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- Discussion Type
- discussion topicLars Brink
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Mar 7th 2024

added pointer to this obituary:

- Hermann Nicolai:
*Lars Brink 1943–2022*, CernCourier (1 Mar 2023):

- Hermann Nicolai:

- Discussion Type
- discussion topiccartesian product
- Category Latest Changes
- Started by Todd_Trimble
- Comments 5
- Last comment by varkor
- Last Active Mar 6th 2024

- Discussion Type
- discussion topicfusion 2-category
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by perezl.alonso
- Last Active Mar 6th 2024

A stub, for the moment just to have a place for recording a couple of references (which were previously at

*fusion category*.

- Discussion Type
- discussion topicspin-statistics theorem
- Category Latest Changes
- Started by Urs
- Comments 19
- Last comment by nLab edit announcer
- Last Active Mar 6th 2024

stub for spin-statistics theorem. Just recording a first few references so far.

- Discussion Type
- discussion topicbiproduct
- Category Latest Changes
- Started by Colin Tan
- Comments 24
- Last comment by J-B Vienney
- Last Active Mar 6th 2024

- Discussion Type
- discussion topicdirected homotopy type theory
- Category Latest Changes
- Started by Urs
- Comments 41
- Last comment by ncfavier
- Last Active Mar 5th 2024

created directed homotopy type theory

- Discussion Type
- discussion topictiny object
- Category Latest Changes
- Started by varkor
- Comments 5
- Last comment by maxsnew
- Last Active Mar 5th 2024

- Discussion Type
- discussion topicweak bialgebra
- Category Latest Changes
- Started by zskoda
- Comments 5
- Last comment by perezl.alonso
- Last Active Mar 5th 2024

To support mentioning weak wreath product in a parallel discussion with Urs, I created a stub for weak bialgebra with redirect weak Hopf algebra.

- Discussion Type
- discussion topicfusion category
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by perezl.alonso
- Last Active Mar 5th 2024

briefly added something to fusion category. See also this blog comment.

- Discussion Type
- discussion topicquantum mechanics
- Category Latest Changes
- Started by Urs
- Comments 31
- Last comment by Urs
- Last Active Mar 5th 2024

in analogy to what I just did at classical mechanics, I have now added some basic but central content to quantum mechanics:

Quantum mechanical systems

States and observables

Spaces of states

Flows and time evolution

Still incomplete and rough. But I have to quit now.

- Discussion Type
- discussion topicoptic (in computer science)
- Category Latest Changes
- Started by David_Corfield
- Comments 10
- Last comment by BryceClarke
- Last Active Mar 5th 2024

- Discussion Type
- discussion topicHayder Salman
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 5th 2024

- Discussion Type
- discussion topicquantum phase transition
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Mar 5th 2024

a stub, for the moment just to make some cross-links work (such as with topological order and topological entanglement entropy)

- Discussion Type
- discussion topiccomplete graph
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Urs
- Last Active Mar 5th 2024

- Discussion Type
- discussion topicRarita-Schwinger field
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 5th 2024

added pointer to:

- Mauricio Valenzuela, Jorge Zanelli,
*The propagating modes of the massless Rarita–Schwinger system*[arXiv:2305.00106]

- Mauricio Valenzuela, Jorge Zanelli,

- Discussion Type
- discussion topicstar-autonomous category
- Category Latest Changes
- Started by Mike Shulman
- Comments 33
- Last comment by Urs
- Last Active Mar 5th 2024

I added to star-autonomous category a mention of “$\ast$-autonomous functors”.

- Discussion Type
- discussion topicworldline formalism
- Category Latest Changes
- Started by Urs
- Comments 18
- Last comment by Urs
- Last Active Mar 5th 2024

created

*worldline formalism*to go with this Physics.SE answer

- Discussion Type
- discussion topicperfect infinity-stack
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Samuel Adrian Antz
- Last Active Mar 5th 2024

I expanded perfect infinity-stack a little. But this deserves further polishing and expanding.

