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Emily Riehl
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Category Theory in Context (Aurora: Dover Modern Math Originals) ペーパーバック – 2016/11/16

英語版 Emily Riehl (著)
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Category theory provides a cross-disciplinary language for mathematics designed to delineate general phenomena, which enables the transfer of ideas from one area of study to another.

Category theory has provided the foundations for many of the twentieth century's most significant advances in pure mathematics. This concise, original text for a one-semester course on the subject is derived from courses that author Emily Riehl taught at Harvard and Johns Hopkins Universities.

The treatment introduces the essential concepts of category theory:
 
  • Categories, functors, natural transformations
  • The Yoneda lemma, limits and colimits
  • Adjunctions, monads, and other topics

Suitable for advanced undergraduates and graduate students in mathematics, the text provides tools for understanding and attacking difficult problems in the following:
 
  • Algebra, number theory
  • Algebraic geometry, and algebraic topology
  • Prerequisites are limited to familiarity with some basic set theory and logic

Drawing upon a broad range of mathematical examples from the categorical perspective, the author illustrates how the concepts and constructions of category theory arise from and illuminate more basic mathematical ideas.

Dover is widely recognized for a magnificent mathematics list featuring such world-class theorists as Paul J. Cohen (Set Theory and the Continuum Hypothesis), Alfred Tarski (Undecidable Theories), Gary Chartrand (Introductory Graph Theory), Hermann Weyl (The Concept of a Riemann Surface), Shlomo Sternberg (Dynamical Systems), and multiple works by C. R. Wylie in geometry, plus Stanley J. Farlow's Partial Differential Equations for Scientists and Engineers.
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  4. 発売日
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  5. 寸法
    15 x 1.5 x 23 cm
  6. ISBN-10
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  7. ISBN-13
    978-0486809038
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著者について

Emily Riehl is Assistant Professor in the Department of Mathematics at Johns Hopkins University. She received her Ph.D. from the University of Chicago in 2011 and was a Benjamin Pierce and NSF Postdoctoral Fellow at Harvard University from 2011–15. She is also the author of Categorical Homotopy Theory.

登録情報

  • 出版社 ‏ : ‎ Dover Publications (2016/11/16)
  • 発売日 ‏ : ‎ 2016/11/16
  • 言語 ‏ : ‎ 英語
  • ペーパーバック ‏ : ‎ 272ページ
  • ISBN-10 ‏ : ‎ 048680903X
  • ISBN-13 ‏ : ‎ 978-0486809038
  • 寸法 ‏ : ‎ 15 x 1.5 x 23 cm
  • カスタマーレビュー:
    5つ星のうち4.7 133

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Emily Riehl

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nicolas desmaziers
5つ星のうち5.0 Interesting book
2024年1月23日に英国でレビュー済み
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Very good book on category theory, but certainly not the first one you should get. You probably need a good level in Algebra, and compared to Awodey's book it's much more difficult. That said, it's very cheap and incredibly well written.
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Carlos G.
5つ星のうち5.0 Bueno pero difícil
2020年10月17日にメキシコでレビュー済み
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Es difícil de leer para estudiantes de licenciatura (como yo), pero presenta muchísimos ejemplos y la teoría se expone de manera muy interesante. Varias veces he quedado asombrado por los resultados que aparecen y cómo generalizan conceptos de diferentes ramas de la matemática simultáneamente. Me encanta!
3人のお客様がこれが役に立ったと考えています
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Antoine Beaudet
5つ星のうち5.0 Better than CWM!
2018年8月11日にカナダでレビュー済み
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Riehl always exhibits plenty of examples of mathematical phenomena BEFORE giving the categorical concept that subsumes them all. The effect is that category theory is seen as revelatory. In reponse to the usual saying that "there are no theorems in category theory," she also gives lots of categorical theorems, in other branches of math as well as in category theory itself.

