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A Handbook of Categorical Algebra, in three volumes, is a detailed account of everything a mathematician needs to know aboutcategory theory. Each volume is self-contained and is accessible to graduate students with a good background in mathematics.Volume 1 is devoted to general concepts.After introducing the terminology and proving the fundamental results concerning limits, adjoint functors and Kan extensions, the categories of fractions are studied in detail; special consideration is paid to the case of localizations. The remainder of the first volume studies various "refinements" of the fundamental concepts of category and functor. ... Read more

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Algebraic theories, introduced as a concept in the 1960s, have been a fundamental step towards a categorical view of general algebra. Moreover, they have proved very useful in various areas of mathematics and computer science. This carefully developed book gives a systematic introduction to algebra based on algebraic theories that is accessible to both graduate students and researchers. It will facilitate interactions of general algebra, category theory and computer science. A central concept is that of sifted colimits - that is, those commuting with finite products in sets. The authors prove the duality between algebraic categories and algebraic theories and discuss Morita equivalence between algebraic theories. They also pay special attention to one-sorted algebraic theories and the corresponding concrete algebraic categories over sets, and to S-sorted algebraic theories, which are important in program semantics. The final chapter is devoted to finitary localizations of algebraic categories, a recent research area. ... Read more

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Aimed at starting researchers in the field, Realizability gives a rigorous, yet reasonable introduction to the basic concepts of a field which has passed several successive phases of abstraction. Material from previously unpublished sources such as Ph.D. theses, unpublished papers, etc. has been molded into one comprehensive presentation of the subject area.

- The first book to date on this subject area
- Provides an clear introduction to Realizability with a comprehensive bibliography
- Easy to read and mathematically rigorous
- Written by an expert in the field ... Read more

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A monograph about the effective topos
This book is intended to provide advanced undergraduate students and Ph.D. students with a concise introduction to the effective topos. Contains introductions to partial combinatory algebras, tripos theory, construction of the effective topos. Synthesizes the existing literature, making a lot of material accessible to all readers. Many applications to recursive analysis, intuitionistic set theory, synthetic domain theory. Variations are also developed, such as Lifschitz realizability, Modified realizability, Kleene-Vesley realizability. ... Read more

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This book investigates propositional intuitionistic and modallogics from an entirely new point of view, covering quite recent andsometimes yet unpublished results. It mainly deals with the structureof the category of finitely presented Heyting and modal algebras,relating it both with proof theoretic and model theoretic facts:existence of model completions, amalgamability, Beth definability,interpretability of second order quantifiers and uniforminterpolation, definability of dual connectives like difference,projectivity, etc. are among the numerous topics which are covered.Dualities and sheaf representations are the main techniques in thebook, together with Ehrenfeucht-Fraisse games and boundedbisimulations. The categorical instruments employed are rich, but aspecific extended Appendix explains to the reader all concepts used inthe text, starting from the very basic definitions to what is neededfrom topos theory. Audience: The book is addressed to a large spectrum ofprofessional logicians, from such different areas as modal logics,categorical and algebraic logic, model theory and universal algebra. ... Read more

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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.
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Excellent book.
Excellent book. The best in its field. I would recommend it, particularly for students. ... Read more

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Goguen categories extend the relational calculus and its categorical formalization to the fuzzy world. Starting from the fundamental concepts of sets, binary relations and lattices, this book introduces several categorical formulations of an abstract theory of relations such as allegories, Dedekind categories and related structures. It is shown that neither theory is sufficiently rich to describe basic operations on fuzzy relations.

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Researchers, teachers and graduate students in algebra and topology--familiar with the very basic notions of category theory--will welcome this categorical introduction to some of the key areas of modern mathematics, without being forced to study category theory. Rather, each of the eight largely independent chapters analyzes a particular subject, revealing the power and applicability of the categorical foundations in each case. ... Read more

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This book investigates propositional intuitionistic and modal logics from an entirely new point of view, covering quite recent and sometimes yet unpublished results. It mainly deals with the structure of the category of finitely presented Heyting and modal algebras, relating it both with proof theoretic and model theoretic facts: existence of model completions, amalgamability, Beth definability, interpretability of second order quantifiers and uniform interpolation, definability of dual connectives like difference, projectivity, etc. are among the numerous topics which are covered. Dualities and sheaf representations are the main techniques in the book, together with Ehrenfeucht-Fraissé games and bounded bisimulations. The categorical instruments employed are rich, but a specific extended Appendix explains to the reader all concepts used in the text, starting from the very basic definitions to what is needed from topos theory.

Audience: The book is addressed to a large spectrum of professional logicians, from such different areas as modal logics, categorical and algebraic logic, model theory and universal algebra.

