Author: Maria Cristina PedicchioPublisher: Cambridge University Press
ISBN: 9780521834148
Category : Mathematics
Languages : en
Pages : 452
Book Description
Publisher Description
Categorical Foundations
Author: Maria Cristina PedicchioPublisher: Cambridge University Press
ISBN: 9780521834148
Category : Mathematics
Languages : en
Pages : 452
Book Description
Publisher Description
Pseudo Limits, Biadjoints, and Pseudo Algebras: Categorical Foundations of Conformal Field Theory
Author: Thomas M. FiorePublisher: American Mathematical Soc.
ISBN: 0821839144
Category : Mathematics
Languages : en
Pages : 171
Book Description
In this paper we develop the categorical foundations needed for working out completely the rigorous approach to the definition of conformal field theory outlined by Graeme Segal. We discuss pseudo algebras over theories and 2-theories, their pseudo morphisms, bilimits, bicolimits, biadjoints, stacks, and related concepts. These 2-categorical concepts are used to describe the algebraic structure on the class of rigged surfaces. A rigged surface is a real, compact, not necessarily connected, two dimensional manifold with complex structure and analytically parametrized boundary components. This class admits algebraic operations of disjoint union and gluing as well as a unit. These operations satisfy axioms such as unitality and distributivity up to coherence isomorphisms which satisfy coherence diagrams.These operations, coherences, and their diagrams are neatly encoded as a pseudo algebra over the 2-theory of commutative monoids with cancellation. A conformal field theory is a morphism of stacks of such structures. This paper begins with a review of 2-categorical concepts, Lawvere theories, and algebras over Lawvere theories. We prove that the 2-category of small pseudo algebras over a theory admits weighted pseudo limits and weighted bicolimits. This 2-category is biequivalent to the 2-category of algebras over a 2-monad with pseudo morphisms. We prove that a pseudo functor admits a left biadjoint if and only if it admits certain biuniversal arrows.An application of this theorem implies that the forgetful 2-functor for pseudo algebras admits a left biadjoint. We introduce stacks for Grothendieck topologies and prove that the traditional definition of stacks in terms of descent data is equivalent to our definition via bilimits. The paper ends with a proof that the 2-category of pseudo algebras over a 2-theory admits weighted pseudo limits. This result is relevant to the definition of conformal field theory because bilimits are necessary to speak of stacks.
Categorical Logic and Type Theory
Author: B. JacobsPublisher: Gulf Professional Publishing
ISBN: 9780444508539
Category : Computers
Languages : en
Pages : 784
Book Description
This book is an attempt to give a systematic presentation of both logic and type theory from a categorical perspective, using the unifying concept of fibred category. Its intended audience consists of logicians, type theorists, category theorists and (theoretical) computer scientists.
Basic Category Theory for Computer Scientists
Author: Benjamin C. PiercePublisher: MIT Press
ISBN: 0262326450
Category : Computers
Languages : en
Pages : 117
Book Description
Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories. Category theory is a branch of pure mathematics that is becoming an increasingly important tool in theoretical computer science, especially in programming language semantics, domain theory, and concurrency, where it is already a standard language of discourse. Assuming a minimum of mathematical preparation, Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories. Four case studies illustrate applications of category theory to programming language design, semantics, and the solution of recursive domain equations. A brief literature survey offers suggestions for further study in more advanced texts. Contents Tutorial • Applications • Further Reading
Proceedings of the Conference on Categorical Algebra
Author: S. EilenbergPublisher: Springer Science & Business Media
ISBN: 3642999026
Category : Mathematics
Languages : en
Pages : 571
Book Description
This volume contains the articles contributed to the Conference on Categorical Algebra, held June 7-12,1965, at the San Diego campus of the University of California under the sponsorship of the United States Air Force Office of Scientific Research. Of the thirty-seven mathemati cians, who were present seventeen presented their papers in the form of lectures. In addition, this volume contains papers contributed by other attending participants as well as by those who, after having planned to attend, were unable to do so. The editors hope to have achieved a representative, if incomplete, cover age of the present activities in Categorical Algebra within the United States by bringing together this group of mathematicians and by solici ting the articles contained in this volume. They also hope that these Proceedings indicate the trend of research in Categorical Algebra in this country. In conclusion, the editors wish to thank the participants and contrib. utors to these Proceedings for their continuous cooperation and encour agement. Our thanks are also due to the Springer-Verlag for publishing these Proceedings in a surprisingly short time after receiving the manu scripts.
Practical Foundations of Mathematics
Author: Paul TaylorPublisher: Cambridge University Press
ISBN: 9780521631075
Category : Mathematics
Languages : en
Pages : 590
Book Description
This book is about the basis of mathematical reasoning both in pure mathematics itself (particularly algebra and topology) and in computer science (how and what it means to prove correctness of programs). It contains original material and original developments of standard material, so it is also for professional researchers, but as it deliberately transcends disciplinary boundaries and challenges many established attitudes to the foundations of mathematics, the reader is expected to be open minded about these things.
Category Theory in Context
Author: Emily RiehlPublisher: Courier Dover Publications
ISBN: 0486820807
Category : Mathematics
Languages : en
Pages : 272
Book Description
Introduction to concepts of category theory — categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads — revisits a broad range of mathematical examples from the categorical perspective. 2016 edition.
Sketches of an Elephant: A Topos Theory Compendium
Author: P. T. JohnstonePublisher: Oxford University Press
ISBN: 9780198515982
Category : Computers
Languages : en
Pages : 836
Book Description
Topos Theory is a subject that stands at the junction of geometry, mathematical logic and theoretical computer science, and it derives much of its power from the interplay of ideas drawn from these different areas. Because of this, an account of topos theory which approaches the subject from one particular direction can only hope to give a partial picture; the aim of this compendium is to present as comprehensive an account as possible of all the main approaches and to thereby demonstrate the overall unity of the subject. The material is organized in such a way that readers interested in following a particular line of approach may do so by starting at an appropriate point in the text.
Basic Category Theory
Author: Tom LeinsterPublisher: Cambridge University Press
ISBN: 1107044243
Category : Mathematics
Languages : en
Pages : 193
Book Description
A short introduction ideal for students learning category theory for the first time.
Topology
Author: Tai-Danae BradleyPublisher: MIT Press
ISBN: 0262359626
Category : Mathematics
Languages : en
Pages : 167
Book Description
A graduate-level textbook that presents basic topology from the perspective of category theory. This graduate-level textbook on topology takes a unique approach: it reintroduces basic, point-set topology from a more modern, categorical perspective. Many graduate students are familiar with the ideas of point-set topology and they are ready to learn something new about them. Teaching the subject using category theory--a contemporary branch of mathematics that provides a way to represent abstract concepts--both deepens students' understanding of elementary topology and lays a solid foundation for future work in advanced topics.