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The correct notion of a Kähler differential for C^∞-rings

See the article Kähler C^∞-differentials of smooth functions are differential 1-forms for motivation and definition and the article smooth differential forms form the free C^∞-DGA on smooth functions for further developments and applications like the Poincaré lemma.

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The correct notion of a derivation for C^∞-rings

See the article Kähler C^∞-differentials of smooth functions are differential 1-forms for motivation and definition and the article smooth differential forms form the free C^∞-DGA on smooth functions for further developments and applications like the Poincaré lemma.

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See the article Kähler C^∞-differentials of smooth functions are differential 1-forms for the necessary background for this article, including the notions of C^∞-ring, C^∞-derivation, and Kähler C^∞-differential.

In algebraic geometry, (algebraic) differential forms on the Zariski spectrum of a [commutative ring $R$ (or a commutative $k$-algebra $R$) can be defined as the free commutative differential graded algebra on $R$.

This definition does not quite work for smooth manifolds: as already explained in the article Kähler C^∞-differentials of smooth functions are differential 1-forms, the notion of a Kähler differential must be refined in order to extract smooth differential 1-forms from the C^∞-ring of smooth functions on a smooth manifold $M$.

Thus, in order to get the algebra of smooth differential forms, the notion of a commutative differential graded algebra must likewise be adjusted.

\begin{definition}
A **commutative differential graded C^∞-ring** is a real commutative differential graded algebra $A$ whose degree 0 component $A_0$ is equipped with a structure of a C^∞-ring in such a way that the degree 0 differential $A_0\to A_1$ is a C^∞-derivation.
\end{definition}

With this definition, we can recover smooth differential forms in a manner similar to algebraic geometry, deducing the following consequence of the Dubuc–Kock theorem for Kähler C^∞-differentials.

\begin{theorem} The free commutative differential graded C^∞-ring on the C^∞-ring of smooth functions on a smooth manifold $M$ is canonically isomorphic to the differential graded algebra of smooth differential forms on $M$. \end{theorem}

The Poincaré lemma becomes a trivial consequence of the above theorem.

\begin{proposition} For every $n\ge0$, the canonical map

$\mathbf{R}[0]\to \Omega(\mathbf{R}^n)$is a quasi-isomorphism of differential graded algebras. \end{proposition}

\begin{proof} (Copied from the MathOverflow answer.) The de Rham complex of a finite-dimensional smooth manifold $M$ is the free C^∞-dg-ring on the C^∞-ring $C^\infty(M)$. If $M$ is the underlying smooth manifold of a finite-dimensional real vector space $V$, then $C^\infty(M)$ is the free C^∞-ring on the vector space $V^*$ (the real dual of $V$). Thus, the de Rham complex of a finite-dimensional real vector space $V$ is the free C^∞-dg-ring on the vector space $V^*$. This free C^∞-dg-ring is the free C^∞-dg-ring on the free cochain complex on the vector space $V^*$. The latter cochain complex is simply $V^*\to V^*$ with the identity differential. It is cochain homotopy equivalent to the zero cochain complex, and the free functor from cochain complexes to C^∞-dg-rings preserves cochain homotopy equivalences. Thus, the de Rham complex of the smooth manifold $V$ is cochain homotopy equivalent to the free C^∞-dg-ring on the zero cochain complex, i.e., $\mathbf{R}$ in degree 0. \end{proof}

- E. J. Dubuc, A. Kock,
*On 1-form classifiers*, Communications in Algebra 12:12 (1984), 1471–1531. doi.

I finally gave this statement its own entry, in order to be able to conveniently point to it:

*embedding of smooth manifolds into formal duals of R-algebras*

gave this reference item some more hyperlinks:

- Michael Atiyah, Ian G. Macdonald,
*Introduction to commutative algebra*, (1969, 1994) $[$pdf, ISBN:9780201407518$]$

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In algebraic geometry, the module of Kähler differentials of a commutative ring $R$ corresponds under the Serre–Swan duality to the cotangent bundle of the Zariski spectrum of $R$.

