A Logical and Algebraic Characterization of Adjunctions Between Generalized Quasi-Varieties
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Willem Blok and Modal Logic M
W. Rautenberg F. Wolter Willem Blok and modal logic M. Zakharyaschev Abstract. We present our personal view on W.J. Blok’s contribution to modal logic. Introduction Willem Johannes Blok started his scientific career in 1973 as an algebraist with an investigation of varieties of interior (a.k.a. closure or topological Boolean or topoboolean) algebras. In the years following his PhD in 1976 he moved on to study more general varieties of modal algebras (and matrices) and, by the end of the 1970s, this algebraist was recognised by the modal logic community as one of the most influential modal logicians. Wim Blok worked on modal algebras till about 1979–80, and over those seven years he obtained a number of very beautiful and profound results that had and still have a great impact on the development of the whole discipline of modal logic. In this short note we discuss only three areas of modal logic where Wim Blok’s contribution was, in our opinion, the most significant: 1. splittings of the lattice of normal modal logics and the position of Kripke incomplete logics in this lattice [3, 4, 6], 2. the relation between modal logics containing S4 and extensions of intuitionistic propositional logic [1], 3. the position of tabular, pretabular, and locally tabular logics in lattices of modal logics [1, 6, 8, 10]. As usual in mathematics and logic, it is not only the results themselves that matter, but also the power and beauty of the methods developed to obtain them. In this respect, one of the most important achievements of Wim Blok was a brilliant demonstration of the fact that various techniques and results that originated in universal algebra can be used to prove significant and deep theorems in modal logic. -
Category Theory for Autonomous and Networked Dynamical Systems
Discussion Category Theory for Autonomous and Networked Dynamical Systems Jean-Charles Delvenne Institute of Information and Communication Technologies, Electronics and Applied Mathematics (ICTEAM) and Center for Operations Research and Econometrics (CORE), Université catholique de Louvain, 1348 Louvain-la-Neuve, Belgium; [email protected] Received: 7 February 2019; Accepted: 18 March 2019; Published: 20 March 2019 Abstract: In this discussion paper we argue that category theory may play a useful role in formulating, and perhaps proving, results in ergodic theory, topogical dynamics and open systems theory (control theory). As examples, we show how to characterize Kolmogorov–Sinai, Shannon entropy and topological entropy as the unique functors to the nonnegative reals satisfying some natural conditions. We also provide a purely categorical proof of the existence of the maximal equicontinuous factor in topological dynamics. We then show how to define open systems (that can interact with their environment), interconnect them, and define control problems for them in a unified way. Keywords: ergodic theory; topological dynamics; control theory 1. Introduction The theory of autonomous dynamical systems has developed in different flavours according to which structure is assumed on the state space—with topological dynamics (studying continuous maps on topological spaces) and ergodic theory (studying probability measures invariant under the map) as two prominent examples. These theories have followed parallel tracks and built multiple bridges. For example, both have grown successful tools from the interaction with information theory soon after its emergence, around the key invariants, namely the topological entropy and metric entropy, which are themselves linked by a variational theorem. The situation is more complex and heterogeneous in the field of what we here call open dynamical systems, or controlled dynamical systems. -
Notes and Solutions to Exercises for Mac Lane's Categories for The
