2 Dimensional Categories
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2 Dimensional Categories
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Author : Niles Johnson
language : en
Publisher: Oxford University Press
Release Date : 2021-01-31
2 Dimensional Categories written by Niles Johnson and has been published by Oxford University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-01-31 with Science categories.
Category theory emerged in the 1940s in the work of Samuel Eilenberg and Saunders Mac Lane. It describes relationships between mathematical structures. Outside of pure mathematics, category theory is an important tool in physics, computer science, linguistics, and a quickly-growing list of other sciences. This book is about 2-dimensional categories, which add an extra dimension of richness and complexity to category theory. 2-Dimensional Categories is an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories, pasting diagrams, lax functors, 2-/bilimits, the Duskin nerve, 2-nerve, internal adjunctions, monads in bicategories, 2-monads, biequivalences, the Bicategorical Yoneda Lemma, and the Coherence Theorem for bicategories. Grothendieck fibrations and the Grothendieck construction are discussed next, followed by tricategories, monoidal bicategories, the Gray tensor product, and double categories. Completely detailed proofs of several fundamental but hard-to-find results are presented for the first time. With exercises and plenty of motivation and explanation, this book is useful for both beginners and experts.
Models Logics And Higher Dimensional Categories
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Author : Bradd T. Hart
language : en
Publisher: American Mathematical Soc.
Release Date :
Models Logics And Higher Dimensional Categories written by Bradd T. Hart and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on with Mathematics categories.
Proceedings of a conference held at Centre de recherches mathematiques of the Universite de Montreal, June 18-20, 2009.
Higher Dimensional Categories From Double To Multiple Categories
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Author : Marco Grandis
language : en
Publisher: World Scientific
Release Date : 2019-09-09
Higher Dimensional Categories From Double To Multiple Categories written by Marco Grandis and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-09-09 with Mathematics categories.
The study of higher dimensional categories has mostly been developed in the globular form of 2-categories, n-categories, omega-categories and their weak versions. Here we study a different form: double categories, n-tuple categories and multiple categories, with their weak and lax versions.We want to show the advantages of this form for the theory of adjunctions and limits. Furthermore, this form is much simpler in higher dimension, starting with dimension three where weak 3-categories (also called tricategories) are already quite complicated, much more than weak or lax triple categories.This book can be used as a textbook for graduate and postgraduate studies, and as a basis for research. Notions are presented in a 'concrete' way, with examples and exercises; the latter are endowed with a solution or hints. Part I, devoted to double categories, starts at basic category theory and is kept at a relatively simple level. Part II, on multiple categories, can be used independently by a reader acquainted with 2-dimensional categories.
Frobenius Algebras And 2 D Topological Quantum Field Theories
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Author : Joachim Kock
language : en
Publisher: Cambridge University Press
Release Date : 2004
Frobenius Algebras And 2 D Topological Quantum Field Theories written by Joachim Kock and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004 with Mathematics categories.
This 2003 book describes a striking connection between topology and algebra, namely that 2D topological quantum field theories are equivalent to commutative Frobenius algebras. The precise formulation of the theorem and its proof is given in terms of monoidal categories, and the main purpose of the book is to develop these concepts from an elementary level, and more generally serve as an introduction to categorical viewpoints in mathematics. Rather than just proving the theorem, it is shown how the result fits into a more general pattern concerning universal monoidal categories for algebraic structures. Throughout, the emphasis is on the interplay between algebra and topology, with graphical interpretation of algebraic operations, and topological structures described algebraically in terms of generators and relations. The book will prove valuable to students or researchers entering this field who will learn a host of modern techniques that will prove useful for future work.
Coherence In Three Dimensional Category Theory
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Author : Nick Gurski
language : en
Publisher: Cambridge University Press
Release Date : 2013-03-21
Coherence In Three Dimensional Category Theory written by Nick Gurski and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-21 with Mathematics categories.
Serves as an introduction to higher categories as well as a reference point for many key concepts in the field.
A Study Of Dynamic Stall Vortex Development Using Two Dimensional Data From The Afdd Oscillating Wing Experiment
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Author : Myung J. Rhee
language : en
Publisher:
Release Date : 2002
A Study Of Dynamic Stall Vortex Development Using Two Dimensional Data From The Afdd Oscillating Wing Experiment written by Myung J. Rhee and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002 with Aerodynamic load categories.
The purpose of this study is to examine the previously unpublished instantaneous pressure data of the Aeroflightdynamics Directorate Two-Dimensional (2D) and Three-Dimensional (3D) Oscillating Wing Experiment to better understand the process of dynamic stall vortex development on the NACA 0015 airfoil. This report presents representative 2D instantaneous pressure data for the upper and lower surfaces of the airfoil at various chordwise locations obtained at specific angles of attack during upstroke and downstroke cycles. Furthermore, the report contains a complete set of plots of instantaneous pressure distributions for the upper surface for all the 2D data sets obtained in the experiment. First, the lift, drag and pitching moment data of various testing conditions are reviewed and analyzed to classify the data both with and without a boundary layer trip into "no stall," "moderate stall," and "deep stall" data. Next, instantaneous pressure distributions on the upper surface of the airfoil are examined for the study of vortex development. The lift and pitching moment data are analyzed to document the dynamic overshoot which delays the development of the stall on the airfoil. Next, the range of angles of attack are selected where the lift and pitching moment data shows significant changes from unsteady flow behavior daring oscillation cycles. Furthermore, based on the unsteady flow characteristics found in each classification of dynamic stall, analysis is continued to identify the conditions where the reduced frequency clearly affects the unsteady flow behavior of the airfoil during the oscillation. This can result in a change of the dynamic stall classification of the airfoil response under various unsteady flow conditions. These conditions are discussed in detail in the comparative studies.
