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We study sheaves in the context of a duality theory for lattice structureendowed with extra operations, and in the context of forcing in a topos. UsingSheaf duality theory of Comer for cylindric algebras, we give a representationtheorem of of distributive bounded lattices expanded by modalities (functionsdistributing over joins) as the continuous sections of sheaves. Ourrepresentation is defined via a contravariant functor from an algebraiccategory to a category of sheaves. We show that if our category is a small site(cartesian closed with a stability condition on pullbacks), then we can definea notion of forcing using this category. In particular, we define fuzzy forcingby interpreting the additional Lukasiewicz conjunction $\otimes$ as induced bya tensor product in the target monodial category of pre-sheaves. We also studytopoi as semantics for higher order logic of many sorted theories in connectionto set theory, and the quasi-topoi based on $MV$ algebras, for fuzzy logic. Weshow that the interpretation of a theory $T$, in this case into $Set_{\Omega}$where $\Omega$ is an almost sub-object classifier in a quasi-topos $CAT$defined from $T$, is completed by defining semantics for $\otimes$, and this isdone similarly to its defining clause in forcing. We give applications tomany-valued logics and various modifications of first order logic andmulti-modal logic, set in an algebraic framework.

Back and forth between algebraic geometry, algebraic logic, sheaves and forcing