DOCSLIB.ORG

Tacl 2019 Abstracts

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Tacl 2019 Abstracts TACL 2019 --------------- ABSTRACTS TACL 2019 Preface Preface This volume contains the papers presented at TACL 2019: Topology, Algebra, and Categories in Logic held June 17{21, 2019 in Nice. The volume includes the abstracts of 91 accepted contributed talks and of 10 invited talks (a few contributions were rejected after a light refereeing process and a couple of papers were withdrawn by the authors themselves due to the impossibility of their attendance at the con- ference). We thank all Program Committee members for their precious work in reading the submitted abstracts and giving useful suggestions to the authors. TACL 2019 is the ninth conference in the series Topology, Algebra, and Categories in Logic (TACL, formerly TANCL). Earlier instalments of the series have been held in Tbilisi (2003), Barcelona (2015), Oxford (2007), Amsterdam (2009), Marseille (2011), Nashville (2013), Ischia (2015), and Prague (2017). The program of the conference TACL 2019 focuses on three interconnecting mathematical themes central to the semantic study of logic and its applications: topological, algebraic, and categorical methods. Our main sponsors are the European Research Council, the CNRS, and the Universit´eC^ote d'Azur. In particular, the conference has received funding from the European Research Council under the European Union's Horizon 2020 research and innovation program through the DuaLL project (grant agreement No. 670624); from the CNRS, which is supporting the TACL 2019 school as one of its Ecoles´ Th´ematiques;from the Universit´eC^oted'Azur through its Inter- national Conferences program, its International Schools program, through support from the interdisciplinary academies R´eseaux,Information, et Soci´et´eNum´erique and Syst`emesCom- plexes, respectively, and finally through the Laboratoire J. A. Dieudonn´eand the Laboratoire I3S, both of the Universit´eC^oted'Azur. We thank the EasyChair development team for providing their conference management platform. May 6, 2019 Silvio Ghilardi Nice Ramon Jansana Mai Gehrke i TACL 2019 Table of Contents Table of Contents Invited talks Simulations of quantum resources and the degrees of contextuality. 1 Samson Abramsky, Rui Soares Barbosa, Martti Karvonen and Shane Mansfield Generalized continuous closure spaces: a topological approach to domain theory . 3 Marcel Erné Possibility Semantics . 5 Wesley Holliday Non-finitely axiomatisable canonical varieties of BAOs with infinite canonical axiomatisations . 6 Agi Kurucz The poset of all logics . 7 Tommaso Moraschini Some Applications of Stone Duality to Automata Theory . 8 Daniela Petrisan Snapshots of duality theory, from 2019 and fifty years earlier . 9 H. A. Priestley Modal Logics of Dependence . 11 Johan van Benthem Logic, Automata, and Model Companions . 13 Samuel J. van Gool Anabelian geometry in model theory setting. 15 Boris Zilber Contributed talks Norm complete abelian `-groups: topological duality . 17 Marco Abbadini, Vincenzo Marra and Luca Spada Norm complete Abelian `-groups: equational axiomatization. 19 Marco Abbadini, Vincenzo Marra and Luca Spada Polyhedral Completeness of Intermediate and Modal Logics . 21 Sam Adam-Day, Nick Bezhanishvili, David Gabelaia and Vincenzo Marra Product of neighborhood frames with additional common modality.. 23 Rajab Aghamov and Andrey Kudinov Predicative Implications: A Topological Approach . 25 Amirhossein Akbar Tabatabai Ranges of functors and elementary classes via topos theory . 27 Peter Arndt ii TACL 2019 Table of Contents When is the frame of nuclei spatial: A new approach . 29 Francisco Avila, Guram Bezhanishvili, Patrick Morandi and Angel Zaldivar 1 Analysis of the Σ1-Fragment of First Order Gödel Logic extended with Propositional Quantifiers . 31 Matthias Baaz and Norbert Preining Enriched Lawvere Theories for Operational Semantics . 33 John Baez and Christian Williams Enriched distributivity over finite commutative residuated lattices . 35 Adriana Balan, Peter Jipsen and Alexander Kurz Proofs and surfaces . 