- Discussion Type
- discussion topicclassifying space
- Category Latest Changes
- Started by Urs
- Comments 44
- Last comment by Samuel Adrian Antz
- Last Active Mar 4th 2024

I noticed that the entry classifying space is in bad shape. I have added a table of contents and tried to structure it slightly, but much more needs to be done here.

I have added a paragraph on standard classifying spaces for topological principal bundles via the geometric realization of the simplicial space associated to the given topological group.

In the section “For crossed complexes” there is material that had been provided by Ronnie Brown which needs to be harmonized with the existing Idea-section. It proposes something like a general axiomatics on the notion of “classifying space” more than giving details on the geometric realization of crossed complexes

- Discussion Type
- discussion topicquasi-Hopf algebra
- Category Latest Changes
- Started by Urs
- Comments 18
- Last comment by perezl.alonso
- Last Active Mar 4th 2024

The Idea-section at

*quasi-Hopf algebra*had been confused and wrong. I have removed it and written a new one.

- Discussion Type
- discussion topicgravitino
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Mar 4th 2024

one more from Meissner & Nicolai, last week:

- Krzysztof A. Meissner, Hermann Nicolai,
*Evidence for a stable supermassive gravitino with charge $2/3$?*[arXiv:2303.09131]

- Krzysztof A. Meissner, Hermann Nicolai,

- Discussion Type
- discussion topicpromonoidal category
- Category Latest Changes
- Started by Mike Shulman
- Comments 20
- Last comment by varkor
- Last Active Mar 4th 2024

I moved the definition of promonoidal categories from Day convolution to promonoidal category, and expanded on it a bit.

- Discussion Type
- discussion topicrepresentable morphism
- Category Latest Changes
- Started by zskoda
- Comments 11
- Last comment by anuyts
- Last Active Mar 4th 2024

New entry representable morphism, in the sense of Grothendieck school. The notion is used at closed immersion of schemes where I just made some changes.

- Discussion Type
- discussion topicsuper Poincaré Lie algebra
- Category Latest Changes
- Started by perezl.alonso
- Comments 3
- Last comment by perezl.alonso
- Last Active Mar 4th 2024

- Discussion Type
- discussion topicdependent linear type theory
- Category Latest Changes
- Started by Urs
- Comments 92
- Last comment by Urs
- Last Active Mar 4th 2024

Following discussion in some other threads, I thought one should make it explicit and so I created an entry

Currently this contains some (hopefully) evident remarks of what “dependent linear type theory” reasonably should be at least, namely a hyperdoctrine with values in linear type theories.

The entry keeps saying “should”. I’d ask readers to please either point to previous proposals for what “linear dependent type theory” is/should be, or criticise or else further expand/refine what hopefully are the obvious definitions.

This is hopefully uncontroversial and should be regarded an obvious triviality. But it seems it might be one of those hidden trivialities which deserve to be highlighted a bit more. I am getting the impression that there is a big story hiding here.

Thanks for whatever input you might have.

- Discussion Type
- discussion topicjet bundle
- Category Latest Changes
- Started by Urs
- Comments 63
- Last comment by perezl.alonso
- Last Active Mar 3rd 2024

stub for jet bundle

- Discussion Type
- discussion topicnormed ring
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Mar 3rd 2024

An edit to

*normed ring*is being discussed in another thread, here.

- Discussion Type
- discussion topicPoincaré-Weyl algebra
- Category Latest Changes
- Started by perezl.alonso
- Comments 1
- Last comment by perezl.alonso
- Last Active Mar 2nd 2024

- Discussion Type
- discussion topicYoshiaki Tanii
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 2nd 2024

- Discussion Type
- discussion topicpiecewise flat spacetime
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Mar 2nd 2024

- Discussion Type
- discussion topicWheeler superspace
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Mar 2nd 2024

- Discussion Type
- discussion topicspin connection
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Mar 2nd 2024

added pointer to:

- Thanu Padmanabhan, §11.6 in:
*Gravitation – Foundations and Frontiers*, Cambridge University Press (2012) [doi:10.1017/CBO9780511807787, spire:852758, toc: pdf]

- Thanu Padmanabhan, §11.6 in:

- Discussion Type
- discussion topicCartan connection
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Urs
- Last Active Mar 2nd 2024

added to Cartan connection

definition

a standard reference

a standard example

- Discussion Type
- discussion topicsuper-Cartan geometry
- Category Latest Changes
- Started by David_Corfield
- Comments 33
- Last comment by Urs
- Last Active Mar 2nd 2024

I see super-Cartan geometry is taking shape. Will Clifford algebras make an appearance in the The super-Klein geometry: super-Minkowski spacetime section?

