A theory of quantal sets

AbstractA quantale is a complete lattice provided with a (generally non commutative) binary multiplication which, in particular, distributes over arbitrary suprema in each variable. That notion is intended to replace locales in the spectral constructions involving non commutative algebraic structures. We propose a corresponding notion of quantale-valued set, which generalizes the classical notions of boolean or heyting-valued sets. We study the category obtained in this way and show that it can be seen as a fibration over the original quantale, whose fibres are toposes