What is a good resource to have a first look at category theory?

Question

In the introduction of Jean-Luc Brylinski's book "Loop Spaces, Characteristic Classes and Geometric Quantization it says that to read this book one should have a basic knowledge of point-set topology, manifolds, differential geometry, graduate algebra, be familiar with basic facts regarding Lie groups, Hilbert spaces  and categories.

So what is a good resource to have a first look at category theory in this context.

The book describes among other things two levels for looking at degree-$3$ cohomology theory, and the more abstract one involves sheaves and groupoids which is where I suspect category theory kicks in (?).

In particular I would be also thankful if somebody could tell me what I need to look at first in the context of the book...

Comments

I am not an expert but I think that is more better to try to read the book directly.  After a first reading you could know what is the necessary background in category theory.

Answer 1

I have never studied categories systematically from a single source, but there are few ones from which I studied the subject.

A very short introduction is given in the following review by Dijkgraaf (Les Houches Lectures on Fields, Strings and Duality): https://arxiv.org/abs/hep-th/9703136.

In the context of Mirror Symmetry (Fukaya category, etc.) the topic is discussed in the following two weighty textbooks on Mirror Symmetry: http://www.claymath.org/library/monographs/cmim01c.pdf, http://www.claymath.org/library/monographs/cmim04.pdf.

Physics-oriented discussion by physicist is presented in this review of supersymmetry by Tachikawa (A Pseudo-mathematical Pseudo-review on $4d$ ${\cal N}=2$ Supersymmetric Quantum Field Theories): http://member.ipmu.jp/yuji.tachikawa/tmp/review-rebooted7.pdf.

My friend mathematician recommended to start from this book by Mac Lane (Categories for the Working Mathematician): http://www.maths.ed.ac.uk/~aar/papers/maclanecat.pdf.

Not-very-thorough discussion from a slightly non-standard view point is given here (Categories for the Practising Physicist): https://arxiv.org/abs/0905.3010.

I hope it will be useful.

Comments

Thanks Andrey, I will have a look at them ...

Answer 2

Leinster's book Basic Category Theory is a gentle but solid introduction, and it has the virtue that it was just recently released for free on arXiv: https://arxiv.org/abs/1612.09375

I've found Awodey's book Category Theory to be of a similar level. Here also is a playlist of videos from 4 lectures of Awodey teaching introductory category theory: https://www.youtube.com/playlist?list=PLGCr8P_YncjVjwAxrifKgcQYtbZ3zuPlb

Answer 3

As a mathematician, I first met categories in the context of algebraic topology (one of the main areas in which they originated) in Hatcher's Algebraic Topology, available as a free PDF on his website here.

For (smooth) manifolds specifically, my introduction was Bott and Tu's Differential Forms in Algebraic Topology.

I found both of these books pretty thorough and clear to read.

Answer 4

Yet another book that could be quite useful as, inter alia, an introduction to category theory specifically geared towards physicists is Mathematical Physics by Robert Geroch. 

Comments

A very good choice. If you liked the approach and if you see all sorts of dualities in maths, you can continue with Olivia Caramello ( on arxiv and amazon )