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- Discussion Type
- discussion topicD=2 Yang-Mills theory
- Category Latest Changes
- Started by Samuel Adrian Antz
- Comments 11
- Last comment by Urs
- Last Active 30 minutes ago

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 52 minutes ago

- Discussion Type
- discussion topicFrobenius manifold
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 53 minutes ago

- Discussion Type
- discussion topicgroup of ideles
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active 13 hours ago

- Discussion Type
- discussion topicMaurer-Cartan form
- Category Latest Changes
- Started by Urs
- Comments 43
- Last comment by Urs
- Last Active 15 hours ago

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 17 hours ago

- Discussion Type
- discussion topicMorava K-theory
- Category Latest Changes
- Started by Jon Beardsley
- Comments 24
- Last comment by nLab edit announcer
- Last Active 18 hours ago

- Created a page Morava K-theory . A lot to add. Will fill out later, with better reference list. Please edit!

- Discussion Type
- discussion topicover category
- Category Latest Changes
- Started by Bubbles
- Comments 57
- Last comment by varkor
- Last Active 18 hours ago

- In codomain fibration one calls the function

C \ (-) : C --> Cat

mapping c to the slice category (C \ c) a pseudofunctor. However I fail to see how this is not functorial.

A morphism f : a --> b is sent to the functor (C \ f) : (C \ a) --> (C \ b) defined by (g : c --> a) |--> (fg : c --> b), and this assignment clearly satisfies composition. It also preserves identity. So what am I missing here?

- Discussion Type
- discussion topicFriedrich Schur
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active 21 hours ago

- Discussion Type
- discussion topicMaría Ronco
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active 21 hours ago

- Discussion Type
- discussion topicMilnor-Moore theorem
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active 21 hours ago

(Today’s arXiv) A homotopification

- Imma Gálvez-Carrillo, María Ronco, Andy Tonks,
*On differential Hopf algebras and $B_\inf$ algebras*, https://arxiv.org/abs/2409.06632

and few more additions.

- Imma Gálvez-Carrillo, María Ronco, Andy Tonks,

- Discussion Type
- discussion topicHausdorff series
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active 23 hours ago

- Discussion Type
- discussion topicgeneralized the
- Category Latest Changes
- Started by David_Corfield
- Comments 77
- Last comment by Tim_Porter
- Last Active 1 day ago

I added the HoTT introduction rule for ’the’, then added a speculative remark on why say things like

The Duck-billed Platypus is a primitive mammal that lives in Australia.

- Discussion Type
- discussion topiclocally internal category
- Category Latest Changes
- Started by nLab edit announcer
- Comments 9
- Last comment by DavidRoberts
- Last Active 1 day ago

- Discussion Type
- discussion topicfunding of the nLab
- Category Latest Changes
- Started by Richard Williamson
- Comments 32
- Last comment by Keith Harbaugh
- Last Active 1 day ago

Describing the arrangements which have been made for funding of the nLab in collaboration with the Topos Institute. The page, linked to from the home page, is intended to be fairly general; specific requests for donations can be made elsewhere.

- Discussion Type
- discussion topicLie group
- Category Latest Changes
- Started by Urs
- Comments 16
- Last comment by Joe Moeller
- Last Active 1 day ago

tried to bring the entry Lie group a bit into shape: added plenty of sections and cross links to other nLab material. But there is still much that deserves to be done.

- Discussion Type
- discussion topicLie category
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 4
- Last comment by Joe Moeller
- Last Active 1 day ago

Created:

## Definition

An internal category object in the category of smooth manifolds in which the source and target maps are submersions.

Sometimes, the smooth manifold of morphisms is allowed to have a boundary, in which case the restrictions of the source and target maps to the boundary are required to be submersions themselves.

## Related concepts

## References

- Žan Grad,
*Fundamentals of Lie categories*, arXiv:2302.05233.

- Žan Grad,

- Discussion Type
- discussion topicfirst uncountable ordinal
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by RodMcGuire
- Last Active 1 day ago

i have split off (copied over) the paragraph on the first uncountable ordinal from

*countable ordinal*to*first uncountable ordinal*, just in order to make it possible to link to “first uncountable ordinal” more directly. Cross-linked with*long line*.

