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`

(η`L=L`

η) and for which η`L`

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?