37 Ðorđe Baralić, Pierre-Louis Curien, Marina Milićević, Jovana Obradović, Zoran Petrić, Mladen Zekić and Rade Živaljević A new logic arising from a scattered Stone space . 39 Guram Bezhanishvili, Nick Bezhanishvili, Joel Lucero-Bryan and Jan van Mill A generalization of Gelfand-Naimark-Stone duality to completely regular spaces . 41 Guram Bezhanishvili, Patrick Morandi and Bruce Olberding The van Benthem characterisation theorem for descriptive models . 43 Nick Bezhanishvili and Tim Henke Nearness Posets . 45 Tristan Bice Towards completeness of logics of information and common belief. 47 Marta Bilkova Counting finite involutive bisemilattices . 49 Stefano Bonzio, Michele Pra Baldi and Diego Valota Difference hierarchies over lattices . 51 Célia Borlido Time-reversal homotopical properties of concurrent systems . 53 Cameron Calk, Eric Goubault and Philippe Malbos A temporal interpretation of intuitionistic quantifiers . 55 Luca Carai and Guram Bezhanishvili Semantic analysis and proof theory for monotone modal logic. 57 Jinsheng Chen, Giuseppe Greco, Alessandra Palmigiano and Apostolos Tzimoulis Completeness properties in abstract algebraic logic . 59 Petr Cintula and Carles Noguera Orders on Groups: an Approach through Spectral Spaces. 61 Almudena Colacito The logic of categories and informational entropy. 63 Willem Conradie, Andrew Craig, Alessandra Palmigiano and Nachoem Wijnberg iii TACL 2019 Table of Contents Goldblatt-Thomason for LE-logics. 65 Willem Conradie, Alessandra Palmigiano and Apostolos Tzimoulis Intuitionistic and classical non-normal modal logics: An embedding. 67 Tiziano Dalmonte, Charles Grellois and Nicola Olivetti Lifting functors from Pos to Pries . 69 Jim de Groot A Sahlqvist theorem for subordination algebras. 71 Laurent De Rudder and Georges Hansoul Extensions of the Stone Duality to the category of zero-dimensional Hausdorff spaces . 73 Georgi Dimov and Elza Ivanova-Dimova Semi-reflective Extensions of Dualities and a New Approach to the Fedorchuk Duality . 74 Georgi Dimov, Elza Ivanova-Dimova and Walter Tholen Extensions of dualities and a new approach to de Vries’ Duality Theorem. 76 Georgi Dimov, Elza Ivanova-Dimova and Walter Tholen Resource Reasoning in Duality-theoretic Form: Stone-type Dualities for Bunched and Separation Logics . 77 Simon Docherty and David Pym Amalgamating poset extensions . 79 Rob Egrot Artin glueings as semidirect products. 81 Peter Faul and Graham Manuell Generic Models for Topological Evidence Logics . 82 Saul Fernandez Gonzalez, Alexandru Baltag and Nick Bezhanishvili On non-distributive lattices with involution . 84 Wesley Fussner Characterization of flat polygonal logics . 86 David Gabelaia, Levan Uridia, Mamuka Jibladze and Evgeny Kuznetsov Undecidabilty methods for residuated lattices . 88 Nikolaos Galatos and Gavin St. John A Glivenko theorem for lattice-ordered groups . 90 José Gil-Férez, Frederik Lauridsen and George Metcalfe Nonclassical first order logics: semantics and proof theory . 92 Giuseppe Greco, Peter Jipsen, Alexander Kurz, M. Andrew Moshier, Alessandra Palmigiano and Apostolos Tzimoulis Algebraic proof theory for LE-logics . 94 Giuseppe Greco, Peter Jipsen, Fei Liang, Alessandra Palmigiano and Apostolos Tzimoulis Proof theory and semantics for structural control . 96 Giuseppe Greco, Apostolos Tzimoulis and Michael Moortgat iv TACL 2019 Table of Contents On the variety of Goedel-MV-algebras. 98 Revaz Grigolia, Antonio Di Nola and Gaetano Vitale Some theorems concerning Grzegorczyk contact lattices . 100 Rafal Gruszczynski and Andrzej Pietruszczak Computing the validity degree in Łukasiewicz logic . 102 Zuzana Hanikova Remarks on the Topos Approach to Quantum Mechanics . 104 John Harding and Chris Heunen Hyper-MacNeille completions of Heyting algebras . 106 John Harding and Frederik Lauridsen Comprehension bicategories and fibrations of toposes . 108 Sina Hazratpour and Steve Vickers Twist products arising from residuated bimodules . 110 Pengfei He, Adam Přenosil and Constantine Tsinakis Relational semantics for extended contact algebras . 112 Tatyana Ivanova Colimits of effect algebras via a reflection . ..
