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started a very stubby
along with very stubby
and a very stubby
Had wanted to do more, but now I am running out of steam. Maybe the stubs inspire somebody to add a little more…
created a brief entry for Bousfield equivalence
I messed up slightly: i had forgotten that there was already a stub titled contact geometry. Now I have created contact manifold with some content that might better be at contact geometry. I should fix this. But not right now.
created Einstein’s equation, only to record a writeup by Gonzalo Reyes which I just came across by chance, who gives a discussion in terms of synthetic differential geometry.
quick entry for phantom map
brief Idea-section at chromatic convergence theorem
stub for telescopic localization,
finally created the category:reference-entry for Lurie’s chromatic lecture. See Chromatic Homotopy Theory
(And as a special service to the community… with lecture titles. ;-)
Lecture 1 Introduction (pdf)
Lecture 2 Lazard’s theorem (pdf)
Lecture 3 Lazard’s theorem (continued) (pdf)
Lecture 4 Complex-oriented cohomology theories (pdf)
Lecture 5 Complex bordism (pdf)
Lecture 6 MU and complex orientations (pdf)
Lecture 7 The homology of MU (pdf)
Lecture 8 The Adams spectral sequence (pdf)
Lecture 9 The Adams spectral sequence for MU (pdf)
Lecture 10 The proof of Quillen’s theorem (pdf)
Lecture 11 Formal groups (pdf)
Lecture 12 Heights and formal groups (pdf)
Lecture 13 The stratification of (pdf)
Lecture 14 Classification of formal groups (pdf)
Lecture 15 Flat modules over (pdf)
Lecture 16 The Landweber exact functor theorem (pdf)
Lecture 17 Phanton maps (pdf)
Lecture 18 Even periodic cohomology theories (pdf)
Lecture 19 Morava stabilizer groups (pdf)
Lecture 20 Bousfield localization (pdf)
Lecture 21 Lubin-Tate theory (pdf)
Lecture 22 Morava E-theory and Morava K-theory (pdf)
Lecture 23 The Bousfield Classes of and (pdf)
Lecture 24 Uniqueness of Morava K-theory (pdf)
Lecture 25 The Nilpotence lemma (pdf)
Lecture 26 Thick subcategories (pdf)
Lecture 27 The periodicity theorem (pdf)
Lecture 28 Telescopic localization (pdf)
Lecture 29 Telescopic vs -localization (pdf)
Lecture 30 Localizations and the Adams-Novikov spectral sequence (pdf)
Lecture 31 The smash product theorem (pdf)
Lecture 32 The chromatic convergence theorem (pdf)
Lecture 33 Complex bordism and -localization (pdf)
Lecture 34 Monochromatic layers (pdf)
Lecture 35 The image of (pdf)
created a stub for cluster decomposition, since I wanted the link elsewhere, but nothing there yet…
I recently created entry Bol loop. Now I made some corrections and treated the notion of a core of a right Bol loop (the term coming allegedly from Russian term сердцевина).
created a disambiguation page: spectral geometry
New article: Tychonoff space.
I gave the book
a category:reference entry and linked to it from a few relevant entries.
started stubs E-∞ geometry, E-∞ scheme.
To be filled with more content, for the moment I just need to be able to use the links.
I am starting entries
for the moment mostly to collect some references. But not much there yet. But if anyone can provide furher hints, that would be welcome.
have split-off quantization of loop groups from loop group
created A Survey of Elliptic Cohomology - elliptic curves with seminar notes on an exposition on elliptic curves.
Am hoping that some kind soul will eventually further go through these seminar notes and copy bits of material to separete entries, where it belongs. Eventually.
have created affine modality
I have created a stub for n-truncation modality and cross-linked with double negation modality.
I gather that double negation = (-1)-truncation in a “predicative context”, but maybe I don’t fully understand yet what predicativity has to do with it.
I have added some more informative Idea-sentences to Adams spectral sequence and to Adams-Novikov spectral sequence. Also added more references.
felt like the nLab should have an entry fraction
have split off an entry stable unitary group from the material at topological K-theory
added the “song of stable homotopy groups” to stable orthogonal group
Popped my head round the door and made a couple of changes to Banach algebra
The first change was to attempt a more lax position on what should constitute a Banach coalgebra: only looking at comonoids in the monoidal category of Banach spaces (geometric or topogical) with projective tensor product would rule out several important examples that have arisen in e.g. abstract harmonic analysis. The existence of different monoidal structures in the category of Banach spaces is a pain, but without it one would miss out on a rich world of examples.
The second was to add, to the list of examples, the celebrated-in-my-world-and-possibly-no-others Arens products on the double dual of a Banach algebra. I’ve made a stab at linking them to the related concepts of tensorial strength and strong monad but would welcome feedback or improvements.
New entry synthetic projective geometry and also sort of disambiguation and history page synthetic geometry making a distinction with synthetic differential geometry.
Stephen Gaito has joined with a question at cardinal number. I tried to reply.
isotope (physics) and isotope (algebra) with redirect for isotopy (algebra). I have read and thought much about isotopies in last couple of weeks, but no time at this point to write much about it into Lab.
Added references to the recent Riehl-Verity papers to quasi-category, adjoint (infinity,1)-functor and monadicity theorem. Any more places it should go?
started Weil conjecture
Chris Schommer-Pries posted a question/suggestion in the query box at semisimple category
have added to the Idea-section at Schubert calculus the following paragraph:
Schubert calculus is concerned with the ring structure on the cohomology of flag varieties and Schubert varieties. Traditionally this was considered for ordinary cohomology (see References – traditional) later also for generalized cohomology theories (see References – In generalized cohomology), notably in complex oriented cohomology theory such as K-theory, elliptic cohomology and algebraic cobordism.
