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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

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- Discussion Type
- discussion topicvirtual equipment
- Category Latest Changes
- Started by anuyts
- Comments 1
- Last comment by anuyts
- Last Active Sep 19th 2024

- Discussion Type
- discussion topicpositroid
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Sep 19th 2024

A combinatorial notion in the study of total positivity.

- Discussion Type
- discussion topicSpin(7)/G₂ is the 7-sphere
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by perezl.alonso
- Last Active Sep 19th 2024

for completeness, to go with the other entries in

*coset space structure on n-spheres – table*

- Discussion Type
- discussion topicbiology
- Category Latest Changes
- Started by nLab edit announcer
- Comments 19
- Last comment by Urs
- Last Active Sep 19th 2024

- Discussion Type
- discussion topicontology log
- Category Latest Changes
- Started by Corbin
- Comments 2
- Last comment by Corbin
- Last Active Sep 18th 2024

- Discussion Type
- discussion topicreal number
- Category Latest Changes
- Started by Urs
- Comments 36
- Last comment by TobyBartels
- Last Active Sep 18th 2024

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 number*is something that may be approximated by rational numbers. Equipped with the operations of addition and multiplication induced from the rational numbers, real numbers form a*number field*, denoted $\mathbb{R}$. The underlying set is the*completion*of 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 line*also known as*the 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)*manifold*is modeled on these Cartesian spaces. These, in turnm are standard models for the notion of space in particular in physics (see*spacetime*), or at least in classical physics. See at*geometry of physics*for more on this.

- Discussion Type
- discussion topicLie's three theorems
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Hurkyl
- Last Active Sep 18th 2024

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]

- Discussion Type
- discussion topicE₁₁
- Category Latest Changes
- Started by Urs
- Comments 15
- Last comment by Urs
- Last Active Sep 17th 2024

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

- Discussion Type
- discussion topicKac-Moody algebra
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Sep 17th 2024

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.

- Discussion Type
- discussion topicenriched bicategory
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by varkor
- Last Active Sep 17th 2024

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

- Discussion Type
- discussion topicexceptional tangent bundle
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 17th 2024

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)

- Discussion Type
- discussion topicstandard model of particle physics
- Category Latest Changes
- Started by Urs
- Comments 16
- Last comment by Urs
- Last Active Sep 17th 2024

**Edit to**: standard model of particle physics by Urs Schreiber at 2018-04-01 01:15:37 UTC.**Author comments**:added textbook reference

- Discussion Type
- discussion topicCary Malkiewich
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Sep 16th 2024

- Discussion Type
- discussion topicE₁₀
- Category Latest Changes
- Started by Samuel Adrian Antz
- Comments 3
- Last comment by Urs
- Last Active Sep 16th 2024

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.)

- Discussion Type
- discussion topicKähler C^∞-differential
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Sep 16th 2024

Created:

## Idea

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

## Definition

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.

## Related concepts

- Discussion Type
- discussion topicC^∞-derivation
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Sep 16th 2024

Created:

## Idea

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

## Definition

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.

## Related concepts

- Discussion Type
- discussion topicsmooth differential forms form the free C^∞-DGA on smooth functions
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Sep 16th 2024

Created:

## Background

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.

## Idea

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}

## Application: the Poincaré lemma

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}

## References

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

- E. J. Dubuc, A. Kock,

- Discussion Type
- discussion topiccommutative algebra
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Dmitri Pavlov
- Last Active Sep 16th 2024

gave this reference item some more hyperlinks:

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

- Michael Atiyah, Ian G. Macdonald,

- Discussion Type
- discussion topicKähler C^∞-differentials of smooth functions are differential 1-forms
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Sep 16th 2024

Created:

## Idea

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}

## Related concepts

## References

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

- E. J. Dubuc, A. Kock,

- Discussion Type
- discussion topicBoris Shoikhet
- Category Latest Changes
- Started by zskoda
- Comments 1
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- Last Active Sep 16th 2024

- Discussion Type
- discussion topichom-groupoid
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
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- Last Active Sep 16th 2024

- Discussion Type
- discussion topicE₉
- Category Latest Changes
- Started by Samuel Adrian Antz
- Comments 2
- Last comment by Urs
- Last Active Sep 16th 2024

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.)

