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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 complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation 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 lie-theory 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 nlab noncommutative noncommutative-geometry number-theory 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 sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

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- Discussion Type
- discussion topicChan-Paton gauge field over D-brane and worldsheet anomaly cancellation
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 18th 2013

I started discussing the

*Chan-Paton gauge field*and how it cancels the Kapustin-part of the Freed-Witten-Kapustin anomaly for the open string.The technical ingredients are now all there, but I need to fill in more glue text to make this readable. Will do so, but might have to interrupt now. I ran a bit out of time here…

- Discussion Type
- discussion topicLiouville-Poincaré cocycle in cohesive homotopy type theory
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by zskoda
- Last Active Jan 17th 2013

It just occurred to me that there is an immediate axiomatization of the Liouville-Poincaré 1-form (the canonical differential 1-form on a cotangent bundle) in differential cohesion.

In fact, it is the special case of a much more general notion: for $A$ any type in differential cohesion the total space $\underset{X}{\sum} \mathcal{O}_X(A)$ of the $A$-valued structure sheaf over any $X$ carries a canonical $A$-cocycle.

For $A = \Omega^1$ the sheaf of 1-forms and $X$ a manifold, this is the traditional Liouville-Poincaré 1-form on $T^* X$.

I made a quick note on that at

*differential cohesion – Liouville-Poincaré cocycle*.Thanks to a conversation with Owen Gwilliam I now also understand how that construction gives the antibracket in the BV-BRST complex. I still need to write that out. Not today though.

- Discussion Type
- discussion topictensor product of abelian categories
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Tim_Porter
- Last Active Jan 17th 2013

quick note on

*Deligne tensor product of abelian categories*

- Discussion Type
- discussion topicPicard 3-group
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 16th 2013

added

*References*to*Picard 3-group*.

- Discussion Type
- discussion topicK-invariant
- Category Latest Changes
- Started by Tim_Porter
- Comments 10
- Last comment by Tim_Porter
- Last Active Jan 16th 2013

Just a comment, I mostly have seen k-invariant, with a lower case k. Does anyone have ‘strong’ feelings about this?

- Discussion Type
- discussion topicextended Lagrangian
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jan 15th 2013

Since I found myself repeatedly referring to it from other $n$Lab entries, I finally put some content into the entry

*extended Lagrangian*.

- Discussion Type
- discussion topicHigher Algebraic Structures and Quantization
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 14th 2013

I needed a Reference-entry for Freed’s old article

*Higher Algebraic Structures and Quantization*, so I created one.

- Discussion Type
- discussion topicstandard model of fundamental physics - table
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 14th 2013

I am experimenting with a table

But I am still experimenting. I need a table with roughly the content as given there, but loads of things still need attention. The table itself omits some details even of that which it manifestly aims to display and doesn’t display at all yet what one might also list under its title.

Please be gentle to this stub for the moment. I need this for some lecture notes elsewhere and right now am only investing a few minutes into this, need to look into other things with higher priority for the moment. But of course eventually we should prettify this.

- Discussion Type
- discussion topicEinstein-Maxwell theory
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 14th 2013

added briefly the definition to

*Einstein-Maxwell theory*

- Discussion Type
- discussion topicscalar curvature
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 14th 2013

stub for

*scalar curvature*

- Discussion Type
- discussion topicGrassmannian
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by TobyBartels
- Last Active Jan 14th 2013

I ended up creating brief entries

and cross-linked them a bit. But nothing much there yet.

- Discussion Type
- discussion topicbig bang
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Jan 11th 2013

created

*big bang*(really, I was just working on

*field (physics)*, which made me create*scalar field*, then*inflaton*, and now this. That’s how it goes.)

- Discussion Type
- discussion topichyperdeterminant
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Jan 10th 2013

New stub hyperdeterminant (I was convinced we had it before, but…no).

- Discussion Type
- discussion topichigher spin structure - table
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 10th 2013

I keep drawing and re-drawing that Whitehead tower again and again. That needs to stop. So I created now an entry with a table, to be included where needed:

*higher spin structure - table*

- Discussion Type
- discussion topicreduction modality -| infinitesimal shape modality -| infinitesimal flat modality
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Mike Shulman
- Last Active Jan 9th 2013

created little entries

- reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality

to go along with the previous entries

- shape modality $\dashv$ flat modality $\dashv$ sharp modality

(whose $n$Forum-discussion is here)

All of this is part of the cohesion - table.