- Discussion Type
- discussion topicPaul Sutcliffe
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by zskoda
- Last Active 1 day ago

brief

`category:people`

-entry for hyperlinking references at*skyrmion*,*atomic nucleus*

- Discussion Type
- discussion topicTakeuchi product
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active 1 day ago

- Discussion Type
- discussion topiccartesian object
- Category Latest Changes
- Started by Evan Patterson
- Comments 5
- Last comment by varkor
- Last Active 1 day ago

- Discussion Type
- discussion topicGabriel-Popescu theorem
- Category Latest Changes
- Started by zskoda
- Comments 3
- Last comment by zskoda
- Last Active 1 day ago

- Discussion Type
- discussion topicperiod
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by zskoda
- Last Active 1 day ago

I tried to brush-up the References at

*period*a little.I have trouble downloading the first one, which is

- M. Kontsevich, Don Zagier, Periods (pdf)

My system keeps telling me that the pdf behind this link is broken. Can anyone see it? (It may well just be my system misbehaving, wouldn’t be the first time…).

- Discussion Type
- discussion topicdual gebra
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by zskoda
- Last Active 2 days ago

- Discussion Type
- discussion topicDavid E. Radford
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active 2 days ago

- Discussion Type
- discussion topicJacob Towber
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active 2 days ago

- Discussion Type
- discussion topicReedy model structure
- Category Latest Changes
- Started by Urs
- Comments 21
- Last comment by anuyts
- Last Active 2 days ago

- Discussion Type
- discussion topicPeripatetic Seminar on Sheaves and Logic
- Category Latest Changes
- Started by valeriadepaiva
- Comments 8
- Last comment by nLab edit announcer
- Last Active 2 days ago

- Discussion Type
- discussion topicpseudo-order
- Category Latest Changes
- Started by nLab edit announcer
- Comments 8
- Last comment by nLab edit announcer
- Last Active 4 days ago

- Discussion Type
- discussion topicdecidable proposition
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by nLab edit announcer
- Last Active 4 days ago

at

*decidable proposition*I found the simple basic idea a bit too deeply hidden in the text. In an attempt to improve on this I have added right before the subsections of the Idea-section this quick preview:External decidability: either $p$ or $\not p$ may be deduced in the metalanguage;

Internal decidability: $p \vee \not p$ may be deduced, hence “$p$ or not $p$” holds in the object language.

Okay?

- Discussion Type
- discussion topicprobicategory
- Category Latest Changes
- Started by varkor
- Comments 1
- Last comment by varkor
- Last Active 4 days ago

- Discussion Type
- discussion topicsubsingleton
- Category Latest Changes
- Started by IngoBlechschmidt
- Comments 3
- Last comment by nLab edit announcer
- Last Active 5 days ago

Recorded at

*subsingleton*that a different nomenclature also exists, in which “subterminal” and “subsingleton” are not synonymous (see for instance Anders Kock in page 2 of Algebras for the Partial Map Classifier Monad).

- Discussion Type
- discussion topicmere proposition
- Category Latest Changes
- Started by Urs
- Comments 20
- Last comment by nLab edit announcer
- Last Active 5 days ago

an entry for

*mere proposition*had been missing. Created a minimum, just so as to satisfy links.

- Discussion Type
- discussion topic?-modality
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by Urs
- Last Active 5 days ago

- Discussion Type
- discussion topicuniqueness quantifier
- Category Latest Changes
- Started by nLab edit announcer
- Comments 9
- Last comment by nLab edit announcer
- Last Active 5 days ago

- Discussion Type
- discussion topicexclusive disjunction
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by nLab edit announcer
- Last Active 5 days ago

moving the following ancient query box out of the entry:

+– {: .query} What about the ’or’ of parental threat? Consider the logician parent who says “Come here or I’ll smack you” to his child and smacks even after obedience as they believe in the inclusive ’or’. -David

That's no different from ’If you don't come here, then I'll smack you.’, which also suggests (but does not state) the converse. And in fact, no parent, logician or otherwise, is actually making the promise implied by the $\neg(p \wedge q)$ clause; if the child comes to such a parent and then kicks the parent in the shin, then the parent will still smack the child. Instead, if you