A fun and elucidating read!
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Amazon Customer
5つ星のうち5.0 A NEW CLASSIC-THE NEW GOLD STANDARD FOR CATEGORIES, MORPHISMS AND DIAGRAM CHASING FOR THE SERIOUS MATHEMATICS STUDENT
2017年8月13日にアメリカ合衆国でレビュー済み
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Category theory is largely an abstract generalization of the entire structure of modern mathematics where functions rather than sets and their elements are the atomic components from which all else is constructed. While I understood the basic definitions and ideas of categories, morphisms, functors, duals and adjoints, being originally trained in rigorous mathematics via naïve set theory, it was very tough for me transition to-as the late, great Joseph Rotman wonderfully described it-“living without elements”. It was very hard to let go of the ideas of mathematics as being woven from collections of various “sizes” and “structures” that were composed of “elements”, functions were a special kind of Cartesian products of such sets,etc. The central idea of category theory is to reverse the logic of this view of mathematics and take functions-more specifically, arrows or morphisms-as the irreducible components from which everything else is synthesized. What did help me understand-in addition to working through several sessions in John’s office-was looking at as many examples of categories, morphisms and functors as I could find. The example of groupoids in particular was helpful to me because it was a simple generalization of the definition of a group. A groupoid is simply a category where all the morphisms are invertible i.e. are isomorphisms. It was really while studying the groupoid and its’ role in topology using Ronald Brown’s wonderful book that the proverbial lightbulb went off over my head: Category theory strips all but the essential properties that link all the objects in a single category and their morphisms. While working with arrows doesn’t always lead to simplifications, since most of the important properties of mathematical objects either depend on the functions between objects rather than the structure of the objects themselves or can be “coded” by functions, categorical language does result in simplifications a great deal of the time.

Category theory has always vexed me even though I think I’ve mastered the basic ideas of it. What I’ve hoped for along the way-not only for myself but future students-is a textbook that would allow a strong mathematics undergraduate or first year graduate student to master this odd but so important subject. The classic mathematical introduction to the subject is, of course, Saunders McLane’s Categories For the Working Mathematician, by one of the subject’s founding fathers. But the title of that book is certainly to be taken literally-MacLane is pitched at advanced graduate students and PhDs whose original training had not included categories. I remember trying to read it as an undergraduate and after 60 pages, I was actually exhausted. Literally mentally exhausted. It’s incredibly dense, packing an enormous amount into each paragraph while providing very little motivation and few examples. While McLane was certainly a master and writes very well, this is not a friendly text on a subject that isn’t easy to begin with. The title is as much a warning to the prospective reader as a description.

It also should be noted there have been a number of advances in the subject since the latest edition of MacLane, such as higher category theory, abstract homotopy theory and higher topos theory. Much more significantly, a huge number of important applications of categories and functors have arisen in a number of fields, from computer science to linguistics to neuroscience. So MacLane no longer represents the frontiers of the subject. There are more far accessible introductions now-some of which I’ll discuss in a future post-but most of these other introductions are not for strong mathematics students. They are introductions for graduate students in other fields like philosophy, linguistics and computer science who need an understanding of category theory to pursue research. The more mathematically oriented of such books could also be used as undergraduate mathematics student texts.

What’s needed for the audience I have in mind is a successor textbook to MacLane which a) covers all the major definitions and theorems of categories and their morphisms with good exercises b) provides a large stock of examples from various fields of mathematics to both motivate and clarify these ideas and c) does (a) and (b) in a comprehensive, well organized and well-written way but isn’t too long winded and (d) assumes a reasonable level of prerequisite knowledge from the students. In other words, we need a text which is pitched at an intermediate level between the mathematics undergraduate/non-mathematics graduate student books and Mac Lane.

Which brings us to this little green volume from Dover Books’ new Aurora line of original textbooks-Category Theory In Context by Emily Riehl.

Riehl is a young mathematician at John Hopkins who’s been developing the lecture notes from which this book is based from courses in category theory she’s taught at both Harvard and John Hopkins for strong undergraduates and first year graduate students since 2015. I’ve seen the early draft versions of the notes since they began at her homepage and I was incredibly excited by what I saw gestating there. When the book was finished and published with surprising speed in an inexpensive Dover paperback-which made me even more excited, since it meant the book would be affordable for most students-I didn’t hesitate to buy it and begin reading it voraciously.

So what’s the verdict, you ask? Is it a good book someone could use to either learn or teach category theory?

Absolutely not. It’s not a good book.

