Book Description
Category theory is a branch of pure mathematics that is becoming an increasingly important tool in theoretical computer science, especially in programming language semantics, domain theory, and concurrency, where it is already a standard language of discourse. Assuming a minimum of mathematical preparation, Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories. Four case studies illustrate applications of category theory to programming language design, semantics, and the solution of recursive domain equations. A brief literature survey offers suggestions for further study in more advanced texts. Benjamin C. Pierce received his doctoral degree from Carnegie Mellon University.

Contents: Tutorial. Applications. Further Reading. ... Read more

Customer Reviews (7)

Good Introduction
I have been reading several different category theory texts recently, and this one was very succinct and accessible.Particularly useful for understanding functional programming.

Basic crib sheet for category theory
Anyone coming to this book from Pierce's "Types and Programming Languages" will be disappointed. While his "Types ..." book is a model of clear exposition, this book reads like a set of notes jotted down on the back on an envelope. The extensive bibliographic sections are more than fifteen years out of date. Much of the material referenced is no longer in print, and recent developments are, of course, not mentioned. Those seeking a very gentle introduction to category theory would do better with the book by Lawvere and Schanuel, who cover more of category theory than Pierce. Mathematically mature computer science readers will find everything they need to know about the subject in Mac Lane's book.

Really expensive for a set of notes...
You can find better introductions to category theory available on the net for free.And I'm not talking about P2P!Try searching for Lambert Meertens, Marten Fokkinga, and Jaap Van Oosten, for example.

If you have some money to spend, get Barr and Wells, Category Theory for Computing Science.It's a great book, *way* better than this!

Too terse
This is a very short book: 70 pages of text + a bibliography.The first 50 pages are about general category theory, and the last 20 pages are specifically for computer scientists.My interest is in general category theory, and I bought this because I have a BS in CS and thought I'd find plenty of familiar examples.Unfortunately this book doesn't have nearly enough examples.I found it easier to skim some undergrad abstract algebra books in the library (groups, rings, vector spaces) and then continuing with category theory intros written for math students.

the best understaning of categories you can get
This book is tiny in volume but large in contents. It does not only provide the definitions of the fundamental concepts but also clear explanations and motivations of why must everything be defined that way, which are not always found in other texts. Plenty of the right examples help you build the right intuitions. The case studies at the end put everything into context and prepare you for CS texts on semantics, type theory, etc.
If you want to UNDERSTAND this wonderful theory read this book! ... Read more

Book Description
This text and reference book on Category Theory, a branch of abstract algebra, is aimed not only at students of Mathematics, but also researchers and students of Computer Science, Logic, Linguistics, Cognitive Science, Philosophy, and any of the other fields that now make use of it.
Containing clear definitions of the essential concepts, illuminated with numerous accessible examples, and providing full proofs of all important propositions and theorems, this book aims to make the basic ideas, theorems, and methods of Category Theory understandable to this broad readership.
Although it assumes few mathematical pre-requisites, the standard of mathematical rigour is not compromised.The material covered includes the standard core of categories; functors; natural transformations; equivalence; limits and colimits; functor categories; representables; Yoneda's lemma;
adjoints; monads. An extra topic of cartesian closed categories and the lambda-calculus is also provided; a must for computer scientists, logicians and linguists! ... Read more

Customer Reviews (1)

It's the glue.
Several years ago I came across an on-line .pdf format of Awodey's manuscript while trying to find a text on Category Theory whose content was not as intense as Mac Lane's `Categories for the Working Mathematician' and it is wonderful to see this book come to fruition.Without a doubt it is true that the available array of Category theoretic texts for mathematicians has been confined to the more abstract texts whose readership is limited to those individuals who are either researching topics integral to Category theory or graduate students of, say Algebraic Topology/Geometry, who utilize Categorical constructs and processes within the confines of their respective fields.

So where does this text fit in?I believe this text can be quantified as "the glue" between Category theoretic texts written for non-mathematicians and the hardcore texts of Mac Lane, Herrlich or Ademek et al.

