93

For a student embarking on a study of algebraic topology, requiring a knowledge of basic category theory, with a long-term view toward higher/stable/derived category theory, ...

Is Mac Lane still the best place to start?

Or has the day arrived when it is possible to directly learn ($\infty$,n)-categories, without first learning ordinary category theory? (So the next generation will be, so to speak, natively derived.) If so, via what route? If not, what's the most efficient path through the classical core material to a modern perspective?

cdouglas
  • 3,083
  • 28
    The title question looks like it's asking the best book to learn category theory from (which anyway may be impossible to answer; different books address different needs), but the actual question seems to be whether you need to learn 1-category theory before some of the more modern theories. My opinion is that you should absolutely learn some 1-category theory first. – Todd Trimble Jul 01 '11 at 11:38
  • 59
    My opinion is that one should learn most of category theory before one actually learns category theory, in the form of examples. As a corollary, the best place to learn category theory is in a good algebra textbook together with a good topology textbook and, for optimal rsults, a good algebraic topology textbook. – Mariano Suárez-Álvarez Jul 01 '11 at 12:17
  • 17
    (The idea of a derived generation makes my cringe a little...) – Mariano Suárez-Álvarez Jul 01 '11 at 12:25
  • I agree with Todd. It depends a bit on where you want to go, but I think Borceux is a better book than Mac Lane for a wide range of possible journeys. It is clean, and emphasises the inspiration and application quite nicely. – James Cranch Jul 01 '11 at 12:34
  • 11
    There's a bit of truth to that, Mariano, although "most of category theory" is an exaggeration, and it doesn't address the OP's concern. My point is that as of the present day, there is a lot of stuff in $(\infty, n)$-category theory which I don't think will make much sense without first having studied a lot of category theory per se. – Todd Trimble Jul 01 '11 at 12:42
  • 2
    I agree with Mariano. In my opinion, the best place to learn category theory is John Baez's TWF! – Qiaochu Yuan Jul 01 '11 at 12:48
  • 5
    I am not sure what is the best place to learn category theory, but I do feel that MacLane's book is now a bit outdated and old-fashioned. It is I think still fine for learning the basic definitions and the theory up to adjunctions, but not the best.

    Altbough non-traditional, I like Bergman's introduction to category theory in his book on universal algebra. It only covers the basic notions, but it does it in a very concrete and motivated way. After reading his book, one can jump directly to Borceux or to the chapter on categories in MacLane-Mordeijk's topos book.

     – Benjamin Steinberg Jul 01 '11 at 13:18
  • 6
    For a modern view on basic category theory, I'm quite fond of Kashiwara&Shapira's "Categories and sheaves" -- tough indeed knowing about algebra, topology and algebraic topology can definitely help understanding the examples. – Julien Puydt Jul 01 '11 at 13:23
  • 14
    @Qiaochu: Baez's TWF will give one a taste of a variety of topics, definitely. But if you want to learn $(\infty, n)$-category, for example if you want to read Jacob Lurie's stuff, just reading the TWF won't cut it. You have to hunker down and really learn category theory. Let's please stay on topic here. – Todd Trimble Jul 01 '11 at 14:08
  • 4
    I prefer Horst Schubert's "Categories" (MR0349793).

    Does there exist a textbook on categories that works with universes by now? Would seem useful to me, in order to get rid of questions like "existence of functor categories", "existence of localisations", "existence of the set of isoclasses".