It’s a GAME CHANGING TEXTBOOK.

Riehl has written a textbook that will not only become a classic in short order, but one that will change the teaching of category theory in universities across the world at every level. It’s richly written, reasonably detailed, crystal clear, completely up to date and wonderfully organized. It’s going to encourage many more math departments to begin to offer category theory to strong undergraduates and graduate students in both regular courses and independent reading seminars. It’s going to make teachers of graduate courses at relatively weak programs much more comfortable using diagram chasing in their presentation, it’s going to land on the required reading lists of the top graduate programs like Harvard and Columbia for the suggested background of applying undergraduates.

In short, it’s going to raise one hell of a noise once people become familiar with it.

I firmly believe Riehl’s book is going to replace MacLane’s as the definitive textbook on the subject for advanced pure mathematics students and it will do so relatively quickly.

Yes, it’s that good.

And now an overview of this future classic.

The book begins with a lengthy Preface, which is actually a wonderful short essay on category theory itself, providing a preview of much of the books content as well as stating important theorems. It’s in this preface that Riehl sets the tone for what follows with her beautiful writing style and wonderfully intuitive priming that sets the table for later rigorous definitions and proofs.

Chapter 1 sets the basic definitions and theorems of category theory: categories, morphisms, functors, natural transformations, abstract vs. concrete categories, duality and opposite categories, covariance and contravariance of functors and natural transformations. This chapter sets the tone for the rest of the book-there may be more examples of basic structures in category theory in this first chapter alone then in all other books on the subject published to date combined. More on that later. It also includes a rigorous discussion of diagrams and a glimpse of higher category theory through 2-categories.

Chapter 2 discusses universal properties of categories as encoded in natural transformations. The ultimate goal of this chapter is to state, prove and illustrate most of the important consequences of Yoneda’s Lemma, which is essentially “The Fundamental Theorem of Category Theory”. Yoneda’s lemma says that if certain conditions are met on a category C and there is a functor F: C→Set where a ɛ C, there exists an isomorphism functor from the category of all natural transformations on F to the image category F (A). This is really a crude statement of the result, which is actually a good deal more sophisticated than this. To state it precisely requires the definition of a functor being represented by an object in C which in turn requires initial and terminal objects in a commutative diagram. These and all the associated machinery is defined and developed here.

Chapter 3 discusses a powerful generalization of topological spaces and their subspaces: limits and colimits in categories. This is where the precise formulation of a commutative diagram really comes into it's own here. A lot of other books shy away from this critical spade work and it many other sources, it really makes limits confusing. Riehl makes you understand in this chapter how difficult it is to develop and understand categorical limits rigorously without it.

Chapter 4 defines and explains the rather subtle but quite important concept of adjunction. More sophisticated concepts of modern category theory hinge heavily on adjunction, so Riehl has her work cut out for her in making this chapter comprehensible. She has a very insightful way of motivating this concept: it can be thought of as the inverse of the forgetful functor’s action on a small category into the category of sets. As the forgetful functor strips all structure away from a defined category to produce a set with no structure, an adjunction takes a “free” construction of sets and builds a specific category with its’ expected structure. This is not an easy concept to for a beginner to grasp. This is where the plethora of examples the author provides in concert with a precise formulation is extremely clarifying.

The rest of the book tackles advanced concepts that a graduate student needs for forays into the current frontiers of modern algebra and topology as well as higher category theory. Chapter 5 discusses monads and their algebras. Chapter 6 discusses Kan extensions and their role as a unifying concept in category theory, where most of the concepts of the previous 5 chapters can be expressed as Kan extensions.

Before I go any deeper into this review, I don’t want to give the impression this is bathroom reading or something you can zip through on the bus on the way to class. It’s anything but. It’s certainly easier and much more accessible then MacLane. But that’s a little like saying the annotated Complete William Shakespeare is much more accessible than the original 15th century English versions. It’s true, but that’s hardly saying it’s an easy read. However, this is category theory. Like its’ brethren subjects in the foundations of mathematics, mathematical logic and axiomatic set theory, unless one gives a very shallow and cursory treatment, there’s really no way to make the subject easy to digest. What a reader can and should expect from a well-written treatment of such a subject is that his or her effort and focus will be rewarded with a deep and thorough understanding of the material that will allow them to study most advanced treatises and research papers on the subject without significant effort.

