DOCSLIB.ORG

Frok Axiokatizatioj to Gbibralizatioj of Sbt Thbory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Frok Axiokatizatioj to Gbibralizatioj of Sbt Thbory FROK AXIOKATIZATIOJ TO GBIBRALIZATIOJ OF SBT THBORY SAXUBL FBIDRICH LOJDOJ SCHOOL OF BCOID_ICS AID POLITICAL SCIBJCB THESIS SUB_ITTED FOR THE DEGRBE OF PH.D. UIIVERSITY OF LOIDOJ JUIB 1987 - 1 - ABSTRACT: The thesis examines the philosophical and foundational significance of Cohen's Independence results. A distinction is made between the mathematical and logical analyses of the ·set· concept. It is argued that topes theory is the natural generalization of the mathematical theory of sets and is the appropriate foundational response to the proble_ raised by Cohen's results. The thesis is divided into three part.. The fir.t i. a discussion of the relationship between ·inforaal· aathe.atical theories and their formal axiomatic realizations this relationship being singularly problematic in the case of set theory. The second part deals with the development of the set concept within the _the_tical approach. In particular Skolem's reformulation of ZerJlelo's notion of "def i nfte properties·. In the third part an account is given of the emergence and development of topes theory. Then the considerations of the first two parts are applied to demonstrate that the shift to topos theory, specifically in its guise of LST (local set theory>, is the appropriate next step in the evolution of the concept of set, within the mathematical approach, in the light of the significance of Cohen's Independence results. - 2 - ACKNOWLEDGEXEITS: I thank: John Vorrall for his supervision; the LSE Philosophy Department especially for the Teaching Assistantship; Gay Woolven and Sue Burrett for lots of things nat least the caffee. I am grateful to: Philippa for much more than the typing; Annette Goldberg for her invaluable help in obtaining the Ii terature; lUke Hallett for his encouragement in the vulnerable early stages; the ather members of the London Pentagon: Youssef Aliabadi, Frank Arntzenius, Andrej Lodynski and Peter Kiine for our many stimulating meetings, and Peter Xilne, in particular, for consistent and considerable help. Throughout my years at LSE John Bell has taught, supported and inspired me. Xy very grateful thanks to him. FOR JlY PAREITS. - 3 - CONTENTS INTRODUCTION 7 PART I: PROBLEMS OF FORMALIZATION CHAPTER 1: Introduction: It is important to analyze the relationship between informal and formal mathematics. 11 CHAPTER 2: Mathematical intuition and informal mathematics. 16 CHAPTER 3: Plausible desiderata for formalization. 19 CHAPTER 4: Modern mathematics and effective classical first order systems. 29 CHAPTER 5: Can we effectively formalize and realize codification and precisification? 36 CHAPTER 6: Axiom systems as implicit definitions. 50 CHAPTER 7: Appendix to Chapter 2. 65 PART II: ZERXELO, SKOLEM, AND DEFIIITE PROPERTIES CHAPTER 1: Introduction. 75 CHAPTER 2: The mathematical and logical approaches to set theory. 80 CHAPTER 3: Zermelo and axiomatization. 90 CHAPTER 4: Zermelo and definite properties. 110 CHAPTER 5: Problems inherent in Zermelo's account of definite properties. 122 CHAPTER 6: Skolem's reformulation of Zermelo's notion of definite property. 134 - 4 - PART III: THE GENERALIZATION OF SET THEORY CHAPTER 1: INTRODUCTION. 158 CHAPTER 2: CATEGORY THEORY AID THE EMERGENCE OF TOPOS THEORY. 2.1: LST 1s set theory. 164 2.11: Category theory and its foundat1onal importance. 168 2.1i.a: The emergence of category theory and the emphasis on structure in modern mathematics. 168 2.1i.b: The development of category theory. 175 2.11.c: Adjoints. 188 2.1i1: The e..rgence of topes theory. 195 2.i11.a: Grothendieck's work. 197 2.ii1.b: Lawvere's work and... 199 2.ii1.c: ...and the development of elementary topos theory. 208 CHAPTER 3: THE ALGEBRAIC VIEW OF SET THEORY. 221 CHAPTER 4: TOPOS THEORY, LST AID DEFINITE PROPERTIES. 4.1: Definite properties and topos theory. 236 4.i1: Internal languages and LST. 246 CHAPTER 5: THE TOPOS EXPLICATION OF FORCING. 5.1: Forcing or/and the continuum hypothesis. 266 5.i1.a: Grothend1eck's generalization of the classical notion of sheaf. 272 5.i1.b: Lawvere and Tierney's generalization of the notion of sheaf to an arbitrary topes. 282 5.ii.c: Forcing and Lawvere's notion of 'variable set'. 290 APPENDIX: A GUIDE TO FORCING (FEIDRICH/KILIE) 295 BIBLIOGRAPHY: 320 - 5 - lOT ATION: lTxP(x) 'for all XI P(x)' 1:xP(x) 'exists an XI such that P(x)' #0. 