What features set this text apart from the others?Simple, it is focused.Let me preface my explanation with the following: I firmly believe in the importance of demonstrating or motivating any given subject through the use of concrete examples and, in particular, through the use of several examples that can be built upon throughout the text.Awodey sees the importance of this and focuses on illuminating the abstractness of Category theory by carefully building on or utilizing Monoids and Posets.Such structures may readily seem un-familiar to some readers but, if they pause long enough to compare what they know with the basic axioms for a given set to be a Monoid/Poset, then they will see that the majority of structures in which they have been working are, in fact, specialized Monoids/Posets.Take for example Groups.Any set possessing an associative binary law of composition all of whose objects satisfy the 3-axioms for a group also trivially satisfy the axioms for a Monoid.This is not to say that Awodey has chosen two basic blocks from which all examples are derived, instead, he motivates each topic with a vast assortment of the standard examples taken from a diverse set of available fields.

So who should read this text?Anyone who wants to learn Category Theory from the ground up but lacks the standard assumed breadth of knowledge, namely, familiarity with Topology, in particular Algebraic Topology, as well as advanced abstract Algebra (inclusive of Module theory).As in any case of defining the readership one would state that their text is readable by the illusive and readily undefined "mathematically mature" student.Personally I would assume that you know how construct logically sound proofs and that you have taken courses in set theory (never given in America) as well as Algebra at the level of, say Hungerford's undergraduate text.Furthermore, and as is the case with anything mathematical, you must be willing to suffer through abstractness and be diligent as well as disciplined enough to work through the exercises.With respect to this last point, Awodey does a remarkable job providing a well thought out set of exercises ranging from simple applications of the material to more advanced exercises that will cause you to pull out your hair and possibly throw the book across the room in sheer agony.

As a final note regarding the overall text, I would even suggest this Awodey's book to more advanced student who lack a firm understanding of Category Theory but who have already suffered through someone else's text.Why?Simple, because Awodey's text will help you `see' and hence understand, at the necessary level, Category Theory.After all, one can not become proficient in anything unless they `see' what it is they are trying to become proficient in.

Finally, I would like to personally thank Mr. Awodey for writing this text and for doing such a remarkable job introducing and motivating a miraculous and awe-inspiring subject.Enjoy! ... Read more

Customer Reviews (1)

Still available
This book is not out of print.

A very nice (and economical) paperback copy of the 3rd edition is still available from the authors at the Centre de Recherches Mathématiques at the Université de Montréal. I wish it were listed here, but it's not; so I mention it.

The text itself is useful if you're coming to category theory from a background other than pure math. ... Read more

Book Description
Set theory, logic and category theory lie at the foundations of mathematics, and have a dramatic effect on the mathematics that we do, through the Axiom of Choice, Gödel's Theorem, and the Skolem Paradox. But they are also rich mathematical theories in their own right, contributing techniques and results to working mathematicians such as the Compactness Theorem and module categories. The book is aimed at those who know some mathematics and want to know more about its building blocks. Set theory is first treated naively an axiomatic treatment is given after the basics of first-order logic have been introduced. The discussion is su pported by a wide range of exercises. The final chapter touches on philosophical issues. The book is supported by a World Wibe Web site containing a variety of supplementary material. ... Read more

Customer Reviews (1)

Poor refrence/class textbook. Good preview, general overview for selected topics in FOM
Having purchased this text (as part of the Springer Undergraduate Mathematics Series), I expected a decent publication. I had purchased and used their Number Theory text (Jones & Jones) and I was satisfied with the overal quality of the problem sets and the exposition.

Like all the SUMS texts, this book provides the solutions (in this case selected solutions and the link to the website that has the rest) for all the excercises found in this textbook. However, in this case, the excercises are much to trivial to be considered a good workout in any of the topics covered in this book.

To account briefly book covers Naive Set Theory, Sentential Logic, 1st Order Predicate Calculus, "Model Theory" (I'll explain the quote later), Ordinal numbers, Aximoatic Set Theory, and Category Theory.

Topics that are missing in a introductory treatment includes Recursion Theory, most of even the basic developments of Model Theory.

This would be fine in it of itself, however, what the text does cover, namely the 1st order Predicate Caculus and Model Theory, is so sparing that one gets a very tiny glimpse of the subject and that is it.

For a SUMS text, the book is suprisingly lacking in rigor and substance. Theorems are still stated and proved yet nothing but the most basic results are displayed. For instance, in 1st order predicate calculus the book introduces the Deduction theorem then right after goes to Soundness and then Completeness.

