- orde stelling is waar met betrekking tot die ultraproduk as, en slegs as, daar ’n “groot” subreeks (van die hoofreeks) bestaan sodat die stelling waar is met be- trekking tot elke struktuur in di´esubreeks. In hierdie tesis word ultraprodukte uit die standarde model-teoretiese oogpunt behandel, sowel as uit die oogpunt van kategorie teorie. Potentie¨eleprobleme met die kategorie-teoretiese ultra- produk word uitgelig, maar dit word geargumenteer dat hul nie so ’n groot probleem veroorsaak as wat dit blyk nie. ’n Algmene weergawe vanLo´sse stelling is bewys vir alle kategorie¨e.D´ıtmaak gebruik van die konsep van in- jektiwiteit van ’n (kompakte) boom. Die bedoeling hiervan is om die waarheid van ’n eerste-orde stelling (onder ’n gegewe toedeling van veranderlikes) te veralgemeen. Hierdie idee ly tot ’n metode om liggame in die kategorie van groepe uit te lig, sowel as ’n nuwe bewys dat elke ultraproduk van liggame weer self ’n liggaam is. Hierdie bewys neem heeltemaal in die kategorie van groepe plaas (tesame met die funktor wat die kategorie van liggame in die kategorie van groepe insluit). Laastens, word dit angevoer dat die reeks van alle (kategorie-teoretiese) ultraprodukte van ’n gegewe reeks bestaan uit die “codigtheids monade” van die funktor wat die kategorie van eindige reekse insluit in die kategorie van oneindige reekse. Hierdie is dan ’n oortuiging dat ultraprodukte redelik natuurlik bestaan, ten minste uit die oogpunt van kategorie-teorie. Stellenbosch University https://scholar.sun.ac.za Acknowledgements I would firstly like to thank my supervisor, Dr Gareth Boxall, for putting up with me throughout this process. He has had to deal with my constant switch- ing of topics and changing of approaches. He has listened to my sometimes incoherent rambling in the process of trying to understand the relevant math- ematics. He has given up many hours meticulously combing through many error-ridden pages. Throughout all of this, he has managed to remain friendly and instructive. Gareth, you are an excellent supervisor, and have gone far beyond what you were strictly required to do. Thank you, also, to Prof. Zurab Janelidze, who provided a great deal of advice and encouragement. Thank you for being patient, even when my ideas were a bit strange, or even downright wrong. Our many sessions discussing mathematics, whether it was category theory or simply philosophy, have been vital to my success. Next, I would like to thank my girlfriend, Lilley. She has been there through all my moments of despondence when the thesis just seemed to loom over my head. She has celebrated with me whenever I completed each chapter, and comforted me when I faced the next. Lilley, you have really been very supportive. To my parents, I extend a most heartfelt gratitude. Though they had to feign interest whenever I mentioned my work, they nevertheless did all they could to help me. I do not think that I would have stood a chance without their assistance both financially and emotionally. Mom, dad, thank you. I love you. I spent countless hours on the phone with my brother, Liam, just chatting late into the night when I should have been sleeping or working on my thesis. Both he and my sister, Juanelle, were great at distracting me when I should be working. But more importantly, they were great for conversation and spending time when I wasn’t working. My time at home helped refresh me when I was emotionally drained, and without them my life would have felt a bit empty. To the Wolfpack, you know who you are! You guys have been great company over the past year, and are the people with whom I’ve spent most of my free time. Thanks for being patient with me, and for being my friends. iv Stellenbosch University https://scholar.sun.ac.za ACKNOWLEDGEMENTS v To the Guild. You guys are great! Sorry the thesis took my time away from you. I’ll see more of you next year! Thank you to the Frank-Wilhelm Bursary Fund for funding my masters degree. I am sincerely grateful for their help. Without them, I would not have been able to afford to study. Finally, I would like to thank the University of Stellenbosch for providing the facilities and the opportunity for my studies, and my entire student experience. I will sincerely miss this university and its campus. Pectora roborant cultus recti. Stellenbosch University https://scholar.sun.ac.za Dedications This thesis is dedicated to my girlfriend’s guide-dog, who understands almost as much about my thesis as she does, but is a much better listener. She has guided me through the hard times I have experienced in my writing; Teska- Monster, you truly are a human’s best friend. vi Stellenbosch University https://scholar.sun.ac.za Contents Declaration i Abstract ii Acknowledgements iv Dedications vi Contents vii 1 Preliminaries 1 1.1 Category Theory . 1 1.1.1 Concepts Assumed and Described . 1 1.1.2 Notation . 2 1.1.3 Subobjects . 3 1.1.4 Monoids and Monads . 4 1.1.5 Ends . 10 1.2 Set Theory . 12 1.2.1 Overview . 12 1.2.2 Notation . 12 1.2.3 Cardinality . 13 1.2.4 Axiom of Choice . 13 1.3 Logic . 13 1.3.1 Overview . 13 1.3.2 Logic Preliminaries . 14 1.3.3 Languages . 14 1.3.4 Structures . 15 1.3.5 Relational Structures . 16 1.3.6 Propositional Logic . 16 1.3.7 First-Order Logic . 19 1.3.8 Models and Theories . 23 1.3.9 Theory of Groups . 25 1.3.10 Groups as Relational Structures . 26 1.3.11 G¨odel’sTheorems and Other Important Theorems of Logic 27 vii Stellenbosch University https://scholar.sun.ac.za CONTENTS viii 1.4 Well-Founded Induction . 28 1.5 Varieties of Algebras . 30 2 Overview 33 2.1 Summary . 33 2.1.1 Ultraproducts . 33 2.1.2 Ultrafilters . 33 2.1.3Lo´s’sTheorem .
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