  In this paper, we present a generalization of Grothendieck pretopologies --
suited for semicartesian categories with equalizers $C$ -- leading to a closed
monoidal category of sheaves, instead of closed cartesian category. This is
proved through a different sheafification process, which is the left adjoint
functor of the suitable inclusion functor but does not preserve all finite
limits. If the monoidal structure in $C$ is given by the categorical product,
all constructions coincide with those for Grothendieck toposes. The motivation
for such generalization stems from a certain notion of sheaves on quantales
that does not form a topos.