Is there a higher super-Cartan way of thinking about what is at 3-category of fermionic conformal nets, about the String 2-group and superstrings, as here about the spin group and fermions.

- Discussion Type
- discussion topicNon-Archimedean Analysis
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 2nd 2024

added ISBN:9783540125464

- Discussion Type
- discussion topicgeometry of physics -- perturbative quantum field theory
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Urs
- Last Active Mar 1st 2024

I am changing the page title – this used to be “A first idea of quantum field theory”, which of course still redirects. The “A first idea…” seemed a good title for when this was an ongoing lecture that was being posted to PhysicsForums. I enjoyed the double meaning one could read into it, but it’s a bad idea to carve such jokes into stone. And now that the material takes its place among the other chapters of

*geometry of physics*, with the web of cross-links becoming thicker, the canonical page name clearly is “perturbative quantum field theory”.

- Discussion Type
- discussion topicL-infinity algebras in physics
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active Mar 1st 2024

added pointer to these two recent references, identifying further $L_\infty$-algebra structure in Feynman amplitudes/S-matrices of perturbative quantum field theory:

Markus B. Fröb,

*Anomalies in time-ordered products and applications to the BV-BRST formulation of quantum gauge theories*(arXiv:1803.10235)Alex Arvanitakis,

*The $L_\infty$-algebra of the S-matrix*(arXiv:1903.05643)

- Discussion Type
- discussion topicKnizhnik-Zamolodchikov equation
- Category Latest Changes
- Started by Urs
- Comments 16
- Last comment by perezl.alonso
- Last Active Mar 1st 2024

finally added the actual definition,

`!include`

-ed from*Knizhnik-Zamolodchikov-Kontsevich construction – definition*(as per the discussion here)

- Discussion Type
- discussion topicArtinian local ring
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by Urs
- Last Active Mar 1st 2024

renaming page to the one used in the Stacks reference, parallel with Artinian ring.

Anonymous

- Discussion Type
- discussion topicW-type
- Category Latest Changes
- Started by Mike Shulman
- Comments 34
- Last comment by Madeleine Birchfield
- Last Active Feb 29th 2024

Created W-type.

- Discussion Type
- discussion topicarithmetic Chern-Simons theory
- Category Latest Changes
- Started by David_Corfield
- Comments 7
- Last comment by nLab edit announcer
- Last Active Feb 29th 2024

- Discussion Type
- discussion topicMinhyong Kim
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Feb 29th 2024

Adding reference

- Hee-Joong Chung, Dohyeong Kim, Minhyong Kim, Jeehoon Park, Hwajong Yoo,
*Entanglement entropies in the abelian arithmetic Chern-Simons theory*[2312.17138]

Anonymouse

- Hee-Joong Chung, Dohyeong Kim, Minhyong Kim, Jeehoon Park, Hwajong Yoo,

- Discussion Type
- discussion topictopological entanglement entropy -- references
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by nLab edit announcer
- Last Active Feb 29th 2024

this is a bare list of references which used to be (and still is) at

*entanglement entropy*. But since the same references are now also needed at*long-range entanglement*, I am putting them in a separate page here, to be`!include`

-ed into both these entries

- Discussion Type
- discussion topicSachdev-Ye-Kitaev model
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Urs
- Last Active Feb 29th 2024

- Discussion Type
- discussion topiclimit
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by varkor
- Last Active Feb 28th 2024

I toiuched the formatting and the hyperlinking of the paragraphs on compatibility of limits with other universal constructions.

Merged the previous tiny subsections on this to a single one, now

*Compatibility with universal constructions*.added the hyperlink to the stand-alone entry

*adjoints preserve (co-)limits*.Will create an analogous stand-alone entry for

*limits commute with limits*.

