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A Small Complete Category
Annals of Pure and Applied Logic 40 (1988) 135-165 135 North-Holland A SMALL COMPLETE CATEGORY J.M.E. HYLAND Department of Pure Mathematics and Mathematical Statktics, 16 Mill Lane, Cambridge CB2 ISB, England Communicated by D. van Dalen Received 14 October 1987 0. Introduction This paper is concerned with a remarkable fact. The effective topos contains a small complete subcategory, essentially the familiar category of partial equiv- alence realtions. This is in contrast to the category of sets (indeed to all Grothendieck toposes) where any small complete category is equivalent to a (complete) poset. Note at once that the phrase ‘a small complete subcategory of a topos’ is misleading. It is not the subcategory but the internal (small) category which matters. Indeed for any ordinary subcategory of a topos there may be a number of internal categories with global sections equivalent to the given subcategory. The appropriate notion of subcategory is an indexed (or better fibred) one, see 0.1. Another point that needs attention is the definition of completeness (see 0.2). In my talk at the Church’s Thesis meeting, and in the first draft of this paper, I claimed too strong a form of completeness for the internal category. (The elementary oversight is described in 2.7.) Fortunately during the writing of [13] my collaborators Edmund Robinson and Giuseppe Rosolini noticed the mistake. Again one needs to pay careful attention to the ideas of indexed (or fibred) categories. The idea that small (sufficiently) complete categories in toposes might exist, and would provide the right setting in which to discuss models for strong polymorphism (quantification over types), was suggested to me by Eugenio Moggi. -
Quantum Duality, Unbounded Operators, And
JOURNAL OF MATHEMATICAL PHYSICS 51, 063511 ͑2010͒ Quantum duality, unbounded operators, and inductive limits ͒ Anar Dosia Middle East Technical University Northern Cyprus Campus, Guzelyurt, KKTC, Mersin 10, Turkey ͑Received 20 October 2009; accepted 7 April 2010; published online 14 June 2010͒ In this paper, we investigate the inductive limits of quantum normed ͑or operator͒ spaces. This construction allows us to treat the space of all noncommutative con- tinuous functions over a quantum domain as a quantum ͑or local operator͒ space of all matrix continuous linear operators equipped with S-quantum topology. In par- ticular, we classify all quantizations of the polynormed topologies compatible with the given duality proposing a noncommutative Arens–Mackey theorem. Further, the inductive limits of operator spaces are used to introduce locally compact and lo- cally trace class unbounded operators on a quantum domain and prove the dual realization theorem for an abstract quantum space. © 2010 American Institute of Physics. ͓doi:10.1063/1.3419771͔ I. INTRODUCTION The operator analogs of locally convex spaces have been started to develop in Ref. 14 by Effros and Webster. The central goal of this direction is to create a theory of quantum polynormed spaces or quantum spaces, which should reflect the “locally convex space chapters” of quantum functional analysis. This theory has been created as the basic language of quantum physics. To have a comprehensive mathematical model of quantum physics, it is necessary to consider the linear spaces of unbounded Hilbert space operators or “noncommutative variable spaces.”19 This is the mathematical background of Heisenberg’s “matrix mechanics.” The quantizations of a variable space have provided functional analysis with the new constructions, methods, and problems.22 Being a new and modern branch of functional analysis, quantum functional analysis can be divided into the normed13,20 and polynormed ͑or locally convex͒ topics,14,25 as in the classical theory. -
Mathematicians Fleeing from Nazi Germany
