Let ${k}$ be a field. The commutative cochain problem over ${k}$ is to assign (contravariantly) functorially, to every simplicial set ${K_\bullet}$, a commutative (in the graded sense) ${k}$-algebra ${A(K_\bullet)}$, which is naturally weakly equivalent to the algebra ${C^*(K_\bullet, k)}$ of singular cochains (with ${k}$-coefficients). We also require that ${A(K_\bullet) \rightarrow A(L_\bullet)}$ is a surjection whenever ${L_\bullet \subset K_\bullet}$. Recall that ${C^*(K_\bullet, k)}$ is an associative algebra, but it is not commutative; the commutativity only appears after one takes cohomology. The commutative cochain problem attempts to find an improvement to ${C^*(K_\bullet, k)}$.

If ${k}$ has finite characteristic, this problem cannot be solved, owing to the existence of nontrivial cohomology operations. (The answers at this MathOverflow question are relevant here.) However, there is a solution for ${k = \mathbb{Q}}$, given by the polynomial de Rham theory. In this post, I will explain this.

First, we need to construct the relevant functor.  By general nonsense, the following are equivalent for any cocomplete category ${\mathcal{C}}$:

1. Cosimplicial objects in ${\mathcal{C} }$.
2. Colimit-preserving functors ${\mathbf{SSet} \rightarrow \mathcal{C}}$.

This follows because a cosimplicial object defines a functor from standard simplices to ${\mathcal{C}}$, and any simplicial set is canonically a colimit of standard simplices.

Let ${\mathbf{DGA}}$ be the category of (noncommutative) dgas over ${k}$. As a result, we can observe the equivalence of the following:

1. Simplicial objects in ${\mathbf{DGA}}$.
2. Contravariant functors ${\mathbf{SSet} \rightarrow \mathbf{DGA}}$ that send colimits to limits.

The same holds when ${\mathbf{DGA}}$ is replaced with the category ${\mathbf{cDGA}}$ of commutative dgas over ${k}$. We can explicitly describe the map in the interesting direction as follows. If we have a simplicial dga ${A_\bullet}$ and want to associate to it a contravariant functor ${A: \mathbf{SSet} \rightarrow \mathbf{DGA}}$, then we let

$\displaystyle A(X_\bullet) = \hom_{\mathbf{SSet}}(X_\bullet, A_\bullet).$

So, explicitly, this means that for each ${n}$-simplex of ${X_\bullet}$, we are given an element of ${A_n}$, and that these glue appropriately. This is a dga.

We shall now give an important example of a simplicial ${\mathbf{cDGA}}$Suppose that ${k}$ has characteristic zero here.

The polynomial de Rham algebra is the following simplicial ${\mathbf{cDGA}}$ ${\mathrm{A}^{PL}_\bullet}$.

1. In level ${n}$, ${\mathrm{A}^{PL}_n}$ is the free graded-commutative algebra over ${k}$ on generators ${x_0, \dots, x_n}$ in degree zero and ${y_0, \dots, y_n}$ in degree one, quotiented by the relations ${\sum x_i = 0}$ and ${\sum y_i = 0}$. (Freeness means that ${\mathrm{A}^{PL}_n}$ is a tensor product of a polynomial ring on the ${\left\{x_i\right\}}$ and an exterior algebra on the ${\left\{y_i\right\}}$.) The differential is given by ${dx_i = y_i}$ and ${dy_i = 0}$.In other words, to borrow intuition from differential geometry, ${\mathrm{A}^{PL}_n}$ is the space of polynomial differential forms on the ${n}$-simplex ${\Delta[n]}$.
2. Suppose given a map ${\phi: [m] \rightarrow [n]}$ of ordered sets. The map ${\phi^*: \mathrm{A}^{PL}_n \rightarrow \mathrm{A}^{PL}_m}$ is given by restriction to a face: algebraically, it sends ${x_i \mapsto \sum_{j:\phi(j) = i } x_j}$ and similarly on the ${y_i}$.

Consequently, there is defined a contravariant functor ${\mathbf{SSet} \rightarrow \mathbf{cDGA}}$ given by ${\mathrm{A}^{PL}_\bullet}$; we write this as ${\mathrm{A}^{PL}}$. So, given ${X_\bullet}$, then ${\mathrm{A}^{PL}(X_\bullet)}$ is the commutative dga whose elements can be described as follows: for every ${n}$-simplex of ${X_\bullet}$, ${\Delta[n]_\bullet \rightarrow X_\bullet}$, there has to be given a form on ${\Delta[n]_\bullet}$; and these have to fit together compatibly based on maps of standard simplices.

We want to show that the polynomial de Rham functor ${\mathrm{A}^{PL}}$ is a solution to the commutative cochain problem.

