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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration 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 kan lie-theory limit 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 newpage nlab nonassociative noncommutative noncommutative-geometry number-theory object 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 simplicial space spin-geometry stable-homotopy-theory string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

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- Discussion Type
- discussion topichadron supersymmetry
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Dec 7th 2021

- Discussion Type
- discussion topicAlan Hatcher > history 2
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Dec 7th 2021

Moved this from Alan Hatcher to make it redirect to Allen Hatcher, hopefully. (Am I doing this right?)

Anonymous

- Discussion Type
- discussion topicsimplicial complex
- Category Latest Changes
- Started by Todd_Trimble
- Comments 38
- Last comment by nLab edit announcer
- Last Active Dec 6th 2021

Made some edits and additions at simplicial complex.

- Discussion Type
- discussion topicadjoint (infinity,1)-functor
- Category Latest Changes
- Started by Urs
- Comments 36
- Last comment by Hurkyl
- Last Active Dec 5th 2021

polished and expanded adjoint (infinity,1)-functor

- Discussion Type
- discussion topicAQFT
- Category Latest Changes
- Started by zskoda
- Comments 21
- Last comment by Tim_Porter
- Last Active Dec 5th 2021

I did not change anything, I would not like to do it without Urs’s consent and some opinion. The entry AQFT equates algebraic QFT and axiomatic QFT. In the traditional circle, algebraic quantum field theory meant being based on local nets – local approach of Haag and Araki. This is what the entry now describes. The Weightman axioms are somewhat different, they are based on fields belonging some spaces of distributions, and 30 years ago it was called field axiomatics, unlike the algebraic axiomatics. But these differences are not that important for the main entry on AQFT. What is a bigger drawback is that the third approach to axiomatic QFT if very different and was very strong few decades ago and still has some followers. That is the

**S-matrix axiomatics**which does not believe in physical existence of observables at finite distance, but only in the asymptotic values given by the S-matrix. The first such axiomatics was due Bogoliubov, I think. (Of course he later worked on other approaches, especially on Wightman’s. Both the Wightman’s and Bogoliubov’s formalisms are earlier than the algebraic QFT.)I would like to say that axiomatic QFT has 3 groups of approaches, and especially to distinguish S-matrix axiomatics from the “algebraic QFT”. Is this disputable ?

- Discussion Type
- discussion topicfinite set
- Category Latest Changes
- Started by David_Corfield
- Comments 8
- Last comment by Mike Shulman
- Last Active Dec 4th 2021

Added

For a treatment in homotopy type theory see

- Dan Frumin, Herman Geuvers, Léon Gondelman, Niels van der Weide,
*Finite Sets in Homotopy Type Theory*, (pdf)

- Dan Frumin, Herman Geuvers, Léon Gondelman, Niels van der Weide,

- Discussion Type
- discussion topiccomputational trilogy
- Category Latest Changes
- Started by Urs
- Comments 54
- Last comment by David_Corfield
- Last Active Dec 4th 2021

created

*computational trinitarianism*, combining a pointer to an exposition by Bob Harper (thanks to David Corfield) with my table logic/category-theory/type-theory.

- Discussion Type
- discussion topicdiscrete fibration
- Category Latest Changes
- Started by zskoda
- Comments 8
- Last comment by nLab edit announcer
- Last Active Dec 4th 2021

In discrete fibration I added a new section on the Street’s definition of a discrete fibration from $A$ to $B$, that is the version

**for spans of internal categories**. I do not really understand this added definition, so if somebody has comments or further clarifications…

- Discussion Type
- discussion topicXiao-Gang Wen
- Category Latest Changes
- Started by David_Corfield
- Comments 2
- Last comment by David_Corfield
- Last Active Dec 4th 2021

Added

- Liang Kong, Tian Lan, Xiao-Gang Wen, Zhi-Hao Zhang, Hao Zheng,
*Algebraic higher symmetry and categorical symmetry – a holographic and entanglement view of symmetry*(arXiv:2005.14178)

Sounds interesting

We would like to mention that a structure similar to categorical symmetry was found previously in AdS/CFT correspondence, [63–65] where a global symmetry $G$ at the high-energy boundary is related to a gauge theory of group $G$ in the low-energy bulk. In this paper, we stress that the categorical symmetry encoded by the bulk $G$-gauge theory not only contains the $G$ symmetry at the boundary, it also contains a dual algebraic higher symmetry $\tilde{G}^{(n-1)}$ at the boundary.

