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 $n$-simplices of the simplicial group as the degree $n$ 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 $X$, there is a means of obtaining the homology of $X$ 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 $X_n$ 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 ${\Delta}$ describe the relations between them. Let ${n \in \mathbb{Z}_{\geq 0}}$. We define

Here ${d^i}$ maps the ordered set ${[n-1]}$ to ${[n]}$ via an inclusion, but where the element ${i}$ in ${[n]}$ is omitted. These are called the coface maps. So one is supposed to think of the coface map is being the string ${0 \rightarrow 1 \rightarrow \dots \rightarrow i-1 \rightarrow i+1 \rightarrow \dots \rightarrow n}$ of ${n-1}$ elements in ${[n]}$. Similarly, we define the codegeneracy maps

The codegeneracy ${s^i}$ is a surjective map, where the elements ${i, i+1}$ are mapped to the same element. One is supposed to think of this as the string of ${n}$ elements ${0 \rightarrow 1 \rightarrow \dots \rightarrow i-1 \rightarrow i \rightarrow i \rightarrow i+1 \rightarrow \dots \rightarrow n-1}$ of elements of ${[n-1]}$.

Lemma 1 (Cosimplicial identities) We have:

Proof: We can think of the map ${d^j d^i: [n-2] \rightarrow [n]}$ as a map that clearly omits ${j}$ from the image. Moreover, ${d^j(i) = i}$ is omitted. Similarly, we see that ${d^i d^{j-1}}$ omits ${i}$ and ${d^i(j-1) = j}$ 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 ${X_\bullet}$ be a simplicial set. There are induced maps

$\displaystyle d_i: X_n \rightarrow X_{n-1}, \quad s_i: X_n \rightarrow X_{n+1}$

for each ${n}$, by applying the functor ${X_\bullet}$ to the ${d^i, s^i}$. These are called the face and degeneracy maps, respectively.

Lemma 3 (Simplicial identities) For any simplicial set ${X_\bullet}$, 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 ${d_i d_j}$ with ${i< j}$, then we can move the “smaller” map ${d_i}$ to the inside of the composition. Another thing to keep in mind is that for a simplicial set ${X_\bullet}$, the degeneracy maps are injective; indeed, they have canonical sections, namely the face maps.

2. Simplicial abelian groups

simplicial abelian group ${A_\bullet}$ is a simplicial object in the category of abelian groups. This means that there are abelian groups ${A_n, n \in \mathbb{Z}_{\geq 0}}$ and group-homomorphisms ${A_n \rightarrow A_m}$ for each map ${[m] \rightarrow [n]}$ in ${\Delta}$.

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 4 The Moore complex of a simplicial abelian group ${A_\bullet}$ is the complex ${A_*}$ which in dimension ${n}$ is ${A_n}$. The boundary map

$\displaystyle \partial: A_n \rightarrow A_{n-1}$

is the map ${\sum_{i=0}^n (-1)^i d_i}$, the alternating sum of the face maps. The simplicial identities easily imply that this is in fact a chain complex. Thus ${A_\bullet \mapsto A_*}$ defines a functor from simplicial abelian groups to chain complexes.

The singular chain complex of a topological space ${X}$ can be obtained by taking the Moore complex of ${\mathbb{Z}[\mathrm{Sing} X_\bullet]}$, where ${\mathbb{Z}[]}$ denotes the operation of taking the free abelian group. (Note that applying ${\mathbb{Z}}$ turns a simplicial set into a simplicial abelian group.) Recall that if ${X_\bullet}$ is a simplicial set, then a simplex ${x \in X_n}$ is called degenerate if it is in the image of one of the degeneracy maps (from ${X_{n-1}}$).

Proposition 5 Let ${A_\bullet}$ be a simplicial abelian group. There is a subcomplex ${DA_* \subset A_*}$ of the Moore complex such that ${DA_n}$ consists of the sums of degenerate simplices in degree ${n}$.

Proof: We need only check that ${DA_*}$ is stable under ${\partial}$. In particular, we have to check that ${\partial(s_i a)}$ is a sum of degeneracies for any ${a \in A_{n-1}}$. Now this is

$\displaystyle \partial (s_i a) = \sum (-1)^j d_j (s_i a) = \sum_{j \neq i, i+1} d_j s_i a ,$

because the terms ${(-1)^i ( d_i s_i a - d_{i+1} s_i a) = (-1)^i (a - a) = 0}$ vanishes in view of the simplicial identities. Moreover, the simplicial identities show that we can move the ${d}$ part inside in the rest of the terms of the summation, potentially changing the subscript of the ${s}$. So ${\partial s_i a}$ belongs to ${DA_{n-1}}$.

Definition 6 Consequently, if ${A_\bullet}$ is a simplicial abelian group, we can consider the chain complex ${(A/DA)_*}$. This is functorial in ${A_\bullet}$, and there is a natural transformation

$\displaystyle A_* \rightarrow (A/DA)_*.$

Nonetheless, in defining the Dold-Kan correspondence, we shall use a different construction (which we will prove is isomorphic to ${(A/DA)_*}$).

Definition 7 If ${A_\bullet}$ is a simplicial abelian group, we define the normalized complex ${NA_*}$ as follows. In dimension ${n}$, ${NA_n}$ consists of the subgroup of ${A_n}$ that is killed by the face maps ${d_i, i < n}$. The differential

$\displaystyle NA_n \rightarrow NA_{n-1}$

is given by ${(-1)^nd_n}$.

It needs to be checked that we indeed have a chain complex. Suppose ${a \in NA_n}$; we must show that ${d_{n-1}d_n a = 0}$. But ${d_{n-1}d_n = d_{n-1} d_{n-1}}$ by the simplicial identities, and we know that ${d_{n-1}}$ kills ${a}$. Thus the verification is clear. We thus have three different ways of obtaining a complex from ${A_\bullet}$. By the way we defined the normalized chain complex, we have natural morphisms

$\displaystyle NA_* \rightarrow A_* \rightarrow (A/DA)_*.$

Our goal is to prove:

Theorem 8 (Dold-Kan) The functor ${A_\bullet \mapsto NA_*}$ defines an equivalence of categories between chain complexes of abelian groups and simplicial abelian groups. Moreover, the three complexes ${NA_*, A_*, (A/DA)_*}$ are all naturally homotopically equivalent (and the first and the last are even isomorphic).

I will start on the proof of this next time.