Talk:Localization of a category
This page needs to include something on Bousfield localization from homotopy theory. - Gauge 18:33, 10 August 2005 (UTC)
[edit] Categories of Fractions
Bousfield seems to special to me to include it in the text; Verdier localization sounds about right. BTW, for the calculus of fractions there are quite some restrictions on the set of morphisms to be inverted; in particular, it is not generally applicable to construct the derived category directly from chain complexes (see Gelfand/Manin, Methods of Homological Algebra). So I'm wondering: does the name "Localization" just refer to the categories of fractions as stated in the article, or is the general concept of adjoining inverses of arbitrary sets of morphisms? - 80.143.125.195 15:05, 18 January 2007 (UTC)
[edit] Definition doesn't make any sense
Based on the heading, localization should be the process of formally adjoining inverses to a specified set of morphisms in a category. The given definition though (coaugmented functor that is idempotent) has nothing to do with this, and doesn't even yield a new category, or at least not the localization that we want. What is going on? If you work through the idempotent coaugmented functor definition in a few simple cases (monoids as 1-element categories, and posets), then you get that in the monoid case, a 'localization' is equivalent to an invertible element of the monoid, and in the poset case, a 'localization' is an order-preserving map f which satisfies x ≤ f(x) for every x. Neither of these have anything to do with what localization ought to be, which in these cases would be the processes of formally adding inverses to a set of elements in a monoid, and collapsing intervals in a poset. — Preceding unsigned comment added by 71.217.2.181 (talk) 23:54, 31 July 2011 (UTC)
