Author: Jiri Adamek,Jiří Adámek (ing.),Horst Herrlich,George E. Strecker
This up-to-date introductory treatment employs the language of category theory to explore the theory of structures. Its unique approach stresses concrete categories, and each categorical notion features several examples that clearly illustrate specific and general cases. A systematic view of factorization structures, this volume contains seven chapters. The first five focus on basic theory, and the final two explore more recent research results in the realm of concrete categories, cartesian closed categories, and quasitopoi. Suitable for advanced undergraduate and graduate students, it requires an elementary knowledge of set theory and can be used as a reference as well as a text. Updated by the authors in 2004, it offers a unifying perspective on earlier work and summarizes recent developments.
Healthy ageing can lead to declines in both perceptual and cognitive functions. Impaired perception, such as that resulting from hearing loss or reduced visual or tactile resolution, increases demands on ‘higher-level’ cognitive functions to cope or compensate. It is possible, for example, to use focused attention to overcome perceptual limitations. Unfortunately, cognitive functions also decline in old age. This can mean that perceptual impairments are exacerbated by cognitive decline, and vice versa, but also means that interventions aimed at one type of decline can lead to improvements in the other. Just as improved cognition can ameliorate perceptual deficits, improving the stimulus can help offset cognitive deficits. For example, making directions and routes easy to follow can help compensate for declines in navigation abilities. In this Topic, we bring together papers from both auditory and visual researchers that address the interaction between perception and cognition in the ageing brain. Many of the studies demonstrate that a broadening of representations or increased reliance on gist underlie perceptual and cognitive age-related declines. There is also clear evidence that impaired perception is associated with poor cognition although, encouragingly, it can also be seen that good perception is associated with better cognition. Compensatory cognitive strategies were less successful in improving perception than might be expected. We also present papers which highlight important methodological considerations that are required when studying the older brain.
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.
Methods in Cognitive Linguistics is an introduction to empirical methodology for language researchers. Intended as a handbook to exploring the empirical dimension of the theoretical questions raised by Cognitive Linguistics, the volume presents guidelines for employing methods from a variety of intersecting disciplines, laying out different ways of gathering empirical evidence. The book is divided into five sections. Methods and Motivations provides the reader with the preliminary background in scientific methodology and statistics. The sections on Corpus and Discourse Analysis, and Sign Language and Gesture describe different ways of investigating usage data. Behavioral Research describes methods for exploring mental representation, simulation semantics, child language development, and the relationships between space and language, and eye movements and cognition. Lastly, Neural Approaches introduces the reader to ERP research and to the computational modeling of language.
First the concepts of [lambda]-presentable objects, locally [lambda]-presentable categories, and [lambda]-accessible categories are discussed in detail. The authors go on to prove that Freyd's essentially algebraic categories are precisely the locally presentable categories. In the final chapter they treat some advanced topics in model theory. For researchers in category theory, algebra, computer science, and model theory, this book will be a necessary purchase.
The series, founded in 1970, publishes works which either combine studies in the history of philosophy with a systematic approach or bring together systematic studies with reconstructions from the history of philosophy. Monographs are published in English as well as in German. The founding editors are Erhard Scheibe (editor until 1991), Günther Patzig (until 1999) and Wolfgang Wieland (until 2003). From 1990 to 2007, the series had been co-edited by Jürgen Mittelstraß.
With Applications to Wreath Products and Graphs. A Handbook for Students and Researchers
Author: Mati Kilp,Ulrich Knauer,Alexander V. Mikhalev
Publisher: Walter de Gruyter
The aim of the series is to present new and important developments in pure and applied mathematics. Well established in the community over two decades, it offers a large library of mathematics including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers wishing to thoroughly study the topic. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany
Categories for the Working Mathematician begins with foundations, illuminating concepts such as category, functor, natural transformation, and duality. It then continues by extensively illustrating these categorical concepts while presenting applications to more advanced topics. This second edition includes many revisions and additions.
The Interplay of Homology with Distributive Lattices and Orthodox Semigroups
Author: Marco Grandis
Publisher: World Scientific
In this book we want to explore aspects of coherence in homological algebra, that already appear in the classical situation of abelian groups or abelian categories. Lattices of subobjects are shown to play an important role in the study of homological systems, from simple chain complexes to all the structures that give rise to spectral sequences. A parallel role is played by semigroups of endorelations. These links rest on the fact that many such systems, but not all of them, live in distributive sublattices of the modular lattices of subobjects of the system. The property of distributivity allows one to work with induced morphisms in an automatically consistent way, as we prove in a 'Coherence Theorem for homological algebra'. (On the contrary, a 'non-distributive' homological structure like the bifiltered chain complex can easily lead to inconsistency, if one explores the interaction of its two spectral sequences farther than it is normally done.) The same property of distributivity also permits representations of homological structures by means of sets and lattices of subsets, yielding a precise foundation for the heuristic tool of Zeeman diagrams as universal models of spectral sequences. We thus establish an effective method of working with spectral sequences, called 'crossword chasing', that can often replace the usual complicated algebraic tools and be of much help to readers that want to apply spectral sequences in any field.
Expert treatment introduces semi-Riemannian geometry and its principal physical application, Einstein's theory of general relativity, using the Cartan exterior calculus as a principal tool. Prerequisites include linear algebra and advanced calculus. 2012 edition.
This helpful "bridge" book offers students the foundations they need to understand advanced mathematics. The two-part treatment provides basic tools and covers sets, relations, functions, mathematical proofs and reasoning, more. 1975 edition.