Not signed in (Sign In)

A discussion forum about contributions to the nLab wiki and related areas of mathematics, physics, and philosophy.

Want to take part in these discussions? Sign in if you have an account, or apply for one below

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

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

- Discussion Type
- discussion topicholographic light front QCD
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Nov 1st 2021

am splitting this off from

*holographic QCD*, since the latter is getting too crowded. Prompted by today’s- Mohammad Ahmady,
*Holographic light-front QCD in B meson phenomenology*(arXiv:2001.00266)

- Mohammad Ahmady,

- Discussion Type
- discussion topicmonad
- Category Latest Changes
- Started by Urs
- Comments 24
- Last comment by varkor
- Last Active Oct 31st 2021

following Zoran’s suggestion I added to the beginning of the Idea-section at monad a few sentences on the general idea, leading then over to the Idea with respect to algebraic theories that used to be the only idea given there.

Also added a brief stub-subsection on monads in arbitrary 2-categories. This entry deserves a bit more atention.

- Discussion Type
- discussion topicGabriella Böhm
- Category Latest Changes
- Started by Tim_Porter
- Comments 2
- Last comment by Urs
- Last Active Oct 31st 2021

- Discussion Type
- discussion topicbialgebroid
- Category Latest Changes
- Started by zskoda
- Comments 6
- Last comment by Tim_Porter
- Last Active Oct 31st 2021

The stub for “associative” bialgebroid. Bialgebroids are to bialgebras what on dual side groupoids are to groups. More references at Hopf algebroids.

- Discussion Type
- discussion topicweak homotopy equivalence
- Category Latest Changes
- Started by Urs
- Comments 26
- Last comment by Urs
- Last Active Oct 31st 2021

I have been trying to polish

*weak homotopy equivalence*by adding formal Definition/Proposition-environements. Also expanded the Idea-section and edited here and there.The following remark used to be in the entry, but I can’t see right now how it makes sense. If I am mixed up, please clarify and I’ll re-insert it into the entry:

It is tempting to try to restate the definition as “$f$ induces an isomorphism $f_*: \pi_n(X,x) \to \pi_n(Y,f(x))$ for all $x \in X$ and $n \geq 0$,” but this is not literally correct; such a definition would be vacuously satisfied whenever $X$ is empty, without regard to what $Y$ might be. If you really want to go this way, therefore, you still must add a clause for $\Pi_{-1}$ (the truth value that states whether a space is inhabited), so the definition is no shorter.

Then, there used to be the following discussion box, which hereby I am moving from there to here. I have added a brief remark on how weak homotopy equivalences are homotopy equivalences after resolution. But maybe it deserves to be further expanded.

[ begin forwarded discussion ]

+–{.query} Is there any reason for calling these ’weak’ homotopy equivalences rather than merely homotopy equivalences? —Toby

Mike: By “these” I assume you mean weak homotopy equivalences of simplicial sets, categories, etc. My answer is yes. One reason is that in some cases, such as as simplicial sets, symmetric sets, and probably cubical sets, there is also a notion of “homotopy equivalence” from which this notion needs to be distinguished. A simplicial homotopy equivalence, for instance, is a simplicial map $f:X\to Y$ with an inverse $g:Y\to X$ and simplicial homotopies $X\times \Delta^1 \to X$ and $Y\times \Delta^1 \to Y$ relating $f g$ and $g f$ to identities.

*Toby*: Interesting. I would have guessed that any weak homotopy equivalence could be strengthened to a homotopy equivalence in this sense, but maybe not.Tim: I think the initial paragraph is somehow back to front from a philosophical point of view, as well as a historical one. Homotopy theory grew out of studying spaces up to homotopy equivalence or rather from studying paths in spaces (and integrating along them). This leads to some invariants such as homology and the fundamental group. Weak homotopy type (and it might be interesting to find out when this term was first used) is the result and then around the 1950s with the development of Whitehead’s approach (CW complexes etc.) the distinction became more interesting between the two concepts.

