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

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

2011-07-01T11:26:34Z

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?

Answer by tp1 for Is Mac Lane still the best place to learn category theory?

2011-07-01T13:28:29Z

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

Answer by Giorgio Mossa for Is Mac Lane still the best place to learn category theory?

2011-07-01T14:46:26Z

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 About the textbook reference: MacLane's "Category theory for working mathematicians" may be a little outdated but I think it is still one of the most complete book of basic category theory second just to Borceux's books.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.

Answer by janed0e for Is Mac Lane still the best place to learn category theory?

2011-07-07T21:14:36Z

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.

Answer by Stephen Britton for Is Mac Lane still the best place to learn category theory?

2011-07-21T10:50:40Z

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

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.

Answer by Julien Puydt for Is Mac Lane still the best place to learn category theory?

2011-09-05T18:22:36Z

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

Answer by Mozibur Ullah for Is Mac Lane still the best place to learn category theory?

2012-12-03T12:39:47Z

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.