The higher-categorical Dold-Kan correspondence

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

1. admits finite limits and colimits.
2. has a zero object: that is, the initial object is also final.
3. 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 examples of cofinality

Let’s do some more examples of cofinality. In the previous post, I erroneously claimed that the map

was cofinal: that is, taking a colimit of an -truncated simplicial object in an -category was the same as taking the colimit of the associated -truncated semisimplicial object. (The claim has since been deleted.) This is false, even when . In fact, the map of categories

is not even a weak homotopy equivalence. (While this is not obvious, one of the statements of “Theorem A” is that a cofinal map is automatically a homotopy equivalence.)

In fact, looks like . This is not contractible: if we take of the nerve, that’s the same as taking the category itself and inverting all the morphisms. So that gives us a free groupoid on two morphisms. However, is contractible. We’ll see that this is true in general for any , but the only become “asymptotically” contractible.

The purpose of this post is to work through a few examples of Theorem A, discussed in the previous post. This will show that the colimit of an -truncated simplicial object can in fact be recovered from the semisimplicial restriction, but in a somewhat more subtle way than one might expect. We will need this lemma in the discussion of the Dold-Kan correspondence.