for completeness, to go with the other entries in *coset space structure on n-spheres – table*

replaced broken link to Witten’s paper with doi

]]>Added work on Ologs and started restructuring the page

rTuyeras

]]>This is for olog-specific stuff which wouldn’t be appropriate for biology.

]]>I looked at *real number* and thought I could maybe try to improve the way the Idea section flows. Now it reads as follows:

]]>A

real numberis something that may be approximated by rational numbers. Equipped with the operations of addition and multiplication induced from the rational numbers, real numbers form anumber field, denoted $\mathbb{R}$. The underlying set is thecompletionof the ordered field $\mathbb{Q}$ of rational numbers: the result of adjoining to $\mathbb{Q}$ suprema for every bounded subset with respect to the natural ordering of rational numbers.The set of real numbers also carries naturally the structure of a topological space and as such $\mathbb{R}$ is called the

real linealso known asthe continuum. Equipped with both the topology and the field structure, $\mathbb{R}$ is a topological field and as such is the uniform completion of $\mathbb{Q}$ equipped with the absolute value metric.Together with its cartesian products – the Cartesian spaces $\mathbb{R}^n$ for natural numbers $n \in \mathbb{N}$ – the real line $\mathbb{R}$ is a standard formalization of the idea of

continuous space. The more general concept of (smooth)manifoldis modeled on these Cartesian spaces. These, in turnm are standard models for the notion of space in particular in physics (seespacetime), or at least in classical physics. See atgeometry of physicsfor more on this.

added publication data for these two items:

Rui Loja Fernandes, Marius Crainic,

*Integrability of Lie brackets*, Ann. of Math.**157**2 (2003) 575-620 [arXiv:math.DG/0105033, doi:10.4007/annals.2003.157.575]Rui Loja Fernandes, Marius Crainic,

*Lectures on Integrability of Lie Brackets*, Geometry & Topology Monographs**17**(2011) 1–107 [arxiv:math.DG/0611259, doi:10.2140/gtm.2011.17.1]

have added a minimum on the level decompositon of the first fundamental rep of $E_{11}$ here.

]]>I have half-heartedly started adding something to *Kac-Moody algebra*. Mostly refrences so far. But I don’t have the time right now to do any more.

have created enriched bicategory in order to help Alex find the appropriate page for his notes.

]]>have added some minimum of references (there were none before)

but I hope to find the time to put some actual content into the entry:

the sequence of exceptional tangent bundles used to be truncated, and the other day I saw (cf. nForum discussion here and here) how to complete it, using recent results.

a pdf note is now here (just 1 page)

]]>I have added

- Paolo Facchi, Giovanni Gramegna, Arturo Konderak,
*Entropy of quantum states*(arXiv:2104.12611)

and added publication details to

- A. P. Balachandran, T. R. Govindarajan, Amilcar R. de Queiroz, A. F. Reyes-Lega,
*Algebraic approach to entanglement and entropy*, Phys. Rev. A 88, 022301 (2013) (arXiv:1301.1300)

and grouped together more discernibly the references on operator-algebraic entropy

]]>**Edit to**: standard model of particle physics by Urs Schreiber at 2018-04-01 01:15:37 UTC.

**Author comments**:

added textbook reference

]]>Fixing a broken link

Natalie Stewart

]]>I have begun cleaning up the entry cycle category, tightening up definitions and proofs. This should render some of the past discussion obsolete, by re-expressing the intended homotopical intuitions (in terms of degree one maps on the circle) more precisely, in terms of “spiraling” adjoints on the poset $\mathbb{Z}$.

Here is some of the past discussion I’m now exporting to the nForum:

]]>The cycle category may be defined as the subcategory of Cat whose objects are the categories $[n]_\Lambda$ which are freely generated by the graph $0\to 1\to 2\to\ldots\to n\to 0$, and whose morphisms $\Lambda([m],[n])\subset\mathrm{Cat}([m],[n])$ are precisely the functors of degree $1$ (seen either at the level of nerves or via the embedding $\mathrm{Ob}[n]_\Lambda\to \mathbf{R}/\mathbf{Z}\cong S^1$ given by $k\mapsto k/(n+1)\,\mathrm{mod}\,\mathbf{Z}$ on the level of objects, the rest being obvious).

The simplex category $\Delta$ can be identified with a subcategory of $\Lambda$, having the same objects but with fewer morphisms. This identification does not respect the inclusions into $Cat$, however, since $[n]$ and $[n]_\Lambda$ are different categories.

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. (Removed two redirects for “E10” from the top and added one for “E10” at the bottom of the page.)

]]>Created:

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.

Created:

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.

Created:

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$]$

Created:

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.)

]]>A person entry.

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