I like to think of ’weak homotopy equivalence’ as being ’observational’, i.e. $f$ is a w.h.e if when we look at it through the observations that we can make of it, it looks to be an ’equivalence’. It is ’top down’. ’Homotopy equivalence’ is more ’constructive’ and ’bottom up’. The idea of simple homotopy theory takes this to a more extreme case, (which is related to Toby’s query and to the advent of K-theory).

With the constructive logical side of the nLab becoming important is there some point in looking at this ’constructive’ homotopy theory as a counter balance to the model category approach which can tend to be very demanding on the set theory it calls on?

On a niggly point, the homotopy group of a space is only defined if the space is non-empty so one of the statements in this entry is pedantically a bit dodgy!

*Toby*: I would say that it has a homotopy group at every point, and this is true even if it is empty. You can only pretend that it has a homotopy group, period, if it's inhabited and path-connected.Anyway, how do you like the introduction now? You could add a more extensive History section too, if you want.

Tim: It looks fine. I would add some more punctuation but I’m a punctuation fanatic!!!

With all these entries I suspect that in a few months time we will feel they need some tender loving care, a bit of Bonsai pruning!! For the moment lets get on to more interesting things.

Do you think some light treatment of simple homotopy theory might be useful,say at a historical level? =–

[ end forwarded discussion ]

- Discussion Type
- discussion topicG(3)
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active Oct 31st 2021

- Discussion Type
- discussion topicNorihiko Minami
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 31st 2021

- Discussion Type
- discussion topicequivariant weak homotopy equivalence
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 31st 2021

- Discussion Type
- discussion topicMichael Barr
- Category Latest Changes
- Started by varkor
- Comments 1
- Last comment by varkor
- Last Active Oct 30th 2021

- Discussion Type
- discussion topicEllen Maycock Parker
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 30th 2021

- Discussion Type
- discussion topiccondensed local contractibility
- Category Latest Changes
- Started by David_Corfield
- Comments 89
- Last comment by Richard Williamson
- Last Active Oct 30th 2021

- Discussion Type
- discussion topicmodule over a monad
- Category Latest Changes
- Started by FinnLawler
- Comments 23
- Last comment by varkor
- Last Active Oct 30th 2021

I've split module over a monad off from algebra for an endofunctor. It still needs work, notably the definition of tensor product of bimodules, but it's late and I'm tired.

Also I added a remark to internal category about internal cats as monads in the bicategory of spans. I'm leading up to talking about internal profunctors and 2-sided fibrations, which Mike has been helping me understand at the café.

- Discussion Type
- discussion topicPoincaré conjecture
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 30th 2021

added a pointer for the higher dimensional case

- M. H. A. Newman, Theorem 7 in:
*The Engulfing Theorem for Topological Manifolds*, Annals of Mathematics Second Series, *84** 3 (1966) 555-571 (jstor:1970460)

(prompted by discussion in another thread, here)

- M. H. A. Newman, Theorem 7 in:

- Discussion Type
- discussion topicholographic entanglement entropy
- Category Latest Changes
- Started by Urs
- Comments 25
- Last comment by Urs
- Last Active Oct 29th 2021

created a stub for

*holographic entanglement entropy*in reaction to this MO question.

- Discussion Type
- discussion topiccategory algebra
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Tim_Porter
- Last Active Oct 29th 2021

- Discussion Type
- discussion topicgroupoid
- Category Latest Changes
- Started by Urs
- Comments 22
- Last comment by Tim_Porter
- Last Active Oct 29th 2021

added to groupoid a section on the description in terms of 2-coskeletal Kan complexes.

- Discussion Type
- discussion topicfinite group
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Urs
- Last Active Oct 29th 2021

had added to

*finite group*two classical references, Atiyah on group cohomology of finite groups, and Milnor on free actions of finite groups on $n$-spheres.What I’d really like to know eventually is the degree-3 group cohomology with coefficients in $U(1)$ for the finite subgroups of $SO(3)$.

