Representation theorem for Heyting algebras?

Solution 1:

Esakia Duality might just be the thing that you are looking for.

For every Heyting algebra $A$ there exists a so called Esakia space $\mathscr{X}=(X, \leq, \mathscr{O})$, which is a certain kind of ordered topological space, such that $A$ is isomorphic to the Heyting algebra of clopen up-sets of $\mathscr{X}$.

This gives a dual equivalence between the category of Heyting algebras and the category of Esakia spaces, very similar to the well-know Stone duality between Boolean algebras and Stone spaces. However the category of Esakia spaces is not a full subcategory of the category of ordered topological spaces and continuous and order preserving functions between them.

Finally you can give a purely topological description of the category of Esakia spaces, in the sense that it is isomorphic to a (non full) subcategory of the category of Spectral spaces and spectral maps see Bezhanishvili et al. 2010 Theorem 7.12.