The double nature, both logical and geometrical, of topos theory is one of the most fascinating aspects of this discipline. Consequently, it might be of some interest that an essentially “geometric” fact such as the associated sheaf theorem admits a proof entirely based on methods of categorical logic.
The idea of this proof comes from a previous paper [V], where the same technique was used in the context of Grothendieck topoi. That paper used a generalized notion of forcing which leads directly from classical Tarski semantics in sets to Kripke-Joyal semantics in an arbitrary Grothendieck topos and gives a precise description of the links between the two. On account of this very close relationship, we thus could establish various basic facts on Grothendieck topoi without an extensive use of categorical methods, simply by viewing given subobjects as interpretations of appropriate formulas written in the formal language that had been used all along.