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 book 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 deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry goodwillie-calculus 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 homology homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology newpage nforum nlab 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 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 topicadjoint action
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Urs
- Last Active Jul 4th 2021

simple definition at adjoint action

- Discussion Type
- discussion topicQuillen equivalence between simplicial groups and reduced simplicial sets
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jul 4th 2021

- Discussion Type
- discussion topicmodel structure on simplicial groups
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jul 4th 2021

added pointer to the original statement in

- Daniel Quillen, p. II 3.7 of:
*Axiomatic homotopy theory*in:*Homotopical Algebra*, Lecture Notes in Mathematics 43, Springer 1967(doi:10.1007/BFb0097438)

- Daniel Quillen, p. II 3.7 of:

- Discussion Type
- discussion topicIntroduction to Stable homotopy theory -- 1-1
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Urs
- Last Active Jul 4th 2021

- Discussion Type
- discussion topicKate Gruher
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 4th 2021

brief

`category:people`

-entry for hyperlinking references at*free loop space of classifying space*

- Discussion Type
- discussion topiccofunctor
- Category Latest Changes
- Started by BryceClarke
- Comments 6
- Last comment by BryceClarke
- Last Active Jul 4th 2021

- Discussion Type
- discussion topicArpon Raksit
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 3rd 2021

brief

`category:people`

-entry for hyperlinking references at*transchromatic character*

- Discussion Type
- discussion topictranschromatic character
- Category Latest Changes
- Started by David_Corfield
- Comments 7
- Last comment by Urs
- Last Active Jul 3rd 2021

I added to transchromatic character

- Nathaniel Stapleton,
*An Introduction to HKR Character Theory*(pdf)

and

- Takeshi Torii,
*HKR characters, p-divisible groups and the generalized Chern character*, (pdf)

- Nathaniel Stapleton,

- Discussion Type
- discussion topicnoncommutative measure theory
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 2
- Last comment by Urs
- Last Active Jul 3rd 2021

Created:

## Idea

According to the duality between geometry and algebra, various categories of commutative algebras and similar objects are equivalent to certain corresponding categories of spaces.

For example, the Gelfand duality establishes a contravariant equivalence of categories between the category of commutative unital C*-algebras and the category of compact Hausdorff topological spaces.

Thus, one can drop the commutativity condition and argue that studying arbitrary C*-algebras amounts to studying noncommutative general topology, a part of noncommutative geometry.

Analogously, in measure theory one works with commutative von Neumann algebras instead, which are equivalent to several other categories: compact strictly localizable enhanced measurable spaces, hyperstonean topological spaces and open maps, hyperstonean locales and open maps, measurable locales (and arbitrary morphisms of locales).

Thus, dropping the commutativity condition results in the category of von Neumann algebras (with ultraweakly continuous *-homomorphisms), and we can view this category as the category of noncommutative measurable spaces.

Thus, noncommutative measure theory is more-or-less the study of von Neumann algebras, more precisely those aspects of them that reproduce known phenomena in measure theory when specialized to the commutative case.

## Examples

The notion of a spatial product of (compact strictly localizable enhanced) measurable spaces can be expressed algebraically as the spatial tensor product of von Neumann algebras. This is quite different from the categorical product of measurable spaces, which corresponds to the Dauns tensor product of von Neumann algebras.

As another example, the theory of L^p-spaces generalizes to von Neumann algebras. This is a work of many authors, including von Neumann, Edward Nelson, Irving Segal, and Uffe Haagerup. It plays a crucial role in Tomita-Takesaki modular theory, which is essentially nothing else than the theory of L^p-spaces where $p$ is a purely imaginary number. In particular, it allows one to classify type III factors.

The notion of a measurable field of Hilbert spaces can be expressed using two equivalent categories: the category of W*-modules over von Neumann algebras, and the category of representations of von Neumann algebras on a Hilbert space.

## References

- Masamichi Takesaki,
*Theory of Operator Algebras I, II, III*, Springer, I: 1979. vii+415 pp. ISBN: 0-387-90391-7; II: 2003. xxii+518 pp. ISBN: 3-540-42914-X; III: 2003. xxii+548 pp. ISBN: 3-540-42913-1.