- Discussion Type
- discussion topicfinite rotation group
- Category Latest Changes
- Started by Urs
- Comments 43
- Last comment by Urs
- Last Active Oct 29th 2021

I am splitting off an entry

*classification of finite rotation groups*from*ADE classification*in order to collect statements and references specific to the classification of finite subgroups of $SO(3)$ and $SU(2)$.Is there a canonical reference for the proof of the classification statement? I find lots of lecture notes that give the proof, but all of them without citing sources or original publications of proofs.

- Discussion Type
- discussion topicspherical space form
- Category Latest Changes
- Started by Urs
- Comments 15
- Last comment by Urs
- Last Active Oct 29th 2021

Presently this entry has much overlap with

*Clifford-Klein space form*and*group actions on spheres*. Eventually the three will diverge.

- Discussion Type
- discussion topicGodement product
- Category Latest Changes
- Started by Eric
- Comments 23
- Last comment by jademaster
- Last Active Oct 28th 2021

Hi,

I was going to add some details to Godement product, but I can’t reproduce what is there and suspect a typo.

For categories $A,B,C$, if $\alpha: F_1\to G_1 : A\to B$ and $\beta: F_2\to G_2 : B\to C$ are natural transformations of functors, the components $(\alpha * \beta)_M$ of the Godement product $\alpha * \beta: F_2\circ F_1\to G_2\circ G_1$ are defined by any of the two equivalent formulas:

$(\beta * \alpha)_M = \beta_{F_2 M}\circ G_1(\alpha_M)$ $(\beta * \alpha)_M = G_2(\alpha_M)\circ\beta_{F_1 M}$Following MacLane (page 42), the natural transformation $\alpha:F_1\Rightarrow G_1$ implies the existence of a morphism $\alpha_M:F_1(M)\to G_1(M)$. This, together with the natural transformation $\beta:F_2\Rightarrow G_2$, implies

$\array{ F_2\circ F_1(M) & \stackrel{F_2(\alpha_M)}{\to} & F_2\circ G_1(M) \\ \beta_{F_1(M)}\downarrow && \downarrow \beta_{G_1(M)} \\ G_2\circ F_1(M) & \stackrel{G_2(\alpha_M)}{\to} & G_2\circ G_1(M) } \,.$I thought the component of the Godement product, i.e. horizontal composition in Cat, should be the diagonal of this diagram so that

$(\beta\circ\alpha)_M = \beta_{G_1(M)}\circ F_2(\alpha_M) = G_2(\alpha_M)\circ \beta_{F_1(M)}.$Is there a typo on the page or am I completely missing the mark?

Note: Interchanging $F_2\leftrightarrow G_1$ in the two formulas on the page would give my two formulas.

- Discussion Type
- discussion topicTadayuki Watanabe
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 28th 2021

- Discussion Type
- discussion topicdiffeomorphism group
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Oct 28th 2021

started a section on the homotopy type of the diffeomorphism group and recorded the case for closed orientable surfaces

- Discussion Type
- discussion topicRonald J. Stern
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 28th 2021

- Discussion Type
- discussion topicRonald Fintushel
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 28th 2021

- Discussion Type
- discussion topicGeorges Vincent
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 28th 2021

brief

`category:people`

-entry for hyperlinking references*group actions on n-spheres*

- Discussion Type
- discussion topicHans Zassenhaus
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 28th 2021

- Discussion Type
- discussion topicGeorge R. Livesay
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 28th 2021

- Discussion Type
- discussion topicSantiago López de Medrano
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 28th 2021

- Discussion Type
- discussion topicinvolution
- Category Latest Changes
- Started by bgm
- Comments 3
- Last comment by Urs
- Last Active Oct 28th 2021

- Discussion Type
- discussion topicAleksandar Mikovic
- Category Latest Changes
- Started by Tim_Porter
- Comments 3
- Last comment by Tim_Porter
- Last Active Oct 28th 2021

- Discussion Type
- discussion topicArtin-Lam induction exponent
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Oct 27th 2021

For the moment just a blind definition, to satisfy links at

*free group actions on n-spheres*

- Discussion Type
- discussion topicTsit-Yuen Lam
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Oct 27th 2021

