The next thing I’d like to do on this blog is to understand the derived -category of an abelian category.
Given an abelian category with enough projectives, this is a stable -category with a special universal property. This universal property is specific to the -categorical case: in the ordinary derived category of an abelian category (which is the homotopy category of ), forming cofibers is not quite the natural process it is in (in which it is a type of colimit), and one cannot expect the same results.
For instance, , and given a triangulated category and a functor taking exact sequences in to triangles in , we might want there to be an extended functor
where is the ordinary (1-categorical) bounded derived category of . We might expect this by the following rough intuition: given an object of we can represent it as obtained from objects in by taking a finite number of cofibers and shifts. As such, we should take the image of to be the appropriate combination of cofibers and shifts in of the images of . Unfortunately, this does not determine a functor because cofibers are not functorial or unique up to unique isomorphism at the level of a trinagulated category.
The derived -category, though, has a universal property which, among other things, makes very apparent the existence of derived functors, and which makes it very easy to map out of it. One formulation of it is specific to the nonnegative case: is obtained from the category of projective objects in by freely adjoining geometric realizations. In other words:
Theorem 1 (Lurie) Let be an abelian category with enough projectives, which form a subcategory . Then has the following property. Let be any -category with geometric realizations; then there is an equivalence
between the -categories of functors and geometric realization-preserving functors .
This is a somewhat strange (and non-abelian) universal property at first sight (though, for what it’s worth, there is another more natural one to be discussed later). I’d like to spend the next couple of posts understanding why this is such a natural universal property (and, for one thing, why projective objects make an appearance); the answer is that it is an expression of the Dold-Kan correspondence. First, we’ll need to spend some time on the actual definition of this category.