Introducing Categories
once we see the formal definition below, it will become clear that mathematical (say, first-order logical) statements, together with proofs of implication, form a category. Even though a “proof” isn’t strictly a structure-preserving map, it still fits with the roughly stated axioms above. One can compose proofs by laying the implications out one after another, this composition is trivially associative, and there is an identity proof. Thus, proofs provide a way to “transform” true statements into true statements, preserving the structure of boolean-valued truth. The section on diagram categeories was fantatsic.