- Liang Kong, Tian Lan, Xiao-Gang Wen, Zhi-Hao Zhang, Hao Zheng,

- Discussion Type
- discussion topicmultisimplicial set
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 4
- Last comment by Dmitri Pavlov
- Last Active Dec 3rd 2021

I created the article multisimplicial set.

It seems that I triggered some bug in Instiki that prevents it from rendering displayed formulas correctly.

The article compiles just fine in TeX, so this is clearly a bug.

In fact, Instiki seems to insert some new text:

definedasthechunk29161400wikichunklinkchunkofthetautologicalfunctor

All this “chunk” and “wikichunklink” stuff is clearly a bug.

- Discussion Type
- discussion topicLiouville theory
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by nLab edit announcer
- Last Active Dec 3rd 2021

started something stubby at

*Liouville theory*, for the moment just so as to record some references and provide for a minimum of cross-links (e.g. with Chern-Simons gravity).(also created a stub for quantum Teichmüller theory in the course of this, but nothing there yet except a pointer to reviews)

- Discussion Type
- discussion topic8-sphere
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by David_Corfield
- Last Active Dec 3rd 2021

- Discussion Type
- discussion topiclocally strongly finitely presentable category
- Category Latest Changes
- Started by varkor
- Comments 10
- Last comment by Dmitri Pavlov
- Last Active Dec 2nd 2021

- Discussion Type
- discussion topicuniversal exceptionalism
- Category Latest Changes
- Started by Urs
- Comments 86
- Last comment by Urs
- Last Active Dec 2nd 2021

started

*universal exceptionalism*related discussion is taking place on g+ here

- Discussion Type
- discussion topicempty 210
- Category Latest Changes
- Started by varkor
- Comments 3
- Last comment by Urs
- Last Active Dec 2nd 2021

- Discussion Type
- discussion topicmusic theory
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Dec 1st 2021

- Discussion Type
- discussion topic4-sphere
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by David_Corfield
- Last Active Dec 1st 2021

added this to the list of references:

Speculative remarks on the possible role of maps from spacetime to the 4-sphere in some kind of quantum gravity via spectral geometry are in

- Ali Chamseddine, Alain Connes, Viatcheslav Mukhanov, section 4 of
*Geometry and the Quantum: Basics*, JHEP 12 (2014) 098 (arXiv:1411.0977)

reviewed in

Alain Connes, section 4 of

*Geometry and the Quantum*, Foundations of Mathematics and Physics One Century After Hilbert. Springer, Cham, 2018. 159-196 (arXiv:1703.02470)Alain Connes, from 58:00 to 1:25:00 in

*Why Four Dimensions and the Standard Model Coupled to Gravity - A Tentative Explanation From the New Geometric Paradigm of NCG*, talk at IHES, 2017 (video recording)

- Ali Chamseddine, Alain Connes, Viatcheslav Mukhanov, section 4 of

- Discussion Type
- discussion topicprobability theory
- Category Latest Changes
- Started by David_Corfield
- Comments 10
- Last comment by David_Corfield
- Last Active Dec 1st 2021

Strangely, we don’t seem to have an nForum discussion for probability theory.

I added a reference there to

- Chris Heunen, Ohad Kammar, Sam Staton, Hongseok Yang,
*A Convenient Category for Higher-Order Probability Theory*, (arXiv:1701.02547)

It replaces the category of measurable spaces, which isn’t cartesian closed, with the category of quasi-Borel spaces, which is. As they point out in section IX, what they’re doing is working with concrete sheaves on an established category of spaces, rather like the move to diffeological spaces.

