A theory of quantal sets
The authors introduce a more elaborate analogue for quantales [\textit{F. Borceux} and \textit{G. van den Bossche}, Order 3, 61-87 (1986; Zbl 0595.18003)] of sheaves of a locale. A Q-set for a quantale Q is defined as a set A with a Q-valued relation \(''='': A\times A\to Q\) which is transitive \((a=a'\&a'=a''\leq a=a'')\) and satisfies a weakened version of symmetry. Q-sets lead to a fibration \(\underset \tilde{} {E} Q\) whose fibers are toposes. The properties of this construction are developed with the aim of showing that it gives an adequate version of sheaves.
scientific article; zbMATH DE number 4122192