Distributive Lattices in a Localic Topos

Abstract

<p>In this thesis we examine various properties of bounded distributive lattices in the topos of sheaves on a locale. We prove that BooShℒ, the category of Boolean algebras in ℘Shℒ, is a reflective subcategory of ℘Sℒ, the category of bounded distributive lattices in Sℒ_ Injective distributive lattices in Sℒ are discussed, and two methods of constructing the injective hull of any lattice in ℘Sℒ are described. We characterize indecomposable injectives in ℘sℒ and show that they are exactly the prime bounded distributive lattices. Simple lattices in Sℒ are described and characterized in terms of the points of ℒ. We examine cogenerating sets in ℘Sℒ and the relationships among simple, prime and cogenerating objects in the category. Finally, we consider the initial object 2ℒ of ℘Shℒ, when it is complete and when a cogenerator; we then prove that any locale is isomorphic to the locale of congruences of 2ℒ.</p>