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This book presents the general theory of categorical closure operators together with examples and applications to the most common categories, such as topological spaces, fuzzy topological spaces, groups, abelian and topological groups. The main aim of the theory, whose origin dates back to the 1980s, is to develop a categorical characterization of the classical basic concepts in topology via the newly introduced concept of categorical closure operator. This permits many topological ideas to be introduced in a topology-free environment and imported afterwards into a new category, which often yields interesting new insights into their structures.

The first part of the book deals with the general theory, starting with basic definitions and gradually moving to more advanced properties. The second part includes applications to the classical concepts of epimorphisms, separation, compactness and connectedness. Every chapter ends with exercises. A comprehensive list of references for the reader who wants to consult original works, and a good index complete the book.

Categorical Closure Operators is self-contained and can be considered as a graduate level text for topics courses in algebra, topology or category theory. The book appeals mainly to graduate students and researchers in categorical topology, and to those interested in categorical methods applied to the most common concrete categories. The reader is expected to have some basic knowledge of algebra, topology and category theory, however, all recurrent categorical concepts are included in a preliminary chapter. ... Read more

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This volume contains carefully selected and refereed paperspresented at the International Workshop on Categorical Topology,held at the University of L'Aquila, L'Aquila, Italy from August 31 toSeptember 4, 1994.
This collection represents a wide range of current developments in thefield, and will be of interest to mathematicians whose work involvescategory theory. ... Read more

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This volume contains selected papers of the International Workshop on "Categorical Methods in Computer Science - with Aspects from Topology" and of the "6th International Data Type Workshop" held in August/September 1988 in Berlin. The 23 papers of this volume are grouped into three parts: Part 1 includes papers on categorical foundations and fundamental concepts from category theory in computer science. Part 2 presents applications of categorical methods to algebraic specification languages and techniques, data types, data bases, programming, and process specifications. Part 3 comprises papers on categorial aspects from topology which mainly concentrate on special adjoint situations like cartesian closeness, Galois connections, reflections, and coreflections which are of growing interest in categorical topology and computer science. ... Read more

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From a Geometrical Point of View explores historical and philosophical aspects of category theory, trying therewith to expose its significance in the mathematical landscape. The main thesis is that Klein s Erlangen program in geometry is in fact a particular instance of a general and broad phenomenon revealed by category theory. The volume starts with Eilenberg and Mac Lane s work in the early 1940 s and follows the major developments of the theory from this perspective. Particular attention is paid to the philosophical elements involved in this development. The book ends with a presentation of categorical logic, some of its results and its significance in the foundations of mathematics.

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In this volume, Lambek and Scott reconcile two different viewpoints of the foundations of mathematics, namely mathematical logic and category theory.In Part I, they show that typed lambda-calculi, a formulation of higher-order logic, and cartesian closed categories, are essentially the same.Part II demonstrates that another formulation of higher-order logic, (intuitionistic) type theories, is closely related to topos theory.Part III is devoted to recursive functions.Numerous applications of the close relationship between traditional logic and the algebraic language of category theory are given. The authors have included an introduction to category theory and develop the necessary logic as required, making the book essentially self-contained.Detailed historical references are provided throughout, and each section concludeds with a set of exercises. ... Read more

Customer Reviews (1)

This is definitely Higher!
I was looking for a book for my girlfriend this Christmas and stumbled upon this one.At first I thought it would be too light but was I ever mistaken!!This book is so high that it would make Jack Kerouac dizzy.It begins with a treatment of basic category theory and ccc's and then goes on to present toposes and intuitionistic type theory.The authors take care to annotate their turnstile with the set of free variables (Hah!I bet you thought I had no idea what this book was about!) so that they can deal with empty types in a reasonable way.The treatment of presheaf models is very lucid and the discussion of internal languages and lambda-calculi is excellent.In fact many papers of Koymans are just exercises from this book worked out.The book is slightly out of date, no treatment of linear logic or symmetric monoidal-closed categories.Overall this book is highly recommended for the beginner and expert alike. ... Read more

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This book provides a comprehensive categorical theory ofclosure operators, with applications to topological and uniformspaces, groups, R-modules, fields and topological groups, aswell as partially ordered sets and graphs. In particular, closureoperators are used to give solutions to the epimorphism andco-well-poweredness problem in many concrete categories. The material is illustrated with many examples and exercises, and openproblems are formulated which should stimulate further research. Audience: This volume will be of interest to graduate studentsand professional researchers in many branches of mathematics andtheoretical computer science. Knowledge of algebra, topology, and thebasic notions of category theory is assumed. ... Read more

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Text consisting of introductory surveys and articles containing original research and complete proofs, focusing on the theoretical and foundational elements in category theory and its applications to other fields. For researchers and graduate students in category theory. DLC: Categories (Mathematics). ... Read more