homological algebra

This book introduces the notion of an effective Kan fibration, a new mathematical structure which can be used to study simplicial homotopy theory. The main motivation is to make simplicial homotopy theory suitable for homotopy type theory. Effective Kan fibrations are maps of simplicial sets equipped with a structured collection of chosen lifts that satisfy certain non-trivial properties. Here it is revealed that fundamental properties of ordinary Kan fibrations can be extended to explicit constructions on effective Kan fibrations. In particular, a constructive (explicit) proof is given that effective Kan fibrations are stable under push forward, or fibred exponentials. Further, it is shown that effective Kan fibrations are local, or completely determined by their fibres above representables, and the maps which can be equipped with the structure of an effective Kan fibration are precisely the ordinary Kan fibrations. Hence implicitly, both notions still describe the same homotopy theory. These new results solve an open problem in homotopy type theory and provide the first step toward giving a constructive account of Voevodsky’s model of univalent type theory in simplicial sets.

This book features a series of lectures that explores three different fields in which functor homology (short for homological algebra in functor categories) has recently played a significant role. For each of these applications, the functor viewpoint provides both essential insights and new methods for tackling difficult mathematical problems.In the lectures by Aurélien Djament, polynomial functors appear as coefficients in the homology of infinite families of classical groups, e.g. general linear groups or symplectic groups, and their stabilization. Djament’s theorem states that this stable homology can be computed using only the homology with trivial coefficients and the manageable functor homology. The series includes an intriguing development of Scorichenko’s unpublished results.The lectures by Wilberd van der Kallen lead to the solution of the general cohomological finite generation problem, extending Hilbert’s fourteenth problem and its solution to the context of cohomology. The focus here is on the cohomology of algebraic groups, or rational cohomology, and the coefficients are Friedlander and Suslin’s strict polynomial functors, a conceptual form of modules over the Schur algebra.Roman Mikhailov’s lectures highlight topological invariants: homotopy and homology of topological spaces, through derived functors of polynomial functors. In this regard the functor framework makes better use of naturality, allowing it to reach calculations that remain beyond the grasp of classical algebraic topology.Lastly, Antoine Touzé’s introductory course on homological algebra makes the book accessible to graduate students new to the field.The links between functor homology and the three fields mentioned above offer compelling arguments for pushing the development of the functor viewpoint. The lectures in this book will provide readers with a feel for functors, and a valuable new perspective to apply to their favourite problems.

This book uncovers mathematical structures underlying natural intelligence and applies category theory as a modeling language for understanding human cognition, giving readers new insights into the nature of human thought. In this context, the book explores various topics and questions, such as the human representation of the number system, why our counting ability is different from that which is evident among non-human organisms, and why the idea of zero is so difficult to grasp. The book is organized into three parts: the first introduces the general reason for studying general structures underlying the human mind; the second part introduces category theory as a modeling language and use it for exposing the deep and fascinating structures underlying human cognition; and the third applies the general principles and ideas of the first two parts to reaching a better understanding of challenging aspects of the human mind such as our understanding of the number system, the metaphorical nature of our thinking and the logic of our unconscious dynamics.