- Discussion Type
- discussion topicIan Hambleton
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 27th 2021

- Discussion Type
- discussion topicMichael Rathjen
- Category Latest Changes
- Started by NikolajK
- Comments 1
- Last comment by NikolajK
- Last Active Oct 27th 2021

- Discussion Type
- discussion topicDaniel Allcock
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 27th 2021

brief

`category:people`

-entry for hyperlinking references at*spherical space form*

- Discussion Type
- discussion topicpolyhedron
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 2
- Last comment by Urs
- Last Active Oct 27th 2021

- Discussion Type
- discussion topicEuler characteristic
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active Oct 27th 2021

In order to accompany the nCafe discussion I have started to add some content to the entry Euler characteristic

- Discussion Type
- discussion topicLefschetz trace formula
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Oct 27th 2021

- Discussion Type
- discussion topicBrouwer's fixed point theorem
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Oct 27th 2021

added pointer to

- B. A. Dubrovin, S. P. Novikov, A. T. Fomenko, corollary 15.3.4 of
*Modern Geometry — Methods and Applications: Part II: The Geometry and Topology of Manifolds*, Graduate Texts in Mathematics 104, Springer-Verlag New York, 1985

- B. A. Dubrovin, S. P. Novikov, A. T. Fomenko, corollary 15.3.4 of

- Discussion Type
- discussion topiccondensed set
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 61
- Last comment by David_Corfield
- Last Active Oct 27th 2021

- Discussion Type
- discussion topicantipode
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 27th 2021

This term used to redirect to

*Hopf algebra*. But we need it also in another sense at*real projective space*and elsewhere, so I am making this here a brief`category:diambiguation`

-entry.

- Discussion Type
- discussion topicAlexei Davydov
- Category Latest Changes
- Started by nLab edit announcer
- Comments 5
- Last comment by Tim_Porter
- Last Active Oct 27th 2021

- Discussion Type
- discussion topicflavour anomaly
- Category Latest Changes
- Started by Urs
- Comments 176
- Last comment by Urs
- Last Active Oct 27th 2021

- Discussion Type
- discussion topicCliff P. Burgess
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 27th 2021

- Discussion Type
- discussion topichigher curvature correction
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Urs
- Last Active Oct 27th 2021

- Discussion Type
- discussion topicgeneralized smooth space
- Category Latest Changes
- Started by nLab edit announcer
- Comments 4
- Last comment by Urs
- Last Active Oct 26th 2021

- Discussion Type
- discussion topicsingular cohesion
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Oct 26th 2021

- Discussion Type
- discussion topicorbifold cohomology
- Category Latest Changes
- Started by Urs
- Comments 104
- Last comment by Urs
- Last Active Oct 26th 2021

added references by Pronk-Scull and by Schwede, and wrote an Idea-section that tries to highlight the expected relation to global equivariant homotopy theory. Right now it reads like so:

On general grounds, since orbifolds $\mathcal{G}$ are special cases of stacks, there is an evident definition of cohomology of orbifolds, given by forming (stable) homotopy groups of derived hom-spaces

$H^\bullet(\mathcal{G}, E) \;\coloneqq\; \pi_\bullet \mathbf{H}( \mathcal{G}, E )$into any desired coefficient ∞-stack (or sheaf of spectra) $E$.

More specifically, often one is interested in viewing orbifold cohomology as a variant of Bredon equivariant cohomology, based on the idea that the cohomology of a global homotopy quotient orbifold

$\mathcal{G} \;\simeq\; X \sslash G \phantom{AAAA} (1)$for a given $G$-action on some manifold $X$, should coincide with the $G$-equivariant cohomology of $X$. However, such an identification (1) is not unique: For $G \subset K$ any closed subgroup, we have

$X \sslash G \;\simeq\; \big( X \times_G K\big) \sslash K \,.$This means that if one is to regard orbifold cohomology as a variant of equivariant cohomology, then one needs to work “globally” in terms of

