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How do I go about teaching myself category theory?

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I have to ask *why* you want to learn category theory. My guess is that you are drawn to it because it's a currently popular buzzphrase in programming language design, where the people who think about how data types ought to work have latched onto category theory as vaguely relevant.

If that's the case, then trying to learn what most mathematicians call "category theory" will be *utterly meaningless* to you. Instead, maybe start by searching the web for "category theory for programmers". There's a book by Bartosz Milewski with that title, which the author has apparently made available online in some places (though of course you can support him by buying it). There also seem to be some YouTube video series on the subject.

The original category theory won't make sense until you have motivated it by learning  the areas of higher math that motivated it, mostly abstract algebra and algebraic topology. But the version that computer scientists have been talking about recently might make sense.
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If you only have studied Calculus previously, the concepts introduced in Category Theory might feel weird and it will be hard to see the motivation behind them (this includes even the very first definitions). Also, you will have a hard time finding examples to work with, since they usually come from various areas of math. I would suggest studying some Linear Algebra (vector spaces and linear transformations stuff), Abstract Algebra or Topology beforehand, but it might still be a bit hard to grasp on the beginning.   
Category Theory is better taken later on because it generalizes some properties one may have encountered during their studies which show up in seemingly very distant parts of math. So I guess it's not the best subject to rush towards. It is, however, certainly still possible to learn it, I just think it will be a lot harder to grasp the material without good examples and without the required mathematical maturity.

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