Partially Ordered Sheaves on a Locale. I (II)

TL;DR

This paper explores the algebraic and order-theoretic structures of sheaves on a locale, introducing concepts like dcposheaves and characterizing their completeness and continuity properties within the sheaf category.

Contribution

It introduces the notion of dcposheaves, provides internal characterizations of various classes of posheaves, and establishes categorical equivalences with sheaf locales.

Findings

Characterization of complete partially ordered sheaves.

Introduction of dcposheaves and their properties.

Equivalence between sheaf categories and slice categories of locales.

Abstract

In this paper, we investigate the order algebraic structure in the category of sheaves on a given locale $X$. Since every localic topos has a generating set formed by its subterminal objects, we define a "point" of a partially ordered sheaf to be a morphism from a subterminal sheaf to the partially ordered sheaf. Using the concept of "points," we investigate the completeness of posheaves systemically. Some internal characterizations of complete partially ordered sheaves and frame sheaves are given. We also give an explicit description of the construction of associated sheaf locales and show directly that the category $Sh(X)$ of sheaves on a locale $X$ is equivalent to the slice category $LH/X$ of locales and local homeomorphisms over $X$. Applying this equivalence, we give characterizations of partially ordered sheaves and complete partially ordered sheaves in terms of sheaf locale respectively. This paper is a continuation of our paper [1]. In this paper, we first introduce the concept of directed complete partially ordered sheaves (shortly dcposheaves) on a given locale. some internal characterizations of dcposheaves and meet continuous dcposheaves are given respectively. We also give characterizations of continuous posheaves and completely distributive posheaves, and show that an algebraic completely distributive posheaf is spatial.