Load more

Recommended publications
  • A Guide to Topology
    i i “topguide” — 2010/12/8 — 17:36 — page i — #1 i i A Guide to Topology i i i i i i “topguide” — 2011/2/15 — 16:42 — page ii — #2 i i c 2009 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 2009929077 Print Edition ISBN 978-0-88385-346-7 Electronic Edition ISBN 978-0-88385-917-9 Printed in the United States of America Current Printing (last digit): 10987654321 i i i i i i “topguide” — 2010/12/8 — 17:36 — page iii — #3 i i The Dolciani Mathematical Expositions NUMBER FORTY MAA Guides # 4 A Guide to Topology Steven G. Krantz Washington University, St. Louis ® Published and Distributed by The Mathematical Association of America i i i i i i “topguide” — 2010/12/8 — 17:36 — page iv — #4 i i DOLCIANI MATHEMATICAL EXPOSITIONS Committee on Books Paul Zorn, Chair Dolciani Mathematical Expositions Editorial Board Underwood Dudley, Editor Jeremy S. Case Rosalie A. Dance Tevian Dray Patricia B. Humphrey Virginia E. Knight Mark A. Peterson Jonathan Rogness Thomas Q. Sibley Joe Alyn Stickles i i i i i i “topguide” — 2010/12/8 — 17:36 — page v — #5 i i The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathematical Association of America was established through a generous gift to the Association from Mary P. Dolciani, Professor of Mathematics at Hunter College of the City Uni- versity of New York. In making the gift, Professor Dolciani, herself an exceptionally talented and successfulexpositor of mathematics, had the purpose of furthering the ideal of excellence in mathematical exposition.
    [Show full text]
  • Quantum Heaps, Cops and Heapy Categories
    Mathematical Communications 12(2007), 1-9 1 Quantum heaps, cops and heapy categories Zoran Skodaˇ ∗ Abstract. A heap is a structure with a ternary operation which is intuitively a group with a forgotten unit element. Quantum heaps are associative algebras with a ternary cooperation which are to Hopf algebras what heaps are to groups, and, in particular, the category of copointed quantum heaps is isomorphic to the category of Hopf algebras. There is an intermediate structure of a cop in a monoidal category which is in the case of vector spaces to a quantum heap about what a coalge- bra is to a Hopf algebra. The representations of Hopf algebras make a rigid monoidal category. Similarly, the representations of quantum heaps make a kind of category with ternary products, which we call a heapy category. Key words: quantum algebra, rings, algebras AMS subject classifications: 20N10, 16A24 Received January 25, 2007 Accepted February 1, 2007 1. Classical background: heaps A reference for this section is [1]. 1.1 Before forgetting: group as heap. Let G be a group. Then the ternary operation t : G × G × G → G given by t(a, b, c)=ab−1c, (1) satisfies the following relations: t(b, b, c)=c = t(c, b, b) (2) t(a, b, t(c, d, e)) = t(t(a, b, c),d,e) Aheap(H, t) is a pair of nonempty set H and a ternary operation t : H × H × H satisfying relation (2). A morphism f :(H, t) → (H,t) of heaps is a set map f : H → H satisfying t ◦ (f × f × f)=f ◦ t.