- Discussion Type
- discussion topicbalanced monoidal category
- Category Latest Changes
- Started by nsnyder
- Comments 3
- Last comment by nsnyder
- Last Active Feb 28th 2024

Balancing doesn’t mention duals anywhere, and makes sense even without duals. I removed an incorrect statement and replaced it with the correct one. Not sure if it needs a reference, but the correct result appears as Lemma 4.20 in https://arxiv.org/pdf/0908.3347.pdf (where it’s attributed to Deligne, but the citation is to Yetter).

- Discussion Type
- discussion topicpre-abelian category
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by nLab edit announcer
- Last Active Feb 28th 2024

I have touched the formatting at

*pre-abelian category*.

- Discussion Type
- discussion topicabelian category
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by nLab edit announcer
- Last Active Feb 28th 2024

I am hereby moving an old query-box discussion from

*abelian category*to here. I suggest that to the extent this reached a conclusion, that conclusion should be moved to the Properties-section of the entry

[begin forwarded discussion]

The following discussion is about whether a pre-abelian category in which (epi,mono) is a factorization system is necessarily abelian.

+–{: .query} Mike: In Categories Work, and on Wikipedia, an abelian category is defined to be (in the terms above) a pre-abelian category such that every monic is a kernel and every epi is a cokernel. This implies that (epi, mono) is an orthogonal factorization system, but I don’t see why the converse should hold, as this seems to assert.

Zoran Skoda It is very late night here in Bonn, so check on my reasoning, but I think that the answer is simple. Let $f: A\to B$. The canonical map $\coker(\ker f)\to \ker (\coker f)$ exists as long as we have additive category admitting kernels and cokernels. The arrow from A to coker (ker f) is epi as every cokernel arrow, and the arrow of $\ker(\coker f) \to B$ is mono. Now canonical arrow in between the two is automatically both mono and epi. For all that reasoning I did not yet assume the axiom on uniquely unique factorization. Now assume it and you get that the canonical map must be isomorphism because it is the unique iso between the two decompositions of $f$: one in which you take epi followed by (the composition of) two monics and another in which you have (the composition of) two epis followed by one monic. Right ?

Now do this for $f$ a monic and you get a decomposition into iso iso kernel and for $f$ an epi and you get the cokernel iso iso as required.

Mike: Why is the canonical comparison map mono and epi? It’s late for me too right now, but I think that maybe a counterexample is the “multiplication by 2” map $\mathbb{Z}\to \mathbb{Z}$ in the category of torsion-free abelian groups.

However, if you assume explicitly that that comparison map is always an isomorphism, then I believe it for the reasons that you gave.

Zoran Skoda I do not see this as a counterexample, as this is not a pre-abelian category, you do not have cokernels in this category ? In a pre-abelian category always the canonical map from coker ker to ker coker has its own kernel 0 and cokernel 0.

Mike: Torsion-free abelian groups are reflective in abelian groups, and therefore cocomplete. In particular, they have cokernels, although those cokernels are not computed as in Ab. In particular, the cokernel of $2:\mathbb{Z}\to\mathbb{Z}$ is 0.

Zoran Skoda Yes, I was thinking of this reflection argument (equivalence of torsion and localization argument), that is why I put question mark above. Now I tried to prove the assertion that in preabelian cat the canonical map has kernel 0 and cokernel 0 and I can’t for more than an hour. But that would mean that for example Gelfand-Manin book is wrong – it has the discussion on A4 axiom and it says exactly this. Popescu makes an example of preabelian category where canonical map is not iso, but emphasises in his example that it is bimorphism. On the other hand, later, he says that preabelian category is abelian iff it is balanced and the canonical map is bimorphism, hence he requires it explicitly. Let me think more…

Zoran Skoda I have rewritten in minimalistic way, leaving just what I can prove, and assuming that you are right and Gelfand-Manin book has one wrong statement (that the canonical map in preabelian category is mono and epi). But let us leave the discussion here for some time, maybe we can improve the question of the difference between preabelian with factorization and abelian.

