I certainly have never thought or heard about this idea of yours, Urs, but it sounds believable to me.

There are some portions of this idea that are new to me, but make me feel I should have already known them.

For example, I know and love the Dold–Kan theorem in this guise:

**Theorem 1**: the category of simplicial abelian groups is isomorphic to the category of nonnegatively graded chain complexes of abelian groups.

In fact there’s nothing terribly special about abelian groups here; the proof just requires some elementary stuff about kernels, direct sums, and so on. So it at least generalizes this far:

**Theorem 2**: given an abelian category $A$, the category of simplicial objects in $A$ is isomorphic to the category of nonnegatively graded chain complexes in $A$.

(I leave it to Zoran Skoda to say what happens when $A$ is semiabelian: people definitely *do* think about simplicial *groups* that are not necessarily abelian.)

Anyway, for some reason I’d never thought about the ‘dual’ Dold–Kan theorem stated in the first sentence of Castiglioni and Cortinas’ paper:

**Theorem 3**: the category of cosimplicial abelian groups is isomorphic to the category of nonnegatively graded cochain complexes of abelian groups.

Which is silly of me, because it follows instantly from Theorem 2 by taking $A = AbGp^{op}$. Indeed, just as instantly, Theorem 2 implies

**Theorem 4**: given an abelian category $A$, the category of cosimplicial objects in $A$ is isomorphic to the category of nonnegatively graded cochain complexes in $A$.

Anyway, I think I almost understand how Theorem 2 paves the way towards the result in the second sentence of Castiglioni and Cortinas’ paper:

**Theorem 5**: the model category of cosimplicial rings is Quillen equivalent to the model category of differential graded rings (nonnegatively graded, with differential of grade $1$).

I bet the point is that both [cosimplicial abelian groups] and [nonnegatively graded cochain complexes of abelian groups] have an ‘obvious tensor product’ making them into a monoidal category, and then:

**Theorem 6**: the category of monoids in [cosimplicial abelian groups] is equivalent to the category of cosimplicial rings.

and

**Theorem 7**: the category of monoids in [nonnegatively graded cochain complexes of abelian groups] is equivalent to the category of differential graded rings (nonnegatively graded, with differential of grade $1$).

Then Theorem 5 would be trivial if
[cosimplicial abelian groups] and [nonnegatively graded cochain complexes of abelian groups] were equivalent as monoidal categories.

And this in turn would be trivial if [simplicial abelian groups] and [nonnegatively graded chain complexes of abelian groups] were equivalent as monoidal categories — taking the opposite category can’t hurt here.

But it’s a famous fact that [simplicial abelian groups] and [nonnegatively graded chain complexes of abelian groups] are **not** quite equivalent as monoidal categories, if we give each one its ‘obvious’ tensor product.

Instead, the equivalence of these two categories only preserves the tensor product ‘up to homotopy’. This is the fact lurking behind the Eilenberg–Zilber theorem: note the phrase ‘$G F$ is chain-homotopic to the identity’ here.

So, [cosimplicial abelian groups] and [nonnegatively graded cochain complexes of abelian groups] are *not* equivalent as monoidal categories, but they are so ‘up to homotopy’. And this is what Castiglioni and Cortinas mention “equivalences of monoidal model categories” in the fourth sentence of their paper.

So, I get the basic idea of Theorem 5, and I think I’ll stop here, since this has taken way too long to write. I’ll try to convince myself that it wasn’t a waste of time by putting this comment on the $n$Lab.

“No, Mom, I’m not wasting my time blogging! I’m contributing to the $n$Lab!”

Personally I would also enjoy seeing that [simplicial abelian groups] and [nonnegatively graded chain complexes of abelian groups] are equivalent as monoidal 2-categories, if this is in fact true. Or as $(\infty,1)$-categories.

## Chern 1944; Re: Question on Synthetic Differential Forms

Didn’t Chern replace a laborious proof using simplicial complexes with his amazing new approach linking curvature invariants to characteristic classes in S.-s. Chern, A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann. of Math. 45 (1944), 747-752. MR 0011027 (6:106a)? Isn’t that what needs to be n-Catgorified?