Asking ChatGPT about coloring diagrams with truth values. When you write A -> B the arrow can be valid or invalid, but you still want to talk about it
Level 1: Morphisms = Proofs, Typed with Validity
You can think of these arrows as typed by a truth value — i.e., each morphism has a color: valid, invalid, plausible, context-sensitive, contradictory-but-derivable, etc.
In this sense, truth is not binary, but becomes a fiber over each morphism: a coloring or modality.
So your category becomes a fibration over a poset of truth values, or a category enriched in truth values — maybe in a Heyting algebra or relevance lattice.
I'm not sure I've ever heard it say "maybe" before, but