I've been trying to read a paper by Krause and came across a strange (to me, of course) notion of localization. After looking around for a long time, and then finding this on his site, I see that there are two notions for localization, both with significant usage online. These are namely Verdier localization and Bousfield localization. Is there a strong motivation to use one over the other? A little bit of context:
I see that Bousfield localization is defined for model categories, and this includes the notion of modules over a ring, among many many others. I don't see a similar restriction for the Verdier localization.
Verdier localization uses the (standard for ''localization'') idea of a multiplicative set S of maps which are formally inverted by a functor Q from a category T to a new category denoted T/S. Hartshorne's Residues and Duality is a reference for this. (BTW, where does the assumption that the pullback of a multiplicative map is multiplicative come from?)
Bousfield localization is stated in several places (such as the Krause reference above) as a Verdier localization composed with a right adjoint for Q, which I understand to mean a functorial way of choosing objects in the isomorphism classes, and maps in the multiplicative subsets of each Hom(A,B). It is also stated in the generality of model categories as needing three distinguished collections of morphisms: namely quasi-isomorphisms and (co)fibrations. What bothers me more is the definition as given in Krause: an exact functor L from a triangulated category T to itself for which there exists a natural transformation η:Id-->L which commutes with L (ηL=Lη) and for which ηL is invertible. As a second, smaller, question, what is encoded by the commutative condition (what would be lost without it?)? I can come up with contrived examples (using the automorphisms of the objects LX) of course, but in what precise way does η really just encode L as a natural transformation?
No comments:
Post a Comment