*global equivariant homotopy theory*, where one considers equivariance with respect to “all compact Lie groups at once”, in a suitable sense.Concretely, in global equivariant homotopy theory the plain orbit category $Orb_G$ of $G$-equivariant Bredon cohomology is replaced by the global orbit category $Orb_{glb}$ whose objects are the delooping stacks $\mathbf{B}G \coloneqq \ast\sslash G$, and then any orbifold $\mathcal{G}$ becomes an (∞,1)-presheaf $y \mathcal{G}$ over $Orb_{glb}$ by the evident “external Yoneda embedding”

$y \mathcal{G} \;\coloneqq\; \mathbf{H}( \mathbf{B}G, \mathcal{G} ) \,.$More generally, this makes sense for $\mathcal{G}$ any orbispace. In fact, as a construction of an (∞,1)-presheaf on $Orb_{glb}$ it makes sense for $\mathcal{G}$ any ∞-stack, but supposedly precisely if $\mathcal{G}$ is an orbispace among all ∞-stacks does the cohomology of $y \mathcal{G}$ in the sense of global equivariant homotopy theory coincide the cohomology of $\mathcal{G}$ in the intended sense of ∞-stacks, in particular reproducing the intended sense of orbifold cohomology.

At least for topological orbifolds this is indicated in (Schwede 17, Introduction, Schwede 18, p. ix-x, see also Pronk-Scull 07)

- Discussion Type
- discussion topicsingular-smooth ∞-groupoid
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 26th 2021

giving this a minimal page for now, to satisfy links at

*singular cohesion*

- Discussion Type
- discussion topicpyknotic set
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 23
- Last comment by DavidRoberts
- Last Active Oct 26th 2021

- Discussion Type
- discussion topicdouble category
- Category Latest Changes
- Started by John Baez
- Comments 17
- Last comment by Mike Shulman
- Last Active Oct 26th 2021

- I added more info on pseudo double categories and double bicategories to double category. I also simplified the picture of a square, which had been bristling with scary unnecessary detail. There's a slight blemish in the left vertical arrow, which I can't see how to fix.

- Discussion Type
- discussion topicArtin representability theorem
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 26th 2021

added pointer to today’s

- Nadia Ott,
*Artin’s theorems in supergeometry*(arXiv:2110.12816)

- Nadia Ott,

- Discussion Type
- discussion topicsupergravity
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Oct 26th 2021

created supergravity

so far just an "Idea" section and a link to D'Auria-Fre formulation of supergravity (which i am busy working on)

- Discussion Type
- discussion topicuniversal Chern-Simons line 3-bundle
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 26th 2021

Thanks to an alert by Dmitri here, I realized that this entry had two spurious relative entries, one titled

- “Chern-Simons 2-gerbe”

which did nothing but point here

and one titled

- “Chern-Simons gerbe”

which did not even do that.

I have cleared these entries and instead made their titles be redirects to here.

- Discussion Type
- discussion topicChern-Simons gerbe > history
- Category Latest Changes
- Started by nLab edit announcer
- Comments 4
- Last comment by Urs
- Last Active Oct 26th 2021

- Discussion Type
- discussion topicChern-Simons 2-gerbe > history
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Oct 26th 2021

created Chern-Simons 2-gerbe

linked to it from principal infinity-bundle and Chern-Simons theory. In fact, I added an Idea-section to Chern-Simons theory.

- Discussion Type
- discussion topicde Donder-Weyl field theory
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Urs
- Last Active Oct 25th 2021

- Discussion Type
- discussion topicJohn Huerta
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active Oct 24th 2021

Looking to add an article I came across

- John Huerta,
*Bundle gerbes on supermanifolds*(arXiv:2012.15813)

I updated the page.

- John Huerta,

- Discussion Type
- discussion topicClifford-Klein space form
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Oct 24th 2021

- Discussion Type
- discussion topiccopower
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Oct 24th 2021

expanded copower:

added an Idea-section, an Example-section, and a paragraph on copowers in higher category theory.