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Journal Articles Annals of Pure and Applied Logic Year : 2009

On principally generated Q-modules in general, and skew local homeomorphisms in particular

Abstract

Ordered sheaves on a small quantaloid Q have been defined in terms of Q-enriched categorical structures; they form a locally ordered category Ord(Q). The free-cocompletion KZ-doctrine on Ord(Q) has Mod(Q), the quantaloid of Q-modules, as category of Eilenberg-Moore algebras. In this paper we give an intrinsic description of the Kleisli algebras: we call them the 'locally principally generated Q-modules'. We deduce that Ord(Q) is biequivalent to the 2-category of locally principally generated Q-modules and left adjoint module morphisms. The example of locally principally generated modules on a locale X is worked out in full detail: relating X-modules to objects of the slice category Loc/X, we show that ordered sheaves on X correspond with 'skew local homeomorphisms into X' (like sheaves on X correspond with local homeomorphisms into X).

Dates and versions

hal-03606835 , version 1 (12-03-2022)

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Hans Heymans, Isar Stubbe. On principally generated Q-modules in general, and skew local homeomorphisms in particular. Annals of Pure and Applied Logic, 2009, 161 (1), pp.43-65. ⟨10.1016/j.apal.2009.05.001⟩. ⟨hal-03606835⟩

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