A category theory for programming languages

The paper first shows how the categorical theory of sequences is useful in explaining properties of the (word) differentiation of Brzozowski (1964). Then, the paper shows how a more general theory of languages, including both procedural and functional languages, should be constructed by using a switch proposition that extends the common equivalence between the general tensor and general hom functors of categories. Lastly, the paper offers a new way of approaching projectivity concepts of general relative homological algebra. The most interesting result is that there is a class of functors so that functor-projectives intersect projectives and injectives trivially, and behave correctly. Moreover, any non-zero abelian group is not Q-projective.