Stem The Battle Between 2 D And 3 D Shapes
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Author : Georgia Beth
language : en
Publisher: Triangle Interactive, Inc.
Release Date : 2022-01-21
Stem The Battle Between 2 D And 3 D Shapes written by Georgia Beth and has been published by Triangle Interactive, Inc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-01-21 with Juvenile Nonfiction categories.
Read about a high-stakes competition for a job as a game designer! Two young game designers-Zak and Posie-both want a job at Phenomtech. They'll need to convince Phenomtech's CEO that their idea for a new game is the better one. It's a battle between old-school 2-D games and up-and-coming 3-D virtual reality games. Which designer will prevail? Let the games begin! Students will be engaged in reading fiction content as they learn 2-D and 3-D shapes. This book seamlessly integrates the teaching of math and reading, and uses real-world examples to teach math concepts. Text features include images, a glossary, an index, captions, and a table of contents to build students' vocabulary and reading comprehension skills as they interact with the text. The rigorous practice problems, math charts and diagrams, and sidebars extend learning and provide multiple opportunities for students to practice what they have learned. Math Talk provides an in-depth problem-solving experience.
Stem The Battle Between 2 D And 3 D Shapes
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Author : Georgia Beth
language : en
Publisher: Teacher Created Materials
Release Date : 2018-01-02
Stem The Battle Between 2 D And 3 D Shapes written by Georgia Beth and has been published by Teacher Created Materials this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-01-02 with Juvenile Nonfiction categories.
Read about a high-stakes competition for a job as a game designer! Two young game designers-Zak and Posie-both want a job at Phenomtech. They'll need to convince Phenomtech's CEO that their idea for a new game is the better one. It's a battle between old-school 2-D games and up-and-coming 3-D virtual reality games. Which designer will prevail? Let the games begin! Students will be engaged in reading fiction content as they learn 2-D and 3-D shapes. This book seamlessly integrates the teaching of math and reading, and uses real-world examples to teach math concepts. Text features include images, a glossary, an index, captions, and a table of contents to build students' vocabulary and reading comprehension skills as they interact with the text. The rigorous practice problems, math charts and diagrams, and sidebars extend learning and provide multiple opportunities for students to practice what they have learned. Math Talk provides an in-depth problem-solving experience.
Stem The Battle Between 2 D And 3 D Shapes 6 Pack
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Author :
language : en
Publisher: Teacher Created Materials
Release Date : 2018-01-02
Stem The Battle Between 2 D And 3 D Shapes 6 Pack written by and has been published by Teacher Created Materials this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-01-02 with Juvenile Fiction categories.
In this exciting math reader, Vikram Patel wants to hire a game designer who will help his company create cutting-edge games. The first contender for the position is Zak, a master videogame designer and architect of shapes. The other competitor is Posie, an artist who is pushing the limits of virtual reality. Students will learn about 2-D and 3-D shapes as they are engaged in reading about this high-stakes competition. This 6-Pack of math readers builds math content knowledge and literacy skills, and uses real-world connections to help students explore math in a meaningful way. Text features such as a glossary, a table of contents, an index, and detailed illustrations will increase understanding and develop academic vocabulary. Let's Explore Math sidebars, the Problem Solving section, and the math charts and diagrams provide extensive opportunities for students to practice what they have learned. The DOK-leveled Math Talk section includes questions that facilitate mathematical discourse, and activities that students can respond to at home or school. This fiction title is sure to captivate readers as they are engaged in learning. This 6-Pack includes six copies of this title and a lesson plan.
Dualizable Tensor Categories
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Author : Christopher L. Douglas
language : en
Publisher: American Mathematical Soc.
Release Date : 2021-06-18
Dualizable Tensor Categories written by Christopher L. Douglas and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-06-18 with Education categories.
We investigate the relationship between the algebra of tensor categories and the topology of framed 3-manifolds. On the one hand, tensor categories with cer-tain algebraic properties determine topological invariants. We prove that fusion categories of nonzero global dimension are 3-dualizable, and therefore provide 3-dimensional 3-framed local field theories. We also show that all finite tensor cat-egories are 2-dualizable, and yield categorified 2-dimensional 3-framed local field theories. On the other hand, topological properties of 3-framed manifolds deter-mine algebraic equations among functors of tensor categories. We show that the 1-dimensional loop bordism, which exhibits a single full rotation, acts as the double dual autofunctor of a tensor category. We prove that the 2-dimensional belt-trick bordism, which unravels a double rotation, operates on any finite tensor category, and therefore supplies a trivialization of the quadruple dual. This approach pro-duces a quadruple-dual theorem for suitably dualizable objects in any symmetric monoidal 3-category. There is furthermore a correspondence between algebraic structures on tensor categories and homotopy fixed point structures, which in turn provide structured field theories; we describe the expected connection between piv-otal tensor categories and combed fixed point structures, and between spherical tensor categories and oriented fixed point structures.