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2-categories 2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions 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 graph graphs gravity 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 infinity integration integration-theory k-theory kan lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal modal-logic model model-category-theory monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics planar 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 topicBaire lattice
- Category Latest Changes
- Started by Daniel Luckhardt
- Comments 2
- Last comment by Daniel Luckhardt
- Last Active 55 minutes ago

- Discussion Type
- discussion topicEuler class
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 2 hours ago

started Euler class

- Discussion Type
- discussion topiclandscape of string theory vacua
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active 3 hours ago

I am starting landscape of string theory vacua -- hah! :-)

- Discussion Type
- discussion topicweak gravity conjecture
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 3 hours ago

started some minimum at

*weak gravity conjecture*

- Discussion Type
- discussion topiceffective quantum field theory
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active 3 hours ago

at effective quantum field theory I have started writing an Idea-section and added more reference

- Discussion Type
- discussion topicquaternionic Kähler manifold > history
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 3 hours ago

finally realized that there were two duplicate entries with similar titles. Removed this one and merged it all into

*quaternion-Kähler manifold*.

- Discussion Type
- discussion topicspecial holonomy table
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 3 hours ago

am starting a

*special holonomy table*, have included it into relevant entries and created some of these entries, edited others.Not really done yet and not really good yet. Hope to improve on it later.

- Discussion Type
- discussion topicSpin(7) manifold
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 3 hours ago

added this statement:

Let $X$ be a closed smooth manifold of dimension 8 with Spin structure. If the frame bundle moreover admits G-structure for

$G = Spin(7) \hookrightarrow Spin(8)$then the Euler class $\chi$, the second Pontryagin class $p_2$ and the cup product-square $(p_1)^2$ of the first Pontryagin class of the frame bundle/tangent bundle are related by

$8 \chi \;=\; 4 p_2 - (p_1)^2 \,.$

- Discussion Type
- discussion topicnormed division algebra Riemannian geometry -- table
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 7 hours ago

- Discussion Type
- discussion topicquaternion-Kähler manifold
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 7 hours ago

- Discussion Type
- discussion topicglobular set
- Category Latest Changes
- Started by Stephan A Spahn
- Comments 4
- Last comment by Richard Williamson
- Last Active 8 hours ago

I added a reference to globular set.

- Discussion Type
- discussion topicquaternionic unitary group
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 8 hours ago

- Discussion Type
- discussion topicSpin(6)
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 8 hours ago

- Discussion Type
- discussion topicC-field tadpole cancellation
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 9 hours ago

starting something here. For the moment this is just the list of references which I had previously recorded at

*RR-field tadpole cancellation*, now joined by a quick Idea-sentence and a minimal statement of the actual cancellation condition

- Discussion Type
- discussion topictadpole cancellation
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 9 hours ago

- Discussion Type
- discussion topicBrouwer's fixed point theorem
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 10 hours ago

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 topicHopf degree theorem
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Urs
- Last Active 10 hours ago

- Discussion Type
- discussion topicAdS-QCD correspondence
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Urs
- Last Active 13 hours ago

- Discussion Type
- discussion topicdifferentiable manifold
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active 13 hours ago

I have re-written the content at

*differentiable manifold*, trying to make it look a little nicer. Also gave*topological manifold*some minimum of content.

- Discussion Type
- discussion topicmodal type theory
- Category Latest Changes
- Started by Urs
- Comments 83
- Last comment by David_Corfield
- Last Active 16 hours ago

created an entry

*modal type theory*; tried to collect pointers I could find to articles which discuss the interpretation of modalities in terms of (co)monads. I was expecting to find much less, but there are a whole lot of articles discussing this. Also cross-linked with*monad (in computer science)*.

- Discussion Type
- discussion topictest category
- Category Latest Changes
- Started by zskoda
- Comments 11
- Last comment by Dmitri Pavlov
- Last Active 17 hours ago

The entry test category which I wrote some time ago, came into the attention of Georges Maltsiniotis who kindly wrote me an email with a kind praise on nlab and noting that his Astérisque treatise on the topic of Grothendieck’s homotopy theory is available online on his web page and that the Cisinski’s volume is sort of a continuation of his Astérisque 301. Georges also suggested that we should emphasise that a big part of the Pursuing Stacks is devoted to the usage of test categories, so I included it into the bibliography and introductory sentence. I hinted to Georges that when unhappy with a state of an nlab entry he could just feel free to edit directly.

