A topos for algebraic quantum theory

ing frames O (X ) coming from a topological space to general frames is a genuine generalization of the concept of a space, as plenty of frames exist tha t are not of the form O (X ). A simple example is the frame Oreg(R) of regular open subsets of R, i.e. of open subsets U with the property ——U = U, where —U is the interior of the complement of U . This may be contrasted with the situation for unital commutative C*-algebras, which, as just recalled, are all of the form C (X ). Moreover, far from obscuring the logical structure of space, the generalization of spaces by frames rather explains and deepens this structure. Indeed, a frame is a complete Heyting algebra, with its intrinsic structure of an intuitionistic propositional logic. Here a Heyting algebra is a distributive lattice L with a map ^ : £ x £ ^ £ satisfying x ^ (y ^ z) iff x A y ^ z, called implication [41, 63, 80]. Every Boolean algebra is a Heyting algebra, but not vice versa; in fact, a Heyting algebra is Boolean iff ——x = x for all x, which is the case iff —x V x = T for all x. Here negation is a derived notion, defined by —x = (x ^ ^ ) . For example, Oreg(R) is Boolean, but O(R) is not. In general, the elements of a Heyting algebra form an intuitionistic propositional logic under the usual logical interpretation of the lattice operations. A Heyting algebra is complete when arbitrary joins (i.e. sups) and meets (i.e. infs) exist. A complete Heyting algebra is essentially the same thing as a frame, for in a frame one may define y ^ z = V{x | x A y ^ z}. Conversely, the infinite distributivity law in a frame is automatically satisfied in a Heyting algebra. The set of subobjects of a given object in a topos forms a complete Heyting algebra (as long as the topos in question is defined “internal to S e ts”), generalizing the fact tha t the set of subsets of a given set is a Boolean algebra. The subobject classifier of such a topos is a complete Heyting algebra as well; in fact, these two statements are equivalent. (Note, however, frame maps do not necessarily preserve the implication ^ defining the Heyting algebra structure, as can