Proposition 13 If ${L_\bullet \hookrightarrow K_\bullet}$ is an injection of simplicial sets, then ${\mathrm{A}^{PL}(K_\bullet ) \rightarrow \mathrm{A}^{PL}(L_\bullet)}$ is surjective.

Proof: This states that any map ${L_\bullet \rightarrow \mathrm{A}^{PL}_\bullet}$ of simplicial sets can be extended to a map ${K_\bullet \rightarrow A_\bullet}$. In other words, the proposition can be phrased as stating that ${\mathrm{A}^{PL}_\bullet}$ (as a simplicial set) is a contractible Kan complex. As a simplicial group, it is a Kan complex. Checking that it is contractible, though, takes some work (it can be done using explicit homotopies). $\Box$

This fact will be very useful to us, as it will allow us to obtain a model for the relative cohomology algebra. Namely, let ${A_\bullet}$ be a simplicial dga which is also a contractible Kan complex, with associated functor ${A}$ (not to be confused with ${\mathrm{A}^{PL}}$, though this is a candidate!). Given ${L_\bullet \subset K_\bullet}$, we define the relative cohomology ring ${A(K_\bullet, L_\bullet)}$ as the kernel of the surjection ${A(K_\bullet) \rightarrow A(L_\bullet)}$. So there is a short exact sequence

$\displaystyle 0 \rightarrow A(K_\bullet, L_\bullet) \rightarrow A(K_\bullet) \rightarrow A(L_\bullet) \rightarrow 0.$

Now, we are interested in a natural weak equivalence between the cochain algebra ${C^*(K_\bullet, k)}$ of a simplicial set and the polynomial de Rham algebra ${\mathrm{A}^{PL}(K_\bullet)}$. Since the cochain algebra is not commutative, we can’t produce a map between the two. However, we shall produce a chain of weak equivalences.

Note first that the map ${K_\bullet \rightarrow C^*(K_\bullet, k)}$ sends colimits to limits. In other words, it can be described by a simplicial ${\mathbf{DGA}}$. This simplicial dga, which we denote ${\mathrm{B}_\bullet}$, is given by ${[n] \mapsto C^*(\Delta[n]_\bullet, k)}$. So

$\displaystyle C^*(K_\bullet, k) = \hom_{\mathbf{SSet}}(K_\bullet, \mathrm{B}_\bullet).$

Let us note that the map ${C^*(K_\bullet, k) \rightarrow C^*(L_\bullet, k)}$ is surjective for any inclusion ${L_\bullet \hookrightarrow K_\bullet}$ of simplicial sets; this means that ${\mathrm{B}_\bullet}$ is also a contractible Kan complex. This is why we can define relative cohomology.

Let ${\mathrm{AB}_\bullet}$ be the simplicial dga given by ${\mathrm{A}^{PL}_\bullet \otimes \mathrm{B}_\bullet}$; this is given degreewise as ${(\mathrm{AB})_n = A_n \otimes_k B_n}$. There is an associated functor from simplicial sets to ${\mathbf{DGA}}$, which we write as ${\mathrm{AB}}$. Because there are natural inclusions of simplicial dgas ${\mathrm{A}^{PL}_\bullet \hookrightarrow \mathrm{AB}_\bullet, \mathrm{B}_\bullet \hookrightarrow \mathrm{AB}_\bullet}$, we have natural transformations of functors

$\displaystyle \mathrm{A}^{PL}(K_\bullet) \hookrightarrow \mathrm{AB}(K_\bullet) \hookleftarrow C^*(K_\bullet, k), k). \ \ \ \ \ (1)$

We shall need:

Lemma 14 Let ${\mathcal{D}_\bullet, \mathrm{E}_\bullet}$ be simplicial dgas, leading to functors ${\mathcal{D}, \mathrm{E}: \mathbf{SSet}^{op} \rightarrow \mathbf{DGA}}$. Suppose given a map of simplicial dgas ${\mathcal{D}_\bullet \rightarrow \mathrm{E}_\bullet}$, leading to a natural transformation of functors ${\mathcal{D} \rightarrow \mathrm{E}}$. Suppose both ${\mathcal{D}_\bullet, \mathrm{E}_\bullet}$ are contractible Kan complexes (many sources say “${\mathcal{D}_\bullet}$ and ${\mathrm{E}_\bullet}$ are extendable”) and that ${\mathcal{D}_n \rightarrow \mathrm{E}_n}$ is a quasi-isomorphism for each ${n}$. Then ${\mathcal{D}(K_\bullet) \rightarrow \mathrm{E}(K_\bullet) }$ is a quasi-isomorphism for each simplicial set ${K_\bullet}$.

Proof: To see this, we then work by induction on the simplices, and then use a limiting argument.

