Categories of embeddings

AbstractWe present a categorical generalisation of the notion of domains, which is closed under (suitable) exponentiation. The goal was originally to generalise Girard's model of polymorphism to Fω. If we specialise this notion in the poset case, we get new cartesian closed categories of domains

Remarks on Morphisms of Spectral Geometries

Having in view the study of a version of Gel'fand-Neumark duality adapted to the context of Alain Connes' spectral triples, in this very preliminary review, we first present a description of the relevant categories of geometrical spaces, namely compact Hausdorff smooth finite-dimensional orientable Riemannian manifolds (or more generally Hermitian bundles of Clifford modules over them); we give some tentative definitions of the relevant categories of algebraic structures, namely "propagators" and "spectral correspondences" of commutative Riemannian spectral triples; and we provide a construction of functors that associate a naive morphism of spectral triples to every smooth (totally geodesic) map. The full construction of spectrum functors (reconstruction theorem for morphisms) and a proof of duality between the previous "geometrical' and "algebraic" categories are postponed to subsequent works, but we provide here some hints in this direction. We also show how the previous categories of "propagators" of commutative C*-algebras embed in the mildly non-commutative environments of categories of suitable Hilbert C*-bimodules, factorizable over commutative C*-algebras, with composition given by internal tensor product.Comment: 9 pages, AMS-LaTeX2e. Reformatted, heavily revised and corrected version, only for arXiv, of a previous review paper published in East-West Journal of Mathematics. The main results presented in this review are now part of F.Jaffrennou PhD thesis "Morphisms of Spectral Geometries" (Mahidol University, June 2014

Coherence and Consistency in Domains (Extended Outline)

Almost all of the categories normally used as a mathematical foundation for denotational semantics satisfy a condition known as consistent completeness. The goal of this paper is to explore the possibility of using a different condition - that of coherence - which has its origins in topology and logic. In particular, we concentrate on those posets whose principal ideals are algebraic lattices and whose topologies are coherent. These form a cartesian closed category which has fixed points for domain equations. It is shown that a universal domain exists. Since the construction of this domain seems to be of general significance, a categorical treatment is provided and its relationship to other applications discussed

Regular R-R and NS-NS BPS black holes

We show in a precise group theoretical fashion how the generating solution of regular BPS black holes of N=8 supergravity, which is known to be a solution also of a simpler N=2 STU model truncation, can be characterized as NS-NS or R-R charged according to the way the corresponding STU model is embedded in the original N=8 theory. Of particular interest is the class of embeddings which yield regular BPS black hole solutions carrying only R-R charge and whose microscopic description can possibly be given in terms of bound states of D-branes only. The microscopic interpretation of the bosonic fields in this class of STU models relies on the solvable Lie algebra (SLA) method. In the present article we improve this mathematical technique in order to provide two distinct descriptions for type IIA and type IIB theories and an algebraic characterization of S*T--dual embeddings within the N=8,d=4 theory. This analysis will be applied to the particular example of a four parameter (dilatonic) solution of which both the full macroscopic and microscopic descriptions will be worked out.Comment: latex, 30 pages. Final version to appear on Int.J.Mod.Phy