Let be a stable -category. For us, this means that we have three important properties:

- admits finite limits and colimits.
- has a zero object: that is, the initial object is also final.
- A square in is a pull-back if and only if it is a push-out.

This is equivalent to the stability of . Actually, stability is usually defined using slightly weaker conditions, and then it takes a little work to show that these stronger ones are implied. We’ll just work with these. Stability can be thought of as a higher-categorical version of being triangulated; a general stable -category has many of the properties (in a higher categorical sense) of the homotopy category of spectra, or the (classical) derived category.

Our goal is to show that in this case, we have an equivalence of -categories

between *simplicial* objects in and *filtered *(nonnegatively) objects in . The idea here is that the geometric realization of a simplicial object comes with a canonical filtration, given by geometric realizing the -truncations for each . This is going to give the associated filtered object. (We don’t know that the geometric realization exists, but the realizations of the truncations will.)

We will actually prove something stronger: for each , there is an equivalence

where is the subcategory of elements . In other words, -truncated simplicial objects are the same as -filtered objects of . (Note that, as a simplicial set, the nerve of is .) These equivalences will be compatible, and taking inverse limits will give the Dold-Kan correspondence. (more…)