*want*to make that promise, then you say ’If you come here, then I won't smack you.’ explicitly. This has a very different tenor (unless you say it in a wink-nudge mafia kind of way), as it's a promise rather than a threat. (I know, it's*only*a promise, which is still different in tenor than a statement that is*both*promise and threat, as an exclusive disjunction would be. But I still hold that your statement is*only*a threat.) Note that a logician child who believes the parent's literal expression would still choose to come if avoiding smacking is the highest priority; but the reason is that refusal guarantees a smack, not that obedience necessarily avoids it. That is why the wise child also throws in a contrite expression and an oral apology, to improve the odds. —TobyI see there’s a literature on the subject including “The Myth of the Exclusive ’Or’” (Mind, 80 (317), 116–121). —David

Also: I argued above that the meaning of ’Come here or I'll smack you’ must be weaker than exclusive disjunction, since the parent will smack the child anyway under some circumstances. However, I agree that it is stronger than inclusive disjunction, but that is because we may go beyond the literal meaning of the words and apply a Gricean implicature. To be specific, if the parent intends to smack the child regardless, then the parent should say ’I'll smack you’ by the Maxim of Quantity, but the parent in fact said something more wordy. Thus we conclude that the parent does not intend to smack the child if the child comes, without ruling out the possibility that the parent will still smack the child for some other reason, as yet unanticipated. —Toby =–

- Discussion Type
- discussion topicaffirmative proposition
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 5 days ago

- Discussion Type
- discussion topicdissolution locale
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active 5 days ago

Created:

## Definition

The

**dissolution locale**$\mathfrak{C}L$ of a locale $L$ is defined as the poset of its sublocales (equivalently: nuclei on $L$) equipped with the relation of reverse inclusion.There is a canonical morphism of locales

$\iota\colon\mathfrak{C}L \to L$such that the map $\iota^*$ sends an open $a\in L$ to the open in $\mathfrak{C}L$ given by the open sublocale of $a$.

## Interpretation

The map $\mathfrak{C}L\to L$ can be considered an analogue of the canonical map $T_d \to T$ for a topological space $T$, where $T_d$ is the underlying set of $T$ equipped with the discrete topology.

In particular,

*discontinuous maps*$L\to M$ could be defined as morphisms of locales $\mathfrak{C}L\to M$, see Picado–Pultr, XIV.7.3.## References

Original reference:

- John R. Isbell, On dissolute spaces, Topology and its Applications 40:1 (1991), 63–70. doi.

Expository account:

- Frames and Locales, see Sections III.3, VI.4-6, and others. The dissolution frame is denoted there by $\mathcal{Sl}(L)^{op}$ (III.3.2) or by $\mathfrak{C}(L)$ (III.5.2) and the dissolution locale is denoted by $\mathfrak{S}(L)$ (XIV.7.2).

- Discussion Type
- discussion topicrefutative proposition
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 5 days ago

- Discussion Type
- discussion topicnonabelian cohomology
- Category Latest Changes
- Started by nLab edit announcer
- Comments 27
- Last comment by Urs
- Last Active 5 days ago

- Discussion Type
- discussion topicstrong negation
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 5 days ago

- Discussion Type
- discussion topicshifted tangent bundle
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active 5 days ago

created shifted tangent bundle because I thought somebody was asking about that on the blog, but now looking more closely I find that maybe nobody asked for that...

- Discussion Type
- discussion topiccompact space
- Category Latest Changes
- Started by Todd_Trimble
- Comments 89
- Last comment by Zhen Lin
- Last Active 5 days ago

As an outcome of recent discussion at Math Overflow here, Mike Shulman suggested some nLab pages where comparisons of different definitions of compactness are rigorously established. I have created one such page: compactness and stable closure. (The importance and significance of the stable closure condition should be brought out better.)

- Discussion Type
- discussion topicaffine logic
- Category Latest Changes
- Started by David_Corfield
- Comments 16
- Last comment by nLab edit announcer
- Last Active 5 days ago

I started a stub at affine logic as I saw the link requested in a couple of places.

- Discussion Type
- discussion topiclinear logic
- Category Latest Changes
- Started by Mike Shulman
- Comments 43
- Last comment by nLab edit announcer
- Last Active 5 days ago

The cut rule for linear logic used to be stated as

If $\Gamma \vdash A$ and $A \vdash \Delta$, then $\Gamma \vdash \Delta$.

I don’t think this is general enough, so I corrected it to

If $\Gamma \vdash A, \Phi$ and $\Psi,A \vdash \Delta$, then $\Psi,\Gamma \vdash \Delta,\Phi$.