[Given the interest in topology around these parts at the moment, we hear of ’C-spaces’ as generalized topological spaces arising from a similar sheaf construction in C. Xu and M. Escardo, “A constructive model of uniform continuity,” in Proc. TLCA, 2013.]

- Chris Heunen, Ohad Kammar, Sam Staton, Hongseok Yang,

- Discussion Type
- discussion topicstereographic projection
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 1st 2021

For no particular reason, I have added another illustrating graphics to the entry, taken from Fig 1.3 in Apostol 1973.

- Discussion Type
- discussion topicanalysis
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active Dec 1st 2021

I noticed that the entry

*analysis*is in a sad state. I now gave it an Idea-section (here), which certainly still leaves room for expansion; and I tried to clean up the very little that is listed at*References – General*

- Discussion Type
- discussion topicTom Apostol
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 1st 2021

- Discussion Type
- discussion topicinteractions of images and pre-images with unions and intersections
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Dec 1st 2021

At

*interactions of images and pre-images with unions and intersections*I have added pointer to Lawvere 69 and there at*Adjointness in Foundations*I added a bit more text and cross-references.

- Discussion Type
- discussion topicgeneric interval
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Urs
- Last Active Dec 1st 2021

- Discussion Type
- discussion topicmonoidal monad
- Category Latest Changes
- Started by Todd_Trimble
- Comments 12
- Last comment by varkor
- Last Active Nov 30th 2021

Started an article on monoidal monad. An earlier redirect had sent it over to Hopf monad which is something that Zoran was working on, but I think it deserves an article to itself, with discussion of the relation to commutative monads, etc. (which I have started).

- Discussion Type
- discussion topicAdS-QCD correspondence
- Category Latest Changes
- Started by Urs
- Comments 71
- Last comment by Urs
- Last Active Nov 30th 2021

- Discussion Type
- discussion topictensor network
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Urs
- Last Active Nov 30th 2021

The term “tensor network”, while essentially just a synonym for “string diagram”, has in recent years become widely used and now fully established in quantum physics, especially in its use for discussion of

*holographic entanglement entropy*. It needs a page of its own, if only to point to*string diagram*while also listing the relevant physics references

- Discussion Type
- discussion topicdifferential form on a supermanifold
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Urs
- Last Active Nov 30th 2021

added pointer to

- R. Catenacci, P.A. Grassi, S. Noja,
*Superstring Field Theory, Superforms and Supergeometry*(arXiv:1807.09563)

- R. Catenacci, P.A. Grassi, S. Noja,

- Discussion Type
- discussion topicmysterious duality
- Category Latest Changes
- Started by David_Corfield
- Comments 35
- Last comment by Urs
- Last Active Nov 30th 2021

Since it was mentioned by Urs on g+, I thought I’d start mysterious duality. Maybe not a great name when someone discovers how it works (as someone claims to have done here).

- Discussion Type
- discussion topicDelta-generated topological space
- Category Latest Changes
- Started by Mike Shulman
- Comments 21
- Last comment by Urs
- Last Active Nov 30th 2021

Stub for Delta-generated space.

- Discussion Type
- discussion topicmodel structure on topological spaces
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Nov 30th 2021

addes some sentences concerning the Quillen model structure to model structure on topological spaces

- Discussion Type
- discussion topicWirthmüller context
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Nov 30th 2021

- Discussion Type
- discussion topictranscendental syntax
- Category Latest Changes
- Started by nLab edit announcer
- Comments 9
- Last comment by Boris Eng
- Last Active Nov 29th 2021

- Discussion Type
- discussion topicHawaiian earring space
- Category Latest Changes
- Started by nLab edit announcer
- Comments 32
- Last comment by R.Arthur
- Last Active Nov 29th 2021

I have changed the title of this article, as well as references to the object within it. Use of the term “Hawaiian Earring” is objected to by Hawaiian mathematicians. Please see these two threads, one by native Hawaiian and math PhD Dr. Marissa Loving, and the other by an expert on the Hawaiian Earring, Dr. Jeremy Brazas.

https://twitter.com/MarissaKawehi/status/1406244897611522049

https://twitter.com/jtbrazas/status/1406652385263501319

I have retitled the article “Shrinking wedge of circles”, which is the name used for this space in Hatcher’s “Algebraic Topology”. I have a retained a note in the body of the article that the space is sometimes referred to as the “Hawaiian earring space”.

