I am now learning localization theory for triangulated catgeory(actually, more general (co)suspend category) in a lecture course. I found Verdier abelianization which is equivalent to universal cohomological functor) is really powerful and useful formalism. The professor assigned many problems concerning the property of localization functor in triangulated category.He strongly suggested us using abelianization functor to do these problems
If we do these problems in triangulated category, we have to work with various axioms TRI to TRIV which are not very easy to deal with. But if we use Verdier abelianization functor, we can turn the whole story to the abelian settings. Triangulated category can be embedded to Frobenius abelian category(projectives and injectives coincide). Triangulated functors become exact functor between abelian categories. Then we can work in abelian category. Then we can easily go back(because objects in triangulated category are just projectives in Frobenius abelian category, we can use restriction functor). In this way, it is much easier to prove something than Verider did in his book.
My question is:
What makes me surprised is that Verdier himself even did not use Abelianization in his book to prove something. I do not know why?(Maybe I miss something)
I wonder whether there are any non-trivial application of Verdier abelianization functor in algebraic geometry or other fields?