Stefan Dawydiak Version 0.3 July 2, 2020 Notes and Exercises from Categories for the Working Mathematician Contents 0 Preface 2 1 Categories, Functors, and Natural Transformations 2 1.1 Functors . .2 1.2 Natural Transformations . .4 1.3 Monics, Epis, and Zeros . .5 2 Constructions on Categories 6 2.1 Products of Categories . .6 2.2 Functor categories . .6 2.2.1 The Interchange Law . .8 2.3 The Category of All Categories . .8 2.4 Comma Categories . 11 2.5 Graphs and Free Categories . 12 2.6 Quotient Categories . 13 3 Universals and Limits 13 3.1 Universal Arrows . 13 3.2 The Yoneda Lemma . 14 3.2.1 Proof of the Yoneda Lemma . 14 3.3 Coproducts and Colimits . 16 3.4 Products and Limits . 18 3.4.1 The p-adic integers . 20 3.5 Categories with Finite Products . 21 3.6 Groups in Categories . 22 4 Adjoints 23 4.1 Adjunctions . 23 4.2 Examples of Adjoints . 24 4.3 Reflective Subcategories . 28 4.4 Equivalence of Categories . 30 4.5 Adjoints for Preorders . 32 4.5.1 Examples of Galois Connections . 32 4.6 Cartesian Closed Categories . 33 5 Limits 33 5.1 Creation of Limits . 33 5.2 Limits by Products and Equalizers . 34 5.3 Preservation of Limits . 35 5.4 Adjoints on Limits . 35 5.5 Freyd's adjoint functor theorem . 36 1 6 Chapter 6 38 7 Chapter 7 38 8 Abelian Categories 38 8.1 Additive Categories . 38 8.2 Abelian Categories . 38 8.3 Diagram Lemmas . 39 9 Special Limits 41 9.1 Interchange of Limits . -
Semisimplicity, Glivenko Theorems, and the Excluded Middle
Semisimplicity, Glivenko theorems, and the excluded middle Adam Pˇrenosil (joint work with Tomáš Láviˇcka) Vanderbilt University, Nashville, USA Online Logic Seminar 29 October 2020 1 / 40 Introduction Semisimplicity is equivalent to the law of the excluded middle (LEM). The semisimple companion and the Glivenko companion of a logic coincide. for compact propositional logics with a well-behaved negation. 2 / 40 Intuitionistic and classical logic How can we obtain classical propositional logic from intuitionistic logic? 1. Classical logic is intuitionistic logic plus the LEM (the axiom ' '). _: 2. Classical consequence is related to intuitionistic consequence through the Glivenko translation (the double negation translation): Γ CL ' Γ IL '. ` () ` :: In other words, CL is the Glivenko companion of IL. 3. The algebraic models of classical logic (Boolean algebras) are precisely the semisimple models of intuitionistic logic (Heyting algebras). In other words, CL is the semisimple companion of IL. 3 / 40 Modal logic S4 and S5 How can we obtain the global modal logic S5 (modal logic of equivalence relations) from the global modal logic S4 (modal logic of preorders)? 1. Modal logic S5 is modal logic S4 plus the axiom ' '. _ : This is just the LEM with disjunction x y and negation x. _ : 2. Consequence in S5 is related to consequence in S4 through the Glivenko translation (the double negation translation): Γ S5 ' Γ S4 '. ` () ` : : This is just the Glivenko translation with negation x. : 3. The algebraic models of S5 are precisely the semisimple models of S4 (Boolean algebras with a topological interior operator). 4 / 40 Semisimplicity: algebraic definition Each algebra A has a lattice of congruences Con A. -
Basic Category Theory and Topos Theory
Basic Category Theory and Topos Theory Jaap van Oosten Jaap van Oosten Department of Mathematics Utrecht University The Netherlands Revised, February 2016 Contents 1 Categories and Functors 1 1.1 Definitions and examples . 1 1.2 Some special objects and arrows . 5 2 Natural transformations 8 2.1 The Yoneda lemma . 8 2.2 Examples of natural transformations . 11 2.3 Equivalence of categories; an example . 13 3 (Co)cones and (Co)limits 16 3.1 Limits . 16 3.2 Limits by products and equalizers . 23 3.3 Complete Categories . 24 3.4 Colimits . 25 4 A little piece of categorical logic 28 4.1 Regular categories and subobjects . 28 4.2 The logic of regular categories . 34 4.3 The language L(C) and theory T (C) associated to a regular cat- egory C ................................ 39 4.4 The category C(T ) associated to a theory T : Completeness Theorem 41 4.5 Example of a regular category . 44 5 Adjunctions 47 5.1 Adjoint functors . 47 5.2 Expressing (co)completeness by existence of adjoints; preserva- tion of (co)limits by adjoint functors . 52 6 Monads and Algebras 56 6.1 Algebras for a monad . 57 6.2 T -Algebras at least as complete as D . 61 6.3 The Kleisli category of a monad . 62 7 Cartesian closed categories and the λ-calculus 64 7.1 Cartesian closed categories (ccc's); examples and basic facts . 64 7.2 Typed λ-calculus and cartesian closed categories . 68 7.3 Representation of primitive recursive functions in ccc's with nat- ural numbers object . -
Sheaves with Values in a Category7