- Discussion Type
- discussion topictangent (infinity,1)-category
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 19 hours ago

added pointer to

- Marc Hoyois,
*Topoi of parametrized objects*, Theory and Applications of Categories, Vol. 34, 2019, No. 9, pp 243-248. (arXiv:1611.02267, tac:34-09)

- Marc Hoyois,

- Discussion Type
- discussion topicExpository article on differential geometry
- Category Latest Changes
- Started by fred53
- Comments 12
- Last comment by fred53
- Last Active 20 hours ago

- I have written a 20 page expository article on differential geometry at an advanced undergraduate level. I wonder if I could post the article on nLab. Here, I copy from the introduction of the paper, and would appreciate any comments.

This twenty page note aims at a clear and quick exposition of some basic concepts and results in differential geometry, starting from the definition of vector fields, and culminating in Hodge theory on Kahler manifolds. Any success comes at the expense of omitting all proofs as well as key tools like sheaf theory (except in passing remarks) and pull back functions and their functorial properties. I have tried and believe to have make the prerequisites few and the exposition simple. Researching for this note helped me consolidate foggy recollections of my decades-old studies, and

I hope it will likewise prove useful to some readers in their learning introductory differential geometry.

I assume the reader knows how real and complex manifolds and occasionally vector bundles are defined, but beyond this the development is self contained. It concentrates on the algebra $\A$ (or $\A_\C$) of smooth real (or complex) valued functions on the manifold, viewing tensors, forms and indeed smooth sections of all vector bundles as $\A$ (or $\A_\C$) modules. Nothing in commutative algebra harder than the concept of module homomorphism (which I call $\A$-linear) and its multilinear counterpart is used, yet this simple language goes a long way to economize our presentation.

The pace is leisurely in the beginning for the benefit of the novice, then picks up a bit in later sections.

The first 6 sections are about real smooth manifolds, sections 7 and 8 discuss real and complex vector bundles over real manifolds, and the final 3 sections are about complex manifolds. I start by first defining vector fields, tensor fields, Lie derivative and then move on to metrics and (Levi-Civita) connections on the tangent bundle and their Riemann, Ricci and scalar curvature. Sec 5 defines differential forms and lists their main properties. Sec 6 discusses Hodge theory and harmonic forms on real manifolds. Sec 7 is about connections and their curvature on real vector bundles and Bianchi identities and Sec 8 presents complex vector bundles on real manifolds and their Chern classes. Sec 9 discusses complex manifolds and the Dolbeault complex and Sec 10 Chern connections on holoromorphic vector bundles. Sec 11 discusses the Hodge decomposition on compact Kahler manifolds.

Beyond whatever left of my college day studies, I have drawn freely from internet sources, including nLab, particularly Wikepedia, as well as some downloadable books and notes. I give no references because aside from my own expository peculiarities, choices, typos, or any errors, the material is textbook standard.

- Discussion Type
- discussion topiclogical topology
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 21 hours ago

a bare minimum, so far essentially just a pointer to Penon’s thesis, prodded by

- David Myers,
*Logical Topology and Axiomatic Cohesion*, talk at*Geometry in Modal Homotopy Type Theory*2019 (pdf slides)

- David Myers,

- Discussion Type
- discussion topicHopf fibration
- Category Latest Changes
- Started by Urs
- Comments 30
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topiccotopology
- Category Latest Changes
- Started by Daniel Luckhardt
- Comments 3
- Last comment by Daniel Luckhardt
- Last Active 1 day ago

- Discussion Type
- discussion topicempty 170
- Category Latest Changes
- Started by nLab edit announcer
- Comments 13
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicSnaith theorem
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

added pointer to

Peter Arndt, section 3.2 of

*Abstract motivic homotopy theory*, thesis 2017 (web, pdf, ArndtAbstractMotivic.pdf:file)exposition: lecture at

*Geometry in Modal HoTT*, 2019 (recording I, recording II)

- Discussion Type
- discussion topiccohesive homotopy type theory
- Category Latest Changes
- Started by Urs
- Comments 29
- Last comment by Urs
- Last Active 1 day ago

I am working on an entry cohesive homotopy type theory.

This started out as material split off from cohesive (infinity,1)-topos, but is expanding now.

- Discussion Type
- discussion topicMotivic homotopy theory
- Category Latest Changes
- Started by Marc Hoyois
- Comments 17
- Last comment by Urs
- Last Active 1 day ago

Created motivic homotopy theory (renamed from A1-homotopy theory).

Still many blanks to fill in…