- Masamichi Takesaki,

- Discussion Type
- discussion topicUffe Haagerup
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Jul 3rd 2021

Created:

Uffe Haagerup was a Danish mathematician. He received a Master’s degree in 1973 from the University of Copenhagen, advised by Gert Pedersen, and went on to become a professor at the University of Odense in 1981 without a doctorate.

His work is concentrated around the field of von Neumann algebras (and noncommutative measure theory). He is responsible for developing many fundamental concepts in this area, including the standard form of a von Neumann algebra (analogous to the L^2-space), noncommutative L^p-spaces, weights and operator valued weights on von Neumann algebras (analogous to (fiberwise) measures), and many others.

- Discussion Type
- discussion topiclocale
- Category Latest Changes
- Started by Urs
- Comments 34
- Last comment by Urs
- Last Active Jul 3rd 2021

added to locale a section relation to toposes stating localic reflection

- Discussion Type
- discussion topicdecalage
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active Jul 3rd 2021

edited decalage a bit

there was the statement that is a "fibration". I made that Kan fibration. Is that right?

- Discussion Type
- discussion topicsimplicial identities
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Jul 3rd 2021

I polished and expanded a bit at

*simplicial identities*.

- Discussion Type
- discussion topicfaithful functor
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Hurkyl
- Last Active Jul 2nd 2021

- Discussion Type
- discussion topicSullivan model of loop space
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active Jul 2nd 2021

I have created

*Sullivan model of free loop space*with the formula and pointers to the literature.

- Discussion Type
- discussion topiccyclic loop space
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Jul 2nd 2021

I need a word for the homotopy quotient $(\mathcal{L}X)/S^1$ of free loop spaces $\mathcal{L}X$ by their canonical circle action. It seems that the only term in use with respect to this is “twisted loop space”, which however usually refers just to the constant loops $(\mathcal{L}_{const}X)//S^1$. Since under nice conditions the derived functions on the $\mathcal{L}Spec(A)/S^1$ is the cyclic homology complex of $A$, I suggest that a good name is “cyclic loop space”. I made a quick note at

*cyclic loop space*, just to fix and disambiguate terminology.

- Discussion Type
- discussion topicUrs Frauenfelder
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 2nd 2021

brief

`category:people`

-entry for hyperlinking references at*cyclic loop space*

- Discussion Type
- discussion topicNancy Hingston
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 2nd 2021

brief

`category:people`

-entry for hyperlinking references at*cyclic loop space*

- Discussion Type
- discussion topicstable splitting of mapping spaces
- Category Latest Changes
- Started by Urs
- Comments 16
- Last comment by Urs
- Last Active Jul 2nd 2021

- Discussion Type
- discussion topicTarmo Uustalu
- Category Latest Changes
- Started by Tim_Porter
- Comments 3
- Last comment by Dmitri Pavlov
- Last Active Jul 1st 2021

- Discussion Type
- discussion topicgeometric realization of categories
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Dmitri Pavlov
- Last Active Jul 1st 2021

- Discussion Type
- discussion topicBorel model structure
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Jul 1st 2021

- Discussion Type
- discussion topiccyclic loop stack
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Jul 1st 2021

to go alongside

*cyclic loop space*and*free loop stack*. Not quite done yet, but need to save

- Discussion Type
- discussion topicMilky Way
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jul 1st 2021

- Discussion Type
- discussion topicalgebra over a monad
- Category Latest Changes
- Started by Richard Williamson
- Comments 10
- Last comment by DavidRoberts
- Last Active Jun 30th 2021

- Discussion Type
- discussion topicchiral perturbation theory
- Category Latest Changes
- Started by Urs
- Comments 18
- Last comment by Grant_Bradley
- Last Active Jun 30th 2021

- Discussion Type
- discussion topicbaryon chiral perturbation theory -- references
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Jun 30th 2021

a bare list of references to be

`!include`

ed into relevant entries (such as*chiral perturbation theory*and*baryon*)

- Discussion Type
- discussion topicextensive category
- Category Latest Changes
- Started by Sam Staton
- Comments 39
- Last comment by Mike Shulman
- Last Active Jun 30th 2021

In the examples section of extensive category, it is stated that the category of affine schemes is infinitary extensive.

