Categories
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Theory of Categories
Categorization is an essential and unavoidable instrumentality for conceptually navigating a world—indeed for being able to conceptualize a world to be navigated. Classification is a pivotal instrument for scientific systemization, featured as a basis for the philosophical understanding of reality since Aristotle, but classificatory concepts of sorts, types and natural kinds inevitably pervade our understanding of ourselves and our position in the social as well as the natural world at all levels. The authors argue that the character, purpose-, context-, and culture-relativity of categories and categorization have been widely misunderstood—that standard philosophical views are substantially correct in some respects but markedly mistaken in others. The book offers a comprehensive survey of basic principles of classification and categorization, a survey of relevant empirical work, and a multitude of illustrative examples accompanied by instructive analysis of ways and means. The work traces wide-ranging implications of the current approach for philosophical problematic and paradox in philosophy of mind, epistemology and metaphysics, philosophy of science, social philosophy and ethics.
The Homotopy Theory of (?,1)-Categories
Author: Julia E. Bergner
language: en
Publisher: Cambridge University Press
Release Date: 2018-03-15
An introductory treatment to the homotopy theory of homotopical categories, presenting several models and comparisons between them.
Tensor Categories
Author: Pavel Etingof
language: en
Publisher: American Mathematical Soc.
Release Date: 2016-08-05
Is there a vector space whose dimension is the golden ratio? Of course not—the golden ratio is not an integer! But this can happen for generalizations of vector spaces—objects of a tensor category. The theory of tensor categories is a relatively new field of mathematics that generalizes the theory of group representations. It has deep connections with many other fields, including representation theory, Hopf algebras, operator algebras, low-dimensional topology (in particular, knot theory), homotopy theory, quantum mechanics and field theory, quantum computation, theory of motives, etc. This book gives a systematic introduction to this theory and a review of its applications. While giving a detailed overview of general tensor categories, it focuses especially on the theory of finite tensor categories and fusion categories (in particular, braided and modular ones), and discusses the main results about them with proofs. In particular, it shows how the main properties of finite-dimensional Hopf algebras may be derived from the theory of tensor categories. Many important results are presented as a sequence of exercises, which makes the book valuable for students and suitable for graduate courses. Many applications, connections to other areas, additional results, and references are discussed at the end of each chapter.