     – Matthias Künzer Jul 01 '11 at 15:49
  • 2
    Just a thought (possibly unrealistic): Learn category theory along with some higher versions concurrently. That is go back and forth between the two subjects. – Spice the Bird Jul 01 '11 at 19:31
  • 4
    @Birdman: ambitious, but not completely unrealistic, no. From what I understand, Harry Gindi has been doing something similar to that; in particular, he has been steadily working his way through some of Lurie's works, and back-filling as needed (I hope he doesn't mind my putting it like that). – Todd Trimble Jul 01 '11 at 20:12
  • 5
    @Todd: True, but I was familiar with a good deal of the un-enriched theory before I even dared to start looking at HTT. That is, I'd already worked through a good deal of Lang's Algebra and read through some of the bits of the Stacks Project. The trouble really is that the theory of n-categories and oo-categories simply will not make sense without understanding 1-categories. Also, reading HTT without a strong intuition for model categories and simplicial sets is a waste of time because a.) the proofs are very terse, and b.) most of the results are of the form M is Quillen-eq to N. – Harry Gindi Jul 22 '11 at 07:49
  • 5
    I consider the first six months I spent struggling with HTT to be worthless, since it would have been far more worthwhile to read something like Hirschhorn's Model Categories and their Localizations, Quillen's Homotopical Algebra, Goerss-Jardine's Simplicial Homotopy theory, and later on, Cisinski's Les Prefaisceaux Comme Modeles des types d'homotopie. Before that, I think another invaluable resource is Ch. 2 and 3 of Vistoli's notes on descent, which is one of the best references on sheaves, presheaves, etc. Also, Mac Lane or Kashiwara-Schapira should still be used for (ctd) – Harry Gindi Jul 22 '11 at 07:56
  • 5
    (ctd) adjoint functors, monads, and Kan extensions. Last but not least, it's worth reading about at least the very basics of enriched category theory. I know that some will protest here, but I don't think that Kelly's Basics of Enriched Category Theory is really a very good source to look at. Instead, there is a paper by Eilenberg and Kelly in the proceedings of the conference on categorical algebra in La Jolla (1967?) that should be read, perhaps in conjunction with Kelly's aforementioned book, since it contains the majority of the proofs that Kelly left out of the book. – Harry Gindi Jul 22 '11 at 08:01
  • 1
    So anyway, in conclusion, it is unreasonable to try to learn about the current theory of (oo,n)-categories at a reasonable pace without knowing any category theory at all. In fact, I cannot think of a single definition of an n-category (strict or weak) that does not use the framework of 1-category theory. – Harry Gindi Jul 22 '11 at 08:04
  • 5
    The point where knowing some oo-category theory will make aspects of 1-category theory more clear, instead of vice versa, is where it comes to 1-category theory that is secretly really a presentation of oo-category theory. This applies for instance to many of the subjects that are announced to constitute part III of Johnstone's book on topos theory ("Sketches of an Elephant"): cohomology theory and homotopy theory internal to a topos or other category. In pure 1-category theory this involves quite a few ad-hoc concepts that seem to come from nowhere... – Urs Schreiber Sep 05 '11 at 19:54
  • 4
    ... but are easily recognized as presentations of conceptually simple and canonical constructions in oo-category theory. Realizing this makes a whole lot of complicated-looking material become very simple. This is where knowing oo-category theory (and its presentations by 1-category theory tools) makes things easier to learn. (Sometimes I am wondering if this fact is correlated to the fact that part III of the Elephant is not yet there...) – Urs Schreiber Sep 05 '11 at 19:57
  • 1
    I think that MacLane CWM is still the better place for start to learn category theory (or for a more introductory level the MAcLAne-Birkhoff "ALgebra", or a Lawrere-shanuel Conceptual Mathematics: A First Introduction to Categories). but I think is wrong start to study "Higher -categiory" theory without a background on the 2-level of category theory: 2-categories, double-categories, bicategories, fibrations, lax-functors and lax-transformation..then monoidal, braids, enriched categories, ecc. this 2 -level is very richt and qualitatively (and no only quantitatively) different. – Buschi Sergio Dec 03 '12 at 14:28
  • 3
    I know this is an old thread, but I wanted to record my disagreement with @MarianoSuárez-Alvarez's comment. He said you should learn most of category theory through examples before beginning formal study of it. I think that the traditional way is much better: you learn through examples only the most basic parts of category theory, namely, categories, functors, natural tranformations, adjunctions, some special limits and colimits including products and coproducts, and the Yoneda lemma. Other than the Yoneda lemma, these are just definitions. After this you start studying category theory. – Omar Antolín-Camarena Apr 08 '14 at 14:05
  • 2
    @OmarAntolín-Camarena,for most people, that is pretty much all of category theory one will learn, ever. If you are going to work on the subject or use it in a way deeper than as a language and a toolbox, of course you need to study it as a separate subject. – Mariano Suárez-Álvarez Apr 08 '14 at 14:17
  • 3
    Oh, I see. I thought you meant something very, very different by the phrase "most of category theory" in your comment, @MarianoSuárez-Alvarez. If you meant what I would call "only the most basic definitions", we are in agreement! – Omar Antolín-Camarena Apr 08 '14 at 14:39
  • I made this CW as it calls for opinions and for a (big) list of texts. – Todd Trimble Apr 24 '15 at 14:22
  • I never could get through Mac Lane. I found it extremely boring and picked up most of what I know from lecture notes, the stacks project, and some other assorted sources. – Harry Gindi Aug 27 '17 at 18:53