'not a' For sets IIY: IH 'I is a subset of Y' XUY Union of X and Y I6Y Intersection of X and Y f·g Composition of the function f with g where cod(g>=dom(f) ifI Restriction of the function f to X rl-a 'there is a proof of a from r' rl=a 'a is a logical consequence of r' - 6 - IITRODUCTIOI In 1963 Paul Cohen proved the independence of the Continuum Hypothesis and the Axiom of Choice from ZF set theory. A major task for contemporary philosophy of mathematics is to analyse the significance of Cohen's results, particularly for the nature and viability of set theory as a foundation for _the_tics. To achieve his independence results Cohen developed the ..thod of forcing. The forcing technique has proved to be an extre ..ly powerful tool for providing independence resul ts across a wide spectrum of _the.tical disciplines. Jlany important open questions in, for example, topology, group theory and measure theory, 1.e. questions generated by concerns within these disciplines rather than set theory, have been shown to depend on the underlying universe of sets presupposed. Thus the foundational import of Cohen's results per.ate. _the_tics - a fact which underscores the importance of the aforementioned task. This thesis is intended as a contribution to this task. Although in the JI1d-sixties several viewe on the foundations of _the_tics were offered in response to the independence resul ts, usually by philosophically lIinded _theaaticians, subsequently, little philosophical work arising directly fro. the.. results was undertaken. Chief a~ngst the vie.,. put forward were the followiD8: a) we should adopt a foraalistic attitude <Robinson 1ge4, Cohen 1971) - 7 - b) we should adopt a second-order axiomatization of set theory (Kreisel 1967) c) we should adopt a constrained relativism i.e. adopt as a foundation the class of consistent extensions of ZF, e.g. ZF+CH, ZF+#CH (Xostowski 1967) d) we should adopt an alternative to set theoretic foundations e.g. <Lawvere 1966, XacLane 1968, 1971) Ky contention is that none of the above are satisfactory as a progressi ve response to the foundational challenge offered by the independence results. Rather, a combination of elements from c) and d) is needed. )(ore specifically, we should: in essence retain a set theoretical foundation, adopt a relativistic view, but rather than Kostowsk1' s version, employ the generalized set theory yielded by topos theory, i.e. local set theory. In this thesis, then, I aim to establish that topos theory is the natural generalization of the mathematical theory of sets and is the appropriate foundational response to the issues raised by Cohen's results. The thesis comprises three parts. The first two parts derive froD the following comment of Bernays on Xostowski's 1967: The first essential thing which e_rges froD his paper is that the results of Paul J. Cohen on the independence of the continuum hypothesis do not directly concern set theory itself, but rather the axiomatization of set theory; and not even Zermelo's original axiomatization. but a sharper axiomatization which allows of strict formalization. [1967 p.1091 - 8 - In part I I discuss the relationship between 'informal' mathematical theories and their formal axiomatic realizations. In part II I turn to the transformation of Zermelo's axioms into ZP, in particular Skolem's reformulation of Zermelo's notion of 'definite properties', i.e. "the sharper axiomatization which allows of strict formalization". Parts I and II provide the appropriate context in which to deterll1ne the philosophical and foundational significance of Cohen's independence results. In part III after an account of the emergence and development of topos theory the considerations of the first two parts are applied to underpin the shift to topas theory. - 9 - PART I: PROBLBXS OF FORMAL IZATIOB. - 10- (1) It is important to analyse the relationship between informal and formal mathematics. Amongst philosophers of mathematics there are to be found widely varying attitudes towards the relationship between formal systems and informal mathematics, the status of axioms, and indeed, the nature of informal mathematics. As examples we may cite the different opinions on these matters evinced by Frege and Hilbert. As well as differing views allOngst philosophers of mathematics, there are often ambiguities, if not outright contradictions, within the work of individual commentators. Of greatest relevance to the relationship in question