Localization in different categories

Abstract

Localization can be thought of as a systematic way of inverting morphisms in a category to construct a new one. This can be formulated accurately in terms of a universal property. In this study, we looked at localization in some special cases such as in a ring for both the commutative and non-commutative case, module over a commutative ring and in a more general setting, which is localization of the homotopy category of complexes to obtain the derived category. Analysis of the different localizations involved investigations into the localizing class, the universal property and Ore conditions imposed in each case and it was shown that localization has almost the same properties for different categories. An analysis in the relationship of localization in different categories was carried out to obtain the general properties such as the localizing class, Ore conditions and the universal property of localization to enable one localize an arbitrary category as provided in this study.