Mathematicians Fleeing from Nazi Germany Mathematicians Fleeing from Nazi Germany Individual Fates and Global Impact Reinhard Siegmund-Schultze princeton university press princeton and oxford Copyright 2009 © by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW All Rights Reserved Library of Congress Cataloging-in-Publication Data Siegmund-Schultze, R. (Reinhard) Mathematicians fleeing from Nazi Germany: individual fates and global impact / Reinhard Siegmund-Schultze. p. cm. Includes bibliographical references and index. ISBN 978-0-691-12593-0 (cloth) — ISBN 978-0-691-14041-4 (pbk.) 1. Mathematicians—Germany—History—20th century. 2. Mathematicians— United States—History—20th century. 3. Mathematicians—Germany—Biography. 4. Mathematicians—United States—Biography. 5. World War, 1939–1945— Refuges—Germany. 6. Germany—Emigration and immigration—History—1933–1945. 7. Germans—United States—History—20th century. 8. Immigrants—United States—History—20th century. 9. Mathematics—Germany—History—20th century. 10. Mathematics—United States—History—20th century. I. Title. QA27.G4S53 2008 510.09'04—dc22 2008048855 British Library Cataloging-in-Publication Data is available This book has been composed in Sabon Printed on acid-free paper. ∞ press.princeton.edu Printed in the United States of America 10 987654321 Contents List of Figures and Tables xiii Preface xvii Chapter 1 The Terms “German-Speaking Mathematician,” “Forced,” and“Voluntary Emigration” 1 Chapter 2 The Notion of “Mathematician” Plus Quantitative Figures on Persecution 13 Chapter 3 Early Emigration 30 3.1. The Push-Factor 32 3.2. The Pull-Factor 36 3.D. -
Rudi Mathematici
Rudi Mathematici Y2K Rudi Mathematici Gennaio 2000 52 1 S (1803) Guglielmo LIBRI Carucci dalla Somaja Olimpiadi Matematiche (1878) Agner Krarup ERLANG (1894) Satyendranath BOSE P1 (1912) Boris GNEDENKO 2 D (1822) Rudolf Julius Emmanuel CLAUSIUS Due matematici "A" e "B" si sono inventati una (1905) Lev Genrichovich SHNIRELMAN versione particolarmente complessa del "testa o (1938) Anatoly SAMOILENKO croce": viene scritta alla lavagna una matrice 1 3 L (1917) Yuri Alexeievich MITROPOLSHY quadrata con elementi interi casuali; il gioco (1643) Isaac NEWTON consiste poi nel calcolare il determinante: 4 M (1838) Marie Ennemond Camille JORDAN 5 M Se il determinante e` pari, vince "A". (1871) Federigo ENRIQUES (1871) Gino FANO Se il determinante e` dispari, vince "B". (1807) Jozeph Mitza PETZVAL 6 G (1841) Rudolf STURM La probabilita` che un numero sia pari e` 0.5, (1871) Felix Edouard Justin Emile BOREL 7 V ma... Quali sono le probabilita` di vittoria di "A"? (1907) Raymond Edward Alan Christopher PALEY (1888) Richard COURANT P2 8 S (1924) Paul Moritz COHN (1942) Stephen William HAWKING Dimostrare che qualsiasi numero primo (con (1864) Vladimir Adreievich STELKOV l'eccezione di 2 e 5) ha un'infinita` di multipli 9 D nella forma 11....1 2 10 L (1875) Issai SCHUR (1905) Ruth MOUFANG "Die Energie der Welt ist konstant. Die Entroopie 11 M (1545) Guidobaldo DEL MONTE der Welt strebt einem Maximum zu" (1707) Vincenzo RICCATI (1734) Achille Pierre Dionis DU SEJOUR Rudolph CLAUSIUS 12 M (1906) Kurt August HIRSCH " I know not what I appear to the world, -
Spectral Measures in Locally Convex Algebras by Helmut H
SPECTRAL MEASURES IN LOCALLY CONVEX ALGEBRAS BY HELMUT H. SCHAEFER The University of Michigan, U.S.A.Q) Introduction The central subject of classical spectral theory has been the spectral representation of self-adjoint and normal operators in Hilbert space; after Hilbert had given a complete treatment of the bounded case, the spectra] theory of unbounded self-adjoint and normal operators was developed by yon Neumann and others. (An excellent account can be found in [23].) Abstractions of algebras of Hermitian and normal operators were considered by various authors, notably Stone [22] who characterized such algebras as algebras of conti- nuous functions on a compact (Hausdorff) space. Investigations by Freudenthal [9] and Nakano [15] (especially papers 1 and 2), also leading to spectral theories, went in a different direction. Generalizations of unitary operators to reflexive Banach spaces were considered by Lorch [13]; Lorch also developed an operational calculus for those operators in reflexive Banach spaces that can be represented by a spectral measure [12]. (Taylor [24] developed such a calculus for closed operators on a Banach space whose spectrum does not cover the plane, but necessarily restricted to functions locally holomorphic on the spectrum. There are some recent results in this direction for operators on a locally convex space [26], [16].) The most extensive work on bounded and unbounded operators in a Banach space is due to Dunford, Schwartz, Bade and others (for a detailed bibliography, see [6] and [8]). Dunford considers operators that have a countably additive resolution of the identity, but may differ from a spectral operator (in the sense of Definition 4, Section 4 of this paper) by a quasi-nilpotent. -
