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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'. –  BBischof Nov 12 '10 at 18:31
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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. –  BBischof Nov 12 '10 at 18:33
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Also, just in case you haven't looked, check out the nlab page on it: ncatlab.org/nlab/show/localization –  BBischof Nov 12 '10 at 18:35
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1 Answer 1

Localisation in category theory is essentially about "throwing away" irrelevant data. Here is an example which might make it clearer.

Consider a topological space $X$. The usual definition of sheaves on $X$ says that they are certain presheaves that satisfy a descent condition, but there is another way of thinking about it that only involves taking stalks. It is obvious that the definition of stalk can be applied verbatim to presheaves, and so by putting all the stalks together we get a functor $L : \textbf{Psh}(X) \to \textbf{Set}^X$, where $\textbf{Set}^X$ denotes the $X$-fold cartesian power of $\textbf{Set}$. Let us declare a morphism $f : A \to B$ of presheaves on $X$ to be a local isomorphism if $L f$ is.

I claim $\textbf{Sh}(X)$ is precisely the category obtained by inverting these local isomorphisms, and the sheafification functor $j^* : \textbf{Psh}(X) \to \textbf{Sh}(X)$ is the localising functor. This can be checked using the fact that $\textbf{Psh}(X)$ admits a calculus of right fractions:

  • $L$ is pushout-preserving, so pushouts of local isomorphisms in $\textbf{Psh}(X)$ are again local isomorphisms.
  • If $f : A \to B$ is a local isomorphism such that $g \circ f = h \circ f$ for some $g, h : B \to C$, then if $\eta_C : C \to j^* C$ is the unit of the sheafification adjunction, then we must have $\eta_C \circ g = \eta_C \circ h$.

Thus every morphism in the localisation of $\textbf{Psh}(X)$ away from local isomorphisms can be represented by a zigzag of the form $A \leftarrow B \to C$, where $A \leftarrow B$ is a local isomorphism, and by playing around with some diagrams we see that in fact any such zigzag is equivalent to one of the form $A \to j^* A \to j^* C \leftarrow C$, where the first and last arrows are the unit morphisms. Since the local isomorphisms are precisely the presheaf morphisms $f$ such that $j^* f$ is an isomorphism, this is enough to prove the claim.

In other words, a sheaf is precisely what you get when you take a presheaf and throw away all the extra data that cannot be detected at the level of stalks. Similarly, a homotopy type is what you get when you take a topological space and throw away everything that cannot be seen by the homotopy groups. But beware: the example of sheaves is very special; in general the localisation of a category need not be a reflective subcategory as it is in the above.

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