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(Co)end Calculus
The language of ends and (co)ends provides a natural and general way of expressing many phenomena in category theory, in the abstract and in applications. Yet although category-theoretic methods are now widely used by mathematicians, since (co)ends lie just beyond a first course in category theory, they are typically only used by category theorists, for whom they are something of a secret weapon. This book is the first systematic treatment of the theory of (co)ends. Aimed at a wide audience, it presents the (co)end calculus as a powerful tool to clarify and simplify definitions and results in category theory and export them for use in diverse areas of mathematics and computer science. It is organised as an easy-to-cite reference manual, and will be of interest to category theorists and users of category theory alike.
Starting Category Theory
One of the central highlights of this work is the exploration of the Yoneda lemma and its profound implications, during which intuitive explanations are provided, as well as detailed proofs, and specific examples. This book covers aspects of category theory often considered advanced in a clear and intuitive way, with rigorous mathematical proofs. It investigates universal properties, coherence, the relationship between categories and graphs, and treats monads and comonads on an equal footing, providing theorems, interpretations and concrete examples. Finally, this text contains an introduction to monoidal categories and to strong and commutative monads, which are essential tools in current r...
(Co)end Calculus
This easy-to-cite handbook gives the first systematic treatment of the (co)end calculus in category theory and its applications.
Graph Transformation
This book constitutes the refereed proceedings of the 14th International Conference on Graph Transformation, ICGT 2021, which took place virtually during June 24-25, 2021. The 14 full papers and 2 tool papers presented in this book were carefully reviewed and selected from 26 submissions. They deal with the following topics: theoretical advances; application domains; and tool presentations.
Category Theory in Physics, Mathematics, and Philosophy
The contributions gathered here demonstrate how categorical ontology can provide a basis for linking three important basic sciences: mathematics, physics, and philosophy. Category theory is a new formal ontology that shifts the main focus from objects to processes. The book approaches formal ontology in the original sense put forward by the philosopher Edmund Husserl, namely as a science that deals with entities that can be exemplified in all spheres and domains of reality. It is a dynamic, processual, and non-substantial ontology in which all entities can be treated as transformations, and in which objects are merely the sources and aims of these transformations. Thus, in a rather surprising way, when employed as a formal ontology, category theory can unite seemingly disparate disciplines in contemporary science and the humanities, such as physics, mathematics and philosophy, but also computer and complex systems science.
Category Theory in Context
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.
Elements of ?-Category Theory
This book develops the theory of infinite-dimensional categories by studying the universe, or ∞-cosmos, in which they live.
Geometry of Characteristic Classes
Characteristic classes are central to the modern study of the topology and geometry of manifolds. They were first introduced in topology, where, for instance, they could be used to define obstructions to the existence of certain fiber bundles. Characteristic classes were later defined (via the Chern-Weil theory) using connections on vector bundles, thus revealing their geometric side. In the late 1960s new theories arose that described still finer structures. Examples of the so-called secondary characteristic classes came from Chern-Simons invariants, Gelfand-Fuks cohomology, and the characteristic classes of flat bundles. The new techniques are particularly useful for the study of fiber bundles whose structure groups are not finite dimensional. The theory of characteristic classes of surface bundles is perhaps the most developed. Here the special geometry of surfaces allows one to connect this theory to the theory of moduli space of Riemann surfaces, i.e., Teichmüller theory. In this book Morita presents an introduction to the modern theories of characteristic classes.