9 Answers9

41

I doubt that someone could learn higher category theory (and more in general higher dimensional algebra) without first studying a little of category theory, mostly because the definition given in such context use a lot of category theoretic machinery. About the textbook reference: MacLane's "Category theory for working mathematicians" may be a little outdated but I think it is still one of the most complete book of basic category theory second just to Borceux's books. Anyway there isn't a best book to learn basic category theory, any person could find a book better than another one, so I suggest you to take a look a some of these books, then choose which one is the best for you:

S. MacLane: Category theory for working mathematicians (I've already said a lot about this)

S. Awodey: Category theory (Peculiar because it has very low prerequisites and it's rich of examples too)

J. Adamek,H. Herrlich, G. Strecker: Abstract and concrete category theory (freely avaible at at this site "http://katmat.math.uni-bremen.de/acc/acc.pdf", maybe the book with the greatest number of examples from topology and algebra)

After you have read one of these book, you could also use Borceux's books and read some more advanced chapter of category theory which aren't discussed in the previous books.

F. Borceux: Handbook of Categorical Algebra 1: Basic Category Theory

F. Borceux: Handbook of Categorical Algebra 2: Categories and Structures

F. Borceux: Handbook of Categorical Algebra 3: Categories of Sheaves

For higher category theory I know just few reference:

Leinster's "Higher Operads Higher Categories" (http://arxiv.org/abs/math/0305049),

and

Lurie's "Higher Topos Theory" (http://arxiv.org/abs/math/0608040)

other good reference in higher category theory and higher dimensional algebra in general are Baez'This week's finds and arxiv articles Higher dimensional algebra*.

Hope this may help.

Giorgio Mossa
  • 3,299
  • 4
    Depending on OP's interests, HTT might not be the place to start on higher category theory. I found "On the Classification of TQFTs" more readable, because Lurie doesn't there try to give all detailed definitions, just outline a theory. "Higher Algebra" is supposed to be a follow-up to HTT, and uses much of the machinery developed there, but I liked it better because it's more about algebra; for an overview of some of this material, Lurie's ICM address isn't bad. – Theo Johnson-Freyd Jul 01 '11 at 15:17
  • 1
    See (math.harvard.edu/~lurie/papers/moduli.pdf) For the ICM address. – Spice the Bird Jul 01 '11 at 19:35
  • 3
    The first chapter of Leinster's Higher operads, higher categories gives a nice and quick introduction to category theory. I learned alot from there. – Spice the Bird Jul 01 '11 at 19:37
  • 4
    I agree 100% with Todd's comments that one definitely needs a solid grounding in 1-category theory before learning higher category theory. As far as a textbook for 1-category theory goes, I'm fond of Awodey's book. ACC is good too, but also rather idiosyncratic (in different ways than Mac Lane). – Mike Shulman Jul 03 '11 at 08:21
  • 3
    I'm a big fan of Borceux's Handbook of Categorical Algebra 1. Rarely have I had a question about categories which it has been unable to answer. – David White Sep 06 '11 at 13:43
26

