Product Description
Category theory is a branch of abstract algebra with incredibly diverse applications. This text and reference book is aimed not only at mathematicians, but also researchers and students of computer science, logic, linguistics, cognitive science, philosophy, and any of the other fields in which the ideas are being applied. 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 assuming 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!

This Second Edition contains numerous revisions to the original text, including expanding the exposition, revising and elaborating the proofs, providing additional diagrams, correcting typographical errors and, finally, adding an entirely new section on monoidal categories. Nearly a hundred new exercises have also been added, many with solutions, to make the book more useful as a course text and for self-study. ... 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

Product 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.

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

Customer Reviews (8)

nice and slim (the text is only ~70 pages)!
I'm still a beginner at category theory, but I'd like to say this is a nice textbook.The examples are easy to follow (mainly basic set theory), for people with a com sci background.

A later section explains CCCs (Cartesian closed categories) and its isomorphism to typed lambda calculus.I don't fully grasp the details but this is a very important result in higher-order logic, particularly because the substitution mechanism of lambda calculus can be modeled by category theory.

A Good Read
This book is not exactly what I would call easy going. I've managed to get through half of it in 7 months. However, I can say, with absolute confidence, that if you do the problems you will learn.

Most everything I've seen on category theory is a confusing mixture of different notations with seemingly identical meanings (but in fact the meanings are totally different). This book is no exception. Often, I have resorted to IRC to sort things out when some notation is simply impenetrable to me. My mathematical training stopped at complex calculus, so this may not apply to you if you've had abstract algebra or something a little more 'meta'.

There seems to be one typographical error, but I am not sure. In the example on the adjunction between products and exponentiation, the right adjoint is listed as "(_)^A x A" but in the diagrams it ends up as "(_)^A". This may be a sensible ellision, but it is not explained anywhere in the text and of it's not easy to find these things on the internet.

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.Try searching for Lambert Meertens, Marten Fokkinga, and Jaap Van Oosten, for example.Or Barr and Wells, Triples, Toposes, and Theories.Or Asperti and Longo.Or watch Eugenia Cheng's videos on YouTube, which are fantastic.

But if you want to buy a book, get Barr and Wells, Category Theory for Computing Science.Unfortunately, you have to order it directly from the University of Montreal.It's a great book, by far the best intro to category theory available, *way* better than this!Then, after that, you can read MacLane... ... Read more

Product 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 supported by a wide range of exercises. The final chapter touches on philosophical issues. The book is supported by a World Wide 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

Product 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 expoitation 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 (8)

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

Product Description
"What is Category Theory?", third volume of the series "Advanced Studies in Mathematics and Logic", collects contributions written by some of the most representative working scientists in this field of research: John L. Bell, Scott Carter, Bob Coecke, David Corfield, Costas A. Drossos, David Ellerman, Solomon Feferman, Ralf Krömer, Jean-Pierre Marquis, Ronald Brown, Tim Porter, Vidhyānāth K. Rao. This publication aims to propose new trends and perspectives through which to analyse the past, the present and the future of Category Theory. ... Read more

Customer Reviews (1)

Category theory
Category theory links with everything.Very interesting stuff, but difficult to read.This is not an introduction but a status report from philosophers of mathematics.
... Read more

Product Description
A modern introduction to the theory of structures via the language of category theory. Unique to this book is the emphasis on concrete categories. Also noteworthy is the systematic treatment of factorization structures, which gives a new, unifying perspective to earlier work and summarizes recent developments. Each categorical notion is accompanied by many examples, usually moving from special cases to more general cases. Comprises seven chapters; the first five present the basic theory, while the last two contain more recent research results in the realm of concrete categories, cartesian closed categories and quasitopoi. The prerequisite is an elementary knowledge of set theory. Contains exercises. ... Read more

Customer Reviews (1)

Easy-looking Text
This book is very well typeset. It looks very attractive to read.