- Discussion Type
- discussion topicgauge field - table
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 7th 2013

started

*gauge field - table*and included it into the relevant entries

- Discussion Type
- discussion topicformally smooth object
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 5th 2013

quickly created

*formally smooth object*.Have to go offline now. More later.

- Discussion Type
- discussion topicroots, weights, etc.
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by TobyBartels
- Last Active Jan 3rd 2013

- Discussion Type
- discussion topicIMU
- Category Latest Changes
- Started by zskoda
- Comments 8
- Last comment by TobyBartels
- Last Active Jan 3rd 2013

- Discussion Type
- discussion topicpolylogarithm
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Dec 31st 2012

New stub polylogarithm and many new links at dilogarithm.

- Discussion Type
- discussion topicAndersen and Ueno's proof that geometric quantization = quantum groups
- Category Latest Changes
- Started by Bruce Bartlett
- Comments 10
- Last comment by Urs
- Last Active Dec 27th 2012

- I added a query box to the Holographic Principle page, referring to the work of Andersen and Ueno which I believe has now made rigorous that geometric quantization of Chern-Simons theory = quantum groups approach ala Reshetikhin-Turaev.

- Discussion Type
- discussion topicNew version of anafunctors paper
- Category Latest Changes
- Started by DavidRoberts
- Comments 20
- Last comment by DavidRoberts
- Last Active Dec 25th 2012

Before I forget, I uploaded a new version of my anafunctors paper to my page David Roberts. In particular, the finer points have been made a lot tighter. I even use technical phrases such as ’enough groupoids’ and ’admits cotensors’! :) It has also been submitted for publication.

- Discussion Type
- discussion topicLipschitz continuity
- Category Latest Changes
- Started by TobyBartels
- Comments 3
- Last comment by TobyBartels
- Last Active Dec 23rd 2012

Here it is: Lipschitz map. I don't know why I wrote it; I just felt like it. There really is much more to say, but I think that I've said enough for now!

- Discussion Type
- discussion topicNoetherian poset
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 21st 2012

stub for

*Noetherian poset*(just needed to be able to point to it, no real content there yet)

- Discussion Type
- discussion topicMonique Hakim
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 20th 2012

created people-entry

*Monique Hakim*and cross-linked with*Structured Spaces*

- Discussion Type
- discussion topicstable factorization system
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Urs
- Last Active Dec 19th 2012

added in an Examples-section to

*stable factorization system*the statement that in an adhesive category, in particular in a topos, the (epi, mono)-factorization is stable.

- Discussion Type
- discussion topicCrans-Gray tensor product
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Dec 18th 2012

added to Crans-Gray tensor product a brief remark on the fact that it gives a biclosed monoidal structure.

- Discussion Type
- discussion topicstructure sheaf
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 18th 2012

I have renamed the relevant section at

*differential cohesion*to*differential cohesion – Structure sheaves*, expanded a bit, and linked to it from*structure sheaf*and other related entries.

- Discussion Type
- discussion topiclocally representable structured (infinity,1)-topos
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Stephan A Spahn
- Last Active Dec 17th 2012

I split off locally representable structured (infinity,1)-topos from generalized scheme .

This is about Lurie’s $\mathcal{G}$-schemes, but I decided to change the title. For one to avoid the continuous conflict of notions of “generalized scheme” that made generalized scheme a mess, but also because it seems quite reasonable terminology to. Would’t you agree?

- Discussion Type
- discussion topictopological M5-brane
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 16th 2012

stub for

*topological M5-brane*. And now I really have to RUN

- Discussion Type
- discussion topictopological membrane
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 16th 2012

I have split off

*topological membrane*from*topological M-theiry/M2-brane*to record a reference. But otherwise no content yet.

- Discussion Type
- discussion topicHitchin functional
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 16th 2012

started

*Hitchin functional*but have to interrupt now in the middle of it. This entry is not in good shape yet.

- Discussion Type
- discussion topicKadeishvili's theorem
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Dec 14th 2012

split off an entry

*Kadeishvili’s theorem*from*A-infinity algebra*

- Discussion Type
- discussion topicM-theory on G2-manifolds
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 14th 2012

expanded the Idea- and the Definition section at

*G2-manifold*(also further at G2). (Still not really complete, though.) Highlighted the relation to 2-plectic geometry.