I just reviewed what I firmly believe will be the book that will replace MacLane as The Gold Standard for introductions to category theory for graduate students: Category Theory in Context by Emily Riehl. It's developed over the last several years from courses in category theory that Riehl has taught at Harvard and John Hopkins University to strong undergraduates and first year graduate students. She posted her evolving notes at her website each time and the first time I saw a rough draft, I knew she was writing something special. The finished text did not disappoint. It's comprehensive, incredibly clear and amazingly rich in examples, including many you've probably never considered. It's a remarkable book and I think it's going to replace MacLane very quickly once it's known to most experts. Best of all, it's much cheaper then MacLane!

If you're interested in category theory, this is the book you want to learn from. I wish it was the one I'd learned from.

My full review can be found here.

Martin Sleziak
  • 4,608
The Mathemagician
  • 467
  • 4
    Emily Riehl's book is great (I've read maybe 80% of it and learned a lot despite having already known some category theory), but she would probably have to add some chapters on abelian and monoidal categories in order for her book to fully substitute Maclane's :) Or has this subject become so separate nowadays that we are no longer counting it into category theory? – darij grinberg Aug 27 '17 at 20:31
  • 1
    @darijgrinberg I wouldn't think so-I think Emily just thought it wasn't important enough to include in a first course. You can't put EVERYTHING into a textbook-that's how Spanier ruined his algebraic topology book.......lol That being said-I'll suggest she include them in the second edition if she writes one. – The Mathemagician Aug 28 '17 at 04:04
  • @darij: Yeah, I'd say they're separate enough and you're better off reading Riehl and then moving on to EGNO's Tensor Categories if you're wanting to know more about monoidal categories. – Noah Snyder Dec 30 '17 at 15:09
  • Thank you, I was getting second thoughts from printing Riehl's book instead of something else. Now they are gone. – Daniel Teixeira Jan 28 '19 at 09:55
25

I third what Mike wrote: "one definitely needs a solid grounding in 1-category theory before learning higher category theory". With that being said, elaborating and expounding upon janed0e's suggestion, what follows are two study plans according to the prior knowledge of the student. Of course, there is no canonical way to approach learning higher category theory, so adjust the readings as needed. Note well, following the modern terminology as developed by Joyal, quasicategories are a model for ($\infty$, 1)-categories. Following the modern terminology as developed by Lurie, the unqualified usage of '$\infty$-category' or '$\infty$-categories' designates '($\infty$, 1)-category' and '($\infty$, 1)-categories', respectively.

Assumption: Student has no knowledge of 1-category theory (or simplicial sets) and wishes to get the flavor of infinity-category theory, without getting bogged down by technical details, in as short a time as can be reasonably expected. The implicit assumption is that the student has a budget of zero dollars.

Possible reading material and sequence with which to read:

0) J. Adamek, H. Herrlich, G. Strecker: Abstract and Concrete Categories: The Joy of Cats
1) G. Friedman: An elementary illustrated introduction to simplicial sets
2) J. Lurie: What is ... an $\infty$-Category?
3) M. Boyarchenko: Notes and Exercises on $\infty$-categories
4) M. Groth: A Short Course on $\infty$-categories(http://www.math.ru.nl/~mgroth/preprints/groth_scinfinity.pdf)

Repeating what Giorgio Mossa wrote, (0) has an abundant number of examples from topology, algebra, and theoretical computer science. As Mike Shulman noted, (0) is rather idiosyncratic. (0) uses the term 'quasicategory' for what Mac Lane called metacategories. See the nLab page metacategory (http://ncatlab.org/nlab/show/metacategory) for further clarification about the terminology clash. (0) can be supplemented with video lectures by the Catsters (http://www.scss.tcd.ie/Edsko.de.Vries/ct/catsters/linear.php) and Wikipedia's Outline of category theory (http://en.wikipedia.org/wiki/Outline_of_category_theory).

Assumption: Student has knowledge of 1-category theory (but not simplicial sets) and wishes to get an in depth experience of infinity-category theory, allowing an 'ample' amount of time.