About the things covered, this book is pretty comprehensive on the topic of categories. However, even with numerous good examples, some more explanations on the concept will definitely be preferred. For example, Yoneda's lemma is presented here as a corollary of a result whose proof is very short. Although the proof is logically presented with clarity, it makes not much sense why someone would have thought of it in the first place.

One problem I think worth mentioning is the proof of the equivalence of three conditions about concrete reflexive subcategories. In the proof, images of morphisms (under some functors) are written the same as the morphisms themselves, making the proof difficult to follow. And because all the proofs before that one are all very clear, it raises some doubt whether this particular proof is erroneous.

That said, I really appreciate authors' intention to make the book seem easy. A lot of examples definitely are useful. I only wish that, through experience, the authors would add some suggestions on how one might think about entities under discussions. (Examples: Categories as generalized monoids => Functors as generalized monoid homomorphisms. Functors as diagrams => Natural transformations as structure-preserving diagram transformations.)

P.S. This book is available online for free but I bought it anyway. ... Read more

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Michael Wedin argues against the prevailing notion that Aristotle's views on the nature of reality are fundamentally inconsistent. According to Wedin's new interpretation, the difference between the early theory of the Categories and the later theory of the Metaphysics reflects the fact that Aristotle is engaged in quite different projects in the two works--the earlier focusing on ontology, and the later on explanation. ... Read more

Customer Reviews (1)

What is The Meaning Of Being?
I read this book for a graduate seminar on Aristotle.
Topic of Metaphysics is Ousia=substance and being.What is the meaning of being?With respect to matter and form, it is primarily about form.Analytically both can be separate and distinct, but not in reality.One can analyze matter by potentiality and actuality.Matter can't answer the question of being without form.Some natural things are always a composite of matter and form, it is the answer to the question of what is ousia or being in nature.Matter by itself can't give us the answer to what a thing is.

Ousia=substance and being.Ousia= Being is the "this" spoken of in primary ousia.This is contrary to Plato.Categories vs. Metaphysics.We can talk of the "being" as quality as "not white."Being spoken of in many ways but only of one thing, i.e., "the focal being."Word being has flexibility.Other flexible words is essence.(the what it is to be).In Greek for Aristotle, a bed is not an Ousia because it is from techne=craft it can have an essence.Ousia is reserved for material things self manufactured in nature.All things are derived from a primary ousia.
This has to do with focal being, health is such a word.When we talk about different aspects of health, it is not a universal definition like Socrates looks for.Aristotle says you can't find it.Thus, the word "being" is just a word in a sense a focal point like the word health, i.e. healthy skin, healthy food, then there is health, for Socrates what is health.Aristotle says no, health is unity by analogy.Aristotle is OK with using examples.Math is not independent knowledge, it is dependent on things math is not a primary existence.Being is neither a universal nor a genus, (genus is animal in hierarchy).It is as though Aristotle wants to say that the primary meaning of being is the "this" the subject, i.e. Socrates not human all by itself, not animal all by itself.

Ousia= Being is the "this" spoken of in primary ousia.This is contrary to Plato.Categories vs. Metaphysics."This" is ontologically primary.Ontological= the most general branch of metaphysics, concerned with the nature of being.

In the categories discussion, he doesn't talk about the distinction between matter and form, it comes later on in the Physics and then the Metaphysics.The "this" is ontologically primary in terms of what the "being" something, what something is.Why would it be wrong to say that primary ousia can't be primary from the standpoint of knowledge, it can't be the distinction between ontological and epistemological?Why would it be wrong to say that the "this" the perceptible encounter wouldn't be primary from the standpoint of knowledge?Because, whatever the categories are whatever the notions of say "horse" the "this" is a horse, the "this" is ontologically primary, but it can't be epistemologically primary because a "this" by itself is just a "this" the question "What is this" called a horse is to involve the categories of knowledge.Therefore, from a knowledge standpoint, secondary ousia, which is things like categories and context, they have primacy in knowledge.However, from the standpoint of "being" the perceptible "this" has primacy.This is just a technical way of distancing him from Plato.In the Metaphysics, the question of form is primary Ousia.Ousia =form in Metaphysics.In Metaphysics, the "this" is simply matter.Aristotle did not give up on Ousia as form.This matter and form is never separated for Aristotle, thus a composite of matter and form is in the Metaphysics.In realm of nature, form and matter can't be separated for Aristotle.If you only talk about matter, you have nothing definable.You never come across things without their form.God is only exception to form and matter together.

