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Is category theory worth learning for the sake of learning it? Can it be used in applied mathematics/probability? I am currently perusing Categories for the Working Mathematician by Mac Lane.

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Pet peeve: it's Mac Lane, with a space and capital letters in both parts. – Arturo Magidin Nov 2 '10 at 3:46
I guess I wrote his original name. – PEV Nov 2 '10 at 3:50
Not quite: the original name lacked the space, but not the capital L. (-; – Arturo Magidin Nov 2 '10 at 3:53
up vote 6 down vote accepted

It depends on whether you are talking about Category Theory as a topic in mathematics (on a par with Geometry or Probability) or Category Theory as a viewpoint on mathematics as a whole.

If the former, the main prerequisite is that you should have encountered a situation where you wanted to move from one type of "thing" to another type of "thing": say from a group to its group ring, or from a space to its ring of functions, or from a manifold to its differential graded algebra.

If the latter, then there are no prerequisites and it is a Very Good thing to do! But if the latter, then reading Mac Lane isn't necessarily the best way to go. However, I'm not sure if there is a textbook (or other) that tries to teach elementary mathematics (of any flavour) from a categorical viewpoint. I try to teach this way, but I've not written a textbook! I wrote a bit more on this in response to a question on MO, I copied my answer here.

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Well, there's Linderholm's Mathematics Made Difficult, which treats elementary arithmetic from a categorical viewpoint -- with tongue firmly embedded in cheek... – Rahul Nov 2 '10 at 13:14
Peter May's "Concise Course in Algebraic Topology" would be an excellent way to learn some category theory while learning other stuff. – Ryan Budney Nov 2 '10 at 14:34
@Ryan, @Rahul: Great! Didn't know of the first, and didn't know that the second was so categorical. I'll add them to my list of books to take a look at. – Loop Space Nov 2 '10 at 20:36

I do not know of any specific applications to probability ("applied mathematics" is a bit too wide a label, though I don't know any specific ones either). I view Category Theory as an interesting unifying language that allows you to speak about ideas that may seem to be isolated from each other in different subjects with the same notions, recognizing the connections between them. The results that I most often actually find useful have to do with adjuctions (specifically, I seem to find very useful the fact that left adjoints commute with colimits and right adjoints commute with limits...).

I would say that it is good to know at least the most general ideas, enough so that you can communicate and recognize those connections; of course, I'm a big fan of knowing stuff "just for the sake of knowing it" so I may not be the best advice giver in that respect.

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This issue has been discussed multiple times in MO. For instance here, with a view towards programming, or here, where I proposed a selection of Mac Lane's book.

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Another obvious link would be the question here on Math.SE:… – BBischof Nov 2 '10 at 15:19

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