It seems the topics of symbol substitutibility, Henkin langauge expansion, and quantifier elemination were totally ommited. These are very important topics, topics no introduction to Mathematical logic should be without, yet they are absent in this text.

Further, the chapter on Model Theory is nothing but theorem throwing at the reader. In the 2nd page andlittle bit after, the reader is introduced to Lowenheim-Skolem Theorem, Compactness, Consistency and a very brief expostion on a Peano Arithemtic system.

This chapter also serves as a brief introduction to incompleteness (perhaps a page introduction at most). But to be honest, even the proofs given for these thoerems are lacking and wouldn't satisfy many students of this subject as a suffecient explanation (let alone a potential refrence).

To demonstrate the unbelievable terseness (and sheer lacking) of this exposition, the book discusses everything on Godel numbers to incompleteness in a span of 3 - 4 pages. Even in "light" introductions, such as Enderton, this development and the accompanying machinary requires an entire chapter to develop (and Cameron has ommited a signifcent amount of the machianry by ommited all of Recursion theory).

The good in this book or perhaps more accurately, the unqiue, are that it does give an introduction to ordinals (usually reserved for Intro. Set Theory books) and a light introduction to Category Theory ("preview" is more fitting for that chapter).

In fact, "Preview" is a very fitting description of this textbook in general. This text cannot hope to serve as anything more then a preview for the subject discussed within those pages. People who wish to develop a working knowledge of this subject should look towards Enderton as a "lighter" introduction (if Enderton is a diet Coke, then this book is certainly water).

I think this text would go well in two scenarios. One, a indivudal who is about to take his firs FOM course and uses this book as a preview durring the summer (or Winter break) before actually taking the course. The second scenario would be to use this text as a followup with Springer's other text Johnson's "Elements of Logic via Numbers and Sets." That text combined with this would serve as a very good "bridge" course to abstract mathematics.

If, however, you are not one of the above mentioned, then I recommend that you consider purchsing one of hte other more establisehd text on Mathematical logic as this book is to light (and in my opinoin to expensive for the amount of material given) to serve as a useful text. Thus, this book may fail totatlly as a textbook for a intro. FOM course, however it can still find some use as a advanced preview for the subject or a companion in a abstract matheamtics bridge course. ... Read more

Book Description
The idea of a "category"--a sort of mathematical universe--has brought about a remarkable unification and simplification of mathematics. Written by two of the best-known names in categorical logic, Conceptual Mathematics is the first book to apply categories to the most elementary mathematics. It thus serves two purposes: first, to provide a key to mathematics for the general reader or beginning student; and second, to furnish an easy introduction to categories for computer scientists, logicians, physicists, and linguists who want to gain some familiarity with the categorical method without initially committing themselves to extended study. ... Read more

Customer Reviews (11)

Great book; whether you should read it depends on you
Many of the reviews evaluate the book from the perspective of graduate students in mathematics want to learn categories, and it's certainly the wrong choice for that purpose.If you think of this as a serious math textbook, then it fails in that goal: significant proofs are the exception rather than the rule; very few, and trivial, exercises; very lacking in depth.

This is a great book because it provides a motivation for investigating categories.It helped me when I was in the position of hearing from a lot of places that subjects I was interested in often used category theory.I tried to read a few "real" books about category theory, and didn't get very far because they did not make the connections I was looking for.I accumulated three or four such books, all with bookmarks at about page 50 to 75.This book taught me relatively little about the theory of categories or the body of knowledge about them, but it provided a wealth of connections between categories and other topics, which made me better able to finish a couple of the real books and figure out what I needed to know there.

My advice, if you're in anything like that situation, is to read this book.Just don't take it too seriously, and don't try to milk more out of it than is really there.Then go learn more about category theory from elsewhere.

Oustanding book: an absolute must-read for any mathematician
In the preface of this book, the author comments that this book has been used successfully in high schools, colleges, graduate schools, and by professors.After reading this book, I can believe it.This book is simply a gem.

Mind you, although this book is very easy to read, some of the concepts contained within it are very abstract and can be very difficult to fully comprehend.While a high school student will surely get something out of this book, it would be hard to understand everything in it without knowing a fair amount of mathematics.