- Discussion Type
- discussion topicJulia Ramos González
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- Started by zskoda
- Comments 1
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- Last Active Sep 16th 2024

- Discussion Type
- discussion topicGrothendieck category
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- Started by zskoda
- Comments 19
- Last comment by zskoda
- Last Active Sep 15th 2024

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 ?

- Discussion Type
- discussion topicAdS-CFT in condensed matter physics
- Category Latest Changes
- Started by Urs
- Comments 26
- Last comment by Urs
- Last Active Sep 15th 2024

- Discussion Type
- discussion topicfree monoid
- Category Latest Changes
- Started by Todd_Trimble
- Comments 10
- Last comment by varkor
- Last Active Sep 15th 2024

I added to the “abstract nonsense” section in free monoid a helpful general observation on how to construct free monoids. “Adjoint functor theorem” is overkill for free monoids over $Set$.

- Discussion Type
- discussion topicSilviu S. Pufu
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 15th 2024

- Discussion Type
- discussion topicgaseous vector space
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active Sep 15th 2024

- Discussion Type
- discussion topicGrothendieck universe
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Thomas Holder
- Last Active Sep 15th 2024

added at

*Grothendieck universe*at References a pointer to the proof that these are sets of $\kappa$-small sets for inaccessible $\kappa$. (also at inaccessible cardinal)

- Discussion Type
- discussion topiclax morphism classifier
- Category Latest Changes
- Started by Tim_Porter
- Comments 3
- Last comment by varkor
- Last Active Sep 14th 2024

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

- Discussion Type
- discussion topiclax morphism
- Category Latest Changes
- Started by Mike Shulman
- Comments 3
- Last comment by varkor
- Last Active Sep 14th 2024

I have created lax morphism, with general definitions and a list of examples. It would be great to have more examples.

- Discussion Type
- discussion topicweak morphism classifier
- Category Latest Changes
- Started by varkor
- Comments 1
- Last comment by varkor
- Last Active Sep 14th 2024

- Discussion Type
- discussion topicset truncation
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active Sep 13th 2024

- Discussion Type
- discussion topicRezk completion
- Category Latest Changes
- Started by nLab edit announcer
- Comments 13
- Last comment by nLab edit announcer
- Last Active Sep 13th 2024

- Discussion Type
- discussion topicDistLat
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Sep 13th 2024

Added related concepts section with links to coherent category, coherent hyperdoctrine, Pos, and Frm

Anonymouse

- Discussion Type
- discussion topicFrm
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by nLab edit announcer
- Last Active Sep 13th 2024

Added table of contents and links to geometric category and geometric hyperdoctrine

Anonymouse

- Discussion Type
- discussion topicframe
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by nLab edit announcer
- Last Active Sep 13th 2024

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.

- Discussion Type
- discussion topicjoin type
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active Sep 13th 2024

- Discussion Type
- discussion topicpositive element
- Category Latest Changes
- Started by TobyBartels
- Comments 2
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- Last Active Sep 13th 2024

- Discussion Type
- discussion topiclocally positive locale
- Category Latest Changes
- Started by nLab edit announcer
- Comments 9
- Last comment by TobyBartels
- Last Active Sep 13th 2024

- Discussion Type
- discussion topicovert space
- Category Latest Changes
- Started by Mike Shulman
- Comments 12
- Last comment by TobyBartels
- Last Active Sep 13th 2024

At overt space there was a remark that since the definition quantifies over “spaces”, the overtness of a single space might depend on the general meaning chosen for “space”, but that no example was known to the author. I added an example involving synthetic topology, which may not be quite what the author of that remark was thinking of, but which I think is interesting.

