Internal Diagrams and Archetypal Reasoning in Category Theory

We can regard operations that discard information, like specializing to a particular case or dropping the intermediate steps of a proof, as projections, and operations that reconstruct information as liftings. By working with several projections in parallel we can make sense of statements like “Set is the archetypal Cartesian Closed Category”, which means that proofs about CCCs can be done in the “archetypal language” and then lifted to proofs in the general setting. The method works even when our archetypal language is diagrammatical, has potential ambiguities, is not completely formalized, and does not have semantics for all terms. We illustrate the method with an example from hyperdoctrines and another from synthetic differential geometry. Mathematics Subject Classification (2010). Primary 18C50; Secondary 03G30. 1. Mental Space and Diagrams My memory is limited, and not very dependable: I often have to rededuce results to be sure of them, and I have to make them fit in as little “mental space” as possible... Different people have different measures for “mental space”; someone with a good algebraic memory may feel that an expression like Frob : Σf (P ∧ f∗Q) ∼= −→ ΣfP ∧Q is easy to remember, while I always think diagramatically, and so what I do is that I remember this diagram, and I reconstruct the formula from it.