I would recommend this book to any mathematician.It is an absolute must-read.The author makes the claim that working through this book will improve your ability to categorize (no pun intended!) your mathematical knowledge so as to better know how to approach problems.From my experience, this claim is true.This book somehow teaches some of the things about problem-solving that many people believe cannot be taught.

This book looks deceptively simple, especially relative to beasts such as MacLane's "Categories for the Working Mathematician".However, I find that I keep coming back to this book, sometimes after several months.In particular, I have found that reading this book has opened the door to understanding some of the advanced mathematics books that previously seemed inaccessible to me, such as Lang's "Algebra".

not a book for self study
I bought this book a year ago and use it to teach myself the basics.it is difficult to go thru the exercises and tests because no answers are given, and at this stage of ignorance, with no one to rely on for verification, I am left not knowing whether I am doing the right thing or thinking the correct way on the concepts presented.Otherwise, the approach of the book is novel and easy to follow.

Objects and maps are everywhere
Excellent book for non-professional mathematicians, like me (I'm a software engineer), who wants to understand modern mathematics and apply its ideas in analysis of complex problems. Lots of pictures and diagrams (compared to terse wording in other mathematical books) really help to understand and master the subject. I think most of negative reviews come from professional mathematicians, but they don't need this book.

Very uneven, but still useful
As a topic in itself, category theory should need not to wait until grad-level to be described just because that may be when category theory's power can really begin to be exploited, but unfortunately, most of the category theory books I have looked at presume that level of mathematics.

Similar to what other reviewers noted, I would also say that this book demonstrates the potential of creating a good high-school/undergrad level intro to category theory. But unfortunately, that potential is not quite realized here.

There are hokey intermittent "conversations with students", as a tool to describe ideas, that are more distraction than aid. Some of the examples given are rather condescending in their simplicity. Yet, at other times the authors seem to breeze through more difficult topics with little or no examples. And the organization seems erratic - there is no clear sense of a gameplan as to where they are leading the reader or how all the concepts fit together.

Functors are surprisingly almost glossed over, as if they were relatively unimportant. There are exercises throughout the book, but with no answers provided, they are not really very helpful.

Having said all that, with some focused effort on the reader's part, the ideas do come forth, and admittedly, the authors do cover a fairly broad spectrum of aspects of category theory. This is certainly a non-trivial topic to try and teach, and an introductory book cannot be faulted for not carrying every notion to the nth-degree of either breadth or depth.

Category Theory is one of those topics that (to me) appears 'ho-hum' until you see it actually applied to various topics. The authors have necessarily had to perform a balancing act between describing concepts while not getting caught up in excessively complex examples. I think this will leave many readers less than satisfied, but realistically, the book would have been twice as long had they really delved deeper into examples (or they would have had to be very terse in the actual descriptions of category theory, which is the choice most authors writing for a more mathematically-inclined audience seem to make - e.g., _Mathematical Physics_ by Geroch (good book!) or _Basic Category Theory for Computer Scientists_ by Pierce).

If you are mathematically astute, you probably will find this book tedious. But if you are not a grad+ math major, then this book may well be worth the effort as a way to begin to learn a very profound and powerful set of tools and concepts. ... Read more

Book Description
This book presents an innovative theory of syntactic categories and the lexical classes they define. It revives the traditional idea that these are to be distinguished notionally (semantically). The author proposes a notation based on semantic features that accounts for the syntactic behavior of classes. The book also presents a case for considering this classification--again in a rather traditional vein--to be basic to determining the syntactic structure of sentences. ... Read more

Book Description
Categories for the Working Mathematician provides 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.The book 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 characterized 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 two new chapters on topics of active interest.One is on symmetric monoidal categories and braided monoidal categories and the coherence theorems for them.The second describes 2-categories and the higher dimensional categories which have recently come into prominence.The bibliography has also been expanded to cover some of the many other recent advances concerning categories. ... Read more

Customer Reviews (9)

Simply Great
Have you ever tried reading Descartes' "Geometry"? It's not a good place to learn about coordinate geometry. I tried. This was almost 10 years ago, but I still remember it pretty well. Ok, so maybe the experience was even a bit traumatic.

Usually when someone works out a theory, it takes a fresh perspective (or two, or ... you get it) to really digest it, and come up with a reasonable way of teaching it to newcomers. It's less evident nowadays, with improved communications technology and such, but people aren't exactly turning to Grothendieck's expositions as their intro to his geometry either. Mac Lane is an exception.