- Discussion Type
- discussion topicline 2-bundle
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Dec 13th 2012

am starting

*line 2-bundle*. I am headed for some discussion of the 2-stack of*super*line 2-bundles, its role as the twistings of K-theory in degrees 0,1, and 2/3 and so forth. But right now it’s still mostly a stub. More in a short while…

- Discussion Type
- discussion topicPicard n-group
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 13th 2012

earlier today I started creating entries

*Picard 2-group*,*Picard 3-group*,*Picard infinity-group*. Then I was interrupted for a long day. I hope I left stuff in a stubby but roughly sensible way…

- Discussion Type
- discussion topicred herring principle and non-associative algebras
- Category Latest Changes
- Started by Tim_Porter
- Comments 7
- Last comment by Mike Shulman
- Last Active Dec 12th 2012

Someone anonymous has deleted a paragraph at red herring principle on non-associative algebra. This seems a bit strange. I am no expert on those beasties but although non-associative algebra includes the study of Lie algebras etc., amongst them are the modules and it seems to me that a module (with trivial multiplication) considered as a Lie algebra is an associative non-associative algebra! The query by Toby further down the entry is relevant but if we assume ‘non-unital’ as well (and that is sometimes done) there is no problem.

There was no post to the Forum. The IP is 2.40.78.132. which is in Trieste it seems.

Should the paragraph be reinstated?

- Discussion Type
- discussion topicSlide-Dynkin diagram
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Dec 10th 2012

The entry Dynkin diagram is a ‘My First Slide’ one. Sometimes this sort of attempt has resulted in a new entry with substance being generated later, but more often nothing happens. Does anyone want to create such an entry?

- Discussion Type
- discussion topicgeometric quantization of non-integral 2-forms
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Dec 10th 2012

brief note on

*geometric quantization of non-integral 2-forms*

- Discussion Type
- discussion topicmodal hyperdoctrine
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 7th 2012

stub for

*modal hyperdoctrine*

- Discussion Type
- discussion topicsemi-Segal space
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Urs
- Last Active Dec 5th 2012

quick note

*semi-Segal space*(just recording a reference of a speaker we had today)

- Discussion Type
- discussion topicFermat theory
- Category Latest Changes
- Started by John Baez
- Comments 11
- Last comment by TobyBartels
- Last Active Dec 4th 2012

- I tried to start an entry called Fermat theory. Unfortunately I screwed up and forgot to capitalized "Fermat". Maybe a lab elf can clean up my mess?

There are also lots of other obvious ways this page could be improved...

- Discussion Type
- discussion topicmodel structure on cellular sets
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 3rd 2012

am creating

*model structure on cellular sets*, on Dimitri Ara’s work.Have also been adding various related brief comments and cross-links to

Search for “Ara” to find the additions.

- Discussion Type
- discussion topiclocally presentable (oo,1)-category
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Dec 3rd 2012

am starting to work again on locally presentable (infinity,1)-category

not much new yet, apart from some polishing. But at least the title I have finally changed, following Mike’s suggestion.

- Discussion Type
- discussion topiccritical loci in differential cohesive homotopy type theory
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 2nd 2012

I have started making notes at

*differential cohesion*on the axiomatic formulation ofSo far just the bare basics. To be expanded…

The basic observation (easy in itself, but fundamental for the concept formation) is that for any differential cohesive homotopy type $X$, the inclusion of the formally étale maps into $X$ into the full slice over $X$ is not only reflective but also co-reflective (since the formally étale maps are the Pi_inf-closed morphisms with the infinitesimal path groupoid functor / de Rham space functor $\Pi_{inf}$ being a left adjoint).

This means that for $G$ any differential cohesive $\infty$-group with the corresponding de Rham coefficient object $\flat_{dR}\mathbf{B}G$ (the universal moduli for flat $\mathfrak{g}$-valued differential forms), the sheaf of flat $\mathfrak{g}$-valued forms over any $X$ is given by the sections of the coreflection of the product projection $X \times \flat_{dR}\mathbf{B}G \to X$ into the formally étale morphisms into $X$.

- Discussion Type
- discussion topicrealizability model
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 2nd 2012

I created an entry

*realizability model*. But I only got to put one single reference into it and now I am forced to go offline.I’ll try to add more later. But maybe somebody here feels inspired to add a brief explanation…

- Discussion Type
- discussion topicHOMFLY
- Category Latest Changes
- Started by Andrew Stacey
- Comments 5
- Last comment by Tim_Porter
- Last Active Dec 1st 2012

Added HOMFLY-PT polynomial. Hope I got the skein diagrams right!

