My friend Tom Leinster has written a great introduction to that wonderful branch of math called category theory!   It's free:

https://arxiv.org/abs/1612.09375

It starts with the basics and it leads up to a trio of related concepts, which are all ways of talking about universal properties.

Huh?  What's a 'universal property'?

In category theory, we try to describe things by saying what they do, not what they're made of.  The reason is that you can often make things out of different ingredients that still do the same thing!  And then, even though they will not be strictly the same, they will be isomorphic: the same in what they do

A universal property amounts to a precise description of what an object does.

Universal properties show up in three closely connected ways in category theory, and Tom's book explains these in detail:

through representable functors (which are how you actually hand someone a universal property),

through limits (which are ways of building a new object out of a bunch of old ones),

through adjoint functors (which give ways to 'freely' build an object in one category starting from an object in another).

If you want to see this vague wordy mush here transformed into precise, crystalline beauty, read Tom's book!  It's not easy to learn this stuff - but it's good for your brain.  It literally rewires your neurons.

Here's what he wrote, over on the category theory mailing list:

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Dear all,

My introductory textbook "Basic Category Theory" was published by Cambridge University Press in 2014.  By arrangement with them, it's now also free online:

https://arxiv.org/abs/1612.09375

It's also freely editable, under a Creative Commons licence.  For instance, if you want to teach a class from it but some of the examples aren't suitable, you can delete them or add your own.  Or if you don't like the notation (and when have two category theorists ever agreed on that?), you can easily change the Latex macros.  Just go the arXiv, download, and edit to your heart's content.

There are lots of good introductions to category theory out there.  The particular features of this one are:

• It's short.
• It doesn't assume much.
• It sticks to the basics.
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