For all I know, I was the one who stuck in that example. But is that statement actually true? I’m having trouble seeing it.

If $S$ is a commutative ring over $R$ (by which I mean under $R$ (-:), does the functor $S \otimes_R -: CAlg_R \to CAlg_R$ preserve arbitrary cartesian products? Because it seems that’s what we basically need for the statement to be true.

- Discussion Type
- discussion topicnerve theorem
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Dmitri Pavlov
- Last Active Jun 30th 2021

created nerve theorem

linked to it from homotopy groups in an (infinity,1)-topos, where it had implicitly been mentioned before, but not made explicit.

Apart from stating the theorem, I wrote a section that explains what’s going on from the nPOV. As far as I can see, at least.

- Discussion Type
- discussion topicmysterious duality
- Category Latest Changes
- Started by David_Corfield
- Comments 24
- Last comment by Urs
- Last Active Jun 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 topicHuan's inertia orbifold
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jun 30th 2021

- Discussion Type
- discussion topicDavid Jaz Myers
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Jun 30th 2021

brief

`category:people`

-entry for hyperlinking references at*logical topology*and*synthetic differential topology*

- Discussion Type
- discussion topicintermediate value theorem
- Category Latest Changes
- Started by nLab edit announcer
- Comments 5
- Last comment by RodMcGuire
- Last Active Jun 29th 2021

- Discussion Type
- discussion topicconnected limit
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by John Baez
- Last Active Jun 29th 2021

- Discussion Type
- discussion topicbiproduct
- Category Latest Changes
- Started by Colin Tan
- Comments 15
- Last comment by Urs
- Last Active Jun 29th 2021

- Discussion Type
- discussion topicopen problem of confinement -- references
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Urs
- Last Active Jun 29th 2021

a bare sub-section with a list of references – to be

`!included`

into relevant entries – mainly at*confinement*and at*mass gap problem*(where this list already used to live)

- Discussion Type
- discussion topicSergei Dubovsky
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 29th 2021

brief

`category:people`

-entry for hyperlinking references at*holographic QCD*and*confinement*and*mass gap problem*

- Discussion Type
- discussion topicAdS-QCD correspondence
- Category Latest Changes
- Started by Urs
- Comments 63
- Last comment by Urs
- Last Active Jun 29th 2021

- Discussion Type
- discussion topiccyclic set
- Category Latest Changes
- Started by Urs
- Comments 21
- Last comment by Urs
- Last Active Jun 29th 2021

Have added to

*cyclic set*a pointer to notes from 1996 by Ieke Moerdijk where the theory classified by the topos of cyclic sets is identified (abstract circles).This is an unpublished note, but on request I have now uploaded it to the nLab

- Ieke Moerdijk,
*Cyclic sets as a classifying topos*, 1996 (pdf)

I have also added a corresponding brief section to

*classifying topos*.By the way, there is an old query box with an exchange between Mike and Zoran at

*cyclic set*. It seems to me that this has been resolved and the query box could be removed (to make the entry read more smoothly). Maybe Mike and/or Zoran could briefly look into this.- Ieke Moerdijk,

- Discussion Type
- discussion topiclax natural transformation
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Mike Shulman
- Last Active Jun 28th 2021

I tried to polish and impove the idea-section at lax natural transformation after pointing to it from MO

- Discussion Type
- discussion topictranscendental syntax
- Category Latest Changes
- Started by nLab edit announcer
- Comments 6
- Last comment by Boris Eng
- Last Active Jun 27th 2021

- Discussion Type
- discussion topicTate K-theory
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Jun 27th 2021

am finally splitting this off from

*Tate curve*

- Discussion Type
- discussion topicfree loop space
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jun 27th 2021

stub for free loop space (in $Top$)

- Discussion Type
- discussion topiccyclic homology
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Urs
- Last Active Jun 27th 2021

added the definition to cyclic homology

next the task is to write out the details for how under the identification of the Hochschild complex with functions on the derived loop space, the cyclic complex is the $S^1$-equivariant functions on the derived loop space.