Algebraic Cocompleteness and Finitary Functors
Logical Methods in Computer Science Volume 17, Issue 2, 2021, pp. 23:1–23:30 Submitted Feb. 27, 2020 https://lmcs.episciences.org/ Published May 28, 2021 ALGEBRAIC COCOMPLETENESS AND FINITARY FUNCTORS JIRˇ´I ADAMEK´ ∗ Department of Mathematics, Czech Technical University in Prague, Czech Republic Institute of Theoretical Computer Science, Technical University Braunschweig, Germany e-mail address: [email protected] Abstract. A number of categories is presented that are algebraically complete and cocomplete, i.e., every endofunctor has an initial algebra and a terminal coalgebra. Example: assuming GCH, the category Set≤λ of sets of power at most λ has that property, whenever λ is an uncountable regular cardinal. For all finitary (and, more generally, all precontinuous) set functors the initial algebra and terminal coalgebra are proved to carry a canonical partial order with the same ideal completion. And they also both carry a canonical ultrametric with the same Cauchy completion. Finally, all endofunctors of the category Set≤λ are finitary if λ has countable cofinality and there are no measurable cardinals µ ≤ λ. 1. Introduction The importance of terminal coalgebras for an endofunctor F was clearly demonstrated by Rutten [Rut00]: for state-based systems whose state-objects lie in a category K and whose dynamics are described by F , the terminal coalgebra νF collects behaviors of individual states. And given a system A the unique coalgebra homomorphism from A to νF assigns to every state its behavior. However, not every endofunctor has a terminal coalgebra. Analogously, an initial algebra µF need not exist. Freyd [Fre70] introduced the concept of an algebraically complete category: this means that every endofunctor has an initial algebra. -
Nakayama Categories and Groupoid Quantization
NAKAYAMA CATEGORIES AND GROUPOID QUANTIZATION FABIO TROVA Abstract. We provide a precise description, albeit in the situation of standard categories, of the quantization functor Sum proposed by D.S. Freed, M.J. Hopkins, J. Lurie, and C. Teleman in a way enough abstract and flexible to suggest that an extension of the construction to the general context of (1; n)-categories should indeed be possible. Our method is in fact based primarily on dualizability and adjunction data, and is well suited for the homotopical setting. The construction also sheds light on the need of certain rescaling automorphisms, and in particular on the nature and properties of the Nakayama isomorphism. Contents 1. Introduction1 1.1. Conventions4 2. Preliminaries4 2.1. Duals4 2.2. Mates and Beck-Chevalley condition5 3. A comparison map between adjoints7 3.1. Projection formulas8 3.2. Pre-Nakayama map 10 3.3. Tensoring Kan extensions and pre-Nakayama maps 13 4. Nakayama Categories 20 4.1. Weights and the Nakayama map 22 4.2. Nakayama Categories 25 5. Quantization 26 5.1. A monoidal category of local systems 26 5.2. The quantization functors 28 5.3. Siamese twins 30 6. Linear representations of groupoids 31 References 33 arXiv:1602.01019v1 [math.CT] 2 Feb 2016 1. Introduction A k-extended n-dimensional C-valued topological quantum field theory n−k (TQFT for short) is a symmetric monoidal functor Z : Bordn !C, from the (1; k)-category of n-cobordisms extended down to codimension k [CS, Lu1] to a symmetric monoidal (1; k)-category C. For k = 1 one recovers the classical TQFT's as introduced in [At], while k = n gives rise to fully Date: October 22, 2018. -
Professor ADDRESS: Department of Mathematics University of Florida
