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

TL;DR

This paper characterizes a special class of Q-modules called locally principally generated modules, relating ordered sheaves on a small quantaloid to skew local homeomorphisms, and establishes categorical equivalences involving these structures.

Contribution

It provides an intrinsic description of Kleisli algebras as locally principally generated Q-modules and links ordered sheaves to skew local homeomorphisms in a detailed locale example.

Findings

Ordered sheaves on a locale correspond to skew local homeomorphisms.

Locally principally generated Q-modules form a biequivalent category to Ord(Q).

Intrinsic description of Kleisli algebras as locally principally generated modules.

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).