Abelian Groups in a Topos of Sheaves on a Locale - Bhutani, Ravender Kiran (My University, 1983)
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.
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.
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.
Divisible groups are characterized here as absolutely pure groups. We discuss the internal adjointness between the tensor product and the internal hom-functor.
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.
Keywords: Mathematics; Mathematics