- Discussion Type
- discussion topicantithesis interpretation
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 5 days ago

moving section on the antithesis interpretation in linear logic to its own page at antithesis interpretation

Anonymouse

- Discussion Type
- discussion topicunified topological space
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 5 days ago

- Discussion Type
- discussion topicGalois theory
- Category Latest Changes
- Started by Urs
- Comments 37
- Last comment by David_Corfield
- Last Active 5 days ago

the entry Galois theory used to be a stub with only some links. I have now added plenty of details.

- Discussion Type
- discussion topicAlexander Grothendieck
- Category Latest Changes
- Started by Urs
- Comments 22
- Last comment by Urs
- Last Active 6 days ago

I have tried to expand a bit the text at the beginning of the category:people entry

*Alexander Grothendieck*, mention more of what his work was about, add more hyperlinks. It could still be much improved, but right now it reads as follows:The french mathematician

**Alexandre Grothendieck**, (in English usually Alexander Grothendieck), has created a work whose influence has shown him to be the greatest pure mathematician of the 20th century; and his ideas continue to be developed in this century.Initially working on topological vector spaces and analysis, Grothendieck then made revolutionary advances in algebraic geometry by developing sheaf and topos theory and abelian sheaf cohomology and formulating algebraic geometry in these terms (locally ringed spaces, schemes). Later topos theory further developed independently and today serves as the foundation also for other kinds of geometry. Notably its homotopy theoretic refinement to higher topos theory serves as the foundation for modern derived algebraic geometry.

Grothendieck’s work is documented in texts known as EGA (with Dieudonné), an early account FGA, and the many volume account SGA of the seminars at l’IHÉS, Bures-sur-Yvette, where he was based at the time. (See the wikipedia article for some indication of the story from there until the early 1980s.)

By the way, in view of the recent objection to referring to people as “famous” in category:people entries: the lead-in sentence here is not due to me, it has been this way all along. One might feel that it should be rephrased, but I leave that to those who feel strongly about it.

- Discussion Type
- discussion topicquantum field theory
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active 6 days ago

wrote an Idea-section at quantum field theory

- Discussion Type
- discussion topicN-∞ operad
- Category Latest Changes
- Started by nLab edit announcer
- Comments 8
- Last comment by Urs
- Last Active 6 days ago

- Discussion Type
- discussion topicseparable monad
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active 6 days ago

- Discussion Type
- discussion topicHeyting field
- Category Latest Changes
- Started by nLab edit announcer
- Comments 4
- Last comment by nLab edit announcer
- Last Active 6 days ago

- Discussion Type
- discussion topicRel
- Category Latest Changes
- Started by John Baez
- Comments 6
- Last comment by mattecapu
- Last Active 7 days ago

- Discussion Type
- discussion topicrepresentable 2-category
- Category Latest Changes
- Started by varkor
- Comments 7
- Last comment by AlexanderCampbell
- Last Active 7 days ago

- Discussion Type
- discussion topicdifferential crossed module
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 7 days ago

added pointer to:

- Johannes Huebschmann,
*On the history of Lie brackets, crossed modules, and Lie-Rinehart algebras*[arXiv:2208.02539]

- Johannes Huebschmann,

- Discussion Type
- discussion topicindex
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active 7 days ago

I gave

*index*an Idea-section.In the course of this I created some stubby auxiliary entries, such as (in rapidly increasing order of stubbieness)

- Discussion Type
- discussion topicDaniel V. Tausk
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 7 days ago

- Discussion Type
- discussion topicAlessandro Portaluri
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 7 days ago

- Discussion Type
- discussion topicMaslov index
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active 7 days ago

Added to

*Maslov index*and to*Lagrangian Grassmannian*the following quick cohomological definition of the Maslov index:

The first ordinary cohomology of the stable Lagrangian Grassmannian with integer coefficients is isomorphic to the integers

$H^1(LGrass, \mathbb{Z}) \simeq \mathbb{Z} \,.$The generator of this cohomology group is called the

$u \in H^1(LGrass, \mathbb{Z}) \,.$*universal Maslov index*Given a Lagrangian submanifold $Y \hookrightarrow X$ of a symplectic manifold $(X,\omega)$, its tangent bundle is classified by a function

$i \;\colon\; Y \to LGrass \,.$The _Maslov index of $Y$ is the universal Maslov index pulled back along this map

$i^\ast u \in H^1(Y,\mathbb{Z}) \,.$