This book seems completely inapproachable. The title is scary. The topic is scary. Open to a random page and try to judge its accessibility: scary. Well, here's the real story: you need to know algebra through modules, and it'd be nice if this algebra background introduced "universals" like abelianization or free modules in a way that involved the diagrams and the unique mappings you get from the given ones. If this stuff makes any sense, you can read this book. It's not that scary. If you're up to the challenge, you might even enjoy it. This is actually my favorite book.

Here's the approach that I feel worked well for me:

- gloss over the set-theoretic foundations at first. Make sure you know the proper class/set and large/small category distinctions, but don't dwell on them much.

- focus on the examples that are familiar, but read through the others too. Mac Lane uses tons of examples to suit a variety of backgrounds, and his presentation is so clear that the theory can often explain the examples.

- trust the author. It may seem like product or comma categories deserve fuller treatment with more motivation. No. Let Mac Lane's 'minimalism' infect your thinking: it's no more complicated than what's on those pages. Make sure you *know* what's there, and you will come to *understand* the material as it is fleshed out through exercises or later writing.

The last point has been the most important for me. This book has been a great lesson in clear thinking, which is of extreme importance in mathematics. Why? It's complicated enough!

Poorly written standard text.
This book has everything you need, but it is written in an abstruse style in my opinion.

A Classic
Well, let us think about this a little bit...You want to learn Category theory, whether for some course or just for the fun of it, and now where do you turn in order to learn the necessary concepts.If you are a mathematician and have some experience, then you turn to the masters, the originators of the given subject and read their work.Sure, being the founder of a given subject does not imply that you are a good expositor and hence are capable of revealing the necessary concepts for the beginner-allow me to inform that Mac Lane is indeed as good as an expositor as he was a mathematician.For any doubters, I point you to the only other text you should read on Category theory, namely, "Category Theory" by Horst Herrlich and compare this text with Mac Lane's.Aside from that, and with respect to the text, for most beginners or interested readers I would suggest the following outline: Read 1.1-6; 2.1-3 & 8 possibly 2.4; all of 3; as for 4 skip section 3; 5.1-5; all of 8.Then, dependent upon your desires and or focus as well as your mathematical ability, it should become obvious which of the remaining topics should be read.Finally, the only other source I would recommend for learning Category theory can be found on-line using the keyword 'Awodey'.Anyways, Enjoy and good luck.

You may not need this unless you major in category theory.
I entirely agree with the reviewer Lucas Wilman.
As a book by the creator of category theory, it has extensively incorpoated relevant items.
However I don't think this is a *must read" unless you major in the subject: you will seldom need more than what is covered in a typical homological algebra course.
My inmpression is this book should be entitled "Categories for the starting/working category theorists".

Classic and worth it
It is difficult to make understand what "is" category theory. Is it a foundational discipline? Is it a discipline studying homomorphisms between algebras? Is it nonsense? Well, in my opinion this book does not help in gaining this kind of understanding. But all the stuff I read which have been written with that purpose in mind did not have any success - perhaps because I am not a mathematician, or perhaps because some concepts in category theory are really too abstract for anyone to give "an intuition" of them (you still can with functors and natural transformations, but try with adjointness...). This said, I found the book wonderful: Every concept is presented neatly. I use it as a reference each time I want a clear and rigorous definition of a concept. Sometimes this rigour helped me in gaining the famous intuition behind the concept. ... Read more

Book Description
Category Theory has, in recent years, become increasingly important and popular in computer science, and many universities now introduce Category Theory as part of the curriculum for undergraduate computer science students. Here, the theory is developed in a straightforward way, and is enriched with many examples from computer science. ... Read more

Customer Reviews (2)

a recommendation of Category Theory texts for CS/IT
In September 1997 we needed a book on Category Theory for our first yearundergraduate class in the B.A. (Mod) honors degree in Information andCommunications Technology (ICT) at the University of Dublin, TrinityCollege, Dublin 2, Ireland. This book was at that time the only one thatsatisfied our requirements. Now we have chosen (Lawvere and Schanuel 1997)in addition. It is our opinion that one ought to start with the latter, amost excellent introduction of great profundity, and, for application tocomputing, use the Walters text. It is hard to beat this combination for afirst year undergraduate course, as far as we know at this time (Sept.98)