- Discussion Type
- discussion topiccyclic space
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jun 27th 2021

it seems this entry was missing among

*cycle category*,*cyclic object*and*cyclotomic spectrum*. Am starting something, but nothing much here yet.

- Discussion Type
- discussion topicholographic entanglement entropy
- Category Latest Changes
- Started by Urs
- Comments 20
- Last comment by Urs
- Last Active Jun 27th 2021

created a stub for

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

- Discussion Type
- discussion topictopological stack
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Jun 27th 2021

Dave Carchedi has been editing/expanding topological stack

- Discussion Type
- discussion topiccyclic category
- Category Latest Changes
- Started by Todd_Trimble
- Comments 19
- Last comment by Urs
- Last Active Jun 27th 2021

I have begun cleaning up the entry cycle category, tightening up definitions and proofs. This should render some of the past discussion obsolete, by re-expressing the intended homotopical intuitions (in terms of degree one maps on the circle) more precisely, in terms of “spiraling” adjoints on the poset $\mathbb{Z}$.

Here is some of the past discussion I’m now exporting to the nForum:

The cycle category may be defined as the subcategory of Cat whose objects are the categories $[n]_\Lambda$ which are freely generated by the graph $0\to 1\to 2\to\ldots\to n\to 0$, and whose morphisms $\Lambda([m],[n])\subset\mathrm{Cat}([m],[n])$ are precisely the functors of degree $1$ (seen either at the level of nerves or via the embedding $\mathrm{Ob}[n]_\Lambda\to \mathbf{R}/\mathbf{Z}\cong S^1$ given by $k\mapsto k/(n+1)\,\mathrm{mod}\,\mathbf{Z}$ on the level of objects, the rest being obvious).

The simplex category $\Delta$ can be identified with a subcategory of $\Lambda$, having the same objects but with fewer morphisms. This identification does not respect the inclusions into $Cat$, however, since $[n]$ and $[n]_\Lambda$ are different categories.

- Discussion Type
- discussion topicexact square
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Jun 26th 2021

- Discussion Type
- discussion topicdynamical system
- Category Latest Changes
- Started by nLab edit announcer
- Comments 4
- Last comment by Urs
- Last Active Jun 26th 2021

- Discussion Type
- discussion topicChurch numeral
- Category Latest Changes
- Started by Mike Shulman
- Comments 2
- Last comment by nLab edit announcer
- Last Active Jun 26th 2021

- Discussion Type
- discussion topicfree loop orbifold
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 26th 2021

Some minimum, to go alongside

*inertia orbifold*.

- Discussion Type
- discussion topicsemi-simplicial set
- Category Latest Changes
- Started by Mike Shulman
- Comments 14
- Last comment by Urs
- Last Active Jun 26th 2021

Created semi-simplicial set, mainly as a repository for some terminological remarks. I would welcome anyone more knowledgeable about the history to correct or improve it!

- Discussion Type
- discussion topicmodel structure on semi-simplicial sets
- Category Latest Changes
- Started by Urs
- Comments 16
- Last comment by Urs
- Last Active Jun 26th 2021

brief entry

*model structure on semi-simplicial sets*, just in order to record a recent note by Benno van den Berg.

- Discussion Type
- discussion topicsemimodel category
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 4
- Last comment by Urs
- Last Active Jun 26th 2021

- Discussion Type
- discussion topicweak model category
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 18
- Last comment by Urs
- Last Active Jun 26th 2021

- Discussion Type
- discussion topicMatthew Spong
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 25th 2021

- Discussion Type
- discussion topicHawaiian earring space
- Category Latest Changes
- Started by nLab edit announcer
- Comments 31
- Last comment by martinescardo
- Last Active Jun 25th 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 topicThomas Dove
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 25th 2021

brief

`category:people`

-entry for hyperlinking references at*Tate K-theory*,*equivariant elliptic cohomology*and*Huan’s inertia orbifold*

- Discussion Type
- discussion topicgroupoid object
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 25th 2021