are the following two views: i) the subject matter of the informal body of mathematics is understood to be a given realm of objects e.g. a universe of abstract sets; ii) the formal system is construed as an implicit definition. Xy aim is to show that these views face severe problems and in certain instances are untenable. It is important to stress, however, that I am not directly concerned with criUcist ng, for example, commitment to realms of abstract mathematical objects, but rather the views concerning the relationship between such realDS and formal systems. We will be able to identify one such view that, given my analysis of the genesis of ZF in Part II, is the DOst viable in the case of that theory. Xoreover. it underlines my claim as to the appropriate foundational response to Cohen's independence results . • -11- Since the work of the 'founding fathers' Cantor, Frege, Russell, Zer:melo and Skolem, we may recognise three clear landmarks for the three interrelated disciplines of foundational studies, mathematical logic and set theory.
Load more

Recommended publications
  • Topos Theory
    Topos Theory Olivia Caramello Sheaves on a site Grothendieck topologies Grothendieck toposes Basic properties of Grothendieck toposes Subobject lattices Topos Theory Balancedness The epi-mono factorization Lectures 7-14: Sheaves on a site The closure operation on subobjects Monomorphisms and epimorphisms Exponentials Olivia Caramello The subobject classifier Local operators For further reading Topos Theory Sieves Olivia Caramello In order to ‘categorify’ the notion of sheaf of a topological space, Sheaves on a site Grothendieck the first step is to introduce an abstract notion of covering (of an topologies Grothendieck object by a family of arrows to it) in a category. toposes Basic properties Definition of Grothendieck toposes Subobject lattices • Given a category C and an object c 2 Ob(C), a presieve P in Balancedness C on c is a collection of arrows in C with codomain c. The epi-mono factorization The closure • Given a category C and an object c 2 Ob(C), a sieve S in C operation on subobjects on c is a collection of arrows in C with codomain c such that Monomorphisms and epimorphisms Exponentials The subobject f 2 S ) f ◦ g 2 S classifier Local operators whenever this composition makes sense. For further reading • We say that a sieve S is generated by a presieve P on an object c if it is the smallest sieve containing it, that is if it is the collection of arrows to c which factor through an arrow in P. If S is a sieve on c and h : d ! c is any arrow to c, then h∗(S) := fg | cod(g) = d; h ◦ g 2 Sg is a sieve on d.
    [Show full text]
  • The Freyd-Mitchell Embedding Theorem States the Existence of a Ring R and an Exact Full Embedding a Ñ R-Mod, R-Mod Being the Category of Left Modules Over R
    The Freyd-Mitchell Embedding Theorem Arnold Tan Junhan Michaelmas 2018 Mini Projects: Homological Algebra arXiv:1901.08591v1 [math.CT] 23 Jan 2019 University of Oxford MFoCS Homological Algebra Contents 1 Abstract 1 2 Basics on abelian categories 1 3 Additives and representables 6 4 A special case of Freyd-Mitchell 10 5 Functor categories 12 6 Injective Envelopes 14 7 The Embedding Theorem 18 1 Abstract Given a small abelian category A, the Freyd-Mitchell embedding theorem states the existence of a ring R and an exact full embedding A Ñ R-Mod, R-Mod being the category of left modules over R. This theorem is useful as it allows one to prove general results about abelian categories within the context of R-modules. The goal of this report is to flesh out the proof of the embedding theorem. We shall follow closely the material and approach presented in Freyd (1964). This means we will encounter such concepts as projective generators, injective cogenerators, the Yoneda embedding, injective envelopes, Grothendieck categories, subcategories of mono objects and subcategories of absolutely pure objects. This approach is summarised as follows: • the functor category rA, Abs is abelian and has injective envelopes. • in fact, the same holds for the full subcategory LpAq of left-exact functors. • LpAqop has some nice properties: it is cocomplete and has a projective generator. • such a category embeds into R-Mod for some ring R. • in turn, A embeds into such a category. 2 Basics on abelian categories Fix some category C. Let us say that a monic A Ñ B is contained in another monic A1 Ñ B if there is a map A Ñ A1 making the diagram A B commute.
