2008/07/31 by Pedro Resende
Mathematics · #math.RA #math.AT #math.CT #msc:06D22 #msc:06F07 #msc:18B25 #msc:18F20 #msc:22A22 #msc:54B40
paper · pdf · doi:10.1016/j.jpaa.2011.05.002
published as Journal of Pure and Applied Algebra 216 (2012) pp. 41-70 · 62 pages. Structure of preprint has changed. It now contains the contents of former arXiv:0807.3859 (withdrawn), and the definition of Q-sheaf applies only to inverse quantal frames (Hilbert Q-modules with enough sections are given no special name for more general quantales)
arxiv created 2011/03/28 · arxiv updated 2011/08/02
Several notions of sheaf on various types of quantale have been proposed and studied in the last twenty five years. It is fairly standard that for an involutive quantale Q satisfying mild algebraic properties the sheaves on Q can be defined to be the idempotent self-adjoint Q-valued matrices. These can be thought of as Q-valued equivalence relations, and, accordingly, the morphisms of sheaves are the Q-valued functional relations. Few concrete examples of such sheaves are known, however, and in this paper we provide a new one by showing that the category of equivariant sheaves on a localic etale groupoid G (the classifying topos of G) is equivalent to the category of sheaves on its involutive quantale O(G). As a means towards this end we begin by replacing the category of matrix sheaves on Q by an equivalent category of complete Hilbert Q-modules, and we approach the envisaged example where Q is an inverse quantal frame O(G) by placing it in the wider context of stably supported quantales, on one hand, and in the wider context of a module theoretic description of arbitrary actions of étale groupoids, both of which may be interesting in their own right.