This small change in name helps to make mathematics a more inclusive and just field, especially in consideration of the historical marginalization and exclusion of indigenous mathematicians. By taking this action, the nLab site can help to spread a change in language more widely, including on other math reference sites.

I hope that this change is readily accepted and approved by the nLab community. Thank you!

Justin Lanier

- Discussion Type
- discussion topicStein duality
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 2
- Last comment by Guest
- Last Active Nov 29th 2021

- Discussion Type
- discussion topiccomplex Lie group
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by nLab edit announcer
- Last Active Nov 28th 2021

noticed that we didn’t have

*complex Lie group*, so I put in a bare minimum. Also a stub for*complex Lie algebra*. Just so that it’s there.

- Discussion Type
- discussion topictotal relation
- Category Latest Changes
- Started by TobyBartels
- Comments 2
- Last comment by TobyBartels
- Last Active Nov 28th 2021

- Discussion Type
- discussion topicorthogonal group
- Category Latest Changes
- Started by Urs
- Comments 22
- Last comment by DavidRoberts
- Last Active Nov 28th 2021

added a table with some homotopy groups in the unstable range to

*orthogonal group – Homotopy groups*

- Discussion Type
- discussion topiccommutative algebraic theory
- Category Latest Changes
- Started by Andrew Stacey
- Comments 21
- Last comment by varkor
- Last Active Nov 28th 2021

Added a diagram at commutative algebraic theory using the totally awesome SVG Editor.

**it now does itex!**This was a ridiculously simple diagram to do.

- Discussion Type
- discussion topicWigner theorem
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Miklós
- Last Active Nov 28th 2021

added pointer to

- Valentine Bargman,
*Note on Wigner’s theorem on symmetry transformations*, Journal of Mathematical Physics 5.7 (1964): 862-868 (doi:10.1063/1.1704188)

- Valentine Bargman,

- Discussion Type
- discussion topicconnection on a cubical set
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 13
- Last comment by Dmitri Pavlov
- Last Active Nov 27th 2021

In the article connection on a cubical set the definition of a connection uses two types of maps, denoted Γ^+_i and Γ^-_i. Roughly, the former corresponds to a map of cubes that takes the minimum of some coordinates, whereas the latter takes the maximum of some coordinates.

However, in the book by Brown-Higgins-Sivera in Definition 13.1.3, page 446, only the maps Γ^-_i are used. There they are denoted simply by Γ_i. The paper by Maltsiniotis about the strict test category of cubes with connection also uses the same definition.

Which definition is correct? What is the reference for nLab’s definition and why does it deviate from the definition of Brown-Higgins-Sivera?

- Discussion Type
- discussion topicmodel structure on cubical sets
- Category Latest Changes
- Started by Urs
- Comments 24
- Last comment by Dmitri Pavlov
- Last Active Nov 27th 2021

In the past we had some discussion here about why simplicial methods find so much more attention than cubical methods in higher category theory. The reply (as far as I am concerned at least) has been: because the homotopy theory = weak oo-groupoid theory happens to be well developed for simplicial sets and not so well developed for cubical sets. Historically this apparently goes back to the disappointment that the standard cubical geometric realization to Top does not behave as nicely as the one on simplicial sets does.

Still, it should be useful to have as much cubical homotopy theory around as possible. Many structures are more naturally cubical than simplicial.

So as soon as the Lab comes up again (we are working on it...) I want to create a page model structure on cubical sets and record for instance this reference here:

Jardine, Cubical homotopy theory: a beginning

- Discussion Type
- discussion topiccubical set
- Category Latest Changes
- Started by Richard Williamson
- Comments 10
- Last comment by Dmitri Pavlov
- Last Active Nov 27th 2021

Made some further tweaks at cubical set. Hopefully the definitions of the boundary functor, and of a horn, are correct now.

Am continuing to work on homotopy groups of a cubical Kan complex.