Riehl absolutely delivers on that here.

If a student is making the effort to seriously read an advanced mathematics textbook, the 2 most important qualities it must have are readability and clarity. It’s especially true if they’re brave enough to try and do so independently without a teacher. This book is brimming with both, which is quite a feat in a book at this level. Riehl’s prose is wonderful and lively, as well as possessing tremendous professional depth. It’s also concise in the best possible sense of the word-there’s virtually no irrelevant expository fat in the book. Everything in the book is important to the presentation, everything opens up a new perspective-however minor-on the material under consideration.

But what’s most striking about the book- indeed, its single most unique attribute and what raises it head and shoulders above most other textbooks written at this level- is the examples. Most practitioners of mathematics fall into 2 camps on explicit, specific examples in a mathematics textbook: One camp is the “Bourbakian” camp-which believes all mathematics, regardless of audience, is done at the highest level of generality and abstraction and that all concrete examples can and should automatically “drop out” as special cases of the powerful machinery. The other camp believes in the converse approach: We should begin with as many specific, concrete examples as possible and then prove general statements as a “big picture” statement encapsulating them all as special cases. I’m firmly in the second camp. Although everyone has different innate processing for mathematics, one doesn’t have to be a cognitive scientist to see humans tend to learn by going from the specific to the general. While the most abstract presentation may benefit experienced mathematicians, I believe for all but the most gifted students, the latter approach would be most beneficial.

When one is dealing with category theory, one is in an interesting predicament. The entire point of the subject is to present mathematics as abstractly as possible and remove all nonessential properties. As a result, any “middle ground” approach isn’t really an option. You basically can either focus on concrete examples of categories and morphisms and build the general structures around them or state and prove everything in full generality very tersely with no or few examples , with many major results shunted to the exercises. The latter approach is what MacLane does-which is what makes it so arid and forbidding. But given the fact MacLane was writing for strong or advanced graduate students or PhDs, this isn’t really surprising.

Riehl takes the former approach with gusto. It is brazenly, unapologetically and passionately example-driven. And it does so without sacrificing the least bit of rigor or abstraction. This is the “context” referred to in the title. All definitions, theorems and proofs are deeply embedded in a framework of examples.

Many, many, many examples.

With a very few possible exceptions such as E.B. Vinberg’s A Course In Algebra , John and Barbara Hubbard’s Vector Calculus,Linear Algebra And Differential Forms and a handful of others, I’ve never seen any textbook with as many beautifully detailed and presented examples as Riehl’s book.

Never.

You’re actually stunned by not only how many examples there are in the book, but how diverse and insightful many of them are. These examples are drawn from virtually all mathematical disciplines and vary enormously in level of difficulty. And how original a number of them are. I certainly haven’t thought of many of them in the full context she presents and I’m willing to bet even professional mathematicians will be seeing many of them for the first time.

She also gives credit “where credit is due” for those who assisted or inspired both this truly incredible number and diversity of examples she’s collected and presented as well as the nearly equinumerous exercises in the text. She credits quite literally dozens of mathematicians she’s communicated with while writing it and past students in her classes. This includes, surprisingly, the aforementioned John Terilla. The mathematical academic world is a small category, indeed!

In fact, I had a lot of trouble picking my favorites to quote from. So I decided to make it easy for myself. I just picked my favorites from the set of examples she presents in Chapter 1. Keep in mind these are just from the first chapter and the density of examples in the book is uniform throughout.

Example 1.1.3. Many familiar varieties of mathematical objects assemble into a category.

i) Set has sets as its objects and functions, with specified domain and codomain, as
its morphisms.
(ii) Top has topological spaces as its objects and continuous functions as its morphisms.

(iii) Set and Top have sets or spaces with a specified basepoint as objects and basepoint preserving (continuous) functions as morphisms.

(iv) Group has groups as objects and group homomorphisms as morphisms. This example lent the general term “morphisms” to the data of an abstract category. The categories Ring of associative and unital rings and ring homomorphisms and Field of fields and field homomorphisms are defined similarly.

