Sets in homotopy type theory

Homotopy type theory may be seen as an internal language for the ∞-category of weak ∞-groupoids. Moreover, weak ∞-groupoids model the univalence axiom. Voevodsky proposes this (language for) weak ∞-groupoids as a new foundation for Mathematics called the univalent foundations. It includes the sets as weak ∞-groupoids with contractible connected components, and thereby it includes (much of) the traditional set theoretical foundations as a special case. We thus wonder whether those ‘discrete’ groupoids do in fact form a (predicative) topos. More generally, homotopy type theory is conjectured to be the internal language of ‘elementary’ of ∞-toposes. We prove that sets in homotopy type theory form a ΠW-pretopos. This is similar to the fact that the 0-truncation of an ∞-topos is a topos. We show that both a subobject classifier and a 0-object classifier are available for the type theoretical universe of sets. However, both of these are large and moreover the 0-object classifier for sets is a function between 1-types (i.e. groupoids) rather than between sets. Assuming an impredicative propositional resizing rule we may render the subobject classifier small and then we actually obtain a topos of sets.

TOPOS-THEORETIC CLASSIFICATION OF QUANTUM EVENTS STRUCTURES IN TERMS OF BOOLEAN REFERENCE FRAMES

We construct a sheaf-theoretic representation of quantum events structures, in terms of Boolean localization systems. These covering systems are constructed as ideals of structure-preserving morphisms of quantum event algebras from varying Boolean domains, identified with physical contexts of measurement. The modeling sheaf-theoretic scheme is based on the existence of a categorical adjunction between presheaves of Boolean event algebras and quantum event algebras. On the basis of this adjoint correspondence, we also prove the existence of an object of truth values in the category of quantum logics, characterized as subobject classifier. This classifying object plays the equivalent role that the two-valued Boolean truth values object plays in classical events structures. We construct the object of quantum truth values explicitly, and furthermore, demonstrate its functioning for the valuation of propositions in a typical quantum measurement situation.

A sheaf representation and duality for finitely presented Heyting algebras

A. M.Pitts in [Pi] proved that is a bi-Heyting category satisfying the Lawvere condition. We show that the embedding Φ: → Sh(P0, J0) into the topos of sheaves, (P0 is the category of finite rooted posets and open maps, J0 the canonical topology on P0) given by H ↦ HA(H, (−)) : P0 → Set preserves the structure mentioned above, finite coproducts, and subobject classifier; it is also conservative. This whole structure on can be derived from that of Sh(P0, J0) via the embedding Φ. We also show that the equivalence relations in are not effective in general. On the way to these results we establish a new kind of duality between and a category of sheaves equipped with certain structure defined in terms of Ehrenfeucht games. Our methods are model-theoretic and combinatorial as opposed to proof-theoretic as in [Pi].