I strongly agree!
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The whole point of category theory is to provide just this perspective. (Contemporary CT research goes off in all kinds of directions, but that's to be expected.) What's needed is very basic category theory, with an eye to addressing the basics maths of interest.
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Yes… I did category theory with G-C Rota, before anyone else in computer science had heard of it :) FWIW, I think it’s not the right thing, although it has the right aim.
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(2) From this point of view, functors between categories simply compile one internal language on another, which is exactly what we need to understand different interpretations of a topic.
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(1) I would disagree. Allegory Theory is best realized as an algebraized and structured approach to Mathematical Logic. In fact, this is the point of it. As touched on in Freyd’s book, the most important allegories arise from formal languages.
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