    [Show full text]
  • On Dimension and Weight of a Local Contact Algebra
    Filomat 32:15 (2018), 5481–5500 Published by Faculty of Sciences and Mathematics, https://doi.org/10.2298/FIL1815481D University of Nis,ˇ Serbia Available at: http://www.pmf.ni.ac.rs/filomat On Dimension and Weight of a Local Contact Algebra G. Dimova, E. Ivanova-Dimovaa, I. D ¨untschb aFaculty of Math. and Informatics, Sofia University, 5 J. Bourchier Blvd., 1164 Sofia, Bulgaria bCollege of Maths. and Informatics, Fujian Normal University, Fuzhou, China. Permanent address: Dept. of Computer Science, Brock University, St. Catharines, Canada Abstract. As proved in [16], there exists a duality Λt between the category HLC of locally compact Hausdorff spaces and continuous maps, and the category DHLC of complete local contact algebras and appropriate morphisms between them. In this paper, we introduce the notions of weight wa and of dimension dima of a local contact algebra, and we prove that if X is a locally compact Hausdorff space then t t w(X) = wa(Λ (X)), and if, in addition, X is normal, then dim(X) = dima(Λ (X)). 1. Introduction According to Stone’s famous duality theorem [43], the Boolean algebra CO(X) of all clopen (= closed and open) subsets of a zero-dimensional compact Hausdorff space X carries the whole information about the space X, i.e. the space X can be reconstructed from CO(X), up to homeomorphism. It is natural to ask whether the Boolean algebra RC(X) of all regular closed subsets of a compact Hausdorff space X carries the full information about the space X (see Example 2.5 below for RC(X)).
    [Show full text]
  • Ordered Groupoids and the Holomorph of an Inverse Semigroup
    ORDERED GROUPOIDS AND THE HOLOMORPH OF AN INVERSE SEMIGROUP N.D. GILBERT AND E.A. MCDOUGALL ABSTRACT. We present a construction for the holomorph of an inverse semi- group, derived from the cartesian closed structure of the category of ordered groupoids. We compare the holomorph with the monoid of mappings that pre- serve the ternary heap operation on an inverse semigroup: for groups these two constructions coincide. We present detailed calculations for semilattices of groups and for the polycyclic monoids. INTRODUCTION The holomorph of a group G is the semidirect product of Hol(G) = Aut(G) n G of G and its automorphism group (with the natural action of Aut(G) on G). The embedding of Aut(G) into the symmetric group ΣG on G extends to an embedding of Hol(G) into ΣG where g 2 G is identified with its (right) Cayley representation ρg : a 7! ag. Then Hol(G) is the normalizer of G in ΣG ([15, Theorem 9.17]). A second interesting characterization of Hol(G) is due to Baer [1] (see also [3] and [15, Exercise 520]). The heap operation on G is the ternary operation defined by ha; b; ci = ab−1c. Baer shows that a subset of G is closed under the heap operation if and only if it is a coset of some subgroup, and that the subset of ΣG that preserves h· · ·i is precisely Hol(G): that is, if σ 2 ΣG, then for all a; b; c; 2 G we have ha; b; ciσ = haσ; bσ; cσi if and only if σ 2 Hol(G).
