The past few posts have been focused on a discussion of Lurie’s version of the Dold-Kan correspondence in stable -categories. I’ve made these posts more detailed than usual: while I’ve been trying to treat such category theory as a black box on this blog, it should be interesting (at least for me) to see how the machines work beneath the surface, in some specific examples. In previous posts, I stated the result, and described an important lemma on the structure of simplicial objects in a stable -category, which depended on the combinatorics of cubes.
The goal of this post is to (finally) prove the result, an equivalence of -categories
valid for any stable -category . As before, the intuition behind this version of the Dold-Kan correspondence is that a simplicial object determines a filtered object by taking successive geometric realizations of the -truncations. The fact that one can go in reverse, and reconstruct the simplicial object from the geometric realizations of the truncations, is specific to the stable case.
1. Formalizing the equivalence
Let’s make this precise. To do so introduces a little extra notation, but the ideas are exactly those as above.
There is a chain of equivalences which is strictly functorial in the stable -category (functorial for functors which are exact, i.e. preserve finite limits and colimits). Namely, let’s take the category . As always, categories such as are considered as simplicial sets via their nerves (this is admittedly a slight abuse of notation). There is a subcategory which consists of all pairs with . Let be the subcategory not including elements with . We can draw and as subcategories of (which we may represent in a 2-dimensional plane) by those elements lying below a “diagonal.”
Our strategy to get from a simplicial object to a filtered object will be, with more precision, as follows:
- A simplicial object determines a functor , and thus a functor . This functor just forgets about the second factor, so it is not the most interesting functor, and in particular one can recover the original functor from it. Actually, does not quite sit inside , so this requires a bit of an argument.
- We left Kan extend to , i.e. to the points for . At each , this is, roughly speaking, a colimit over all with : that is, the geometric realization of the -truncation of .
- We restrict to . This just remembers the filtered object.
We have four relevant categories and morphisms between them:
where all the morphisms are given by forgetful functors of some sort. The primes denote conditions which will be specified shortly. Namely, sends a simplicial object to a functor —a functor which pays no attention to the second factor. The functor is retriction (an inverse to the left Kan extension mentioned above, except is functorial in a 1-categorical sense). Here is ordinary restriction.
Anyway, the advantage of writing everything out this way is that it is very rigorous: we are working with natural transformations of simplicial sets, in the ordinary 1-categorical sense. The -categorical claim is that all the are equivalences, when is stable.
Here are the conditions on the “primes:”
- refers to functors which only see the first variable: that is, they take all maps which are identity on the first variable to equivalences.
- refers to functors whose restrictions to satisfy the previous condition and which are left Kan extensions of their restrictions to (i.e., the associated filtered object comes from taking partial geometric realizations).
The following result will complete the proof of the Dold-Kan correspondence.
Proposition 12 Each of the is an equivalence of -categories (for stable).
Let’s start by noting that is an equivalence for formal reasons, because we only restricted to functors which were left Kan extensions of the restriction to (and these left Kan extensions can always be made since we have finite colimits). What we really need to do is to handle .
2. The third equivalence
Let’s next show that is an equivalence. This is the harder part, and it requires an induction: it will show that the filtered object in uniquely determines a whole element of .
Let be the truncated part of consisting of elements with or with . We can think of as the union,
Let’s consider the -category consisting of functors which satisfy a truncated version of the condition above: they see only the first variable and are left Kan extensions at each .
Proposition 13 is a trivial Kan fibration.
In particular, taking the inverse limit, we find that
is an equivalence.
Proof: We can think of as being obtained from by adding the object . Actually, we are also adding a bunch of other objects like . So let be obtained by just adding . If one starts with a functor
to extend it to to satisfy the “prime” condition, one has to first extend to (that is, at the object ) in such a way that it is a left Kan extension at , and then extend by left Kan extension to the points for (this just means by sending these points to the same image as in ).
Now we showed in a previous post that choosing a functor such that it is a left Kan extension at is the same thing as right Kan extending along . In other words, this is a recaptiulation of the fact that once one has a -truncated simplicial object (that’s the functor when one forgets about ) and a desired colimit for a -skeleton (that’s where goes) then it uniquely extends (via a right Kan extension process) to a -truncated simplicial object.
It follows from all this that there is (up to unique isomorphism) one way of extending a functor to so as to satisfy the primed conditions: it has to be a right Kan extension followed by a left Kan extension. Stated more precisely, this shows in fact that one has a trivial Kan fibration as stated.
3. The first equivalence
Next, we have to show that is an equivalence. In other words, to give a simplicial object
is equivalent to giving a morphism which only sees the first factor, informally. This is easier since no use of the stability of is made—this is something valid for every -category .
Anyway, one problem is that the map doesn’t admit a section. However, if we truncate, and restrict to pairs with , we get categories , and the associated map
does admit a section (given by sending , and so we can actually regard as a (full) subcategory of .
It will suffice to show that
for each . We can do this by observing that the “prime” condition on is precisely saying that the functor is a right Kan extension of its restriction to (regarded as a subcategory), and the functor in this direction can be identified with right Kan extension. This establishes the equivalence above, and shows (by taking homotopy inverse limits) that is an equivalence as desired.
This completes the proof of the Dold-Kan correspondence, and shows that there is a functorial (for exact functors) chain of three equivalences between simplicial and filtered objects in any stable -category.
This is one of the first genuinely non-formal results I’ve learned in higher category theory — it’s one that is simply different from the 1-categorical case, and which uses the inherently higher-categorical notion of stability. There is the Dold-Kan correspondence for abelian categories, but it relies on somewhat hard-to-motivate constructions, and takes values in chain complexes. This version of the Dold-Kan correspondence is based on an extremely natural functor, familiar once one knows that a geometric realization admits a skeletal filtration. What’s not clear to me yet is whether one can (in some manner) derive the classical version from this version.