Partially ordered objects in a topos

Abstract

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], S. 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 POℇ, is infinitely complete and has exponential objects. This paper would be our first step in investigating partially ordered objects within a topos.