Ousia as form and essence.The essence of a thing is "what" it is, it gives us knowledge.Definition= essence.Bronze can't be essence of circle, the form is important, not the matter.
Can't use abstract math to explain a human.When it comes to knowledge, we must emphasize the ousia as form.It isn't that first you have material things, and then the mind adds form to it, whatever the particular thing is, it always was that form.Then when we learn about it, we actually just discover what the thing is.Therefore, it is a process of coming to understand the universal, the essence, but that was always there in the thing, it just needed to be done.So what he is emphasizing in the Metaphysics is the idea of ousia as form, as some kind of essence, but never separated from matter!

Ousia --1.Grammatically basic.2.Ousia As Ontologically basic, something that exists in its own right.The 1st example is how humans speak, the 2nd example is how things really are, both are both side of the same coin.

Principle of Noncontradiction
Arche= principle, beginning and rule.Aristotle thought that this was the firmest of all principles.It is impossible for the same thing to both belong and not to belong to the same thing at the same time to the same thing in the same respect.An important governing thought in Western philosophy.A thing is what it is, it can't be equal to its opposite.Aristotle thought reality was organized this way.It has to do with both knowledge and being.Aristotle states that if this principle is true then it is the firmest of all principles both for knowledge and reality.In the same respect, what does it mean?It shifts depending on circumstances.From standpoint of knowledge and reality principle of noncontradiction is stable.The three factors of the principle are: the same thing, in the same time, in the same respect, is what Aristotle is calling the principle of noncontradiction.In order for knowledge to be reliable, these factors are in play.Can't be going up and down a hill at the same time.1 of 3 factors has changed, time.A "hill" is both up and down but meaningless unless you think in relation of motion.Aristotle believes when it comes to knowledge and reality the principle of noncontradiction is most basic and most fundamental and evident principle, because without it we can't communicate or think about things.Aristotle explains well how we lead our life by the principle a very pragmatic explanation.This is a principle we live by as humans thus, no one can deny it!
If you talk about change as a potentiality, you have a way of solving the puzzle.This actually serves as a slap at Renee Descartes in the future wondering if he is conscious or in a dream state.All philosophy stems from wonder and puzzlement.Aristotle makes distinction between worthy puzzles or useless ones.

Emphasis between primary and secondary being, Ousia.
For Aristotle Ousia or being is not just a thing, many ways being can be understood.Primary Ousia is things perceptible in nature.Secondary Ousia or being is sometimes being is how we understand things, i.e., big or small, etc, this is how we talk about things.He stretches the way Ousia in many ways.Matter can't be primary being like atomists, nor form alone like Platonists.However, when we analyze beings, we can use secondary being.Idea of "is" or "being" will shift depending on what you are talking about.The term "being" has plurality to it, depending on how we regard it (like using a hammer as a paperweight).Even though Metaphysics emphasizes form, it is "this form."Primary thing is the "this."

He wants to move away from Plato's idea that we can separate matter from form.A things essence is going to be the ultimate answer to the question of what is being.However, a things essence can't be separated from its statement of thing, it is almost as though that this essence is going to mean the definition of a thing, "what it is."Then in some respects, it has the characteristics of a secondary being.If you want to know what is the big deal about the perceptible "this," the primary ousia?Again, and again, the best way you can get a handle on that is he is critiquing Plato!He wants to move away from Plato's idea that it is possible to understand beings apart from the material world.Aristotle does make certain commitments; he makes certain commitments to the idea that the primary sense of being must be used in nature that are evident to us.