    [Show full text]
  • Notes on Categorical Logic
    Notes on Categorical Logic Anand Pillay & Friends Spring 2017 These notes are based on a course given by Anand Pillay in the Spring of 2017 at the University of Notre Dame. The notes were transcribed by Greg Cousins, Tim Campion, L´eoJimenez, Jinhe Ye (Vincent), Kyle Gannon, Rachael Alvir, Rose Weisshaar, Paul McEldowney, Mike Haskel, ADD YOUR NAMES HERE. 1 Contents Introduction . .3 I A Brief Survey of Contemporary Model Theory 4 I.1 Some History . .4 I.2 Model Theory Basics . .4 I.3 Morleyization and the T eq Construction . .8 II Introduction to Category Theory and Toposes 9 II.1 Categories, functors, and natural transformations . .9 II.2 Yoneda's Lemma . 14 II.3 Equivalence of categories . 17 II.4 Product, Pullbacks, Equalizers . 20 IIIMore Advanced Category Theoy and Toposes 29 III.1 Subobject classifiers . 29 III.2 Elementary topos and Heyting algebra . 31 III.3 More on limits . 33 III.4 Elementary Topos . 36 III.5 Grothendieck Topologies and Sheaves . 40 IV Categorical Logic 46 IV.1 Categorical Semantics . 46 IV.2 Geometric Theories . 48 2 Introduction The purpose of this course was to explore connections between contemporary model theory and category theory. By model theory we will mostly mean first order, finitary model theory. Categorical model theory (or, more generally, categorical logic) is a general category-theoretic approach to logic that includes infinitary, intuitionistic, and even multi-valued logics. Say More Later. 3 Chapter I A Brief Survey of Contemporary Model Theory I.1 Some History Up until to the seventies and early eighties, model theory was a very broad subject, including topics such as infinitary logics, generalized quantifiers, and probability logics (which are actually back in fashion today in the form of con- tinuous model theory), and had a very set-theoretic flavour.
    [Show full text]
  • Classifying Categories the Jordan-Hölder and Krull-Schmidt-Remak Theorems for Abelian Categories
    U.U.D.M. Project Report 2018:5 Classifying Categories The Jordan-Hölder and Krull-Schmidt-Remak Theorems for Abelian Categories Daniel Ahlsén Examensarbete i matematik, 30 hp Handledare: Volodymyr Mazorchuk Examinator: Denis Gaidashev Juni 2018 Department of Mathematics Uppsala University Classifying Categories The Jordan-Holder¨ and Krull-Schmidt-Remak theorems for abelian categories Daniel Ahlsen´ Uppsala University June 2018 Abstract The Jordan-Holder¨ and Krull-Schmidt-Remak theorems classify finite groups, either as direct sums of indecomposables or by composition series. This thesis defines abelian categories and extends the aforementioned theorems to this context. 1 Contents 1 Introduction3 2 Preliminaries5 2.1 Basic Category Theory . .5 2.2 Subobjects and Quotients . .9 3 Abelian Categories 13 3.1 Additive Categories . 13 3.2 Abelian Categories . 20 4 Structure Theory of Abelian Categories 32 4.1 Exact Sequences . 32 4.2 The Subobject Lattice . 41 5 Classification Theorems 54 5.1 The Jordan-Holder¨ Theorem . 54 5.2 The Krull-Schmidt-Remak Theorem . 60 2 1 Introduction Category theory was developed by Eilenberg and Mac Lane in the 1942-1945, as a part of their research into algebraic topology. One of their aims was to give an axiomatic account of relationships between collections of mathematical structures. This led to the definition of categories, functors and natural transformations, the concepts that unify all category theory, Categories soon found use in module theory, group theory and many other disciplines. Nowadays, categories are used in most of mathematics, and has even been proposed as an alternative to axiomatic set theory as a foundation of mathematics.[Law66] Due to their general nature, little can be said of an arbitrary category.
