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

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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...

edited Feb 11, 2017

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.

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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.

answered Feb 11, 2017 by (815 points)
edited Feb 11, 2017

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

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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

answered Feb 12, 2017 by (0 points)
edited Feb 12, 2017 by JohnnyMo
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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.

answered Feb 12, 2017 by (10 points)
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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.

answered Nov 16, 2018 by (40 points)

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 )

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