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology Vol. 3, pp. l-18, Pergamon Press. 196s. F’rinted in Great Britain SHEAVES WITH VALUES IN A CATEGORY7 JOHN W. GRAY (Received 3 September 1963) $4. IN’IXODUCTION LET T be a topology (i.e., a collection of open sets) on a set X. Consider T as a category in which the objects are the open sets and such that there is a unique ‘restriction’ morphism U -+ V if and only if V c U. For any category A, the functor category F(T, A) (i.e., objects are covariant functors from T to A and morphisms are natural transformations) is called the category ofpresheaces on X (really, on T) with values in A. A presheaf F is called a sheaf if for every open set UE T and for every strong open covering {U,) of U(strong means {U,} is closed under finite, non-empty intersections) we have F(U) = Llim F( U,) (Urn means ‘generalized’ inverse limit. See $4.) The category S(T, A) of sheares on X with values in A is the full subcategory (i.e., morphisms the same as in F(T, A)) determined by the sheaves. In this paper we shall demonstrate several properties of S(T, A): (i) S(T, A) is left closed in F(T, A). (Theorem l(i), $8). This says that if a left limit (generalized inverse limit) of sheaves exists as a presheaf then that presheaf is a sheaf. -
Abelian Categories
Abelian Categories Lemma. In an Ab-enriched category with zero object every finite product is coproduct and conversely. π1 Proof. Suppose A × B //A; B is a product. Define ι1 : A ! A × B and π2 ι2 : B ! A × B by π1ι1 = id; π2ι1 = 0; π1ι2 = 0; π2ι2 = id: It follows that ι1π1+ι2π2 = id (both sides are equal upon applying π1 and π2). To show that ι1; ι2 are a coproduct suppose given ' : A ! C; : B ! C. It φ : A × B ! C has the properties φι1 = ' and φι2 = then we must have φ = φid = φ(ι1π1 + ι2π2) = ϕπ1 + π2: Conversely, the formula ϕπ1 + π2 yields the desired map on A × B. An additive category is an Ab-enriched category with a zero object and finite products (or coproducts). In such a category, a kernel of a morphism f : A ! B is an equalizer k in the diagram k f ker(f) / A / B: 0 Dually, a cokernel of f is a coequalizer c in the diagram f c A / B / coker(f): 0 An Abelian category is an additive category such that 1. every map has a kernel and a cokernel, 2. every mono is a kernel, and every epi is a cokernel. In fact, it then follows immediatly that a mono is the kernel of its cokernel, while an epi is the cokernel of its kernel. 1 Proof of last statement. Suppose f : B ! C is epi and the cokernel of some g : A ! B. Write k : ker(f) ! B for the kernel of f. Since f ◦ g = 0 the map g¯ indicated in the diagram exists. -
On Iterated Semidirect Products of Finite Semilattices
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 142, 2399254 (1991) On Iterated Semidirect Products of Finite Semilattices JORGE ALMEIDA INIC-Centro de Mawmdtica, Faculdade de Ci&cias, Universidade do Porro, 4000 Porte, Portugal Communicated by T. E. Hull Received December 15, 1988 Let SI denote the class of all finite semilattices and let SI” be the pseudovariety of semigroups generated by all semidirect products of n finite semilattices. The main result of this paper shows that, for every n 2 3, Sl” is not finitely based. Nevertheless, a simple basis of identities is described for each SI”. We also show that SI" is generated by a semigroup with 2n generators but, except for n < 2, we do not know whether the bound 2n is optimal. ‘(” 1991 Academic PKSS, IW 1. INTRODUCTION Among the various operations which allow us to construct semigroups from simpler ones, the semidirect product has thus far received the most attention in the theory of finite semigroups. It plays a special role in the connections with language theory (Eilenberg [S], Pin [9]) and in the Krohn-Rodes decomposition theorem [S]. Questions dealing with semidirect products of finite semigroups are often properly dealt with in the context of pseudovarieties. For pseudovarieties V and W, their semidirect product V * W is defined to be the pseudovariety generated by all semidirect products S * T with SE V and T E W. It is well known that this operation on pseudovarieties is associative and that V * W consists of all homomorphic images of subsemigroups of S * T with SE V and TEW [S]. -
Lie-Butcher Series, Geometry, Algebra and Computation
