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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 topicuniversal covering space
- Category Latest Changes
- Started by Urs
- Comments 26
- Last comment by Urs
- Last Active Dec 26th 2021

edited universal covering space

moved my proof to here that it is the homotopy fiber of the morphism of topological groupoids;

then created a new section universal covering spaces in an (oo,1)-topos where I talk about an idea (the evident idea) for what one should say here.

Also propose a definition for Whitehead tower in an (oo,1)-topos this way.

the entry Postnikov tower in an (infinity,1)-category following JL is under way...

- Discussion Type
- discussion topicfundamental theorem of covering spaces
- Category Latest Changes
- Started by Bartek
- Comments 5
- Last comment by Urs
- Last Active Dec 26th 2021

- Discussion Type
- discussion topicExt
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Bartek
- Last Active Dec 26th 2021

tried to edit Ext a bit. But this needs to be expanded, eventually.

- Discussion Type
- discussion topiccovering space
- Category Latest Changes
- Started by Urs
- Comments 66
- Last comment by Urs
- Last Active Dec 25th 2021

I added to covering space a section In terms of homotopy fibers that explains the universal covering space as the homotopy fiber/principal oo-bundle classified by the cocycle that is the constant path inclusion of topological groupoids.

To fit this into the entry, I added some new sections and restructured slightly. Todd and David should please have a look.

What I just added is essentially what David Roberts says in various query boxes, notably in what is currently the last query box. Back then we talked about the "Roberts-Schreiber construction" or whatnot, but really what this is is just the standard way to compute homotopy fibers in the oo-category of oo-groupoids.

I suspect that Todd's bar construction described there can similarly be understood as being nothing but another way to compute the more abstractly defined homotopy pullback in concrete terms. I'll have to think about this, though. But probably Tim Porter or Mike Shulman will immediately recognize this as the relevant bar construction of homotopy pullbacks in homotopy coherent category theory.

- Discussion Type
- discussion topichomotopy theory and algebraic topology -- references
- Category Latest Changes
- Started by Urs
- Comments 18
- Last comment by Urs
- Last Active Dec 25th 2021

following discussion here I am starting an entry with a bare list of references (sub-sectioned), to be

`!include`

-ed into the References sections of relevant entries (mainly at*homotopy theory*and at*algebraic topology*) for ease of updating and syncing these lists.The organization of the subsections and their items here needs work, this is just a start. Let’s work on it.

I’ll just check now that I have all items copied, and then I will

`!include`

this entry here into*homotopy theory*and*algebraic topology*. It may best be*viewed*withing these entries, because there – but not here – will there be a table of contents showing the subsections here.

- Discussion Type
- discussion topicuniversal vector bundle
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by jim stasheff
- Last Active Dec 25th 2021

- Discussion Type
- discussion topicderivation
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by nLab edit announcer
- Last Active Dec 25th 2021

added to derivation at the very end in the exampls section a discussion of derivations on smooth functions (and how they are vector fields) and f derivations on continuous functions (and how they are trivial).

- Discussion Type
- discussion topicJiannis K. Pachos
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 25th 2021

- Discussion Type
- discussion topicJames Robin Wootton
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 25th 2021

- Discussion Type
- discussion topicSeth Lloyd
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 25th 2021

- Discussion Type
- discussion topicbraid group statistics
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Dec 25th 2021

stub for

*braid group statistics*(again, for the moment mainly in order to record a reference)

- Discussion Type
- discussion topicconformal field theory
- Category Latest Changes
- Started by Tim_van_Beek
- Comments 37
- Last comment by Urs
- Last Active Dec 25th 2021

I began to add a definition of conformal field theory using the Wightman resp. Osterwalder-Schrader axiomatic approach. My intention is to define and explain the most common concepts that appear again and again in the physics literature, but are rarely defined, like “primary field” or “operator product expansion”.

(I remember that I asked myself, when I first saw an operator product expansion, if the existence of one is an axiom or a theorem, I don’t remember reading or hearing an answer of that until I looked in the book by Schottenloher).

- Discussion Type
- discussion topicstate-field correspondence
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Urs
- Last Active Dec 25th 2021

- Discussion Type
- discussion topicexotic brane
- Category Latest Changes
- Started by Luigi
- Comments 7
- Last comment by Urs
- Last Active Dec 24th 2021

- Discussion Type
- discussion topic3d toric code
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 24th 2021

a stub entry, for the moment just to record some references and to satisfy links at

*defect brane*

- Discussion Type
- discussion topicdefect brane
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Dec 24th 2021

- Discussion Type
- discussion topicMasaki Shigemori
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 24th 2021

- Discussion Type
- discussion topicnatural deduction
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active Dec 24th 2021

- Discussion Type
- discussion topicδ-ring
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Dec 24th 2021

Created:

\tableofcontents

## Definition

(Definition 2.1 in Bhatt–Scholze.)