    [Show full text]
  • Second-Order Algebraic Theories (Extended Abstract)
    Second-Order Algebraic Theories (Extended Abstract) Marcelo Fiore and Ola Mahmoud University of Cambridge, Computer Laboratory Abstract. Fiore and Hur [10] recently introduced a conservative exten- sion of universal algebra and equational logic from first to second order. Second-order universal algebra and second-order equational logic respec- tively provide a model theory and a formal deductive system for lan- guages with variable binding and parameterised metavariables. This work completes the foundations of the subject from the viewpoint of categori- cal algebra. Specifically, the paper introduces the notion of second-order algebraic theory and develops its basic theory. Two categorical equiva- lences are established: at the syntactic level, that of second-order equa- tional presentations and second-order algebraic theories; at the semantic level, that of second-order algebras and second-order functorial models. Our development includes a mathematical definition of syntactic trans- lation between second-order equational presentations. This gives the first formalisation of notions such as encodings and transforms in the context of languages with variable binding. 1 Introduction Algebra started with the study of a few sample algebraic structures: groups, rings, lattices, etc. Based on these, Birkhoff [3] laid out the foundations of a general unifying theory, now known as universal algebra. Birkhoff's formalisation of the notion of algebra starts with the introduction of equational presentations. These constitute the syntactic foundations of the subject. Algebras are then the semantics or model theory, and play a crucial role in establishing the logical foundations. Indeed, Birkhoff introduced equational logic as a sound and complete formal deductive system for reasoning about algebraic structure.
    [Show full text]
  • MTH 304: General Topology Semester 2, 2017-2018
    MTH 304: General Topology Semester 2, 2017-2018 Dr. Prahlad Vaidyanathan Contents I. Continuous Functions3 1. First Definitions................................3 2. Open Sets...................................4 3. Continuity by Open Sets...........................6 II. Topological Spaces8 1. Definition and Examples...........................8 2. Metric Spaces................................. 11 3. Basis for a topology.............................. 16 4. The Product Topology on X × Y ...................... 18 Q 5. The Product Topology on Xα ....................... 20 6. Closed Sets.................................. 22 7. Continuous Functions............................. 27 8. The Quotient Topology............................ 30 III.Properties of Topological Spaces 36 1. The Hausdorff property............................ 36 2. Connectedness................................. 37 3. Path Connectedness............................. 41 4. Local Connectedness............................. 44 5. Compactness................................. 46 6. Compact Subsets of Rn ............................ 50 7. Continuous Functions on Compact Sets................... 52 8. Compactness in Metric Spaces........................ 56 9. Local Compactness.............................. 59 IV.Separation Axioms 62 1. Regular Spaces................................ 62 2. Normal Spaces................................ 64 3. Tietze's extension Theorem......................... 67 4. Urysohn Metrization Theorem........................ 71 5. Imbedding of Manifolds..........................
    [Show full text]
  • Chapter 7 Separation Properties
    Chapter VII Separation Axioms 1. Introduction “Separation” refers here to whether or not objects like points or disjoint closed sets can be enclosed in disjoint open sets; “separation properties” have nothing to do with the idea of “separated sets” that appeared in our discussion of connectedness in Chapter 5 in spite of the similarity of terminology.. We have already met some simple separation properties of spaces: the XßX!"and X # (Hausdorff) properties. In this chapter, we look at these and others in more depth. As “more separation” is added to spaces, they generally become nicer and nicer especially when “separation” is combined with other properties. For example, we will see that “enough separation” and “a nice base” guarantees that a space is metrizable. “Separation axioms” translates the German term Trennungsaxiome used in the older literature. Therefore the standard separation axioms were historically named XXXX!"#$, , , , and X %, each stronger than its predecessors in the list. Once these were common terminology, another separation axiom was discovered to be useful and “interpolated” into the list: XÞ"" It turns out that the X spaces (also called $$## Tychonoff spaces) are an extremely well-behaved class of spaces with some very nice properties. 2. The Basics Definition 2.1 A topological space \ is called a 1) X! space if, whenever BÁC−\, there either exists an open set Y with B−Y, CÂY or there exists an open set ZC−ZBÂZwith , 2) X" space if, whenever BÁC−\, there exists an open set Ywith B−YßCÂZ and there exists an open set ZBÂYßC−Zwith 3) XBÁC−\Y# space (or, Hausdorff space) if, whenever , there exist disjoint open sets and Z\ in such that B−YC−Z and .
