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 {<n}). 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

Let us return to the main goal.

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

 

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