The next couple of posts will cover the Dold-Kan correspondence, which establishes an equivalence of categories between simplicial abelian groups and chain complexes. While this will not be strictly necessary for the introduction of the cotangent complex, it is a sufficiently important fact that it seems worth a digression.

As far as I can tell, the Dold-Kan correspondence is a fairly technical result, and I’m not sure I have any good intuition for why one should expect it to work. But at least one can say the following: given a simplicial abelian group (that is, a contravariant functor from the simplex category to the category of abelian groups), one can form a chain complex in a fairly easy manner: just take the -simplices of the simplicial group as the degree part of the complex, and define the boundary using the alternating sum of the simplicial boundary maps (defined below); this is the classical computation that one does in introductory algebraic topology, of showing that the singular chain complex is indeed a complex.

So it’s natural that you would get a chain complex from a simplicial abelian group. Except, as it turns out, this is the **wrong **functor for the Dold-Kan correspondence; it is, however, close, being right up to (natural) homotopy.

The other bit of intuition that I’ve heard is the following. Given a topological space , there is a means of obtaining the homology of as the homotopy groups of the infinite symmetric product; this is the so-called Dold-Thom theorem. (See this, for instance.) The Dold-Kan correspondence is in some sense the simplicial analog of this. The infinite symmetric product is much like the abelianization functor from simplicial sets to simplicial abelian groups (that applies the free abelian group pointwise). Now, it will come out of the Dold-Kan correspondence that the so-called “simplicial homotopy groups” of a simplicial abelian group are going to be the same thing as the homology of the associated chain complex. This is a rather loose analogy, and my understanding is that one cannot derive Dold-Thom from Dold-Kan.

**1. The simplicial identities**

The classical approach of defining a simplicial set was less fancy than the modern approach of a presheaf on the simplex category. Namely, one simply considered a sequence of sets and suitable face and degeneracy maps between them, satisfying a bunch of identities that were basically a way of giving a presentation of the simplex category. I personally find this approach somewhat off-putting and for a long time resolutely avoided any thought of simplicial identities; it seems, however, that one needs to play around with them a bit to go through the Dold-Kan correspondence. Thus, at the risk of being hopelessly inappropriate for a blog setting, I’ll describe these morphisms.

We shall define certain important morphisms in the simplex category describe the relations between them. Let . We define

Here maps the ordered set to via an inclusion, but where the element in is omitted. These are called the *coface maps.* So one is supposed to think of the coface map is being the string of elements in . Similarly, we define the *codegeneracy maps*

The codegeneracy is a surjective map, where the elements are mapped to the same element. One is supposed to think of this as the string of elements of elements of .

Lemma 1 (Cosimplicial identities)We have:

*Proof:* We can think of the map as a map that clearly omits from the image. Moreover, is omitted. Similarly, we see that omits and from the image. Since both maps are injective, the first assertion is clear. The second assertion can be proved similarly. We now describe identities involving the codegeneracies.

Lemma 2 (More cosimplicial identities)We have:

We omit the verification, which is easy. Let be a simplicial set. There are induced maps

for each , by applying the functor to the . These are called the *face* and *degeneracy* maps, respectively.

Lemma 3 (Simplicial identities)For any simplicial set , we have

*Proof:* This is now clear from the cosimplicial identities. One way to think about this is that “the smaller map can be moved to the inside.” For instance, if we have with , then we can move the “smaller” map to the inside of the composition. Another thing to keep in mind is that for a simplicial set , the degeneracy maps are *injective*; indeed, they have canonical sections, namely the face maps.

**2. Simplicial abelian groups**

A *simplicial abelian group* is a simplicial object in the category of abelian groups. This means that there are abelian groups and group-homomorphisms for each map in .

**1.1. Three different complexes**

Following \cite{GJ}, we are going to define several ways of making a chain complex from a simplicial abelian group. They will all have the same homotopy type, but one of them will be the most convenient for the Dold-Kan correspondence.

Definition 4TheMoore complexof a simplicial abelian group is the complex which in dimension is . The boundary map

is the map , the alternating sum of the face maps. The simplicial identities easily imply that this is in fact a chain complex. Thus defines a functor from simplicial abelian groups to chain complexes.

The *singular chain complex* of a topological space can be obtained by taking the Moore complex of , where denotes the operation of taking the free abelian group. (Note that applying turns a simplicial set into a simplicial abelian group.) Recall that if is a simplicial set, then a simplex is called *degenerate* if it is in the image of one of the degeneracy maps (from ).

Proposition 5Let be a simplicial abelian group. There is a subcomplex of the Moore complex such that consists of the sums of degenerate simplices in degree .

*Proof:* We need only check that is stable under . In particular, we have to check that is a sum of degeneracies for any . Now this is

because the terms vanishes in view of the simplicial identities. Moreover, the simplicial identities show that we can move the part inside in the rest of the terms of the summation, potentially changing the subscript of the . So belongs to .

Definition 6Consequently, if is a simplicial abelian group, we can consider the chain complex . This is functorial in , and there is a natural transformation

Nonetheless, in defining the Dold-Kan correspondence, we shall use a different construction (which we will prove is isomorphic to ).

Definition 7If is a simplicial abelian group, we define thenormalizedcomplex as follows. In dimension , consists of the subgroup of that is killed by the face maps . The differential

is given by .

It needs to be checked that we indeed have a chain complex. Suppose ; we must show that . But by the simplicial identities, and we know that kills . Thus the verification is clear. We thus have three different ways of obtaining a complex from . By the way we defined the normalized chain complex, we have natural morphisms

Our goal is to prove:

Theorem 8 (Dold-Kan)The functor defines an equivalence of categories between chain complexes of abelian groups and simplicial abelian groups. Moreover, the three complexes are all naturally homotopically equivalent (and the first and the last are even isomorphic).

I will start on the proof of this next time.

May 6, 2011 at 1:49 pm

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May 11, 2011 at 11:33 pm

Personally I don’t think of Dold-Kan as a technical result. It arises from our ability to take a simplex and break it into a bunch of “chains”.

For example, think of a 1-simplex as an arrow with a tail and a head. This can be broken up into one 0-chain x (the tail of the arrow) and one 1-chain f (the arrow itself). The head is then x + df.

Similarly a 2-simplex or triangle is determined by a 0-chain (the starting corner), two 1-chains (two arrows, one going from the starting corner to the second corner, and another going from the second corner to the third), and one 2-chain (the “body” of the 2-simplex). The other parts of the 2-simplex can be worked out from these, and the rule dd = 0 makes it all consistent.

You’ll note the sequences of numbers here (one, one for the 1-simplex, and one, two, one for the 2-simplex) are rows of Pascal’s triangle. That’s how it goes: in general an n-simplex is determined by n choose 0 0-chains, n choose 1 1-chains, n choose 2 2-chains and so on.

I’m sorry that my explanation is so sketchy and incoherent; a good one requires a lot of pictures.

May 12, 2011 at 8:45 am

Dear John, thanks for the explanation! This is a nice way of thinking about it.

May 15, 2011 at 7:11 pm

[…] chain complexes. Here the analogy extends again: by the Dold-Kan correspondence (which I recently talked about), this is equivalent to the category of simplicial objects in that category. The appropriate […]