- Discussion Type
- discussion topic(oo,1)-quasitopos
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Mike Shulman
- Last Active Nov 30th 2012

started (infinity,1)-quasitopos

- Discussion Type
- discussion topiccomputational consistency
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 30th 2012

not sure why, but reading

- Peter Dybjer,
*Thoughts on Martin-Löf’s Meaning Explanations*(pdf)

made me look at

- Alexandre Miquel,
*The experimental effectiveness of mathematical proof*(pdf)

which seems to be about something deep and important that eventually I’d like to better grasp (but don’t yet), and that made me create

*computational consistency*.But I admit that I don’t really know what I am doing, in this case. So I’ll stop.

- Peter Dybjer,

- Discussion Type
- discussion topicdescent along torsors
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Nov 29th 2012

New entries descent along a torsor and Schneider’s descent theorem. Some changes and literature additions to a number of related entries.

- Discussion Type
- discussion topicaction functional
- Category Latest Changes
- Started by TobyBartels
- Comments 22
- Last comment by Urs
- Last Active Nov 28th 2012

Added some formulas and a manifestly relativistic version to action functional.

I have also been reverting JA's changes to variant conventions of spelling and grammar.

- Discussion Type
- discussion topic(infinity,n)-topos / (infinity,n)-sheaf
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Urs
- Last Active Nov 28th 2012

started stubs for

*(infinity,n)-sheaf*and*(infinity,n)-topos*; for the moment mostly as receptors and donors of cross-links, only.

- Discussion Type
- discussion topicinternal (infinity,1)-category
- Category Latest Changes
- Started by Urs
- Comments 60
- Last comment by DavidRoberts
- Last Active Nov 28th 2012

I am starting an entry internal (infinity,1)-category about complete Segal-like things.

This is prompted by me needing a place to state and prove the following assertion: a cohesive $\infty$-topos is an “absolute distributor” in the sense of Lurie, hence a suitable context for internalizing $(\infty,1)$-categories.

But first I want a better infrastructure. In the course of this I also created a “floating table of contents”

and added it to the relevant entries.

- Discussion Type
- discussion topiccore in a 2-category
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Mike Shulman
- Last Active Nov 28th 2012

I have taken the liberty of moving one more bit of Mike’s stuff on 2-topos theory from his personal web to the main $n$Lab, namely

*core in a 2-category*.(I am trying to fill what used to be the gray links in the proof at

*2-topos – In terms of internal categories*).

- Discussion Type
- discussion topicn-localic 2-topos
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Mike Shulman
- Last Active Nov 28th 2012

I have taken the liberty of moving one more bit of Mike’s stuff on 2-topos theory from his personal web to the main $n$Lab, namely

*n-localic 2-topos*.(I am trying to fill what used to be the gray links in the proof at

*2-topos – In terms of intenral categories*).

- Discussion Type
- discussion topicBn-geometry
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Todd_Trimble
- Last Active Nov 27th 2012

note on

*Bn-geometry*

- Discussion Type
- discussion topicweak omega-groupoid
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 27th 2012

Put a link to

- Thorsten Altenkirch, Ondrej Rypácek,
*A Syntactical Approach to Weak $\omega$-Groupoids*(pdf)

into

*weak omega-groupoid*… only trouble being that this entry doesn’t exist yet but redirects to*infinity-groupoid*, which otherwise has no references currently ?!-o . Somebody should take care of editing this a bit. But it won’t be me right now.- Thorsten Altenkirch, Ondrej Rypácek,

- Discussion Type
- discussion topicroot
- Category Latest Changes
- Started by Todd_Trimble
- Comments 1
- Last comment by Todd_Trimble
- Last Active Nov 26th 2012

I made some modifications to the definition section of root, and added the theorem that finite multiplicative subgroups of a field are cyclic. While I was at it, I added a bit to quaternion.

- Discussion Type
- discussion topicZuckerman induction
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Nov 26th 2012

I have split off a brief entry

*Zuckerman induction*from*cohomological induction*(since the basic version is not necessarily derived).

- Discussion Type
- discussion topicA-infinity space
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by zskoda
- Last Active Nov 25th 2012

In email discussion with somebody I wanted to point to the $n$Lab entry

*A-infinity space*only to notice that there is not much there. I have now spent a minute adding just a tiny little bit more…