As a generalization of DHR analysis, the superselection sectors are studied in the absence of the spectrum condition for the reference representation. Considering a net of local observables in 4-dimensional Minkowski spacetime, we associate to a set of representations, that are local excitations of a reference representation fulfilling Haag duality, a symmetric tensor C∗-category B(A) of bimodu...

Wheelchairs, trolley chairs, and potty chairs are essential for people with disabilities, patients, elderly. These forms of equipments specially designed to support specific needs their users, such as moving, bathing, resting, etc. Since the function equipment is limitted, patients multiple problems may need more than one single be properly supported, which make caring processes complicated bec...

In general, the sheaf criterion on the étale topology may be difficult to verify directly, as a scheme will in general have many étale covers. It is clear that a necessary condition for a presheaf F to be a sheaf on Xet is that it be a sheaf with respect to Zariski covers (i.e., its restriction to Xzar is a sheaf), and that it be a sheaf with respect to one-piece étale covers (V → U) such that ...

This is a version from 29 Sept 2003 of the paper published under the same name in Theoretical Computer Science 316 (2004) 297–321. The double powerlocale P(X) (found by composing, in either order, the upper and lower powerlocale constructions PU and PL) is shown to be isomorphic in [Loc,Set] to the double exponential SS X where S is the Sierpiński locale. Further PU (X) and PL(X) are shown to b...

The Kochen-Specker theorem has been discussed intensely ever since its original proof in 1967. It is one of the central no-go theorems of quantum theory, showing the non-existence of a certain kind of hidden states models. In this paper, we first offer a new, non-combinatorial proof for quantum systems with a type In factor as algebra of observables, including I∞. Afterwards, we give a proof of...

Nominal sets, presheaf categories, and named sets have successfully served as models of the state space of process calculi featuring resource generation. More recently, automata built in such categories have been studied as acceptors of languages of finite words over infinite alphabets. In this paper we investigate automata whose state spaces are nominal sets, and that accept infinite words. Th...

In this paper, we will attempt to establish a connection between quantum set theory, as developed by Ozawa, Takeuti and Titani (see, for example, [13], [12], [10]), and topos quantum theory, as developed by Isham, Butterfield and Döring, amongst others (see, for example, [8], [6], [3]). Towards this end, we will study algebraic valued set-theoretic structures whose truth values correspond to th...

We introduce a general context involving a presheaf A and a subpresheaf B of A. We show that all previously considered cases of local analysis of generalized functions (defined from duality or algebraic techniques) can be interpretated as the B-local analysis of sections of A. But the microlocal analysis of the sections of sheaves or presheaves under consideration is dissociated into a ”frequen...

These are some lecture notes for a course presenting the cubical set model of type theory, first in Copenhagen, December 2014, and then in Paris, February 2015. We describe a particular presheaf model of type theory. This description can also be seen as an operational semantics of a purely syntactical type system. It involves a nominal extension of λ-calculus. We use a generalization of the Kan...

Terms are a concise representation of tree structures. Since they can be naturally defined by an inductive type, they offer data structures in functional programming and mechanised reasoning with useful principles such as structural induction and structural recursion. However, for graphs or ”tree-like” structures – trees involving cycles and sharing – it remains unclear what kind of inductive s...