    [Show full text]
  • Diagram Chasing in Abelian Categories
    Diagram Chasing in Abelian Categories Toni Annala Contents 1 Overview 2 2 Abelian Categories 3 2.1 Denition and basic properties . .3 2.2 Subobjects and quotient objects . .6 2.3 The image and inverse image functors . 11 2.4 Exact sequences and diagram chasing . 16 1 Chapter 1 Overview This is a short note, intended only for personal use, where I x diagram chasing in general abelian categories. I didn't want to take the Freyd-Mitchell embedding theorem for granted, and I didn't like the style of the Freyd's book on the topic. Therefore I had to do something else. As this was intended only for personal use, and as I decided to include this to the application quite late, I haven't touched anything in chapter 2. Some vague references to Freyd's book are made in the passing, they mean the book Abelian Categories by Peter Freyd. How diagram chasing is xed then? The main idea is to chase subobjects instead of elements. The sections 2.1 and 2.2 contain many standard statements about abelian categories, proved perhaps in a nonstandard way. In section 2.3 we dene the image and inverse image functors, which let us transfer subobjects via a morphism of objects. The most important theorem in this section is probably 2.3.11, which states that for a subobject U of X, and a morphism f : X ! Y , we have ff −1U = U \ imf. Some other results are useful as well, for example 2.3.2, which says that the image functor associated to a monic morphism is injective.
    [Show full text]
  • Toposes Toposes and Their Logic
    Building up to toposes Toposes and their logic Toposes James Leslie University of Edinurgh [email protected] April 2018 James Leslie Toposes Building up to toposes Toposes and their logic Overview 1 Building up to toposes Limits and colimits Exponenetiation Subobject classifiers 2 Toposes and their logic Toposes Heyting algebras James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Products Z f f1 f2 X × Y π1 π2 X Y James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Coproducts i i X 1 X + Y 2 Y f f1 f2 Z James Leslie Toposes e : E ! X is an equaliser if for any commuting diagram f Z h X Y g there is a unique h such that triangle commutes: f E e X Y g h h Z Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Equalisers Suppose we have the following commutative diagram: f E e X Y g James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Equalisers Suppose we have the following commutative diagram: f E e X Y g e : E ! X is an equaliser if for any commuting diagram f Z h X Y g there is a unique h such that triangle commutes: f E e X Y g h h Z James Leslie Toposes The following is an equaliser diagram: f ker G i G H 0 Given a k such that following commutes: f K k G H ker G i G 0 k k K k : K ! ker(G), k(g) = k(g).
    [Show full text]
  • Weekly Homework 10
    Weekly Homework 10 Instructor: David Carchedi Topos Theory July 12, 2013 July 12, 2013 Problem 1. Epimorphisms and Monomorphisms in a Topos Let E be a topos. (a) Let m : A ! B be a monomorphism in E ; and let φm : B ! Ω be a map to the subobject classifier of E classifying m; and similarly denote by φB the map classifying the maximal subobject of B (i.e. idB). Denote by m0 : E = lim (B Ω) ! B − ⇒ 0 the equalizer diagram for φm and φB: Show that m and m represent the same subobject of B. Deduce that a morphism in a topos is an isomorphism if and only if it is both a monomorphism and an epimorphism. (b) Let f : X ! Y be a map of sets. Denote by X ×Y X ⇒ X the kernel pair of f and by a Y ⇒ Y Y X the cokernel pair of f: Show that the coequalizer of the kernel pair and the equalizer of the cokernel pair both coincide with the the set f (X) : Deduce that for f : X ! Y a map in E ; ! a X ! lim Y Y Y ! Y − ⇒ X and X ! lim (X × X X) ! Y −! Y ⇒ are both factorizations of f by an epimorphism follows by a monomorphism. (c) Show that the factorization of a morphism f in E into an epimorphism followed by a monomorphism is unique up to isomorphism. Deduce that f : X ! Y is an epimor- phism, if and only if the canonical map lim (X × X X) ! Y − Y ⇒ is an isomorphism.
