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.

Throughout its history as a social science, demography has been associated with an exclusively quantitative orientation for studying social problems. As a result, demographers tend to analyse population issues scientifically through sets of fixed social categories that are divorced from dynamic relationships and local contexts and processes. This volume questions these fixed categories in two ways. First, it examines the historical and political circumstances in which such categories had their provenance, and, second, it reassesses their uncritical applications over space and time in a diverse range of empirical case studies, encouraging throughout a constructive interdisciplinary dialogue involving anthropologists, demographers, historians, and sociologists. This volume seeks to examine the political complexities that lie at the heart of population studies by focusing on category formation, category use, and category critique. It shows that this takes the form of a dialectic between the needs for clarity of scientific and administrative analysis and the recalcitrant diversity of the social contexts and human processes that generate population change. The critical reflections of each chapter are enriched by meticulous ethnographic fieldwork and historical research drawn from every continent. This volume, therefore, exemplifies a new methodology for research in population studies, one that does not simply accept and re-use the established categories of population science but seeks critically and reflexively to explore, test, and re-evaluate their meanings in diverse contexts. It shows that for demography to realise its full potential it must urgently re-examine and contextualize the social categories used today in population research.

Mood, modality and evidentiality are popular and dynamic areas in linguistics. Re-Assessing Modalising Expressions – Categories, co-text, and context focuses on the specific issue of the ways language users express permission, obligation, volition (intention), possibility and ability, necessity and prediction linguistically. Using a range of evidence and corpus data collected from different sources, the authors of this volume examine the distribution and functions of a range of patterns involving modalising expressions as predominantly found in standard American English, British English or Hong Kong English, but also in Japanese. The authors are particularly interested in addressing (co-)textual manifestations of modalising expressions as well as their distribution across different text-types and thus filling a gap research was unable to plug in the past. Thoughts on categorising or re-categorising modalising expressions initiate and complement a multi-perspectival enterprise that is intended to bring research in this area a step forward.

To make sense of the world, we’re always trying to place things in context, whether our environment is physical, cultural, or something else altogether. Now that we live among digital, always-networked products, apps, and places, context is more complicated than ever—starting with "where" and "who" we are. This practical, insightful book provides a powerful toolset to help information architects, UX professionals, and web and app designers understand and solve the many challenges of contextual ambiguity in the products and services they create. You’ll discover not only how to design for a given context, but also how design participates in making context. Learn how people perceive context when touching and navigating digital environments See how labels, relationships, and rules work as building blocks for context Find out how to make better sense of cross-channel, multi-device products or services Discover how language creates infrastructure in organizations, software, and the Internet of Things Learn models for figuring out the contextual angles of any user experience

This book is an attempt to give a systematic presentation of both logic and type theory from a categorical perspective, using the unifying concept of fibred category. Its intended audience consists of logicians, type theorists, category theorists and (theoretical) computer scientists.

In the past decade community groups have been portrayed as the solution to many social problems. Yet the role of ‘below the regulatory radar’ community action has received little research attention and thus is poorly understood in terms of both policy and practice. Focusing on self-organised community activity, this book offers the first collection of papers developing theoretical and empirically grounded knowledge of the informal, unregistered, yet largest, part of the voluntary sector. The collection includes work from leading academics, activists, policy makers and practitioners offering a new and coherent understanding of community action ‘below the radar’. The book is part of the Third Sector Research Series which is informed by research undertaken at the Third Sector Research Centre, funded by the Economic and Social Research Council and Barrow Cadbury Trust.

An array of general ideas useful in a wide variety of fields. Starting from the foundations, this book illuminates the concepts of category, functor, natural transformation, and duality. It then turns to adjoint functors, which provide a description of universal constructions, an analysis of the representations of functors by sets of morphisms, and a means of manipulating direct and inverse limits. These categorical concepts are extensively illustrated in the remaining chapters, which include many applications of the basic existence theorem for adjoint functors. The categories of algebraic systems are constructed from certain adjoint-like data and characterised by Beck's theorem. After considering a variety of applications, the book continues with the construction and exploitation of Kan extensions. This second edition includes a number of revisions and additions, including new chapters on topics of active interest: symmetric monoidal categories and braided monoidal categories, and the coherence theorems for them, as well as 2-categories and the higher dimensional categories which have recently come into prominence.

Despite the wealth of empirical research currently available on the interrelationships of gender and labor, we still know comparatively little about the forms of classification and categorization that have helped shape these social phenomena over time. Categories in Context seeks to enrich our understanding of how cognitive categories such as status, law, and rights have been produced, comprehended, appropriated, and eventually transformed by relevant actors. By focusing on specific developments in France and Germany through a transnational lens, this volume produces insights that can be applied to a wide variety of political, social, and historical contexts.

This book develops abstract homotopy theory from the categorical perspective with a particular focus on examples. Part I discusses two competing perspectives by which one typically first encounters homotopy (co)limits: either as derived functors definable when the appropriate diagram categories admit a compatible model structure, or through particular formulae that give the right notion in certain examples. Emily Riehl unifies these seemingly rival perspectives and demonstrates that model structures on diagram categories are irrelevant. Homotopy (co)limits are explained to be a special case of weighted (co)limits, a foundational topic in enriched category theory. In Part II, Riehl further examines this topic, separating categorical arguments from homotopical ones. Part III treats the most ubiquitous axiomatic framework for homotopy theory - Quillen's model categories. Here, Riehl simplifies familiar model categorical lemmas and definitions by focusing on weak factorization systems. Part IV introduces quasi-categories and homotopy coherence.