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So I know that when introducing sheaf cohomology, there are two main approaches via derived categories, and a perhaps more "down to earth" method of resolving by acyclic, fine, soft, sheaves. I'm curious exactly how these two approaches are related and what the benefits are of one over the other.

For a nice topological space $X$, in the category $\rm{Sh}(X)$ of sheaves on $X$, we can search for certain objects in the category like fine, acyclic, soft, sheaves and find resolutions $0 \to \mathcal{F} \to \mathcal{C}^{0} \to \mathcal{C}^{1} \to \ldots$ of a sheaf $\mathcal{F}$. The sheaf cohomology is then given by $H^{q}(X, \mathcal{F}) \simeq H^{q}(X, \mathcal{C}^{*}(X))$.

To my very naive eye, in the derived categories approach, we can think of these resolutions as quasi-isomorphisms $\mathcal{F} \simeq [\mathcal{C}^{0} \to \mathcal{C}^{1} \to \ldots]$ in the derived category $\mathcal{D}\rm{Sh}(X)$ of sheaves on $X$. Of course, they give rise to isomorphic cohomology.

What I'm unsure about, is one of these methods superior to the other? Is the method using resolutions merely a "shadow" of something much more powerful and general in derived categories? In my naiveté, it seems like in both methods you have to get your hands dirty and search for possibly elusive objects in some category: acyclic objects in $\rm{Sh}(X)$, or quasi-isomorphisms in $\mathcal{D}\rm{Sh}(X)$.

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    $\begingroup$ I like to believe that the principle of conservation of work applies even in pure mathematics. The derived category approach is elegant but is difficult to understand concretely; the resolution approach is concrete but difficult to understand elegantly. $\endgroup$
    – Zhen Lin
    Commented Apr 4, 2016 at 19:08
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    $\begingroup$ I can regurgitate the following philosophy: The injective resolution is the "true" object, cohomology is the shadow. (Cohomology is easier to compute with in general, and suffices for many applications...) This is explained well here (see Whiteheads theorem on the second page): arxiv.org/pdf/math/0501094v1.pdf $\endgroup$
    – Elle Najt
    Commented Apr 4, 2016 at 19:39
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    $\begingroup$ Technically speaking, I guess the derived category aspect has the advantage that it makes sense even when the category does not have enough injectives/projectives. $\endgroup$
    – Captain Lama
    Commented Apr 5, 2016 at 10:45

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From my point of view, these two approaches are essentially the same.

If you view it as a derived functor in the derived category, then you will get a good universal property by the definition of derived functors in derived category. The definition given by using resolution is just telling you how to compute it.

So I think to define or construct a derived functor, it is easy to do in the language of derived category (or model category in the non-abelian setting). Then to show that this derived functor can be computed by using some kinds of resolution. After that you can forget about the abstract definition of that functor if you want, and always think it as a functor constructed by a resolution.

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