Possible reading material and sequence with which to read:

0) P. G. Goerss and J. F. Jardine: Simplicial Homotopy Theory (http://dodo.pdmi.ras.ru/~topology/books/goerss-jardine.pdf)
1) J. Lurie: What is ... an $\infty$-Category?
(http://www.ams.org/notices/200808/tx080800949p.pdf)
2) M. Boyarchenko: Notes and Exercises on $\infty$-categories (http://www.math.uchicago.edu/~mitya/langlands/quasicategories.pdf)
3) M. Groth: A Short Course on $\infty$-categories
(http://www.math.uni-bonn.de/~mgroth/InfinityCategories.pdf)
4) J. Lurie: On the Classification of Topological Field Theories (http://arxiv.org/abs/0905.0465)
5) C. Simpson: Homotopy Theory of Higher Categories
(http://hal.archives-ouvertes.fr/docs/00/44/98/26/PDF/main.pdf)
6) J. Lurie: Higher Topos Theory
(http://www.math.harvard.edu/~lurie/papers/highertopoi.pdf)

(4) may be a more readable than (6), since (4) is an expository paper that gives an informal account of the classification of topological field theories using the technology of ($\infty$, n)-categories. (4) can be nicely supplemented by Lurie's video lecture series on "Topological Quantum Field Theories and the Cobordism Hypothesis" (http://lab54.ma.utexas.edu:8080/video/lurie.html), as well as the corresponding notes for said lecture (http://www.ma.utexas.edu/users/plowrey/dev/rtg/notes/perspectives_TQFT_notes.html).

(5) offers a broad perspective of current research in higher category theory.

(6) develops in detail the vast generalization of 1-category theory to ($\infty$, 1)-category theory. For further roadmaps on learning higher category theory, look at this nForum discussion on reading Lurie's Higher Topos Theory (http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=2748&page=1#Item_0).

Hope this helps.

Patrick Wingert
  • 3
Stephen Britton
  • 141
  • 2
  • 4
10

You may find this helpful:

"A Short Course on Infinity-Categories" by Moritz Groth

http://www.math.uni-bonn.de/~mgroth/InfinityCategories.pdf

You'll first need to learn homotopy theory.

Reference [GJ99](Simplicial Homotopy Theory) in the above link could be a place to start.

janed0e
  • 71
  • 2
7

I would start from "Sets for mathematics", and then going to MacLane.

tp1
  • 41
  • 1
6

I found the Catsters on YouTube divinely useful.

John Baez, in his not so weekly blog, inspiring.

The n-category cafe, to keep you going.

Eugenia Cheng's notes on category theory was tremendously useful.

Eventually, Mac Lane began to make sense, as did Borceux; but oh, ever so slowly.

Sets for mathematicians is pretty.

And the n-lab is a great resource, but mostly dazzles my eyes...

And yes, 1-category theory is definitely best to start with, and be familiar with; but keep an eye on the higher grounds too.

Arnaud D.
  • 555
  • 4
  • 15
Mozibur Ullah
  • 2,043
5

Best paper to get a feel for Category Theory is "When is one thing equal to some other thing" by Barry Mazur. The paper can be obtained at--

http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf

Alex M.
  • 5,217
Viswakarma
  • 21
5

I'm a fan of Kashiwara and Schapira's "Categories and sheaves"... they do cover a lot of material ; you can see the table of contents here: http://books.google.com/books/about/Categories_and_sheaves.html?id=K-SjOw_2gXwC

Julien Puydt
  • 2,013
4

I recommend for a first reading on category theory:

Martin Brandenburg, Einführung in die Kategorientheorie

It is really an excellent exposition with some nice perspectives on the concepts,
supported by plenty of examples. :)

(Note that it is provided in German only.)

A lot of thanks to Konrad S. for suggesting to me the reference!
(Such a pity, that I just missed the chance to meet the author in person..)

Ben McKay
  • 25,490
C-star-W-star
  • 598