nLab differential graded C^∞-ring

Definition

A connective differential graded $\mathrm{C}^\infty$-ring is a (homologically graded with nonnegative degrees) real commutative differential graded algebra $A$ equipped with a structure of a C^∞-ring on $A_0$.

A coconnective differential graded $\mathrm{C}^\infty$-ring is a (cohomologically graded with nonnegative degrees) real commutative differential graded algebra $A$ equipped with a structure of a C^∞-ring on $A_0$ such that the differential $d\colon A_0\to A_1$ in degree 0 is a C^∞-derivation.

In the unbounded case, Carchedi–Roytenberg proposed the following definition:

An unbounded differential graded $\mathrm{C}^\infty$-ring is a (homologically graded with arbitrary degrees) real commutative differential graded algebra $A$ equipped with a structure of a C^∞-ring on $A_0^c=\ker(A_0\to A_{-1})$.

Remark

Every coconnective differential graded C^∞-ring in the sense defined above is also an unbounded differential graded C^∞-ring concentrated in nonpositive homological degrees, since the kernel of a C^∞-derivation is a C^∞-ring. The converse is false: an unbounded differential graded C^∞-ring concentrated in nonpositive homological degrees has a C^∞-ring structure on its 0-cocycles only, which is not enough to reconstruct a C^∞-structure on the whole degree 0 part or ensure that the degree 0 differential is a C^∞-derivation. The stronger condition is essential for some theorems about coconnective differential graded C^∞-rings, such as the one that states that smooth differential forms form the free C^∞-DGA on smooth functions.

Remark

In fact, having a C^∞-structure in degree 0 together an ordinary algebra structure in higher degrees is sufficient to define a C^∞-structure for nonhomogeneous elements of mixed degrees: given a smooth function $f(x_1, \ldots, x_n)$ and some nonhomogeneous even elements $b_1, \ldots, b_n$ (i.e., formal sums of homogeneous elements of even degrees), set $b_i = c_i + d_i$, where $c_i$ has degree 0 and $d_i$ is a sum of homogeneous elements of positive cohomological degree. Now set

$$f(b_1, \ldots, b_n) = \sum_m (\partial_m f)(c_1, \ldots, c_n) \prod_k d_k^{m_k} / m_k!,$$

where $m$ is a multi-index and $\partial_m f$ is the partial derivative of $f$, which can be evaluated on $c_1, \ldots, c_n$ using the C^∞-structure. This formula yields a (possibly infinite) formal sum of homogeneous elements. Nonhomogeneous elements of odd degrees can be incorporated in the picture by taking $f\in \mathrm{C}^\infty(\mathbf{R}^m)\otimes\Lambda(\mathbf{R}^n)$ (i.e., $f$ has $m$ even and $n$ odd variables) and modifying the formula accordingly. This yields an equivalent notion of differential graded C^∞-rings. The price to pay is having to manipulate infinite formal sums of homogeneous elements, so we choose the equivalent definition given above, even though it may seem slightly ad hoc on the first sight.

Properties

Differential graded C^∞-rings can be equipped with the model structure transferred from the projective model structure on chain complexes via the forgetful functor.

Restricting to connective differential graded C^∞-rings, the resulting model structure is Quillen equivalent to the model category of simplicial C^∞-rings equipped with the model structure transferred along the forgetful functor to simplicial sets equipped with the Kan–Quillen model structure. The right adjoint functor is the normalized chains functor, which sends a simplicial C^∞-rings to its normalized chains equipped with the induced structure of a differential graded C^∞-ring. This is analogous to the monoidal Dold–Kan correspondence.

In this form, the statement was first proved in Taroyan 2023. Similar, but not entirely equivalent results can be found in Nuiten 2018, Remark 2.2.11, which uses a homotopy coherent variant of C^∞-rings and does not explicitly identify the adjoint functors.

References

The essential ingredients (Kähler C^∞-differentials and C^∞-derivations) appear in

• E. J. Dubuc, A. Kock, On 1-form classifiers, Communications in Algebra 12:12 (1984), 1471–1531. doi:10.1080/00927878408823064, (author pdf)

The earliest known occurrence of differential graded C^∞-rings is in the paper

• Domenico Fiorenza, Urs Schreiber, Jim Stasheff, Čech cocycles for differential characteristic classes: an ∞-Lie theoretic construction, Advances in Theoretical and Mathematical Physics 16:1 (2012), 149-250. arXiv, doi

where at the bottom of page 28 in arXiv version 1 one reads:

The underlying algebra in degree 0 can be generalized to an algebra over some Lawvere theory. In particular in a proper setup of higher differential geometry, we would demand $CE(\mathfrak{g})_0$ to be equipped with the structure of a C^∞-ring.

Additional references:

On unbounded differential graded rings for arbitrary Fermat theories (including C^∞-rings):

• David Carchedi, Dmitry Roytenberg, Homological Algebra for Superalgebras of Differentiable Functions, arXiv.

On the equivalence of connective differential graded C^∞-rings and simplicial C^∞-rings via the normalized chains functor:

• Gregory Taroyan, Equivalent models of derived stacks, arXiv.

For a similar statement, see also

• Joost Nuiten, Lie algebroids in derived differential topology, PhD dissertation, 2018, doi.