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- Discussion Type
- discussion topicInitiality Project - Type Theory - Pi-types
- Category Latest Changes
- Started by Mike Shulman
- Comments 32
- Last comment by atmacen
- Last Active Mar 30th 2019

- Discussion Type
- discussion topicInitiality Project - Type Theory
- Category Latest Changes
- Started by Mike Shulman
- Comments 233
- Last comment by atmacen
- Last Active Mar 29th 2019

- Discussion Type
- discussion topicmodel structure on simplicial presheaves
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Tim_Porter
- Last Active Mar 29th 2019

fianlly added the details of Dugger’s description of cofibrant objects in the projective model structure on simplicial presheaves in the section Cofibrant objects.

- Discussion Type
- discussion topicorder-type
- Category Latest Changes
- Started by Alec Rhea
- Comments 8
- Last comment by Alec Rhea
- Last Active Mar 28th 2019

- Discussion Type
- discussion topictangent bundle category
- Category Latest Changes
- Started by David_Corfield
- Comments 21
- Last comment by BenMacAdam
- Last Active Mar 28th 2019

Added another reference.

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

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

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

- Discussion Type
- discussion topicAgnès Beaudry
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Mar 28th 2019

added entry for Agnès Beaudry getting rid of a grey link.

- Discussion Type
- discussion topicCisinski model structure
- Category Latest Changes
- Started by Urs
- Comments 21
- Last comment by Tim_Porter
- Last Active Mar 28th 2019

created Cisinski model structure

- Discussion Type
- discussion topicSpin(7)
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active Mar 27th 2019

added pointer to

- Veeravalli Varadarajan,
*Spin(7)-subgroups of SO(8) and Spin(8)*, Expositiones Mathematicae, 19 (2001): 163-177 (pdf)

- Veeravalli Varadarajan,

- Discussion Type
- discussion topicfracture theorem
- Category Latest Changes
- Started by Urs
- Comments 33
- Last comment by Urs
- Last Active Mar 27th 2019

Finally added to

*fracture theorem*the basic statement of the “arithmetic fracture square”, hence the following discussion.

The number theoretic statement is the following:

+– {: .num_prop #ArithmeticFractureSquare}

###### Proposition

The integers $\mathbb{Z}$ are the fiber product of all the p-adic integers $\underset{p\;prime}{\prod} \mathbb{Z}_p$ with the rational numbers $\mathbb{Q}$ over the rationalization of the former, hence there is a pullback diagram in CRing of the form

$\array{ && \mathbb{Q} \\ & \swarrow && \nwarrow \\ \mathbb{Q}\otimes_{\mathbb{Z}}\underset{p\;prime}{\prod} \mathbb{Z}_p && && \mathbb{Z} \\ & \nwarrow && \swarrow \\ && \underset{p\;prime}{\prod} \mathbb{Z}_p } \,.$Equivalently this is the fiber product of the rationals with the integral adeles $\mathbb{A}_{\mathbb{Z}}$ over the ring of adeles $\mathbb{A}_{\mathbb{Q}}$

$\array{ && \mathbb{Q} \\ & \swarrow && \nwarrow \\ \mathbb{A}_{\mathbb{Q}} && && \mathbb{Z} \\ & \nwarrow && \swarrow \\ && \mathbb{A}_{\mathbb{Z}} } \,.$=–

In the context of a modern account of categorical homotopy theory this appears for instance as (Riehl 14, lemma 14.4.2).

+– {: .num_remark}

###### Remark

Under the function field analogy we may think of

$Spec(\mathbb{Z})$ as an arithmetic curve over F1;

$\mathbb{A}_{\mathbb{Z}}$ as the ring of functions on the formal disks around all the points in this curve;

$\mathbb{Q}$ as the ring of functions on the complement of a finite number of points in the curve;

$\mathbb{A}_{\mathbb{Q}}$ is the ring of functions on punctured formal disks around all points, at most finitely many of which do not extend to the unpunctured disk.

Under this analogy the arithmetic fracture square of prop. \ref{ArithmeticFractureSquare} says that the curve $Spec(\mathbb{Z})$ has a cover whose patches are the complement of the curve by some points, and the formal disks around these points.

This kind of cover plays a central role in number theory, see for instance thr following discussions:

=–

- Discussion Type
- discussion topicSU(3)
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 27th 2019

- Discussion Type
- discussion topicFrom classical field theory to perturbative quantum field theory
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 27th 2019

the book is finally out: springer:book/9783030047375

The proof of the fundamental theorem 3.9.4 of renormalization via addition of counterterms is in “essential parts” (footnote 118) taken from the $n$Lab page

*geometry of physics – perturbative quantum field theory*

- Discussion Type
- discussion topicLie theory
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Mar 27th 2019

- Discussion Type
- discussion topicstring theory
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Mar 27th 2019

I have added to string theory a new section Critical strings and quantum anomalies.