Fix a prime $p$. A

$\delta(xy)=x^p \delta(y)+y^p \delta(x) + p\delta(x)\delta(y),$**δ-ring**is a pair $(R,δ)$, where $R$ is a commutative ring and $\delta\colon R\to R$ is a map of underlying sets such that $\delta(0)=0$, $\delta(1)=0$,and

$\delta(x+y)=\delta(x)+\delta(y)+(x^p+y^p-(x+y)^p)/p.$## Properties

If $(R,\delta)$ is a δ-ring, then the map $\phi\colon R\to R$ given by $\phi(x)=x^p + p\delta(x)$ is a ring homomorphism that lifts the Frobenius endomorphism on $R/p$.

For $p$-torsionfree rings, the above correspondence between δ-structures and lifts of the Frobenius endomorphism on $R/p$ to $R$ is bijective. This motivates the identities in the definition of a δ-structure.

## Related entries

## References

- Bhargav Bhatt, Peter Scholze,
*Prisms and prismatic cohomology*, arXiv.

- Bhargav Bhatt, Peter Scholze,

- Discussion Type
- discussion topicsuper 3-brane in 8d
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 23rd 2021

stub entry for the last remaining item in the brane scan, for the moment just to record references

- Discussion Type
- discussion topicmembrane
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Dec 23rd 2021

- Discussion Type
- discussion topicsuper 2-brane in 5d
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 23rd 2021

stub entry for one more item in the brane scan, for the moment just to record references

- Discussion Type
- discussion topicbrane scan
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Dec 23rd 2021

I gave the brane scan table a genuine $n$Lab incarnation and included it at

*Green-Schwarz action functional*and at*brane*.

- Discussion Type
- discussion topic3-brane in 6d
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Dec 23rd 2021

I am trying to give more of the entries of the brane scan in low ambient dimension their proper names.

Next to the little string in $D = 6$ the brane scan says that there is a Green-Schwarz action functional for a 3-brane $\sigma$-model in $D = 6$. This has been first written down in

- James Hughes, Jun Liu, Joseph Polchinski,
*Supermembranes*, Physics Letters B Volume 180, Issue 4, 20 November 1986, Pages 370–374

but it seems to go by no specific name apart from “the 3-brane in 6d”. So I created a stub entry with that title,

*3-brane in 6d*.- James Hughes, Jun Liu, Joseph Polchinski,

- Discussion Type
- discussion topicGeorgios Zikos
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 23rd 2021

- Discussion Type
- discussion topicLayla Hormozi
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 23rd 2021

- Discussion Type
- discussion topicNick E. Bonesteel
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 23rd 2021

- Discussion Type
- discussion topicRanee K. Brylinski
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 23rd 2021

- Discussion Type
- discussion topicGoong Chen
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 23rd 2021

- Discussion Type
- discussion topicquantum computation
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Urs
- Last Active Dec 23rd 2021

stub for

*quantum computation*

- Discussion Type
- discussion topicSteven H. Simon
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 23rd 2021

- Discussion Type
- discussion topicbraid group representations -- references
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Dec 23rd 2021

a bare list of references, to be

`!include`

-ed both at*braid group*and at*topological quantum computation*, for ease of updating

- Discussion Type
- discussion topicZheng Yin
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 23rd 2021

- Discussion Type
- discussion topicDirac-Born-Infeld action
- Category Latest Changes
- Started by Urs
- Comments 26
- Last comment by Urs
- Last Active Dec 23rd 2021

added pointer to Schwarz 01

- Discussion Type
- discussion topicnearby homomorphisms from compact Lie groups are conjugate
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Urs
- Last Active Dec 22nd 2021

- Discussion Type
- discussion topicDaniel C. Cohen
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 22nd 2021

- Discussion Type
- discussion topicIvan Marin
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 22nd 2021