And have added references on Schubert calculus for generalized cohomology.
I gave the following an category:reference-entry
and linked to it from various relevant entries.
stub for reductive group
… need not be , but it shouldn't be larger; remarks about this are now at Banach algebra (and also at JB-algebra).
stub for Kac character formula, for the moment just so as to record the citations.
We don't actually need essentially bounded functions as such, since measurable functions should only be almost-everywhere defined by default, but there they are.
There is a deliberately ambiguous stub at finite-dimensional space.
We might collect there all of the nice things about finite-dimensional spaces (for various notions of ’space’).
Marc Hoyois created formally real field back in August (which was never announced here), and now I've created formally real algebra (and linked them to one another).
Stub symplectic integrator, just a list of basic references so far, redirecting aslo multisymplectic integrator.
have added a paragraph tangent infinity-category – Tangent infinity topos meant to extract the argument from Joyal’s “Notes on Logoi” that the tangent -category of an -topos is an -topos. Then a remark on how this should imply that the tangent -topos of a cohesive topos is itself cohesive over the tangent base -topos.
I am not making any claims tonight, just sketching an argument. Hope to come back to it tomorrow when I am awake again.
I have started a stub for smooth super infinity-groupoids, with the evident definition and observation that this is cohesive, but nothing else so far. To be worked on. (similar to locally-contractible infinity-groupoid)
I’ve constructed the page p-divisible group since I need it for my height of a variety page. I have to admit that I’m incredibly embarrassed that no matter how many times I look up the words “directed” “inductive” “projective” “limit” “colimit” etc I never seem to use them correctly. All of the systems are as I showed I thought this corresponded to directed, inductive, or colimit, but when I looked up inductive limit in the nlab it seemed to be indicating the opposite, so maybe some of the uses are wrong.
In the References-section at 2-sheaf I have added three “classical” references:
in the 1970s Grothendieck, Giraud and then Bunge usually considered “2-sheaves” – namely category-valued stacks – by default. Also there is a good body of work on 2-sheaves realized as internal categories in the underlying 1-sheaf topos. I have added a pointer to Joyal-Tierney’s Strong stacks so far, but I think much more literature exists in this direction.
But if one goes this internalization-route at all, what one should really do is, I think, consider weak internal categories in the (2,1)-topos over the underlying site.
Has this been studied at all? Does anyone know how 2-categories of weak internal categories in -toposes relate to 2-toposes? At least under nice conditions these should be equivalent, I guess. But I want to understand this better.
brief entry differentiable (infinity,1)-category
That is, Jordan–Banach algebras (although there is actually a distinction between these).
Lagrange inversion, redirecting also Lagrange inversion formula and Lagrange inversion theorem, previously wanted at Lambert W-function, noncommutative symmetric function and at Faà di Bruno formula.
I don't know why we never had endofunction, but we didn't; now we do.
I have made functional and operator primarily about the meanings of these in higher-order logic, where these terms are used exclusively and unqualified. I have accordingly split off linear functional from functional; linear operator (redirecting to linear map) was already separate from operator (which was only for disambiguation). I have also checked each incoming link to functional or operator (or a plural form) to link instead to linear functional or linear operator when appropriate.
That said, there are such things as nonlinear functionals and operators on abstract vector spaces, things which are also not functionals or operators in the type-theoretic sense. Possibly we would want pages such as nonlinear functional and nonlinear operator to cover these. (Compare nonassociative algebra, which covers a topic more general than what is covered at associative algebra but also could not be covered at simply algebra.)
I did not know what to do with the phrase ‘various discretised versions are interesting in finite geometries as well as numerical analysis’. Are these linear functionals, type-theoretic functionals, both, or neither?
The revolution will not be televised, but it will be wikified at power-associative algebra.
I created polarization identity and added some disambiguation to polarization.
With our “String Geometry Network” we have another meeting in October at the Max-Planck Institute for Mathematics in Bonn.
In each such meeting we have, besides research talks and discussion sessions, a kind of “reading course”, something to get us all on the same page of some topic.
This time the idea is to talk about higher supergeometry and “super-string geometry”, if you wish. I am preparing some notes to go with this, and naturally I got inclined to prepare them on the nLab. They will be developing here in the entry
Currently there is just an introduction and then a session outline with just a few linked keywords. I’ll be developing this as days go by. Depending on which reactions I get, there might be drastic revisions, or just incremental extension. We’ll see.
started Yetter model, still a stub so far. Tim, I trust you will add references?! :-)
New entry: Reedy category with fibrant constants.
Circumstances prompted me to write a kind of pamphlete pointing out some aspects that seem worth taking notice of have not found much appreciation yet:
This surveys how basic theorems about the standard foundation of quantum mechanics imply an accurate geometric incarnation of the “phase space in quantum mechanics” by an order-theoretic structure that combines with an algebraic structure to a ringed topos, the “Bohr topos”. While the notion of Bohr topos has been motivated by the Kochen-Specker theorem, the point here is to highlight that taking into account further theorems about the standard foundations of quantum mechanics, the notion effectively follows automatically and provides an accurate and useful description of the geometry of “quantum phase space” also in quantum field theory.
created Clebsch-Gordan coefficients
I added Sinh’s thesis plus a link to a scan to 2-group
brief entry “daseinisation”
(Note: I am not embracing the term, I just happen to want to record that somebody proposed it.)