CURRICULUM VITAE NAME: Krishnaswami Alladi PRESENT POSITION: Professor ADDRESS: Department of Mathematics University of Florida Gainesville, Florida 32611 Ph: (352) 294-2290 e-mail: alladik@ufl.edu BORN: October 5, 1955 Trivandrum, India DEGREES: Ph.D., U.C.L.A. - 1978; Advisor-E.G. Straus M.A., U.C.L.A. - 1976 B.Sc., Madras University, India - 1975 RESEARCH INTERESTS: Number Theory - Analytic Number Theory, Sieve Methods, Probabilistic Number Theory, Diophantine Approximations, Partitions, q-hypergeometric identities PROFESSIONAL EXPERIENCE: T.H. Hildebrandt Research Assistant Professor, University of Michigan, Ann Arbor, Michigan, 1978-1981. Visiting Member, Institute for Advanced Study, Princeton, New Jersey, 1981-1982. Visiting Associate Professor, University of Texas, Austin, Texas, 1982-1983. Visiting Associate Professor, University of Hawaii, Honolulu, Hawaii, 1984-1985. Associate Professor, Institute of Mathematical Sciences, Madras, India, 1981-1986. Associate Professor, University of Florida, 1986-1989. Professor, University of Florida, 1989-present. Visiting Professor, Pennsylvania State University, University Park, Pennsylvania, in 1992-93 (on sabbatical from University of Florida) and in Fall 1994. Chairman, Department of Mathematics, University of Florida, 1998-2008 AWARDS AND HONORS: Chancellor's Fellowship for the Ph.D., 1975-78, UCLA. Alumni Medal for one of 5 best Ph.D. theses (all subjects) at UCLA in 1978. CLAS Research Awards at University of Florida thrice: 1987, 1994, 2000 TIP Award for distinguished teaching at the University of Florida in 1994-95. STEP/SPP Awards for distinguished performance in the rank of Full Professor at the University of Florida twice: 2001, 2010. Elected Fellow of the American Mathematical Society in 2012-13 during inaugural year of the Fellows Program of the AMS. -
Subspaces and Quotients of Topological and Ordered Vector Spaces
Zoran Kadelburg Stojan Radenovi´c SUBSPACES AND QUOTIENTS OF TOPOLOGICAL AND ORDERED VECTOR SPACES Novi Sad, 1997. CONTENTS INTRODUCTION::::::::::::::::::::::::::::::::::::::::::::::::::::: 1 I: TOPOLOGICAL VECTOR SPACES::::::::::::::::::::::::::::::: 3 1.1. Some properties of subsets of vector spaces ::::::::::::::::::::::: 3 1.2. Topological vector spaces::::::::::::::::::::::::::::::::::::::::: 6 1.3. Locally convex spaces :::::::::::::::::::::::::::::::::::::::::::: 12 1.4. Inductive and projective topologies ::::::::::::::::::::::::::::::: 15 1.5. Topologies of uniform convergence. The Banach-Steinhaus theorem 21 1.6. Duality theory ::::::::::::::::::::::::::::::::::::::::::::::::::: 28 II: SUBSPACES AND QUOTIENTS OF TOPOLOGICAL VECTOR SPACES ::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 39 2.1. Subspaces of lcs’s belonging to the basic classes ::::::::::::::::::: 39 2.2. Subspaces of lcs’s from some other classes :::::::::::::::::::::::: 47 2.3. Subspaces of topological vector spaces :::::::::::::::::::::::::::: 56 2.4. Three-space-problem for topological vector spaces::::::::::::::::: 60 2.5. Three-space-problem in Fr´echet spaces:::::::::::::::::::::::::::: 65 III: ORDERED TOPOLOGICAL VECTOR SPACES :::::::::::::::: 72 3.1. Basics of the theory of Riesz spaces::::::::::::::::::::::::::::::: 72 3.2. Topological vector Riesz spaces ::::::::::::::::::::::::::::::::::: 79 3.3. The basic classes of locally convex Riesz spaces ::::::::::::::::::: 82 3.4. l-ideals of topological vector Riesz spaces ::::::::::::::::::::::::: -
Internal Groupoids and Exponentiability
VOLUME LX-4 (2019) INTERNAL GROUPOIDS AND EXPONENTIABILITY S.B. Niefield and D.A. Pronk Resum´ e.´ Nous etudions´ les objets et les morphismes exponentiables dans la 2-categorie´ Gpd(C) des groupoides internes a` une categorie´ C avec sommes finies lorsque C est : (1) finiment complete,` (2) cartesienne´ fermee´ et (3) localement cartesienne´ fermee.