The Platonist in Aristotle says if the mind desires and is naturally inclined to pursue knowledge and he gives us a map how does it acquire knowledge.The Platonist in Aristotle says in the Metaphysics that if all there is, is matter and form then there is always an element of elusiveness in things because matter cannot fully deliver how we know things.When he gets to the question of the Divine, he does so because he believes that the natural desire of the mind can know that it will not have a final resting place with respect to just composite things.Especially since these composite things are always changing because nature is the realm of movement and change and the idea of form will at least give us access to how we can know changing things and actuality and potentiality.Changing things will always have this element of excess, beyond the minds capacity to grasp.

His talk of the Divine is the idea that there is something in reality that will satisfy the minds' desire for the ultimate stable resting point.If change were the last word, the mind could never come to rest.This is what Heraclitus argued for, Aristotle didn't like it.He wants to grasp the final.For him the Divine is satisfaction for the mind to grasp reality.
Uber Ousia.Aristotle here is talking about 2 senses of eternity.

1. Endless time.
2. Timelessness.1st is never begins, never ends this is eternity or infinity.2nd is in order to understand whole world there has to be something, the unmoved mover.

Ideas of potentiality and actuality criticizes Platonic idea.Potentiality has idea of negation in it.Thus, a thing in nature always has actuality; we are always on the move.Divine is pure form and actuality without matter and potentiality.Ontology now moves to theology.This is his theological science.(Theology in the Metaphysics is speaking about God for Aristotle).In reality, composite of form and matter is always in motion until it ends.Any actualization has potentiality it is prior.Actuality is prior to potentiality; this is his ultimate metaphysical statement.Two ways Aristotle proves this idea.1st is human reproduction brings us into being.Our parents actually reproduced us.2nd is God the ultimate sense of actuality prior to potentiality.

Talking about other philosopher's ideas.Hesiod question of the Gods in poetry, night comes before day, thus we don't have access in the "dark" symbolic of precedence of something unknowable, and Aristotle doesn't like it.Thus, for him he has the unmoved mover.
The pure actuality of the Divine is Aristotle's nominee for the principal that explains why there is this movement in the first place.Limitation in nature is matter which is unstable but all things in nature strive to their potential.Thus, you have pure actuality of Divine.God is Prime mover or final cause not efficient cause for Aristotle.

Rational and non-rational potentiality.This is how Aristotle recognizes the phenomenology of human thought.What rational means here is human drama of seeking what might or not work out.Now rational is stable when you heat water it boils no other potentiality.Thus, non-rational movement is very regular.Human reason is precarious we may not use potentiality to reach actuality.When we practice medicine, it might not work out.

Theoria=contemplation.There are three kinds of ousia, all are a study of secondary ousia in some way.

1. Physics-study of material and moveable.
2. Mathematical-study of ousia that is non-moving, (1+1=2 always), but is derived from matter.
3. Theology is study of ousia that is non-moving and non-material.

This is scheme of understanding the nature of understanding something.3rd level is big for Aristotle.1st two levels have limitations to them.We begin from wonder (ignorance) philosophy is to illuminate wonder with answers.He doesn't deny Greek deities but the way poets depict them is deficient.

Movement is a way of understanding change we see this in the Physics.Movement is actualization of potential.Psuche=soul which is the word he uses for life.Things in nature that are alive.Soma=body.Plato separates soul from body, Aristotle doesn't.Aristotle's text De Anima is on "The Soul" is a philosophical biological treatise.We have three-part soul, plant, animal and human all are part of this.

I recommend Aristotle's works to anyone interested in obtaining a classical education, and those interested in philosophy.Aristotle is one of the most important philosophers and the standard that all others must be judged by.


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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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Categories and sheaves appear almost frequently in contemporary advanced mathematics. This book covers categories, homological algebra and sheaves in a systematic manner starting from scratch and continuing with full proofs to the most recent results in the literature, and sometimes beyond. The authors present the general theory of categories and functors, emphasizing inductive and projective limits, tensor categories, representable functors, ind-objects and localization.