Really I was beginning to work on a new entry twisted spin^c structure (not done yet) and then I found that a summary discussion along the above lines had been missing.

- Discussion Type
- discussion topicstring theory FAQ
- Category Latest Changes
- Started by Urs
- Comments 77
- Last comment by Urs
- Last Active Mar 27th 2019

added to

*string theory FAQ*two new paragraphs:Prompted by the MO discussion

- Discussion Type
- discussion topichomotopy pullback
- Category Latest Changes
- Started by Urs
- Comments 16
- Last comment by Urs
- Last Active Mar 27th 2019

prompted by a question by email, I have expanded at homotopy pullback the section on Concrete constructions by listing and discussing the precise conditions under which ordinary pullbacks are homotopy pullbacks.

Most of this information is scattered around elswehere on the $n$Lab (such as at homotopy limit and right proper model category) and I had wrongly believed that it was already collected here. But it wasn’t.

- Discussion Type
- discussion topicsymmetric monoidal category
- Category Latest Changes
- Started by Urs
- Comments 26
- Last comment by Richard Williamson
- Last Active Mar 27th 2019

added to symmetric monoidal category a new Properties-section As models for connective spectra with remarks on the theorems by Thomason and Mandell.

- Discussion Type
- discussion topicfree functor
- Category Latest Changes
- Started by anqurvanillapy
- Comments 1
- Last comment by anqurvanillapy
- Last Active Mar 26th 2019

- Discussion Type
- discussion topiccontinuous poset
- Category Latest Changes
- Started by Mike Shulman
- Comments 15
- Last comment by Daniel Luckhardt
- Last Active Mar 26th 2019

Did we really not have continuous poset?

- Discussion Type
- discussion topicstandard model of particle physics
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Mar 26th 2019

**Edit to**: standard model of particle physics by Urs Schreiber at 2018-04-01 01:15:37 UTC.**Author comments**:added textbook reference

- Discussion Type
- discussion topicduality in physics
- Category Latest Changes
- Started by Urs
- Comments 67
- Last comment by Urs
- Last Active Mar 26th 2019

needed to be able to point to

*duality in physics*, so I created an entry. For the moment just a glorified redirect.

- Discussion Type
- discussion topicM5-brane
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Urs
- Last Active Mar 26th 2019

I have added to

*M5-brane*a fairly detailed discussion of the issue with the fractional quadratic form on differential cohomology for the dual 7d-Chern-Simons theory action (from Witten (1996) with help of Hopkins-Singer (2005)).In the new section

*Conformal blocks and 7d Chern-Simons dual*.

- Discussion Type
- discussion topicsphere
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Mar 26th 2019

- Discussion Type
- discussion topicabstract elementary class
- Category Latest Changes
- Started by Mike Shulman
- Comments 4
- Last comment by Mike Shulman
- Last Active Mar 26th 2019

- Discussion Type
- discussion topichomotopy sphere
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Mar 26th 2019

- Discussion Type
- discussion topicSketch
- Category Latest Changes
- Started by David_Corfield
- Comments 5
- Last comment by nLab edit announcer
- Last Active Mar 25th 2019

I pasted in something Mike wrote on sketches and accessible models to sketch. But now it needs tidying up, and I’m wondering if it might have been better placed at accessible category. Alternatively we start a new page on sketch-theoretic model theory. Ideas?

- Discussion Type
- discussion topicSL(2,O)
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 25th 2019

started some minimum, so that our table

*exceptional spinors and division algebras – table*can point somewhere

- Discussion Type
- discussion topicSL(2,O)
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 25th 2019

started some minimum, so that our table

*exceptional spinors and division algebras – table*can point somewhere

- Discussion Type
- discussion topicGromoll-Meyer sphere
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Mar 25th 2019

- Discussion Type
- discussion topicG2 manifold
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Mar 25th 2019

- Discussion Type
- discussion topicheterotic string theory on CY3-manifolds
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 25th 2019

was looking for the right place to record

- Andreas Braun, Sakura Schaefer-Nameki,
*Compact, Singular G2-Holonomy Manifolds and M/Heterotic/F-Theory Duality*, JHEP04(2018)126 (arXiv:1708.07215)

and noticed that

*heterotic string theory on CY3-manifolds*just redirected to*supersymmetry and Calabi-Yau manifolds*, which made no good sense. So I am splitting it off as a stand-alone entry hereby.- Andreas Braun, Sakura Schaefer-Nameki,