(v) For a fixed unital but not necessarily commutative ring R, Mod Ris the category of
left R-modules and R-module homomorphisms. This category is denoted by Vect k
when the ring happens to be a field k and abbreviated as Ab in the case of Mod Z, as
a Z-module is precisely an abelian group.

(vi) Graph has graphs as objects and graph morphisms (functions carrying vertices to
vertices and edges to edges, preserving incidence relations) as morphisms. In the
variant DirGraph, objects are directed graphs, whose edges are now depicted as
arrows, and morphisms are directed graph morphisms, which must preserve sources
and targets.

(vii) Man has smooth (i.e., infinitely differentiable) manifolds as objects and smooth maps
as morphisms.

(viii) Meas has measurable spaces as objects and measurable functions as morphisms.

(ix) Poset has partially-ordered sets as objects and order-preserving functions as morphisms.

(x) Ch R has chain complexes of R-modules as objects and chain homomorphisms as
morphisms.

(xi) For any signature σ specifying constant, function, and relation symbols, and for
any collection of well-formed sentences T in the first-order language associated to
σ, there is a category Model T whose objects are σ-structures that model T, i.e., sets
equipped with appropriate constants, relations, and functions satisfying the axioms
T. Morphisms are functions that preserve the specified constants, relations, and
functions, in the usual sense. Special cases include (iv), (v), (vi), (ix), and (x).

(iii) A poset (P; ≤_) (or, more generally, a preorder ) may be regarded as a category. The
elements of P are the objects of the category and there exists a unique morphism
x → y if and only if x ≤ y. Transitivity of the relation “≤” implies that the required
composite morphisms exist. Reflexivity implies that identity morphisms exist.

(iv) In particular, any ordinal α = { β|β <α} defines a category whose objects are the
smaller ordinals. For example, 0 is the category with no objects and no morphisms. 1
is the category with a single object and only its identity morphism. 2 is the category
with two objects and a single non-identity morphism, conventionally depicted as
0 → 1. is the category freely generated by the graph 0 → 1 →2 → 3 →…. in the sense that every non-identity morphism can be uniquely factored as a composite of morphisms in the displayed graph; a precise definition of the notion of free generation is given in Example 4.1.13.

The examples above are quoted from 2 pages in Riehl’s book.

That’s right, 2 PAGES.

That’s how incredibly dense in examples the book is. While I was reading it, I couldn’t help wondering why no one had written such an example-driven account of category theory before. The subject is a framework for all mathematics. As a result, its’ threads are woven throughout the entire fabric of modern mathematics. So you’d think such an approach would be very-forgive the pun-natural. It may have to do with the fact that until very recently, this was considered a subject at the very apex of mathematical abstraction. As a result, its serious indoctrination was left to very advanced graduate students and research mathematicians. MacLane is clearly written for exactly this purpose. But as stated before, this is no longer an accurate picture of the subject because of the emergence of the many applications mentioned. In all honesty, even if this were not the case, one would think the realization of the interconnected web of categories and morphism woven throughout mathematics would have resulted in such an example-driven approach being written sooner. It’s a bit of a mystery to me why it’s taken so long for one to be written.

But Riehl has now finally written one-and at an advanced level, no less. As clichéd as it is, once again the immortal poem by Robert Frost’s last passage is most appropriate as postscript for Riehl’s success here:

Two roads diverged in a wood, and I—
I took the one less traveled by,
And that has made all the difference.

A few words on the exercises. While there’s quite a few of them, most are straightforward and not too difficult-they either derive further examples that are consequences of the ones given in the text or prove corollaries of the results proven in the chapter. They add more depth to both the exercises and the major results without being necessary to the flow of the text. While one does want to encourage students in mathematics courses-particularly advanced ones-to do as many exercises as possible, I don’t think making the exercises an essential part of the flow of the text is mandatory for learning as long. If the subject matter is particularly challenging, as is the case here, this will frustrate the beginner. We want to encourage students to push through difficult material. I’m pretty sure putting major substantial results for the beginner to prove by themselves does the opposite. We should reserve such exercises for experienced graduate students. However, on the opposite end of the Gauss bell curve, we shouldn’t make exercises so easy they’re useless as practice or learning tools either. Riehl does a very good job of pitching exercises that are just difficult enough to be instructive, but not so difficult the student will throw the book across the room after 7 hours on a single exercise. This makes the exercises just as informative as the examples and there are nearly as many.

