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Author: Saunders Mac Lane
Publisher: Springer Science & Business Media
ISBN: 1475747217
Category : Mathematics
Languages : en
Pages : 320
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Book Description
An array of general ideas useful in a wide variety of fields. Starting from the foundations, this book illuminates the concepts of category, functor, natural transformation, and duality. It then turns to adjoint functors, which provide a description of universal constructions, an analysis of the representations of functors by sets of morphisms, and a means of manipulating direct and inverse limits. These categorical concepts are extensively illustrated in the remaining chapters, which include many applications of the basic existence theorem for adjoint functors. The categories of algebraic systems are constructed from certain adjoint-like data and characterised by Beck's theorem. After considering a variety of applications, the book continues with the construction and exploitation of Kan extensions. This second edition includes a number of revisions and additions, including new chapters on topics of active interest: symmetric monoidal categories and braided monoidal categories, and the coherence theorems for them, as well as 2-categories and the higher dimensional categories which have recently come into prominence.
Author: Saunders Mac Lane
Publisher: Springer Science & Business Media
ISBN: 1475747217
Category : Mathematics
Languages : en
Pages : 320
Get Book Here
Book Description
An array of general ideas useful in a wide variety of fields. Starting from the foundations, this book illuminates the concepts of category, functor, natural transformation, and duality. It then turns to adjoint functors, which provide a description of universal constructions, an analysis of the representations of functors by sets of morphisms, and a means of manipulating direct and inverse limits. These categorical concepts are extensively illustrated in the remaining chapters, which include many applications of the basic existence theorem for adjoint functors. The categories of algebraic systems are constructed from certain adjoint-like data and characterised by Beck's theorem. After considering a variety of applications, the book continues with the construction and exploitation of Kan extensions. This second edition includes a number of revisions and additions, including new chapters on topics of active interest: symmetric monoidal categories and braided monoidal categories, and the coherence theorems for them, as well as 2-categories and the higher dimensional categories which have recently come into prominence.
Author: Nick Gurski
Publisher: Cambridge University Press
ISBN: 1107034892
Category : Mathematics
Languages : en
Pages : 287
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Book Description
Serves as an introduction to higher categories as well as a reference point for many key concepts in the field.
Author: Niles Johnson
Publisher: Oxford University Press (UK)
ISBN: 0198871376
Category : Computers
Languages : en
Pages : 636
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Book Description
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.
Author: Robert Gordon
Publisher: American Mathematical Soc.
ISBN: 0821803441
Category : Mathematics
Languages : en
Pages : 94
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Book Description
This work defines the concept of tricategory as the natural 3-dimensional generalization of bicategory. Trihomomorphism and triequivalence for tricategories are also defined so as to extend the concepts of homomorphism and biequivalence for bicategories.
Author: Tom Leinster
Publisher: Cambridge University Press
ISBN: 0521532159
Category : Mathematics
Languages : en
Pages : 451
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Book Description
Foundations of higher dimensional category theory for graduate students and researchers in mathematics and mathematical physics.
Author: Michael Fullan
Publisher: Corwin Press
ISBN: 150630172X
Category : Education
Languages : en
Pages : 114
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Book Description
Complex times call for clear solutions—If initiative overload and fragmentation are keeping your best plans from becoming reality, it’s time to start leading differently. The key to bringing about the kind of successful and sustainable change you need is the Coherence Framework, a dynamic, customizable road map made up of four essential components: Focused direction to build collective purpose Cultivating collaborative cultures while clarifying individual and team roles Deepening learning to accelerate improvement and foster innovation Securing accountability from the inside out Coherence provides the insights and tools to drive effective leadership. Now you can gain a deeper understanding of Coherence with The Taking Action Guide to Building Coherence in Schools, Districts, and Systems. Coherence is a book that demands action – it moves from the narrative of fixing one teacher at a time, to asking about the coherence of the system (be it school, national, or world issues). Fullan and Quinn create an important narrative about direction, working together, deepening learning, and securing accountability. The book sparkles with examples of coherence in action, it makes no excuses for employing the wrong levers of change. This is the blueprint for a new vocabulary of education action; it shows where we need to go next, and is another example of Fullan at the top of his game. John Hattie Director, Melbourne Education Research Institute and Author of Visible Learning "School systems that struggle are riddled with incoherence—mismatched strategies, competing cultures, and illogical initiatives. Fullan and Quinn explain clearly how coherence can solve the problem. Based on solid research and lessons drawn from effective practice, Coherence provides a comprehensive model to guide educators as they learn and lead their way to better schools." Susan Moore Johnson Jerome T. Murphy Research Professor Harvard Graduate School of Education
Author: Alan Watkins
Publisher: Kogan Page Publishers
ISBN: 0749470062
Category : Business & Economics
Languages : en
Pages : 352
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Book Description
Every business leader faces innumerable challenges every working day, each one taking their toll on precious energy levels and the ability to respond and react positively in a commercial environment. Coherence recognizes the key factors that take their toll on a leader's effectiveness and ability to lead, and provides the reader with unique solutions designed to improve physiological factors that impact on core competencies. Problems today cannot be solved with yesterday's level of thinking. CEOs fail and leaders burn out because our thinking has not sped up or powered up. The author not only recognizes that leaders have the potential for limitless processing power, but shows them how to access it, taking them back to fundamentals and, quite literally, to the heart of who we are and how we function successfully. By showing leaders how to be 'younger, smarter, healthier and happier' Coherence gives every decision maker the power to make influential decisions under pressure and achieve sustainable success at every level.
Author: Emily Riehl
Publisher: Courier Dover Publications
ISBN: 0486820807
Category : Mathematics
Languages : en
Pages : 273
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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.
Author: Paul Thagard
Publisher: MIT Press
ISBN: 9780262700924
Category : Psychology
Languages : en
Pages : 332
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Book Description
This book is an essay on how people make sense of each other and the world they live in. Making sense is the activity of fitting something puzzling into a coherent pattern of mental representations that include concepts, beliefs, goals, and actions. Paul Thagard proposes a general theory of coherence as the satisfaction of multiple interacting constraints, and discusses the theory's numerous psychological and philosophical applications. Much of human cognition can be understood in terms of coherence as constraint satisfaction, and many of the central problems of philosophy can be given coherence-based solutions. Thagard shows how coherence can help to unify psychology and philosophy, particularly when addressing questions of epistemology, metaphysics, ethics, politics, and aesthetics. He also shows how coherence can integrate cognition and emotion.
Author: Donald Yau
Publisher: Springer Nature
ISBN: 3030612031
Category : Mathematics
Languages : en
Pages : 250
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Book Description
This monograph introduces involutive categories and involutive operads, featuring applications to the GNS construction and algebraic quantum field theory. The author adopts an accessible approach for readers seeking an overview of involutive category theory, from the basics to cutting-edge applications. Additionally, the author’s own recent advances in the area are featured, never having appeared previously in the literature. The opening chapters offer an introduction to basic category theory, ideal for readers new to the area. Chapters three through five feature previously unpublished results on coherence and strictification of involutive categories and involutive monoidal categories, showcasing the author’s state-of-the-art research. Chapters on coherence of involutive symmetric monoidal categories, and categorical GNS construction follow. The last chapter covers involutive operads and lays important coherence foundations for applications to algebraic quantum field theory. With detailed explanations and exercises throughout, Involutive Category Theory is suitable for graduate seminars and independent study. Mathematicians and mathematical physicists who use involutive objects will also find this a valuable reference.