    [Show full text]
  • Foundations of Algebraic Theories and Higher Dimensional Categories
    Foundations of Algebraic Theories and Higher Dimensional Categories Soichiro Fujii arXiv:1903.07030v1 [math.CT] 17 Mar 2019 A Doctor Thesis Submitted to the Graduate School of the University of Tokyo on December 7, 2018 in Partial Fulfillment of the Requirements for the Degree of Doctor of Information Science and Technology in Computer Science ii Abstract Universal algebra uniformly captures various algebraic structures, by expressing them as equational theories or abstract clones. The ubiquity of algebraic structures in math- ematics and related fields has given rise to several variants of universal algebra, such as symmetric operads, non-symmetric operads, generalised operads, and monads. These variants of universal algebra are called notions of algebraic theory. Although notions of algebraic theory share the basic aim of providing a background theory to describe alge- braic structures, they use various techniques to achieve this goal and, to the best of our knowledge, no general framework for notions of algebraic theory which includes all of the examples above was known. Such a framework would lead to a better understand- ing of notions of algebraic theory by revealing their essential structure, and provide a uniform way to compare different notions of algebraic theory. In the first part of this thesis, we develop a unified framework for notions of algebraic theory which includes all of the above examples. Our key observation is that each notion of algebraic theory can be identified with a monoidal category, in such a way that theories correspond to monoid objects therein. We introduce a categorical structure called metamodel, which underlies the definition of models of theories.
    [Show full text]
  • DEFINITIONS and THEOREMS in GENERAL TOPOLOGY 1. Basic
    DEFINITIONS AND THEOREMS IN GENERAL TOPOLOGY 1. Basic definitions. A topology on a set X is defined by a family O of subsets of X, the open sets of the topology, satisfying the axioms: (i) ; and X are in O; (ii) the intersection of finitely many sets in O is in O; (iii) arbitrary unions of sets in O are in O. Alternatively, a topology may be defined by the neighborhoods U(p) of an arbitrary point p 2 X, where p 2 U(p) and, in addition: (i) If U1;U2 are neighborhoods of p, there exists U3 neighborhood of p, such that U3 ⊂ U1 \ U2; (ii) If U is a neighborhood of p and q 2 U, there exists a neighborhood V of q so that V ⊂ U. A topology is Hausdorff if any distinct points p 6= q admit disjoint neigh- borhoods. This is almost always assumed. A set C ⊂ X is closed if its complement is open. The closure A¯ of a set A ⊂ X is the intersection of all closed sets containing X. A subset A ⊂ X is dense in X if A¯ = X. A point x 2 X is a cluster point of a subset A ⊂ X if any neighborhood of x contains a point of A distinct from x. If A0 denotes the set of cluster points, then A¯ = A [ A0: A map f : X ! Y of topological spaces is continuous at p 2 X if for any open neighborhood V ⊂ Y of f(p), there exists an open neighborhood U ⊂ X of p so that f(U) ⊂ V .
    [Show full text]
  • Categorical Models of Type Theory
    Categorical models of type theory Michael Shulman February 28, 2012 1 / 43 Theories and models Example The theory of a group asserts an identity e, products x · y and inverses x−1 for any x; y, and equalities x · (y · z) = (x · y) · z and x · e = x = e · x and x · x−1 = e. I A model of this theory (in sets) is a particularparticular group, like Z or S3. I A model in spaces is a topological group. I A model in manifolds is a Lie group. I ... 3 / 43 Group objects in categories Definition A group object in a category with finite products is an object G with morphisms e : 1 ! G, m : G × G ! G, and i : G ! G, such that the following diagrams commute. m×1 (e;1) (1;e) G × G × G / G × G / G × G o G F G FF xx 1×m m FF xx FF m xx 1 F x 1 / F# x{ x G × G m G G ! / e / G 1 GO ∆ m G × G / G × G 1×i 4 / 43 Categorical semantics Categorical semantics is a general procedure to go from 1. the theory of a group to 2. the notion of group object in a category. A group object in a category is a model of the theory of a group. Then, anything we can prove formally in the theory of a group will be valid for group objects in any category. 5 / 43 Doctrines For each kind of type theory there is a corresponding kind of structured category in which we consider models.