- Discussion Type
- discussion topicKnizhnik-Zamolodchikov equation
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Dec 22nd 2021

finally added the actual definition,

`!include`

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

- Discussion Type
- discussion topicanti de Sitter spacetime
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active Dec 22nd 2021

earlier today I had created a stub for anti de Sitter spacetime

- Discussion Type
- discussion topicequivariant homotopy theory
- Category Latest Changes
- Started by Urs
- Comments 28
- Last comment by Urs
- Last Active Dec 22nd 2021

expanded the discussion at equivariant homotopy theory

expanded the statement of the classical Elmendorf theorem

added the statement of the general Elmendorf theorem in general model categories

added remarks on G-equivariant oo-stacks, as special cases of this

- Discussion Type
- discussion topicYunze Lu
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 22nd 2021

- Discussion Type
- discussion topicequivariant ordinary cohomology
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Dec 22nd 2021

- Discussion Type
- discussion topicrational equivariant stable homotopy theory
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Dec 22nd 2021

I have started

*rational equivariant stable homotopy theory*, but so far there is nothing but references.

- Discussion Type
- discussion topicJeremy Hahn
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 21st 2021

- Discussion Type
- discussion topicLukas Brantner
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 21st 2021

- Discussion Type
- discussion topiciterated loop space
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Dec 21st 2021

- Discussion Type
- discussion topicdihedral group
- Category Latest Changes
- Started by Tim_Porter
- Comments 6
- Last comment by Urs
- Last Active Dec 21st 2021

- Discussion Type
- discussion topicequivariant K-theory
- Category Latest Changes
- Started by Urs
- Comments 26
- Last comment by Urs
- Last Active Dec 21st 2021

added to

*equivariant K-theory*comments on the relation to the operator K-theory of crossed product algebras and to the ordinary K-theory of homotopy quotient spaces (Borel constructions). Also added a bunch of references.(Also finally added references to Green and Julg at

*Green-Julg theorem*).This all deserves to be prettified further, but I have to quit now.

- Discussion Type
- discussion topicbundle gerbe
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active Dec 21st 2021

i polished the definition in bundle gerbe and then reorganized the former material on “Interpretations” in a new section

that first shows how to get a shifted central extension of groupoids form the bundle gerbe, and then demonstrates that this is the total space of a principal 2-bundle

- Discussion Type
- discussion topictaboo
- Category Latest Changes
- Started by Mike Shulman
- Comments 13
- Last comment by NikolajK
- Last Active Dec 20th 2021

Inspired by a discussion with Martin Escardo, I created taboo.

- Discussion Type
- discussion topicFibonacci numbers
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Dec 20th 2021

- Discussion Type
- discussion topiclocal field
- Category Latest Changes
- Started by Todd_Trimble
- Comments 26
- Last comment by Urs
- Last Active Dec 20th 2021

Created local field. I discovered there’s no number field and no Pontryagin duality.

- Discussion Type
- discussion topicring of integers
- Category Latest Changes
- Started by Urs
- Comments 26
- Last comment by Hurkyl
- Last Active Dec 20th 2021

needed to point to

*ring of integers*of a number field. The term used to redirect just to*integers*. I have split it off now with a minimum of content. Have to rush off now.

- Discussion Type
- discussion topicDerek Krepski
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 20th 2021

- Discussion Type
- discussion topic2-trivial model structure
- Category Latest Changes
- Started by nLab edit announcer
- Comments 5
- Last comment by Richard Williamson
- Last Active Dec 20th 2021

- Discussion Type
- discussion topicD-topology
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Dec 20th 2021

am splitting this off from

*diffeological space*as a stand-alone entry, for ease of hyperlinking

- Discussion Type
- discussion topicJ. W. S. Cassels
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 19th 2021

- Discussion Type
- discussion topicAlbrecht Fröhlich
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 19th 2021

- Discussion Type
- discussion topicalgebraic number theory
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 19th 2021

added more publication data and links to:

- J. W. S. Cassels, Albrecht Fröhlich (eds.),
*Algebraic number theory*, Acad. Press 1967, with many reprints; Fröhlich, Cassels, Birch, Atiyah, Wall, Gruenberg, Serre, Tate, Heilbronn, Rouqette, Kneser, Hasse, Swinerton-Dyer, Hoechsmann, systematic lecture notes from the instructional conference at Univ. of Sussex, Brighton, Sep. 1-17, 1965 (ISBN:9780950273426, pdf, errata pdf by Kevin Buzzard)

also, I have fixed the order of the editor’s names

- J. W. S. Cassels, Albrecht Fröhlich (eds.),

- Discussion Type
- discussion topicD=2 CFT as functorial field theory -- references
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Dec 19th 2021

a bare list of references, to be

`!include`

-ed into lists of references of relevant entries (such as*2d CFT*,*2d SCFT*,*conformal cobordism category*,*modular functor*and maybe elsewhere)