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

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This work presents a modern approach to a remarkable algebraic technique. It should be of interest to both the mathematical historian and the working specialist in commutative algebra, number theory and algebraic geometry. ... 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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Intended for category theorists and logicians familiar with basic category theory, this book focuses on categorical model theory, which is concerned with the categories of models of infinitary first order theories, called accessible categories. The starting point is a characterization of accessible categories in terms of concepts familiar from Gabriel-Ulmer's theory of locally presentable categories. Most of the work centers on various constructions (such as weighted bilimits and lax colimits), which, when performed on accessible categories, yield new accessible categories. These constructions are necessarily 2-categorical in nature; the authors cover some aspects of 2-category theory, in addition to some basic model theory, and some set theory. One of the main tools used in this study is the theory of mixed sketches, which the authors specialize to give concrete results about model theory. Many examples illustrate the 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 it sets model theory in categorical terms, opening the door for further work along these lines. Requiring a basic background in category theory, this book will provide readers with an understanding of model theory in categorical terms, familiarity with 2-categorical methods, and a useful tool for studying toposes and other categories. ... Read more

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Ontology was once understood to be the philosophical inquiry into the structure of reality: the analysis and categorization of ‘what there is’. Recently, however, a field called ‘ontology’ has become part of the rapidly growing research industry in information technology. The two fields have more in common than just their name.

Theory and Applications of Ontology is a two-volume anthology that aims to further an informed discussion about the relationship between ontology in philosophy and ontology in information technology. It fills an important lacuna in cutting-edge research on ontology in both fields, supplying stage-setting overview articles on history and method, presenting directions of current research in either field, and highlighting areas of productive interdisciplinary contact.

Theory and Applications of Ontology: Computer Applications presents ontology in ways that philosophers are not likely to find elsewhere. The volume offers an overview of current research in ontology, distinguishing basic conceptual issues, domain applications, general frameworks, and mathematical formalisms. It introduces the reader to current research on frameworks and applications in information technology in ways that are sure to invite reflection and constructive responses from ontologists in philosophy.

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The book covers elementary aspects of category theory and topos theory.It has few mathematical prerequisites, and uses categorical methods throughout, rather than beginning with set theoretical foundations.It works with key concepts such as Cartesian closedness, adjunctions, regular categories, and the internal logic of a topos. Full statements and elementary proofs are given for the central theorems, including the fundamental theorem of toposes, the sheafification theorem, and the construction of Grothendieck toposes over any topos as base.Three chapters discuss applications of toposes in detail, namely to sets, to basic differential geometry, and to recursive analysis.The intended readership consists of graduate-level students in mathematics, computer science, logic, and category theory. ... Read more

Customer Reviews (1)

Very good if you need no slack
I've been working my way through McLarty's book off and on for several months now.It is a tremendously clear and well-organized book, and you can learn a lot from it. HOWEVER: it is a "math book" in the strictest sense of the word. Exposition is kept to a bare minimum, and you have to actually work your way through the material (AND the exercises, since many of the definitions are given in them) in order to learn anything. He could have easier doubled or tripled the amount of exposition and still have produced a lean, mean textbook. This is a really good book if you need to learn category theory and you already know why. The only extensive example is a short chapter on group theory. After reading his article on category theory in the Routledge encyclopedia of philosophy I expected rather more in the way of theorizing. Be that as it may: everything you need to know about categories and toposes is in here, and nothing else. The best math book I've read in a long time. ... Read more

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This book reviews recent results on low-dimensional quantumfield theoriesand their connection with quantum grouptheory and the theory of braided, balanced tensorcategories. It presents detailed, mathematicallypreciseintroductions to these subjects and then continues with newresults. Among the main results are a detailed analysis oftherepresentation theory of U (sl ), for q a primitiveroot of unity, and asemi-simple quotient thereof, aclassfication of braided tensor categoriesgenerated by anobject of q-dimension less than two, and an application ofthese results to the theory of sectors in algebraic quantumfieldtheory. This clarifies the notion of "quantizedsymmetries" in quantum fieldtheory. The reader is expectedto be familiar with basic notions and resultsin algebra.The book is intended for research mathematicians,mathematical physicists and graduate students. ... Read more