    [Show full text]
  • Arxiv:1909.05807V1 [Math.RA] 12 Sep 2019 Cs T
    MODULES OVER TRUSSES VS MODULES OVER RINGS: DIRECT SUMS AND FREE MODULES TOMASZ BRZEZINSKI´ AND BERNARD RYBOLOWICZ Abstract. Categorical constructions on heaps and modules over trusses are con- sidered and contrasted with the corresponding constructions on groups and rings. These include explicit description of free heaps and free Abelian heaps, coprod- ucts or direct sums of Abelian heaps and modules over trusses, and description and analysis of free modules over trusses. It is shown that the direct sum of two non-empty Abelian heaps is always infinite and isomorphic to the heap associated to the direct sums of the group retracts of both heaps and Z. Direct sum is used to extend a given truss to a ring-type truss or a unital truss (or both). Free mod- ules are constructed as direct sums of a truss. It is shown that only free rank-one modules are free as modules over the associated truss. On the other hand, if a (finitely generated) module over a truss associated to a ring is free, then so is the corresponding quotient-by-absorbers module over this ring. 1. Introduction Trusses and skew trusses were defined in [3] in order to capture the nature of the distinctive distributive law that characterises braces and skew braces [12], [6], [9]. A (one-sided) truss is a set with a ternary operation which makes it into a heap or herd (see [10], [11], [1] or [13]) together with an associative binary operation that distributes (on one side or both) over the heap ternary operation. If the specific bi- nary operation admits it, a choice of a particular element could fix a group structure on a heap in a way that turns the truss into a ring or a brace-like system (which becomes a brace provided the binary operation admits inverses).
    [Show full text]
  • An Outline of Algebraic Set Theory
    An Outline of Algebraic Set Theory Steve Awodey Dedicated to Saunders Mac Lane, 1909–2005 Abstract This survey article is intended to introduce the reader to the field of Algebraic Set Theory, in which models of set theory of a new and fascinating kind are determined algebraically. The method is quite robust, admitting adjustment in several respects to model different theories including classical, intuitionistic, bounded, and predicative ones. Under this scheme some familiar set theoretic properties are related to algebraic ones, like freeness, while others result from logical constraints, like definability. The overall theory is complete in two important respects: conventional elementary set theory axiomatizes the class of algebraic models, and the axioms provided for the abstract algebraic framework itself are also complete with respect to a range of natural models consisting of “ideals” of sets, suitably defined. Some previous results involving realizability, forcing, and sheaf models are subsumed, and the prospects for further such unification seem bright. 1 Contents 1 Introduction 3 2 The category of classes 10 2.1 Smallmaps ............................ 12 2.2 Powerclasses............................ 14 2.3 UniversesandInfinity . 15 2.4 Classcategories .......................... 16 2.5 Thetoposofsets ......................... 17 3 Algebraic models of set theory 18 3.1 ThesettheoryBIST ....................... 18 3.2 Algebraic soundness of BIST . 20 3.3 Algebraic completeness of BIST . 21 4 Classes as ideals of sets 23 4.1 Smallmapsandideals . .. .. 24 4.2 Powerclasses and universes . 26 4.3 Conservativity........................... 29 5 Ideal models 29 5.1 Freealgebras ........................... 29 5.2 Collection ............................. 30 5.3 Idealcompleteness . .. .. 32 6 Variations 33 References 36 2 1 Introduction Algebraic set theory (AST) is a new approach to the construction of models of set theory, invented by Andr´eJoyal and Ieke Moerdijk and first presented in [16].
    [Show full text]