- Discussion Type
- discussion topicconnected relation
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Nov 27th 2021

- Discussion Type
- discussion topic2-crossed complex
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Nov 27th 2021

- Discussion Type
- discussion topicJoão Faria Martins
- Category Latest Changes
- Started by Tim_Porter
- Comments 5
- Last comment by Tim_Porter
- Last Active Nov 27th 2021

- Discussion Type
- discussion topic2-crossed module
- Category Latest Changes
- Started by DavidRoberts
- Comments 3
- Last comment by Tim_Porter
- Last Active Nov 27th 2021

- Discussion Type
- discussion topiccrossed square
- Category Latest Changes
- Started by jin
- Comments 3
- Last comment by Tim_Porter
- Last Active Nov 27th 2021

- Discussion Type
- discussion topicDaniel Conduché
- Category Latest Changes
- Started by Tim_Porter
- Comments 2
- Last comment by Tim_Porter
- Last Active Nov 27th 2021

- Discussion Type
- discussion topicFrançois Conduché
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Nov 27th 2021

- Discussion Type
- discussion topictangent bundle category
- Category Latest Changes
- Started by David_Corfield
- Comments 22
- Last comment by Urs
- Last Active Nov 27th 2021

Added another reference.

I was chatting with Robin Cockett yesterday at SYCO1. In a talk Robin claims to be after

The algebraic/categorical foundations for differential calculus and differential geometry.

It would be good to see how this approach compares with differential cohesive HoTT.

- Discussion Type
- discussion topicBaez's law
- Category Latest Changes
- Started by varkor
- Comments 2
- Last comment by Urs
- Last Active Nov 26th 2021

This appears to be called Stigler’s law of eponymy on Wikipedia. Is there good reason to have a separate term?

- Discussion Type
- discussion topicF. Conduché > history
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Nov 26th 2021

- Discussion Type
- discussion topicGelfand duality
- Category Latest Changes
- Started by Urs
- Comments 15
- Last comment by nLab edit announcer
- Last Active Nov 26th 2021

I am splitting off Gelfand duality from Gelfand spectrum. Want to state the actual equivalence theorem here. But just a moment…

- Discussion Type
- discussion topiccoproduct
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by R.Arthur
- Last Active Nov 25th 2021

tried to improve the entry coproduct a little

- Discussion Type
- discussion topicdirect sum
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by R.Arthur
- Last Active Nov 25th 2021

I have touched the formatting at

*direct sum*and then expanded a little:Added a paragraph to the Idea-section such that something familiar is mentioned right at the beginning;

Expanded on the example of direct sums in $Ab$ by drawing the cocone diagrams and explicitly mentioning the universal property.

Mentioned the relation to formal linear combinations.

Mentioned the examples of direct sums of modules.

- Discussion Type
- discussion topicR.Arthur
- Category Latest Changes
- Started by R.Arthur
- Comments 3
- Last comment by R.Arthur
- Last Active Nov 25th 2021

- Discussion Type
- discussion topickernel
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by nLab edit announcer
- Last Active Nov 25th 2021

added below the very first definition at kernel a remark that spells out the universal property more explicitly. Also added mentioning of some basic examples.

- Discussion Type
- discussion topiclinearly compact module
- Category Latest Changes
- Started by zskoda
- Comments 5
- Last comment by Hurkyl
- Last Active Nov 24th 2021

New entry linearly compact module with many redirects, just to record some references.

New book entry Abelian categories with applications to rings and modules, to record its contents.

- Discussion Type
- discussion topicquantum contextuality
- Category Latest Changes
- Started by nonemenon
- Comments 1
- Last comment by nonemenon
- Last Active Nov 24th 2021

- Discussion Type
- discussion topicJohn McKay
- Category Latest Changes
- Started by David_Corfield
- Comments 2
- Last comment by David_Corfield
- Last Active Nov 24th 2021

Added

- Yang-Hui He, John McKay,
*Kashiwa Lectures on “New Approaches to the Monster”*, (arXiv:2106.01162)

- Yang-Hui He, John McKay,