Basic Category Theory
Tom Leinster
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Prerequisites: Some general background in algebra or topology is helpful for motivation.
An excellent first introduction to category theory, this text is full of useful examples and exercises for the reader. This book is very short, being less than 200 pages, but thoroughly covers standard core material that often appears in other places in mathematics. A good choice if you want to quickly learn basic category theory for applications in other areas.
Category Theory in Context
Emily Riehl
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Prerequisites: A strong background in algebra or topology is helpful.
Another classic introduction to category theory. This book goes further and faster than Leinster’s text above, having additional chapters on monads and kan extensions. This is the main reference for the category theory module at Warwick (MA4M6 Category Theory).
Category Theory
Steve Awodey
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Prerequisites: Some background in algebra.
Another, more modern, introductory text for category theory. In comparison to the previous two texts above, this textbook contains many more examples from logic and computer science, and is suitable for a much wider audience. Despite the lighter prerequisites, full proofs of important results are still included with all the details.
Apart from the standard material covered in Leinster’s text, this book has additional chapters on cartesian closed categories (though not general monoidal categories or monads) and the lambda calculus, which logicians and computer scientists will enjoy.
Categories for the Working Mathematician
Saunders Mac Lane
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Prerequisites: A very strong background in algebraic topology and algebraic geometry.
A classic reference text for category theory, written by one of the co-founders of the subject. It is detailed and comprehensive, but its age is rather noticable; much notation and terminology is outdated, and differs from many other more modern texts.
This is certainly a reference text, and not an ideal introduction to category theory for a beginner – it is for the Working Mathematician after all. However, long proofs that are omitted in other sources will often be present here, albeit often in very terse terms.