Lie–Butcher series, Geometry, Algebra and Computation Hans Z. Munthe-Kaas and Kristoffer K. Føllesdal Abstract Lie–Butcher (LB) series are formal power series expressed in terms of trees and forests. On the geometric side LB-series generalizes classical B-series from Euclidean spaces to Lie groups and homogeneous manifolds. On the algebraic side, B-series are based on pre-Lie algebras and the Butcher-Connes-Kreimer Hopf algebra. The LB-series are instead based on post-Lie algebras and their enveloping algebras. Over the last decade the algebraic theory of LB-series has matured. The purpose of this paper is twofold. First, we aim at presenting the algebraic structures underlying LB series in a concise and self contained manner. Secondly, we review a number of algebraic operations on LB-series found in the literature, and reformulate these as recursive formulae. This is part of an ongoing effort to create an extensive software library for computations in LB-series and B-series in the programming language Haskell. 1 Introduction Classical B-series are formal power series expressed in terms of rooted trees (con- nected graphs without any cycle and a designated node called the root). The theory has its origins back to the work of Arthur Cayley [5] in the 1850s, where he realized that trees could be used to encode information about differential operators. Being forgotten for a century, the theory was revived through the efforts of understanding numerical integration algorithms by John Butcher in the 1960s and ’70s [2, 3]. Ernst Hairer and Gerhard Wanner [15] coined the term B-series for an infinite formal se- arXiv:1701.03654v2 [math.NA] 27 Oct 2017 ries of the form H. -
Free Split Bands Francis Pastijn Marquette University, [email protected]
Marquette University e-Publications@Marquette Mathematics, Statistics and Computer Science Mathematics, Statistics and Computer Science, Faculty Research and Publications Department of 6-1-2015 Free Split Bands Francis Pastijn Marquette University, [email protected] Justin Albert Marquette University, [email protected] Accepted version. Semigroup Forum, Vol. 90, No. 3 (June 2015): 753-762. DOI. © 2015 Springer International Publishing AG. Part of Springer Nature. Used with permission. Shareable Link. Provided by the Springer Nature SharedIt content-sharing initiative. NOT THE PUBLISHED VERSION; this is the author’s final, peer-reviewed manuscript. The published version may be accessed by following the link in the citation at the bottom of the page. Free Split Bands Francis Pastijn Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, WI Justin Albert Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, WI Abstract: We solve the word problem for the free objects in the variety consisting of bands with a semilattice transversal. It follows that every free band can be embedded into a band with a semilattice transversal. Keywords: Free band, Split band, Semilattice transversal 1 Introduction We refer to3 and6 for a general background and as references to terminology used in this paper. Recall that a band is a semigroup where every element is an idempotent. The Green relation is the least semilattice congruence on a band, and so every band is a semilattice of its -classes; the -classes themselves form rectangular bands.5 We shall be interested in bands S for which the least semilattice congruence splits, that is, there exists a subsemilattice of which intersects each -class in exactly one element. -
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. -
Section I.9. Free Groups, Free Products, and Generators and Relations
I.9. Free Groups, Free Products, and Generators and Relations 1 Section I.9. Free Groups, Free Products, and Generators and Relations Note. This section includes material covered in Fraleigh’s Sections VII.39 and VII.40. We define a free group on a set and show (in Theorem I.9.2) that this idea of “free” is consistent with the idea of “free on a set” in the setting of a concrete category (see Definition I.7.7). We also define generators and relations in a group presentation. Note. To define a free group F on a set X, we will first define “words” on the set, have a way to reduce these words, define a method of combining words (this com- bination will be the binary operation in the free group), and then give a reduction of the combined words. The free group will have the reduced words as its elements and the combination as the binary operation. If set X = ∅ then the free group on X is F = hei. Definition. Let X be a nonempty set. Define set X−1 to be disjoint from X such that |X| = |X−1|. Choose a bijection from X to X−1 and denote the image of x ∈ X as x−1. Introduce the symbol “1” (with X and X−1 not containing 1). A −1 word on X is a sequence (a1, a2,...) with ai ∈ X ∪ X ∪ {1} for i ∈ N such that for some n ∈ N we have ak = 1 for all k ≥ n. The sequence (1, 1,...) is the empty word which we will also sometimes denote as 1.