What are the best books and lecture notes on category theory?
[Math] Good books and lecture notes about category theory.
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Related Solutions
Read "Category Theory" by Steve Awodey. It is a rigorous introduction to category theory (goes as far as adjoints, some monads, Yoneda, ... ) which intentionally does NOT include examples that only a maths major can understand. Instead, its examples are drawn from logic, lambda calculus, etc.
In no particular order:
- Algebraic number theory notes by Sharifi: http://math.arizona.edu/~sharifi/algnum.pdf
- Dalawat's first course in local arithmetic: http://arxiv.org/abs/0903.2615
- Intro to top grps: http://www.mat.ucm.es/imi/documents/20062007_Dikran.pdf
- Representation theory resources: http://www.math.columbia.edu/~khovanov/resources/
- Classical invariant theory: http://jones.math.unibas.ch/~kraft/Papers/KP-Primer.pdf
- CRing project: http://people.fas.harvard.edu/~amathew/CRing.pdf - The notes are huge & has many authors - including MSE's Zev, Akhil (no longer active) & Darij. Check the ToC.
- Partitions bijections, a survey: http://www.math.ucla.edu/~pak/papers/psurvey.pdf
- Hidden subgroup problem (review, open stuff): http://arxiv.org/abs/quant-ph/0411037
- Spirit of moonshine: http://www.math.harvard.edu/theses/senior/booher/booher.pdf
- Vertex operator algebras and modular forms: http://arxiv.org/abs/0909.4460
- Categorified algebra & quantum mechanics: http://arxiv.org/abs/math/0601458
- Exponential sums over finite fields: http://www.math.ethz.ch/~kowalski/exp-sums.pdf
- Gauss sums: http://math.mit.edu/~brubaker/houghexpsums05.pdf
- Adeles over $\Bbb Q$: https://www.maths.nottingham.ac.uk/personal/ibf/text/gl1.pdf, followed by automo reps over GL(1,A) https://www.maths.nottingham.ac.uk/personal/ibf/text/gl2.pdf
- Invariant thry: http://www.win.tue.nl/~jdraisma/teaching/invtheory0910/lecturenotes11.pdf
- Species: http://www.newton.ac.uk/programmes/CSM/Abstract3/Species_intro.pdf
- FLT: http://www.math.mcgill.ca/darmon/pub/Articles/Expository/05.DDT/paper.pdf
- Categorical concepts: http://www.math.harvard.edu/~eriehl/266x/survey.pdf
- Groups, Rings, Fields (Lenstra): http://websites.math.leidenuniv.nl/algebra/topics.pdf, which is part of algebra notes: http://websites.math.leidenuniv.nl/algebra/
If we're going to mention Hatcher (famous to me for the algebraic topology notes), we might as well also mention a few other books that are online, like Algebra chapter 0, Stanley's insane first volume of Enumerative Combinatorics (which reminds me: generatingfunctionology). Also I don't see topology without tears mentioned. The sheer number of books and notes on differential geometry and lie theory is mind-boggling, so I'll have to update later with the juicier ones.
Let's not forget the AMS notes online back through 1995 - they're very nice reading as well.
Best Answer
Categories for the Working mathematician by Mac Lane
Categories and Sheaves by Kashiwara and Schapira