Let us fix ${n}$, and suppose that if ${W_\bullet}$ is any simplicial set whose nondegenerate simplices are in dimension ${< n}$, then ${\mathcal{D}(W_\bullet) \rightarrow \mathrm{E}(W_\bullet)}$ is a quasi-isomorphism. Note that this implies that if ${m \leq n}$, then ${\mathcal{D}(\Delta[m]_\bullet, \partial \Delta[m]_\bullet) \rightarrow \mathrm{E}(\Delta[m]_\bullet, \partial \Delta[m]_\bullet)}$ is a quasi-isomorphism. Indeed, this follows because ${\mathcal{D}(\partial \Delta[m]_\bullet) \rightarrow \mathrm{E}(\partial \Delta[m]_\bullet)}$ is (by the inductive hypothesis), and ${\mathcal{D}(\Delta[m]_\bullet) \rightarrow \mathrm{E}(\Delta[m]_\bullet)}$ is (by the original hypothesis), so we can apply the five-lemma.

Now let ${K_\bullet}$ be a simplicial set whose nondegenerate simplices are in dimensions ${\leq n}$. We are going to show that ${\mathcal{D}(K_\bullet) \rightarrow \mathrm{E}(K_\bullet)}$ is a quasi-isomorphism. This will imply that the natural map is a quasi-isomorphism for any simplicial set concentrated in finitely many dimensions. There is a set ${A}$, and a push-out diagram ${K_\bullet = \bigsqcup_A \Delta[m]_\bullet \sqcup_{\bigsqcup_A \partial \Delta[n]_\bullet} L_\bullet}$ for ${L_\bullet}$ a simplicial set with nondegenerate simplices in dimensions ${< n}$. We have then a diagram

Here we have used the extendability. By induction, the rightmost map is a weak equivalence (since ${L_\bullet}$ has nondegenerate simplices in dimension ${). We want to argue that the leftmost map is as well; this will follows because ${\mathcal{D}(K_\bullet, L_\bullet) = \prod_A \mathcal{D}(\Delta[n]_\bullet, \partial \Delta[n]_\bullet)}$ and similarly for ${\mathrm{E}}$.

Thus the lemma is true for any ${K_\bullet}$ with nondegenerate simplices in only finitely many dimensions, by the above inductive argument. Let ${K_\bullet}$ now be any simplicial set. We have a filtration

$\displaystyle \mathrm{sk}_1 K_\bullet \subset \mathrm{sk}_2 K_\bullet \subset \dots \subset K_\bullet$

such that ${K_\bullet}$ is the filtered colimit of the ${\mathrm{sk}_i K_\bullet}$. Here each ${\mathrm{sk}_i}$ is the ${i}$-skeleton functor, and consequently each ${\mathrm{sk}_i K_\bullet}$ is concentrated in a finite number of dimensions. In particular,

$\displaystyle \mathcal{D}(\mathrm{sk}_i K_\bullet) \rightarrow \mathrm{E}(\mathrm{sk}_i K_\bullet)$

is a quasi-isomorphism for each ${i}$. Note that each of the maps ${\mathcal{D}(\mathrm{sk}_i K_\bullet) \rightarrow \mathcal{D}(\mathrm{sk}_{i-1} K_\bullet)}$ is surjective, and ${\mathcal{D}(K_\bullet)}$ is the inverse limit of these; similarly for ${\mathrm{E}}$. Since these are homotopy inverse limits by the above surjectivity, we find that ${\mathcal{D}(K_\bullet) \rightarrow \mathrm{E}(K_\bullet)}$ is a quasi-isomorphism. $\Box$

Theorem 15 Both the above inclusions in (1) are weak equivalences. In particular, ${\mathrm{A}^{PL}}$ is a solution to the commutative cochain algebra.
Proof: By the lemma, we have to verify that the three simplicial dgas are “extendable” (i.e. contractible Kan complexes), and the maps are quasi-isomorphisms in each dimension. We have already seen that ${\mathrm{A}^{PL}_\bullet}$ and ${\mathrm{B}_\bullet}$ are contractible Kan complexes. We have to show the same for ${\mathrm{AB}_\bullet}$.
But because ${A_\bullet}$ and ${B_\bullet}$ are contractible and ${\mathrm{AB}_\bullet}$ is a tensor product, we can tensor the contracting homotopies together to see that ${\mathrm{AB}_\bullet}$ is contractible. Finally, we have to check that ${\mathrm{A}^{PL}_n }$ and ${\mathrm{B}_n}$ are quasi-isomorphic to ${\mathrm{AB}_n}$ for each ${n}$, but this follows from the Künneth formula: we know (by direct computation) that ${\mathrm{A}^{PL}_n}$ is acyclic except in dimension zero, where it has a ${k}$; we know by singular cohomology that ${\mathrm{B}_n}$ is acyclic except in dimension zero, where it has a ${k}$; and now we can apply the Künneth formula to ${\mathrm{AB}_n = \mathrm{A}^{PL}_n \otimes_k \mathrm{B}_n}$. $\Box$