Quantale Valued Sets: Categorical Constructions and Properties

This work mainly concerns the—here introduced—category of \(\mathscr {Q}\) $\mathcal{Q}$ -sets and functional morphisms, where \(\mathscr {Q}\) $\mathcal{Q}$ is a commutative semicartesian quantale. We prove it enjoys all limits and colimits, that it has a classifier for regular subobjects (a sort of truth-values object), which we characterize and give explicitly. Moreover: we prove it to be \(\kappa \) $\kappa$ -locally presentable, (where \(\kappa =max\{|\mathscr {Q}|^+, \aleph _0\}\) $\kappa =max\{|\mathcal{Q}{|}^{+},{\aleph }_0\}$ ); we also describe a hierarchy of monoidal structures in this category.