In this paper we study the category of partially ordered objects in a topos E . The definition of a partially ordered object or internal poset is taken from [10], ϵ. Mac Lane and I. Moerdijk, Sheaves in geometry and logic, 1992. We will define the concept of monotone morphism between internal posets and then study the resulting category. We will show that the category of partially ordered objects in a topos, denoted PO ϵ, is finitely complete and has exponential objects. This paper would be our first step in investigating partially ordered objects within a topos.

中文翻译：

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拓扑中的部分有序对象

在本文中，我们研究了拓扑 E 中偏序对象的类别。部分有序对象或内部偏序集的定义取自 [10], ϵ。Mac Lane 和 I. Moerdijk，几何和逻辑中的 Sheaves，1992。我们将定义内部偏序集之间的单调态射的概念，然后研究结果类别。我们将证明拓扑中偏序对象的范畴，记为 PO ϵ，是有限完备的并且具有指数对象。这篇论文将是我们研究拓扑中部分有序对象的第一步。