I just love the opening two sentences of Tom Leinster’s 2014 introductory book, which still seem about as good a minimal sketch of what category theory is up to as you could hope for:
Category theory takes a bird’s eye view of mathematics. From high in the sky, details become invisible, but we can spot patterns that were impossible to detect from ground level.
Taking a bird’s eye view, and thereby seeing better how things hang together, is the sort of intellectual enterprise that appeals to philosophers. Hence — according to me — the interest of category theory for those interested in the philosophy of mathematics.
I have praised Leinster’s book here before (I think it is terrific, and it worked well with a student reading group of mathematicians last year). The reason for mentioning it again is that he has now, with the kind agreement of CUP, made the book freely available at this arXiv page.
(If you are new here, and this post is of interest, then you’ll also want to look at my category theory page.)