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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• Created Baire lattice abstracting the Baire category theorem.

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

• 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.

• added statement of the definition

• added mentioning of the first two exceptional isos $Sp(1) \simeq Spin(3)$ and $Sp(2) \simeq Spin(5)$

• am giving this its own little entry (in line with Spin(2), Spin(3), Spin(4), Spin(5)). For the moment just as to record a nice reference for the exceptional iso $Spin(6) \simeq SU(4)$

• 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

• splitting this off as a minimum disambiguation entry, in order to better organize the material

• 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
• giving this its own entry, for ease of referencing

• starting something. Not done yet but need to save

• 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).

• 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.

• 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.
• Corrected/expanded historical remarks and references.

• This is fj test page for putting an article on nLab

fj