On sheaves on semicartesian quantales and their truth values

In this paper, we introduce a new definition of sheaves on semicartesian quantales, providing first examples and categorical properties. We note that our sheaves are similar to the standard definition of a sheaf on a locale; however, we prove that in general it is not an elementary topos—since the lattice of external truth values of $Sh(Q)$, $Sub(1)$, is canonically isomorphic to the quantale $Q$—placing this paper as part of a greater project towards a monoidal (not necessarily cartesian) closed version of elementary topos. To start the study the logical aspects of the category of sheaves we are introducing, we explore the nature of the ‘internal truth value objects’ in such sheaves categories. More precisely, we analyse two candidates for subobject classifier for different subclasses of commutative and semicartesian quantales.