´ Parmi les exemples auxquels on s’interesse´ on trouve, en particulier, (1) les espaces topologiques, (2) les espaces com- pactement engendres,´ (3) les ensembles, respectivement. Nous considerons´ aussi les morphismes pseudo-exponentiables dans les categories´ “pseudo- slice” Gpd(C)==B. Comme ces dernieres` sont les categories´ de Kleisli d’une monade T sur la categorie´ “slice” stricte sur B, nous pouvons appliquer un theor´ eme` gen´ eral´ de Niefield [17] qui dit que si T Y est exponentiable dans une 2-categorie´ K, alors Y est pseudo-exponentiable dans la categorie´ de Kleisli KT . Par consequent,´ nous verrons que Gpd(C)==B est pseudo- cartesienne´ fermee,´ lorsque C est la categorie´ des espaces compactement en- gendres´ et chaque Bi est faiblement de Hausdorff, et Gpd(C) est localement pseudo-cartesienne´ fermee´ quand C est la categorie´ des ensembles ou une categorie´ localement cartesienne´ fermee´ quelconque. Abstract. We study exponentiable objects and morphisms in the 2-category Gpd(C) of internal groupoids in a category C with finite coproducts when C is: (1) finitely complete, (2) cartesian closed, and (3) locally cartesian closed. The examples of interest include (1) topological spaces, (2) com- pactly generated spaces, and (3) sets, respectively. We also consider pseudo- exponentiable morphisms in the pseudo-slice categories Gpd(C)==B. -
Ann I J V 2 E 5 R S R
ANN I J V 2 E 5 R S R J A STUDENT NNEWSPAPERE W S 1 Z3 25 billion alumni backbone Art a memorial killed CPS They might be gone people can donate both money and of alumna but they certainly are not forgotten services By Laura Hosmer Alumni in feet are on the minds Weve seen a dramatic in- As friends of Susan Con well a of college administrators even more crease in the number of volunteers 1991 graduate of Ohio Wesleyan often these days playing an in- said Richard Tantillo director of deal with the shock and pain of her creasingly large role in keeping development at Rensselaer Poly- X murder many reflect upon the art affordable education alive and well technic Institute in Troy NY If work that Conwell left behind For during times of financial difficulty alumni are involved in another way some Conwell s work remains as a They are absolutely essen- first it makes them more passionate memorial to her tial Thats the backbone of it all when they get to fund- raising s According to an article that said Charlotte Heartt director of Volunteer services offered by appeared in the Columbus Dispatch development at Smith College in alumni at various schools include Conwell and a friend Robert Scheid Northampton Mass recruiting serving on legislative were killed in Helena Ark while Statistics agree The 1991 committees to lobby state govern- their boat was moored for repairs Voluntary Support of Education ments serving on college steering Conwell had been sexually assulted survey published by the Council committees and other boards and I Both were stabbed -
A Characterization of Concrete Quasitopoi by Injectivity
Journal of Pure and Applied Algebra 68 (1990) l-9 North-Holland A CHARACTERIZATION OF CONCRETE QUASITOPOI BY INJECTIVITY J. ADAMEK Faculty of Electrical Engineering, Suchbdtarova 2, Praha 6, Czechoslovakia H. HERRLICH FeldhGuser Str. 69, 2804 Lilienthal, FRG Received 9 June 1986 To our friend Bernhard Banaschewski, who demonstrates convincingly that-contrary to popular belief-it is possible to simultaneously get older, more creative and forceful as a mathematician and live a good life as well Concrete quasitopoi are characterized as the injectives among concrete, finitely complete categories. Quasitopos hulls are shown to be the injective hulls. Introduction Banaschewski and Bruns [5] have characterized Mac Neille completions of posets as injective hulls in the category Pos. This result has been generalized in [6], [7] and [l l] to the quasicategory Cat(X) of concrete categories over the base-category X (and concrete functors over X): injective ejects of Cat(X) are precisely the initially complete categories, and in- jective hulls are the Mac Neille completions introduced in [7]. Injectivity of K means (here and below) that for each full embedding A + B, which is a mor- phism in the given quasicategory (here Cat(X)), each morphism A+K can be extended to a morphism B -+ K. If X is the one-morphism category, then Cat(X) is the quasicategory of partially ordered classes, and hence, the latter result is a direct generalization of that in [5]. Recently, injectivity has been studied in the quasicategory Cat,(X) of concrete categories over X having finite, concrete [preserved by the forgetful functor] products (and concrete functors, preserving finite products).