Let be a field. The *commutative cochain problem* over is to assign (contravariantly) functorially, to every simplicial set , a commutative (in the graded sense) -algebra , which is naturally weakly equivalent to the algebra of singular cochains (with -coefficients). We also require that is a surjection whenever . Recall that 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 .

If 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 , 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 :

- Cosimplicial objects in .
- Colimit-preserving functors .

This follows because a cosimplicial object defines a functor from *standard* simplices to , and any simplicial set is canonically a colimit of standard simplices.

Let be the category of (noncommutative) dgas over . As a result, we can observe the equivalence of the following:

- Simplicial objects in .
- Contravariant functors that send colimits to limits.

The same holds when is replaced with the category of *commutative* dgas over . We can explicitly describe the map in the interesting direction as follows. If we have a simplicial dga and want to associate to it a contravariant functor , then we let

So, explicitly, this means that for each -simplex of , we are given an element of , and that these glue appropriately. This is a dga.

We shall now give an important example of a simplicial . *Suppose that has characteristic zero here.*

The **polynomial de Rham algebra** is the following simplicial .

- In level , is the free graded-commutative algebra over on generators in degree zero and in degree one, quotiented by the relations and . (Freeness means that is a tensor product of a polynomial ring on the and an exterior algebra on the .) The differential is given by and .In other words, to borrow intuition from differential geometry, is the space of polynomial differential forms on the -simplex .
- Suppose given a map of ordered sets. The map is given by restriction to a face: algebraically, it sends and similarly on the .

Consequently, there is defined a contravariant functor given by ; we write this as . So, given , then is the commutative dga whose elements can be described as follows: for every -simplex of , , there has to be given a form on ; and these have to fit together compatibly based on maps of standard simplices.

We want to show that the polynomial de Rham functor is a solution to the commutative cochain problem.

Proposition 13If is an injection of simplicial sets, then is surjective.

*Proof:* This states that any map of *simplicial sets* can be extended to a map . In other words, the proposition can be phrased as stating that (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).

This fact will be very useful to us, as it will allow us to obtain a model for the relative cohomology algebra. Namely, let be a simplicial dga which is also a contractible Kan complex, with associated functor (not to be confused with , though this is a candidate!). Given , we define the *relative cohomology ring* as the kernel of the surjection . So there is a short exact sequence

Now, we are interested in a natural weak equivalence between the cochain algebra of a simplicial set and the polynomial de Rham algebra . 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 sends colimits to limits. In other words, it can be described by a simplicial . This simplicial dga, which we denote , is given by . So

Let us note that the map is surjective for any inclusion of simplicial sets; this means that is also a contractible Kan complex. This is why we can define relative cohomology.

Let be the simplicial dga given by ; this is given degreewise as . There is an associated functor from simplicial sets to , which we write as . Because there are natural inclusions of simplicial dgas , we have natural transformations of functors

We shall need:

Lemma 14Let be simplicial dgas, leading to functors . Suppose given a map of simplicial dgas , leading to a natural transformation of functors . Suppose both are contractible Kan complexes (many sources say “ and are extendable”) and that is a quasi-isomorphism for each . Then is a quasi-isomorphism for each simplicial set .

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

Let us fix , and suppose that if is any simplicial set whose nondegenerate simplices are in dimension , then is a quasi-isomorphism. Note that this implies that if , then is a quasi-isomorphism. Indeed, this follows because is (by the inductive hypothesis), and is (by the original hypothesis), so we can apply the five-lemma.

Now let be a simplicial set whose nondegenerate simplices are in dimensions . We are going to show that 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 , and a push-out diagram for a simplicial set with nondegenerate simplices in dimensions . We have then a diagram

Here we have used the extendability. By induction, the rightmost map is a weak equivalence (since has nondegenerate simplices in dimension ). We want to argue that the leftmost map is as well; this will follows because and similarly for .

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

such that is the filtered colimit of the . Here each is the -skeleton functor, and consequently each is concentrated in a finite number of dimensions. In particular,

is a quasi-isomorphism for each . Note that each of the maps is surjective, and is the inverse limit of these; similarly for . Since these are *homotopy *inverse limits by the above surjectivity, we find that is a quasi-isomorphism.

Let us return to the main goal.

Theorem 15Both the above inclusions in (1) are weak equivalences. In particular, 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 and are contractible Kan complexes. We have to show the same for .

But because and are contractible and is a tensor product, we can tensor the contracting homotopies together to see that is contractible. Finally, we have to check that and are quasi-isomorphic to for each , but this follows from the Künneth formula: we know (by direct computation) that is acyclic except in dimension zero, where it has a ; we know by singular cohomology that is acyclic except in dimension zero, where it has a ; and now we can apply the Künneth formula to .

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