    [Show full text]
  • Topos Theory
    Topos Theory Olivia Caramello Local operators For further reading Topos Theory Lecture 11: Local operators Olivia Caramello Topos Theory The concept of local operator Olivia Caramello Local operators For further Definition reading • Let E be a topos, with subobject classifier > : 1 ! Ω.A local operator (or Lawvere-Tierney topology) on E is an arrow j :Ω ! Ω in E such that the diagrams ^ 1 Ω Ω × Ω / Ω > j > j j×j j Ω / ΩΩ / ΩΩ × Ω / Ω j j ^ commute (where ^ :Ω × Ω ! Ω is the meet operation of the internal Heyting algebra Ω). • Let E be an elementary topos. A universal closure operation on E is a closure operation c on subobjects which commutes with pullback (= intersection) of subobjects. 2 / 6 Topos Theory Universal closure operations I Olivia Caramello Local operators For further reading Definition Let c be a universal closure operation on an elementary topos E . • A subobject m : a0 ! a in E is said to be c-dense if c(m) = ida, and c-closed if c(m) = m. • An object a of E is said to be a c-sheaf if whenever we have a diagram f 0 b0 / a m b where m is a c-dense subobject, there exists exactly one arrow f : b ! a such that f ◦ m = f 0. • The full subcategory of E on the objects which are c-sheaves will be denoted by shc(E ). 3 / 6 Topos Theory Universal closure operations II Olivia Caramello Local operators For further reading Theorem For any elementary topos E , there is a bijection between universal closure operations on E and local operators on E .
    [Show full text]
  • Category Theory Course
    Category Theory Course John Baez September 3, 2019 1 Contents 1 Category Theory: 4 1.1 Definition of a Category....................... 5 1.1.1 Categories of mathematical objects............. 5 1.1.2 Categories as mathematical objects............ 6 1.2 Doing Mathematics inside a Category............... 10 1.3 Limits and Colimits.......................... 11 1.3.1 Products............................ 11 1.3.2 Coproducts.......................... 14 1.4 General Limits and Colimits..................... 15 2 Equalizers, Coequalizers, Pullbacks, and Pushouts (Week 3) 16 2.1 Equalizers............................... 16 2.2 Coequalizers.............................. 18 2.3 Pullbacks................................ 19 2.4 Pullbacks and Pushouts....................... 20 2.5 Limits for all finite diagrams.................... 21 3 Week 4 22 3.1 Mathematics Between Categories.................. 22 3.2 Natural Transformations....................... 25 4 Maps Between Categories 28 4.1 Natural Transformations....................... 28 4.1.1 Examples of natural transformations........... 28 4.2 Equivalence of Categories...................... 28 4.3 Adjunctions.............................. 29 4.3.1 What are adjunctions?.................... 29 4.3.2 Examples of Adjunctions.................. 30 4.3.3 Diagonal Functor....................... 31 5 Diagrams in a Category as Functors 33 5.1 Units and Counits of Adjunctions................. 39 6 Cartesian Closed Categories 40 6.1 Evaluation and Coevaluation in Cartesian Closed Categories. 41 6.1.1 Internalizing Composition................. 42 6.2 Elements................................ 43 7 Week 9 43 7.1 Subobjects............................... 46 8 Symmetric Monoidal Categories 50 8.1 Guest lecture by Christina Osborne................ 50 8.1.1 What is a Monoidal Category?............... 50 8.1.2 Going back to the definition of a symmetric monoidal category.............................. 53 2 9 Week 10 54 9.1 The subobject classifier in Graph.................
