Modules on involutive quantales: canonical Hilbert structure, applications to sheaf theory

arXiv:0809.4336v2 [math.CT] 11 Jun 2009 Modules on involutive quantales: canonical Hilbert structure, applications to sheaf theory Hans Heymans∗and Isar Stubbe† Written: August 2008 Submitted: 10 September 2008 Revised: 18 May 2009 Abstract We explain the precise relationship between two module-theoretic descriptions of sheaves on an involutive quantale, namely the description via so-called Hilbert structures on modules and that via so-called principally generated modules. For a principally generated module satisfying a suitable symmetry condition we observe the existence of a canonical Hilbert structure. We prove that, when working over a modular quantal frame, a module bears a Hilbert structure if and only if it is principally generated and symmetric, in which case its Hilbert structure is necessarily the canonical one. We indicate applications to sheaves on locales, on quantal frames and even on sites. 1 Introduction Jan Paseka [1999, 2002, 2003] introduced the notion of Hilbert module on an involutive quantale: it is a module equipped with an inner product. This provides for an order-theoretic notion of “inner product space”, originally intended as a generalisation of complete lattices with a duality. Recently, Pedro Resende and Elias Rodrigues [2008] applied this definition to a locale X and further defined what it means for a Hilbert X-module to have a Hilbert basis. These Hilbert X-modules with Hilbert basis describe, in a module-theoretic way, the sheaves on X. At the same time, the present authors defined the notion of (locally) principally generated module on a quantaloid [Heymans and Stubbe, 2009]. Our aim too was to describe “sheaves as modules”, albeit sheaves on quantaloids in the sense of [Stubbe, 2005b]. In this formulation the ordinary sheaves on a locale X are described as locally principally generated X-modules whose locally principal elements satisfy an extra “openness” condition. Whereas Hilbert locale modules easily generalise to modules on involutive quantales, the principally generated quantaloid modules straightforwardly specialise to involutive quantales. ∗Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, 2020 Antwer- pen, Belgium, hans.heymans@ua.ac.be †Postdoctoral Fellow of the Research Foundation Flanders (FWO), Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, 2020 Antwerpen, Belgium, isar.stubbe@ua.ac.be 1 Thus we have two module-theoretic approaches to sheaves on involutive quantales: in this note we explain the precise relationship between them. This work can be summarised as follows: After some preliminary definitions we show in Section 2 that any principally generated module on an involutive quantale comes with a canoncial (pre-)inner product. In Section 3 we first present the notion of Hilbert basis for modules on an involutive quantale [Resende, 2008]. After introducing a suitable notion of symmetry for such modules, termed principal symmetry, we prove that a module is principally generated and principally symmetric if and only if it admits a canonical Hilbert structure (= canonical inner product plus canonical Hilbert basis). When working over a modular quantal frame it is a fact, as we prove in Section 4, that a module bears a Hilbert structure if and only if it is principally generated and principally symmetric, in which case the given inner product is necessarily the canonical one (admitting the canonical Hilbert basis). That is to say, in this case the only possible (and thus the only relevant) Hilbert structure is the canonical one. We illustrate all this module-theory with many examples. In the final Section 5 we draw some conclusions from our work. We explain all new results in this paper in a self-contained manner in the language of quantale modules, focussing on the purely order-theoretic aspects. However, in some examples, partic- ularly those concerned with sheaf theory in one way or another, we freely use material from the references without recalling much of the details. Thus, the reader who is mainly interested in order theory can safely skip those examples; but the reader who is also interested in the applications to sheaf theory will most likely have to have a quick look at the cited papers too, insofar as the notions involved are not already familiar to her or him. 2 Canonical inner product We begin by recalling some definitions. Throughout this paper, Q = (Q, W, ◦, 1) stands for a quantale, i.e. a monoid in the monoidal category Sup of complete lattices and maps that preserve arbitrary suprema. Explicitly, a quantale Q consists of a complete lattice (Q, W) equipped with a binary operation Q × Q / Q: (f, g) 7→f ◦g and a constant 1 ∈Q such that f ◦(g ◦h) = (f ◦g) ◦h, 1 ◦f = f = f ◦1 and ( _ i∈I fi) ◦( _ j∈J gj) = _ i∈I _ j∈J (fi ◦gj) for all f, g, h, fi, gj ∈Q. (Some call this a unital quantale, but since we shall not encounter “non-unital quantales” in this work we drop that adjec

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