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Lurie's book, higher topos theory describes a new notion of a triangulated category, which is apparently much more natural than the usual definition. Obviously by now a great deal of work has been done on triangulated categories, has anyone translated any of this work into this new language?

That is should one go straight to his book, or become reasonably acquainted with triangulated categories first, in the hope that by then people will have become more familiar with that language by then?

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You should probably become familiar with the derived category. I recommend The Heart of Cohomology by Kato for this, but there are a number of other books. – David Corwin Dec 2 '12 at 4:16
«Corrupted shadow» is a description that can be avoided with much gain. – Mariano Suárez-Alvarez Dec 2 '12 at 4:24
For triangulated and derived categories I would recommend the relevant sections in the book "Sheaves on manifolds" by Schapira and Kashiwara. – Adeel Khan Dec 2 '12 at 6:18
In fact, if you only care about triangulated categories and can read French, then Verdier's original thesis is also quite nice (available on the home page of G. Maltsinoitis). Of course it covers derived categories also but I found the exposition a bit confusing in some places. – Adeel Khan Dec 2 '12 at 6:22
Of course, there's also Neeman's book, available online: – Todd Trimble Dec 2 '12 at 12:17

It is not necessary but it is a good idea to learn about more "classical" definitions. A good place is the stacks project: where you will find a chapter on derived categories.

Behrang Noohi has written a nice survey:"Lectures on derived and triangulated categories" available on arXiv.

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