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In the derived category $D(C)$ of an abelian category $C$, one formally inverts quasi-isomorphisms. In the context of model categories, one inverts weak equivalences.

What does one gain by doing so?

Is there a big-picture way to think about what is being done?

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    I don't think I really understand your question. What is the point of localization in my opinion, is to consider objects the same in a subcategory. Take for instance algebraic topology. We are working in a model category with weak equivalences given by homotopy equivalence. Since we consider all our algebraic topology constructions up to homotopy, it makes sense to consider the homotopy category. In terms of why we think about derived categories, well once again we want to say that some complexes are the 'same'.2010-11-12
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    Another interesting point from my perspective is the following. If you like algebraic geometry and localization of the spectrum, then when we move to Noncommutative algebraic geometry, our spectrum is a category, once again localization is our friend. This time, that localization is a localization of categories.2010-11-12
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    Also, just in case you haven't looked, check out the nlab page on it: http://ncatlab.org/nlab/show/localization2010-11-12

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