    [Show full text]
  • ELEMENTARY TOPOI: SETS, GENERALIZED Contents 1. Categorical Basics 1 2. Subobject Classifiers 6 3. Exponentials 7 4. Elementary
    ELEMENTARY TOPOI: SETS, GENERALIZED CHRISTOPHER HENDERSON Abstract. An elementary topos is a nice way to generalize the notion of sets using categorical language. If we restrict our world to categories which satisfy a few simple requirements we can discuss and prove properties of sets without ever using the word \set." This paper will give a short background of category theory in order to prove some interesting properties about topoi. Contents 1. Categorical Basics 1 2. Subobject Classifiers 6 3. Exponentials 7 4. Elementary Topoi 9 5. Lattices and Heyting Algebras 10 6. Factorizing Arrows 12 7. Slice Categories as Topoi 13 8. Lattice Objects and Heyting Algebra Objects 17 9. Well-Pointed Topoi 22 10. Concluding Remarks 22 Acknowledgments 23 References 23 1. Categorical Basics Definition 1.1. A category, C is made up of: • Objects: A; B; C; : : : • Morphisms: f; g; h; : : : This is a generalization of \functions" between objects. If f is a mor- phism in a category C then there are objects A; B in C such that f is a morphism from A to B, as below. We call A = dom(f) (the \domain") and B = cod(f) (the \codomain"). f A / B Date: August 5, 2009. 1 2 CHRISTOPHER HENDERSON If we have morphisms f : A ! B and g : B ! C then there exists an arrow h : A ! C such that the following diagram commutes: f A / B @@ @@ g h @@ @ C Here we are only asserting some rule of composition. We denote h = g ◦ f. This rule of composition must satisfy associativity, so that for f : A ! B; g : B ! C; h : C ! D, we have that: h ◦ (g ◦ f) = (h ◦ g) ◦ f Lastly, for every object A in C, there is a morphism idA : A ! A such that for any other morphisms g : A ! B and h : B ! A, we have that: g = g ◦ idA; h = idA ◦ h Remark 1.2.
    [Show full text]
  • Category Theory
    Category theory Helen Broome and Heather Macbeth Supervisor: Andre Nies June 2009 1 Introduction The contrast between set theory and categories is instructive for the different motivations they provide. Set theory says encouragingly that it’s what’s inside that counts. By contrast, category theory proclaims that it’s what you do that really matters. Category theory is the study of mathematical structures by means of the re- lationships between them. It provides a framework for considering the diverse contents of mathematics from the logical to the topological. Consequently we present numerous examples to explicate and contextualize the abstract notions in category theory. Category theory was developed by Saunders Mac Lane and Samuel Eilenberg in the 1940s. Initially it was associated with algebraic topology and geometry and proved particularly fertile for the Grothendieck school. In recent years category theory has been associated with areas as diverse as computability, algebra and quantum mechanics. Sections 2 to 3 outline the basic grammar of category theory. The major refer- ence is the expository essay [Sch01]. Sections 4 to 8 deal with toposes, a family of categories which permit a large number of useful constructions. The primary references are the general category theory book [Bar90] and the more specialised book on topoi, [Mac82]. Before defining a topos we expand on the discussion of limits above and describe exponential objects and the subobject classifier. Once the notion of a topos is established we look at some constructions possible in a topos: in particular, the ability to construct an integer and rational object from a natural number object.
    [Show full text]
  • Basic Category Theory
    Basic Category Theory TOMLEINSTER University of Edinburgh arXiv:1612.09375v1 [math.CT] 30 Dec 2016 First published as Basic Category Theory, Cambridge Studies in Advanced Mathematics, Vol. 143, Cambridge University Press, Cambridge, 2014. ISBN 978-1-107-04424-1 (hardback). Information on this title: http://www.cambridge.org/9781107044241 c Tom Leinster 2014 This arXiv version is published under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International licence (CC BY-NC-SA 4.0). Licence information: https://creativecommons.org/licenses/by-nc-sa/4.0 c Tom Leinster 2014, 2016 Preface to the arXiv version This book was first published by Cambridge University Press in 2014, and is now being published on the arXiv by mutual agreement. CUP has consistently supported the mathematical community by allowing authors to make free ver- sions of their books available online. Readers may, in turn, wish to support CUP by buying the printed version, available at http://www.cambridge.org/ 9781107044241. This electronic version is not only free; it is also freely editable. For in- stance, if you would like to teach a course using this book but some of the examples are unsuitable for your class, you can remove them or add your own. Similarly, if there is notation that you dislike, you can easily change it; or if you want to reformat the text for reading on a particular device, that is easy too. In legal terms, this text is released under the Creative Commons Attribution- NonCommercial-ShareAlike 4.0 International licence (CC BY-NC-SA 4.0). The licence terms are available at the Creative Commons website, https:// creativecommons.org/licenses/by-nc-sa/4.0.
    [Show full text]