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.)
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.
Let’s try to motivate why such a result might be reasonable. To give an -filtered object of (by which I simply mean an element of ), is the same as giving an -filtered object of together with a morphism out of the colimit of the -filtered object (i.e., of the last element). That is, we have an equivalence
Stated alternatively, to give a functor is the same as specifying a restricted functor and a colimit of (together with appropriate compatibility data).
If we expect an equivalence of the form (1), then we should expect that to give an -truncated simplicial object is the same as giving an -truncated object together with a specification of a colimit for the full thing. In other words, we find:
Goal: To give an -truncated simplicial object is the same as giving an -truncated simplicial object together with a desired colimit of (along with some compatibility data).
So, once we’re given an -truncated simplicial object, there should be a unique choice of a degree element that will make the -truncated colimit what we want. We will make this precise in the following manner. Let be the category of finite ordered sets (so in other words, is included), and let be defined to be finite ordered sets of cardinality .
Observe that has as initial object, and consequently we can regard elements of
as consisting of -truncated simplicial objects in , together with an extra object (the image of ). In -language, we can say that
is the cone on .
Definition 7 An augmented -truncated simplicial object of is an element of .
The upshot of all this is that if we expect the Dold-Kan correspondence to hold, then to give an augmented -truncated simplicial object in should be equivalent to giving an -truncated simplicial object of . In fact, an “augmented -filtered object” is precisely the same as an -filtered object. So we should expect an equivalence
2. Comparing augmentations and truncations
Intuitively, the functor from to consists of restricting to the -skeleton, plus taking the colimit (of the whole -truncated thing) to be the augmentation in degree . The functor in the other direction will realize the degree element as an appropriate limit of the -truncation plus the augmentation: the interplay between these limits and colimits is precisely where the stability condition enters.
To be very precise, we’re going to construct the equivalence (2) as a chain of equivalences. Namely, we note that augmented -truncated simplicial objects map to both categories: we have a diagram
The main lemma is going to be that that a suitable subcategory of -truncated augmented objects maps equivalently under to both ends. We will show:
Proposition 8 Let be stable. For an -truncated augmented simplicial object , the following are equivalent:
- is a left Kan extension of its restriction to (i.e., forgetting the augmentation).
- is a right Kan extension of its restriction to (i.e., to an -truncated augmented simplicial object).
This is precisely the techincal tool which makes the proof go. The first condition (that is a left Kan extension) states that the augmentation comes from the colimit over the whole -truncated thing. The second condition states that the object in degree is recovered from the -truncation plus the augmentation. The proposition states that we can go either way.
The proposition takes some work with cofinality, and a little effort with stable -categories, to prove. But once it’s proved, we can obtain the desired equivalence (2) fairly easily. Namely, we want to show that augmented -truncated simplicial objects and -truncated simplicial objects are equivalent. We consider the diagram
where means functors that satisfy either (i.e., both) of the conclusions of the proposition. For formal reasons, the two maps are equivalences: given a subcategory of an -category , to give a functor out of is the same as giving a functor out of which is a left (or right) Kan extension of . At least, if enough limits or colimits exist in the target -category, which is true in this case.
3. Proof of the proposition
Let be stable, and fix an -truncated augmented simplicial object . To say that is a left Kan extension of its restriction to is to say that is the colimit of the -truncated thing . In other words, if we regard as the cone on , then the diagram
is a colimit diagram, exhibiting as the colimit of . (To be precise, it is a diagram out of a conewhich is a colimit. A colimit is the whole diagram, not just the image of the cone point.)
Now in the previous post, we saw that there was another model for up to cofinality: namely, . This is the opposite to the category of nonempty finite ordered sets equipped with an injective map into , or equivalently the opposite to the poset of nonempty subsets of . (Warning: My argument was incorrect; a correct one is in Lurie’s book.)
So, to say that is a colimit diagram is to say that the diagram
is a colimit diagram. Here the comes from adding a cone point for the empty set: that’s a terminal object in , and the cone point goes to . In other words, this is the opposite to the poset of allsubsets of : in other words, . We’re somehow going to have to turn this into a condition that will involve limits—that is, right Kan extensions.
Note now that this object comes with a symmetry, and we can talk about it being either a colimit diagram (using the last vertex) or being a limit diagram (using the first vertex). When , this is a square, and we can talk about a square being a pull-back or a push-out. In the stable setting, though the two are equivalent! This turns out to generalize.
Lemma 9 A diagram is a limit diagram if and only if it is a colimit diagram (for stable).
was a colimit diagram. However, because the target category is , this is equivalent to (3) being a limit diagram. In other words, that’s equivalent to the image of the initial object (which is the identity map ) being determined as a limit of all the other objects. That is, needs to be a limit of all the other objects in some sense.
It’s now easy to believe that this condition is equivalent to ‘s being a right Kan extension. However, there is a bit of checking to do. So far, we’ve said that was a left Kan extension if and only if was a limit diagram at the cone point . In other words, this is saying that
This is a complicated condition, but it is saying that is the limit over all injections of equipped with a map to (or rather, the opposite to that).
To turn this into a condition on right Kan extensions, we have to replace by the category
where we don’t just consider injective maps. This is what it means for to be a right Kan extension at : has to be the limit of the diagram of all things that maps to in .
To see this, we use a cofinality argument again:
Proposition 10 The map is right cofinal.
Here “right cofinal” is the dual to cofinal: it means that the map of opposite categories is cofinal. In other words,
is cofinal. To see this, in turn, we have to show (by Theorem A) that given an object in the second category, the category of diagrams
is contractible. Here the map is required to be injective. But this category has an initial object, which comes from the image of in .
Let’s put this all together:
- To say that is a left Kan extension is, by definition, equivalent to saying that has colimit .
- By a cofinality lemma, this is equivalent to saying that is a colimit diagram (at the cone point ). Note that this is a cubical diagram.
- By a lemma about stable -categories, this in turn is equivalent to saying that is a limit diagram at . In other words, that is the limit of .
- Using another cofinality argument, this is equivalent to saying that is the limit of .
- This in turn states precisely that is a right Kan extension at .
This completes the proof of the (somewhat technical) lemma we need for the Dold-Kan correspondence, modulo the lemma on cubical diagrams.