A Very comprehensive textbook for beginners computer sci.
The Book begins with the plain definition of a category, as does any other book. However, it points out a category as a kind of (abstract) Data Type. Distributive Categories are discussed as a milestone for developing the basic concepts in computation, as those of imperative programs and Data Types. The Book has a lot of examples (from computation) and the author took care of drawning conclusions from them before develop an abstract framework. The concepts of automata and automata with inputs are shown (the later in a functorial category). Grammars and Graphs are discussed as well. The book has a very good introduction to the concept of freeness and adjunctions. Its latest chapter treats the computational category theory in the context of Knuth-Bendix procedure. The exercises present in the book are great !! They guide the student gradualy into deeper questions without any frustation. There are very easy exercises which have the only goal of finding out ones undersating of a new definition. ... Read more

Book Description

This book provides a gentle, software engineering oriented introduction to category theory. Assuming only a minimum of mathematical preparation, this book explores the use of categorical constructions from the point of view of the methods and techniques that have been proposed for the engineering of complex software systems: object-oriented development, software architectures, logical and algebraic specification techniques, models of concurrency, inter alia. After two parts in which basic and more advanced categorical concepts and techniques are introduced, the book illustrates their application to the semantics of CommUnitya language for the architectural design of interactive systems.

Book Description

Categories and sheaves, which emerged in the middle of the last century as an enrichment for the concepts of sets and functions, appear almost everywhere in mathematics nowadays.

Customer Reviews (1)

Well, it is what it is!
I'm a math and physics double major, and I've been interested in category theory for a while. This book does exactly what it intends to do, which the authors state in the preface is to "...present categories, homological algebra and sheaves in a systematic and exhaustive manner starting from scratch and continuing with full proofs to an exposition of the most recent results in the literature and sometimes beyond."

It's a typical advanced math book...i.e. a seeming grocery list of definitions and lemmas, then theorems and proofs.

An advanced undergraduate may be interested, but it's a bit abstract (as all advanced math is!).

The first chapter is a decent introduction to categories. I would advise learning linear algebra and abstract algebra before reading this chapter, and of course read this chapter before the rest of the book!

Personally, without applications to reality, learning new math leaves me asking "So what?" The book has mathematical applications of category theory to "cast" abstract algebra and linear algebra in the language of categories instead of using the language of set theory.

So if you don't know linear algebra and abstract algebra, you'll be left asking "So what?"

Further, a lot of these concepts that are presented are hard...not in the sense "Solving this equation is hard!" But in the sense that it's deep, so it's hard like "Reading Hegel is hard!"

Overall, I think it's a great book and worth it's money. I wouldn't advise getting it without good knowledge of abstract algebra (since then the notion of a category of groups, for example, would be meaningless without knowledge of what a group is!) or linear algebra (which helps with the notion of morphisms, etc.).

Just my two cents... ... Read more

Book Description

Category theory is a general mathematical theory of structures and of structures of structures. It occupied a central position in contemporary mathematics as well as computer science. This book describes the history of category theory whereby illuminating its symbiotic relationship to algebraic topology, homological algebra, algebraic geometry and mathematical logic and elaboratively develops the connections with the epistemological significance.

Book Description
This textbook explains the basic principles of categorical type theory and the techniques used to derive categorical semantics for specific type theories. It introduces the reader to ordered set theory, lattices and domains, and this material provides plenty of examples for an introduction to category theory, which covers categories, functors, natural transformations, the Yoneda lemma, cartesian closed categories, limits, adjunctions and indexed categories. Four kinds of formal system are considered in detail, namely algebraic, functional, polymorphic functional, and higher order polymorphic functional type theory. For each of these the categorical semantics are derived and results about the type systems are proved categorically. Issues of soundness and completeness are also considered. Aimed at advanced undergraduates and beginning graduates, this book will be of interest to theoretical computer scientists, logiciansand mathematicians specializing in category theory. ... Read more

Customer Reviews (1)

Excellent introduction to categories for computer scientists
The book gives you all of the cateogry theory you need to study type theory. The examples are from domains that are comfortable for computer scientists. The difficult proofs are given in great detail, while other books often gloss over the details. ... Read more

