Abelian Groups in a Topos of Sheaves on a Locale

Abstract

<p>This thesis is devoted to the study of Abelian Groups in the topos Shℒ of sheaves on a locale ℒ. The main topics considered are: injectivity, essential extensions of torsion groups, divisibility, purity, internal hom-functor, tensor product and flatness.</p> <p>We derive some general results about these notions. Also, we prove the Baer Criterion for injectivity in AbShℒ. For a well-ordered locale ℒ, we describe the injective hulls in AbShℒ and for some special locales we characterize the injectives in AbShℒ.</p> <p>We further discuss essential extensions of torsion groups and show amongst other things, that a first countable Hausdorff space X is discrete iff essential extensions in AbShX preserve torsion.</p> <p>Divisible groups are characterized here as absolutely pure groups. We discuss the internal adjointness between the tensor product and the internal hom-functor.</p> <p>Finally, we consider the notion of flatness, and show that the flat groups in AbShℒ are characterized the same way as in Ab, that is, flat = torsion free, and that A is flat in AbShℒ iff A* = [A,P] is an injective group, where P is an injective cogenerator.</p>