Let be an abelian category with enough projectives. In the previous post, we described the definition of the *derived -category* of . As a simplicial category, this consisted of bounded-below complexes of projectives, and the space of morphisms between two complexes was obtained by taking the *chain complex* of maps between and turning that into a space (by truncation and the Dold-Kan correspondence).

Last time, we proved most of the following result:

Theorem 5is a stable -category whose suspension functor is given by shifting by . has a -structure whose heart is , and the homotopy category of is the usual derived category.

Note for instance that this means that sits as a full subcategory inside : that is, there is a full subcategory (the “heart”) of (spanned by those complexes homologically concentrated in degree zero).

This heart has the property that the mapping spaces in are discrete, and the functor

restricts to an equivalence ; one can prove this by examining the chain complex of maps between two complexes homologically concentrated in degree zero. The inverse to this equivalence runs , and it sends an element of to a projective resolution. This is functorial in the -categorical sense.

Most of the above theorem is exactly the same as the description of the ordinary derived category of (i.e., the homotopy category of ), The goal of this post is to describe what’s special to the -categorical setting: that there is a universal property. I will start with the universal property for the subcategory .

Theorem 6is the -category obtained from (the projective objects) by freely adding geometric realizations.

The purpose of this post is to sketch a proof of the above theorem, and to explain what it means. (more…)