Focusing on the unifying power of Category Theory, this book explores its pervasive influence across various fields, including Mathematics, theoretical Computer Science, and theoretical Physics. It highlights how Category Theory connects different branches of study, fostering a deeper understanding of their foundational principles.
Coherence in homological algebra is examined through the lens of abelian groups and categories, highlighting the significance of lattices of subobjects and semigroups of endorelations. The book presents a 'Coherence Theorem' that emphasizes the importance of distributivity in ensuring consistency within homological systems, contrasting it with the potential inconsistencies in non-distributive structures like bifiltered chain complexes. Additionally, it introduces 'crossword chasing' as a simplified method for working with spectral sequences, offering a practical alternative to traditional algebraic approaches.
Focusing on a different approach to higher dimensional categories, this book explores double categories, n-tuple categories, and multiple categories, along with their weak and lax versions. It presents an alternative framework to the more commonly studied globular forms, providing insights into the structure and applications of these categories. This study aims to deepen the understanding of category theory by expanding the range of categorical forms considered in mathematical research.
Focusing on the unifying nature of Category Theory, this book serves as an essential resource for students and researchers in mathematics, theoretical computer science, and physics. It provides a comprehensive introduction to fundamental concepts such as universal properties, limits, adjoint functors, and monads. Designed for those new to the subject, it can effectively be used as a textbook for an introductory course, facilitating a deeper understanding of the interconnectedness of various mathematical branches.
Algebraic Topology is a system and strategy of partial translations, aiming to reduce difficult topological problems to algebraic facts that can be more easily solved. The main subject of this book is singular homology, the simplest of these translations. Studying this theory and its applications, we also investigate its underlying structural layout - the topics of Homological Algebra, Homotopy Theory and Category Theory which occur in its foundation. This book is an introduction to a complex domain, with references to its advanced parts and ramifications. It is written with a moderate amount of prerequisites - basic general topology and little else - and a moderate progression starting from a very elementary beginning. A consistent part of the exposition is organised in the form of exercises, with suitable hints and solutions. It can be used as a textbook for a semester course or self-study, and a guidebook for further study.