A Gentle Introduction to Category Theory (Feb 2016 version) by Peter Smith.
From the preface:
This Gentle Introduction is work in progress, developing my earlier ‘Notes onBasic Category Theory’ (2014–15).
The gadgets of basic category theory fit together rather beautifully in mul-tiple ways. Their intricate interconnections mean, however, that there isn’t asingle best route into the theory. Different lecture courses, different books, canquite appropriately take topics in very different orders, all illuminating in theirdifferent ways. In the earlier Notes, I roughly followed the order of somewhatover half of the Cambridge Part III course in category theory, as given in 2014by Rory Lucyshyn-Wright (broadly following a pattern set by Peter Johnstone;see also Julia Goedecke’s notes from 2013). We now proceed rather differently.The Cambridge ordering certainly has its rationale; but the alternative orderingI now follow has in some respects a greater logical appeal. Which is one reasonfor the rewrite.
Our topics, again in different arrangements, are also covered in (for example)Awodey’s good but uneven Category Theory and in Tom Leinster’s terrific – and appropriately titled – Basic Category Theory. But then, if there are some rightly admired texts out there, not to mention various sets of notes on category theory available online (see here), why produce another introduction to category theory?
I didn’t intend to! My goal all along has been to get to understand what light category theory throws on logic, set theory, and the foundations of mathematics. But I realized that I needed to get a lot more securely on top of basic category theory if I was eventually to pursue these more philosophical issues. So my earlier Notes began life as detailed jottings for myself, to help really fix ideas: and then – as can happen – the writing has simply taken on its own momentum. I am still concentrating mostly on getting the technicalities right and presenting them in apleasing order: I hope later versions will contain more motivational/conceptual material.
What remains distinctive about this Gentle Introduction, for good or ill, is that it is written by someone who doesn’t pretend to be an expert who usually operates at the very frontiers of research in category theory. I do hope, however,that this makes me rather more attuned to the likely needs of (at least some)beginners. I go rather slowly over ideas that once gave me pause, spend more time than is always usual in motivating key ideas and constructions, and I have generally aimed to be as clear as possible (also, I assume rather less background mathematics than Leinster or even Awodey). We don’t get terribly far: however,I hope that what is here may prove useful to others starting to get to grips with category theory. My own experience certainly suggests that initially taking things at a rather gentle pace as you work into a familiarity with categorial ways of thinking makes later adventures exploring beyond the basics so very much more manageable.
Check the Category Theory – Reading List, also by Peter Smith, to make sure you have the latest version of this work.
Be an active reader!
If you spot issues with the text:
Corrections, please, to ps218 at cam dot ac dot uk.
At the category theory reading page Peter mentions having retired after forty years in academia.
Writing an introduction to category theory! What a great way to spend retirement!
(Well, different people have different tastes.)