On Isomorphism of "Functional" Intersection and Union Types

Type isomorphism is useful for retrieving library components, since a function in a library can have a type different from, but isomorphic to, the one expected by the user. Moreover type isomorphism gives for free the coercion required to include the function in the user program with the right type. The present paper faces the problem of type isomorphism in a system with intersection and union types. In the presence of intersection and union, isomorphism is not a congruence and cannot be characterised in an equational way. A characterisation can still be given, quite complicated by the interference between functional and non functional types. This drawback is faced in the paper by interpreting each atomic type as the set of functions mapping any argument into the interpretation of the type itself. This choice has been suggested by the initial projection of Scott's inverse limit lambda-model. The main result of this paper is a condition assuring type isomorphism, based on an isomorphism preserving reduction.Comment: In Proceedings ITRS 2014, arXiv:1503.0437

A spin-statistics theorem for quantum fields on curved spacetime manifolds in a generally covariant framework

A model-independent, locally generally covariant formulation of quantum field theory over four-dimensional, globally hyperbolic spacetimes will be given which generalizes similar, previous approaches. Here, a generally covariant quantum field theory is an assignment of quantum fields to globally hyperbolic spacetimes with spin-structure where each quantum field propagates on the spacetime to which it is assigned. Imposing very natural conditions such as local general covariance, existence of a causal dynamical law, fixed spinor- or tensor-type for all quantum fields of the theory, and that the quantum field on Minkowski spacetime satisfies the usual conditions, it will be shown that a spin-statistics theorem hols: If for some spacetimes the corresponding quantum field obeys the "wrong" connection between spin and statistics, then all quantum fields of the theory, on each spacetime, are trivial.Comment: latex2e, 1 figure, 32 page

TQFT's and gerbes

We generalize the notion of parallel transport along paths for abelian bundles to parallel transport along surfaces for abelian gerbes using an embedded Topological Quantum Field Theory (TQFT) approach. We show both for bundles and gerbes with connection that there is a one-to-one correspondence between their local description in terms of locally-defined functions and forms and their non-local description in terms of a suitable class of embedded TQFT's.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-14.abs.htm

A classical approach to TQFT's

We present a general framework for TQFT and related constructions using the language of monoidal categories. We construct a topological category C and an algebraic category D, both monoidal, and a TQFT functor is then defined as a certain type of monoidal functor from C to D. In contrast with the cobordism approach, this formulation of TQFT is closer in spirit to the classical functors of algebraic topology, like homology. The fundamental operation of gluing is incorporated at the level of the morphisms in the topological category through the notion of a gluing morphism, which we define. It allows not only the gluing together of two separate objects, but also the self-gluing of a single object to be treated in the same fashion. As an example of our framework we describe TQFT's for oriented 2D-manifolds, and classify a family of them in terms of a pair of tensors satisfying some relations.Comment: 72 pages, 7 figure

Algebraic Quantum Theory on Manifolds: A Haag-Kastler Setting for Quantum Geometry

Motivated by the invariance of current representations of quantum gravity under diffeomorphisms much more general than isometries, the Haag-Kastler setting is extended to manifolds without metric background structure. First, the causal structure on a differentiable manifold M of arbitrary dimension (d+1>2) can be defined in purely topological terms, via cones (C-causality). Then, the general structure of a net of C*-algebras on a manifold M and its causal properties required for an algebraic quantum field theory can be described as an extension of the Haag-Kastler axiomatic framework. An important application is given with quantum geometry on a spatial slice within the causally exterior region of a topological horizon H, resulting in a net of Weyl algebras for states with an infinite number of intersection points of edges and transversal (d-1)-faces within any neighbourhood of the spatial boundary S^2.Comment: 15 pages, Latex; v2: several corrections, in particular in def. 1 and in sec.

Homogeneous coordinates for algebraic varieties

We associate to every divisorial (e.g. smooth) variety $X$ with only constant invertible global functions and finitely generated Picard group a $Pic(X)$-graded homogeneous coordinate ring. This generalizes the usual homogeneous coordinate ring of the projective space and constructions of Cox and Kajiwara for smooth and divisorial toric varieties. We show that the homogeneous coordinate ring defines in fact a fully faithful functor. For normal complex varieties $X$ with only constant global functions, we even obtain an equivalence of categories. Finally, the homogeneous coordinate ring of a locally factorial complete irreducible variety with free finitely generated Picard group turns out to be a Krull ring admitting unique factorization.Comment: 30 page