- Discussion Type
- discussion topicdisjunction
- Category Latest Changes
- Started by David_Corfield
- Comments 6
- Last comment by nLab edit announcer
- Last Active Sep 13th 2024

- Discussion Type
- discussion topictype theory
- Category Latest Changes
- Started by Mike Shulman
- Comments 105
- Last comment by Urs
- Last Active Sep 13th 2024

I incorporated some of my spiel from the blog into the page type theory.

- Discussion Type
- discussion topicflat space holography
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Urs
- Last Active Sep 13th 2024

- Discussion Type
- discussion topicGraeme Segal
- Category Latest Changes
- Started by Andrew Stacey
- Comments 13
- Last comment by Urs
- Last Active Sep 13th 2024

There has GOT to be a better photograph than that! Is there anyone here in Oxford? Can they go and get a picture for us?

- Discussion Type
- discussion topicIsbell duality - table
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by zskoda
- Last Active Sep 12th 2024

the table didn’t have the basic examples, such as

*Gelfand duality*and*Milnor’s exercise*. Added now.

- Discussion Type
- discussion topiccomodule
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by zskoda
- Last Active Sep 12th 2024

- Discussion Type
- discussion topiceffective epimorphism
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- Started by nLab edit announcer
- Comments 5
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- Last Active Sep 12th 2024

- Discussion Type
- discussion topicdescent
- Category Latest Changes
- Started by Tim_Porter
- Comments 6
- Last comment by anuyts
- Last Active Sep 12th 2024

I made some very minor changes to the introduction at descent. I hesitate to do more but at present the discussion does not seem that readable to me. Can someone look at it to see what they think? The intro seems to plunge in deep very quickly and so the ‘idea’ of descent as that of gluing local information together, does not come across to me. The article is lso quite long and perhaps needs splitting up a bit.

- Discussion Type
- discussion topicdominion
- Category Latest Changes
- Started by mattecapu
- Comments 2
- Last comment by mattecapu
- Last Active Sep 12th 2024

- Discussion Type
- discussion topicdisplay map
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by mattecapu
- Last Active Sep 12th 2024

Added some content to display map from Taylor’s book. Not very deep, mostly as a reference to the respective section for me.

- Discussion Type
- discussion topicFrobenius manifold
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by zskoda
- Last Active Sep 12th 2024

- Discussion Type
- discussion topiccategory of elements
- Category Latest Changes
- Started by Mike Shulman
- Comments 9
- Last comment by nLab edit announcer
- Last Active Sep 12th 2024

I added to category of elements an argument for why $El$ preserves colimits.

- Discussion Type
- discussion topicYi-zhuang You
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Sep 12th 2024

- Discussion Type
- discussion topicArtan Sheshmani
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Sep 12th 2024

- Discussion Type
- discussion topicD=2 Yang-Mills theory
- Category Latest Changes
- Started by Samuel Adrian Antz
- Comments 11
- Last comment by Urs
- Last Active Sep 12th 2024

Created basic outline with some important connections. Yang-Mills measure, after all the main concept which makes this special case interesting, and references will be added later.

Edit: Crosslinked D=2 Yang-Mills theory on related pages: D=2 QCD, D=4 Yang-Mills theory, D=5 Yang-Mills theory.

- Discussion Type
- discussion topicYassir Dinar
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 12th 2024

- Discussion Type
- discussion topicgroup of ideles
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active Sep 11th 2024

- Discussion Type
- discussion topicMaurer-Cartan form
- Category Latest Changes
- Started by Urs
- Comments 43
- Last comment by Urs
- Last Active Sep 11th 2024

wrote Maurer-Cartan form

the first part is the standard story, but I chose a presentation which I find more insightful than the standard symbol chains as on Wikipedia.

then there is a section on Maurer-Cartan forms on oo-Lie groups and how that reduces to the standard story for ordinary Lie groups.

The detailed statements and proofs of this second part are at Lie infinity-groupoid in the new section The canonical form on a Lie oo-group that is just a Lie group.

- Discussion Type
- discussion topicdistributive lattice
- Category Latest Changes
- Started by nLab edit announcer
- Comments 8
- Last comment by nLab edit announcer
- Last Active Sep 11th 2024