# Is Mac Lane still the best place to learn category theory?

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?

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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 1 '11 at 11:38
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-Alvarez Jul 1 '11 at 12:17
(The idea of a derived generation makes my cringe a little...) – Mariano Suárez-Alvarez Jul 1 '11 at 12:25
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 1 '11 at 12:42
@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 1 '11 at 14:08

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. 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.

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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 1 '11 at 15:17
See (math.harvard.edu/~lurie/papers/moduli.pdf) For the ICM address. – Spice the Bird Jul 1 '11 at 19:35
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 1 '11 at 19:37
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 3 '11 at 8:21
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 6 '11 at 13:43

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.

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.

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.

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"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.

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

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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 Chengs notes on category theory was tremendously useful

Eventually, Maclane began to make sense, as did Borceaux; 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.

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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

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