Book Description
The book is an original treatise designed to defend an original, non-Aristotelian theory of categories. Chisholm argues that there are necessary things and contingent things; necessary things being things that are not capable of coming into being or passing away. He defends the argument from design, and thus includes the category of necessary substance (God).Further contentions of the essay are that attributes are also necessary beings, but not necessary substances, and that human beings are contingent substances but may not be material substances. ... Read more

Customer Reviews (1)

Not Chisholm's best work
This is the last book the excellent 20th century philosopher Roderick Chisholm wrote. It contains many of the final thoughts he had on metaphysical matters and for that reason is interesting. However, Chisholm was losing his sight by the time this book came about, and he wasn't able to edit it very well. Apparently he was a very visual writer and needed to see his thoughts written out to evaluate them and put them together coherently. This book jumps around a lot, even within chapters. Pieces of it are good on their own, but it doesn't fit together well, and substantive subject changes are frequent without any completion to the previous line of thought. It's probably worth reading for anyone who enjoys Chisholm's work. There's probably much benefit from such a pursuit. However, be aware of the deficiences of this book. ... Read more

Book Description
The two volumes of Architectural Theory bring together the fundamental elements of architecture and present them in a new and accessible format. The books define the areas of knowledge necessary for successful design and criticism and, for the first time in the history of architectural literature, integrate all the concepts to form a balanced and comprehensive whole. Volume One, A History of the Categories in Architecture and Philosophy, establishes the framework of architectural theory. The author presents a systematic analysis of what constitutes 'good' architecture in the West, tracing the history of architectural theory through the metaphysics of ancient Greece, the doctrines of early and medieval Christianity, up to the concepts and 'categories' of modern philosophy. The twentieth century has seen more building and more analysis of building than any other. Volume Two, Principles of Twentieth-century Architectural Theory Arranged by Category, focuses on the recent fragmentation of architectural theory into distinct doctrines. Formalism, minimalism, mannerism, functionalism, rationalism, brutalism, positivism, romanticism, expressionism, classicism, constructivism, organicism, modernism, futurism, radicalism, deconstructivisim, historicism, post-modernism - each movement has influenced the shape of architectural thinking over the last century. Principles of Twentieth-century Architectural Theory Arranged by Category analyses each in turn and places each in context. The volumes are liberally illustrated with representative buildings of the period and include a glossary of terms, a thesaurus, an annotated guide to further reading as well as diagrammatic links connecting themes across both volumes. The two volumes, whether studied together or individually, will prove invaluable to students of architecture and related disciplines. ... Read more

Book Description
The Oxford Topology Symposium was held in June 1989. Since techniques from topology and category theory have been used increasingly by theoretical computer scientists in recent years, it was decided to hold a special session at the symposium which would be devoted to the application of these topics in computer science. By holding this session in the context of the topology symposium, the organisers hoped to achieve a cross-fertilization between the communities they brought together - giving one a course of new problems with a more practical flavour, and the other a source of solutions and ideas. The session itself proved successful, attracting a large audience of mathematicians as well as computer scientists.The organizing committee decided to produce two separate proceedings for the conference. All those who had presented papers, plus a very few others, were invited to submit papers for these proceedings of the special session on topology and category theory in computer science. ... Read more

Book Description
Intended for category theorists and logicians familiar with basiccategory theory, this book focuses on categorical model theory,which is concerned with the categories of models of infinitaryfirst order theories, called accessible categories. The startingpoint is a characterization of accessible categories in terms ofconcepts familiar from Gabriel-Ulmer's theory of locallypresentable categories. Most of the work centers on variousconstructions (such as weighted bilimits and lax colimits),which, when performed on accessible categories, yield newaccessible categories. These constructions are necessarily2-categorical in nature; the authors cover some aspects of2-category theory, in addition to some basic model theory, andsome set theory. One of the main tools used in this study is thetheory of mixed sketches, which the authors specialize to giveconcrete results about model theory. Many examples illustratethe extent of applicability of these concepts. In particular,some applications to topos theory are given.

Perhaps the book's most significant contribution is the way itsets model theory in categorical terms, opening the door forfurther work along these lines. Requiring a basic background incategory theory, this book will provide readers with anunderstanding of model theory in categorical terms, familiaritywith 2-categorical methods, and a useful tool for studyingtoposes and other categories. ... Read more