In contrast, the module of Kähler differentials of the commutative real algebra of smooth functions on a smooth manifold $M$ receives a canonical map from the module of smooth sections of the cotangent bundle of $M$ that is quite far from being an isomorphism.

An example illustrating this point is $M=\mathbf{R}$, since in the module of (traditionally defined) Kähler differentials of $C^\infty(M)$ we have $d(exp(x))\ne exp dx$, where $\exp\colon\mathbf{R}\to\mathbf{R}$ is the exponential function. That is to say, the traditional algebraic notion of a Kähler differential is unable to deduce that $\exp'=\exp$ using the Leibniz rule.

However, this is not a defect in the conceptual idea itself, but merely a failure to use the correct formalism. The appropriate notion of a ring in the context of differential geometry is not merely a commutative real algebra, but a more refined structure, namely, a C^∞-ring.

This notion comes with its own variant of commutative algebra. Some of the resulting concepts turn out to be exactly the same as in the traditional case. For example, ideals of C^∞-rings and modules over C^∞-rings happen to coincide with ideals and modules in the traditional sense. Others, like derivations, must be defined carefully, and definitions that used to be equivalent in the traditional algebraic context need not remain so in the context of C^∞-rings.

Observe that a map of sets $d\colon A\to M$ (where $M$ is an $A$-module) is a derivation if and only if for any real polynomial $f(x_1,\ldots,x_n)$ the chain rule holds:

$d(f(a_1,\ldots,a_n))=\sum_i {\partial f\over\partial x_i}(x_1,\ldots,x_n) dx_i.$Indeed, taking $f(x_1,x_2)=x_1+x_2$ and $f(x_1,x_2)=x_1 x_2$ recovers the additivity and Leibniz property of derivations, respectively.

Observe also that $f$ is an element of the free commutative real algebra on $n$ elements, i.e., $\mathbf{R}[x_1,\ldots,x_n]$.

If we now substitute C^∞-rings for commutative real algebras, we arrive at the correct notion of a derivation for C^∞-rings:

```
A __C^∞-derivation__ of a [[C^∞-ring]] $A$ is a map of sets $A\to M$ (where $M$ is a [[module]] over $A$) such that the following chain rule holds for every smooth function $f\in\mathrm{C}^\infty(\mathbf{R}^n)$:
$$d(f(a_1,\ldots,a_n))=\sum_i {\partial f\over\partial x_i}(x_1,\ldots,x_n) dx_i,$$
where both sides use the structure of a [[C^∞-ring]] to evaluate a smooth real function on a collection of elements in $A$.
```

The module of Kähler C^∞-differentials can now be defined in the same manner as ordinary Kähler differentials, using C^∞-derivations instead of ordinary derivations.

\begin{theorem} (Dubuc, Kock, 1984.) The module of Kähler C^∞-differentials of the C^∞-ring of smooth functions on a smooth manifold $M$ is canonically isomorphic to the module of sections of the cotangent bundle of $M$. \end{theorem}

- E. J. Dubuc, A. Kock,
*On 1-form classifiers*, Communications in Algebra 12:12 (1984), 1471–1531. doi.

Person stub.

]]>Categories enriched over groupoid form strict (2,1) categories. Edited for clarity.

Mark Williams

]]>Used unicode subscripts for indices of exceptional Lie groups including title and links. When not linked, usual formulas are used. See discussion here. Links will be re-checked after all titles have been changed. (Added redirect for “E9” at the bottom of the page.)

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]]>I strongly disagree with the statement in Grothendieck category that the Grothendieck category is small. The main examples like ${}_R Mod$ are not! What did the writer of that line have in mind ?

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The entry lax morphism classifier was started two yeats ago, is actually empty!

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]]>Created a stub page for this concept, which surprisingly didn’t exist yet.

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]]>I have added some things to frame. Mostly duplicating things said elsewhere (at locale and at (0,1)-topos), but I need these statements to be at *frame* itself.