A Sheaf Representation and Duality for Finitely Presenting Heyting Algebras

A. M.Pitts in [Pi] proved that is a bi-Heyting category satisfying the Lawvere condition. We show that the embedding Φ: → Sh ( P 0 , J 0 ) into the topos of sheaves, ( P 0 is the category of finite rooted posets and open maps, J 0 the canonical topology on P 0 ) given by H ↦ HA ( H , (−)) : P 0 → Set preserves the structure mentioned above, finite coproducts, and subobject classifier; it is also conservative. This whole structure on can be derived from that of Sh ( P 0 , J 0 ) via the embedding Φ. We also show that the equivalence relations in are not effective in general. On the way to these results we establish a new kind of duality between and a category of sheaves equipped with certain structure defined in terms of Ehrenfeucht games. Our methods are model-theoretic and combinatorial as opposed to proof-theoretic as in [Pi].