# How to formulate principle of duality in projective geometry in terms of category theory?

In projective geometry, the principle of duality states that any theorem that holds for an incidence structure $$(P, L, I)$$, where $$P$$ are the points, $$L$$ are the lines and $$I \subseteq P \times L$$ is the incidence relation, also holds for the dual incidence structure $$(L, P, I^*)$$, where $$I^*$$ is the converse relation of $$I$$.

In category theory, the principle of duality states that a categorical theorem has a dual, which holds for the dual category obtained by reversing all the arrows.

How can the first one be specified in such a way that it derives from the second one?

• Not sure if this is what you are after, but you can make a category from $(P,L,I)$ by letting $P\cup L$ be the objects and the only nontrivial morphisms are dictated by members of $I$ (so $(p,l)\in I$ makes an arrow $p\to l$). Then its dual category is $(L,P,I^*)$. Feb 21, 2021 at 11:57