As wonderful as the book is, I do have some minor quibbles with it. No textbook is perfect and this one is no different. (Not even the author of a text should consider it perfect. Which should be the real reason authors should write later editions, namely to create a finite sequence of editions that converges to perfection as n → + ∞. Sadly, it usually the last reason even considered by publishers.)

Firstly, I wish in a book this wonderfully researched, that Riehl had incorporated more historical notes. The book actually opens with a detailed account of the birth of category theory-which began as most new branches of mathematics begin, with an attempt to solve a specific problem. In category theory’s case, it grew from MacLane and Samuel Eilenberg’s attempt in 1941 to give a general formulation of the universal coefficient theorem of algebraic topology. This is a wonderfully detailed and researched vignette one rarely sees in textbooks and is to be heartily praised. Unfortunately, it’s a relative anomaly in the book. Another particularly nice example: Section 2.2 on the statement of the Yoneda lemma begins with a quote from a paper by MacLane on how he named the lemma during Yoneda’s visit to France. I wish she’d expanded considerably on this quote in the chapter (although she does include full references in a footnote). I wish she’d inserted many narrative sidebars like the one that opens the book, embedding this difficult material into a rich historical context. The history of scientific and mathematical thought is an amazing tapestry filled with amazing people and events lush with life. I always encourage teachers to mine it as thoroughly as possible. Not only does this bring the subject to life, it gives full answers to the many “why” questions students have. Sooner or later, some student is going to ask you, “Why do we care?” It’s really hard for them to dispute the significance of a subject when you make them walk the path of those who created it-when you show them what lead them to create these concepts. (Of course, there’s always going to be the apathetic sociopath student who’s just passing time until they get into Harvard law or medical school, but we all have to deal with those.) There are some narrative detours like this in Riehl-I just wish there’d been quite a few more.

My major quibble with the book is one I have with a lot of mathematics textbooks, particularly advanced ones. It’s particularly a problem with this one. In fact, I’ll state it as an axiom for future reference:

The Axiom of Textbook Prerequisites: Let M be the minimum actual prerequisites for an average student being able to read and understand a given presentation of a subject in a mathematics textbook. Let A be the author’s stated prerequisites. Then usually:
A <<<<<<< M

Riehl claims in the introduction that only a bare knowledge of basic set theory and logic is needed for the book. Meanwhile,you can't get past the first page without talking about group extensions.

Ok, throwing away that ridiculous claim, what do I think is the bare minimum background needed to be able to effectively read the book i.e. study and understand most of it? I think the student at absolute minimum would need:

1) A decent undergraduate course in abstract algebra, one that contains all the basic definitions and theorems of groups, rings and fields,

2) A good year-long undergraduate course in linear algebra up to and including diagonalization, the Cayley-Hamilton theorem and the Jordan form and

3) A good undergraduate course in topology that goes beyond basic point set topology to the fundamental group and basic homotopy in low dimensional spaces.
But these are both minor quibbles and both are easily fixed. I believe Riehl has written the Great American Category Theory Book for the serious mathematics students at all levels. She is to be heartily commended on writing such a remarkable work and Dover is to be commended for making it available at such a reasonable price. Frankly, if this was the only original work published in Dover’s Aurora series, the line would be worth having. Here’s hoping in the future it produces many more affordable jewels like this one and Riehl's text-and hopefully many equally affordable revised editions-finds their way into the hands of many talented students both now and in the future as a result!
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5つ星のうち5.0 Excelente libro
2018年11月30日にスペインでレビュー済み
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Muy buen libro para profundizar en teoría de categorías, aunque no lo recomendaría para un primer contacto con la materia (Awodey es un buen comienzo).
Si bien la impresión no es de buena calidad (bajo gramaje de las hojas), no se puede pedir más por un precio tan bajo. También es de destacar que la autora haya tenido la gentileza de dejar disponible el pdf gratis para descargar desde su página web.
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