Mike: I refactored the page to make clear what we know and what we don’t, and include some examples. Maybe someone will come along and give us a counterexample or a proof. I wonder what the epimorphisms are in the category of torsion-free abelian groups, and in particular whether it is balanced (since if so, it would be a counterexample).

Mike: Okay, it’s obvious: the epimorphisms in $tfAb$ are the maps whose cokernel (in $Ab$) is torsion. Thus $2:\mathbb{Z}\to\mathbb{Z}$ is monic and epic, so $tfAb$ is not balanced. And since $2:\mathbb{Z}\to\mathbb{Z}$ is its own canonical map, that canonical map

*is*monic and epic in $tfAb$, so this isn’t a counterexample.*Zoran*: http://www.uni-trier.de/fileadmin/fb4/INF/TechReports/semi-abelian_categories.pdf says at one place that Palamodov’s version of semi-abelian category is preabelian + canonical morphism is epi and mono. =–[end forwarded discussion]

- Discussion Type
- discussion topicdark matter
- Category Latest Changes
- Started by Urs
- Comments 36
- Last comment by nLab edit announcer
- Last Active Feb 28th 2024

stub for

*dark matter*

- Discussion Type
- discussion topicAntonio Gallerati
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Madeleine Birchfield
- Last Active Feb 28th 2024

- Discussion Type
- discussion topicRuggero Noris
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 28th 2024

- Discussion Type
- discussion topicRiccardo Matrecano
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 28th 2024

- Discussion Type
- discussion topicM5-brane
- Category Latest Changes
- Started by Urs
- Comments 24
- Last comment by Urs
- Last Active Feb 28th 2024

I have added to

*M5-brane*a fairly detailed discussion of the issue with the fractional quadratic form on differential cohomology for the dual 7d-Chern-Simons theory action (from Witten (1996) with help of Hopkins-Singer (2005)).In the new section

*Conformal blocks and 7d Chern-Simons dual*.

- Discussion Type
- discussion topicHiggs bundle
- Category Latest Changes
- Started by zskoda
- Comments 18
- Last comment by perezl.alonso
- Last Active Feb 28th 2024

- Ludmil Katzarkov, Dmitri Orlov, Tony Pantev,
*Notes on Higgs bundles and D-branes*, (delivered as a lecture by Tony Pantev at winter school at Guanajuato 2013 link) draft pdf

added to references at Higgs bundle.

- Ludmil Katzarkov, Dmitri Orlov, Tony Pantev,

- Discussion Type
- discussion topicSylvain Douteau
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Feb 28th 2024

Created an entry on Sylvain Douteau, who works on stratified spaces and their homotopy theory.

- Discussion Type
- discussion topicstratified space
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Tim_Porter
- Last Active Feb 28th 2024

I have created

*stratified space*in order to collect some references

- Discussion Type
- discussion topicdisplayed category
- Category Latest Changes
- Started by Mike Shulman
- Comments 11
- Last comment by BryceClarke
- Last Active Feb 28th 2024

New page displayed category.

- Discussion Type
- discussion topicJoshua Lackman
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 28th 2024

- Discussion Type
- discussion topicpath integral
- Category Latest Changes
- Started by Urs
- Comments 25
- Last comment by Urs
- Last Active Feb 28th 2024

Mike Stay kindly added the standard QM story to path integral.

I changed the section titles a bit and added the reference to the Baer-Pfaeffle article on the QM path integral. Probably the best reference there is on this matter.

- Discussion Type
- discussion topicBousfield localization of triangulated categories
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Feb 28th 2024

I split the material so far at "Bousfield localization" into

and made Bousfield localization itself a dismbiguation page.

Then I edited Bousfield localization of triangulated categories a bit and added some references.

I expect it should be true that the (left) Bousfield localization of a model category whose homotopy category is a triangulated category corresponds to the Bousfield localization of that triangulated homotopy category, but I don't discuss a statement to that extent yet.

I'd be grateful for explicit pointers to the literature on this. I haven't had time yet to look much even at the literature that I do list as BLoc for triang cats. Am too busy at the moment with BLoc for mod cats.