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Wrote a quick note at centrally extended groupoid and interlinked a little, for the moment just motivated by having the link point somwhere.
am starting foliation of a Lie algebroid
stub for double Lie algebroid
as mentioned in another thread, I have expanded the Idea-section at polarization in order to highlight the relation to canonical momenta (which I also edited accordingly).
I keep making links to positive number, so now I filled them.
felt like making a terminological note on phase and phase space in physics (and linked to it from the relevant entries).
If anyone has more information on the historical origin of the term “phase space”, please let me know.
started a dismabiguation page for phase. Feel invited to add further meanings.
Just in case you see me editing in the Recently Revised list and are wondering:
I have created and have started to fill some content into semiclassical state. But I am not done yet and the entry is not in good shape yet. So don’t look at yet it unless in a mood for fiddling and editing.
I started an entry classical-to-quantum notions - table for inclusion in “Related concepts”-sections in the relevant entries.
This is meant to clean up the existing such “Related concepts”-lists. But I am not done yet with the cleaning-up…
New entry semiclassical approximation. It requires a careful choice of references. The ones at the wikipedia article are catastrophically particular, 1-dimensional, old and non-geometric and hide the story more than reveal. Stub Maslov index containing the main references for Maslov index.
I created Galois topos following Dubuc’s article.
But I must be missing something about the notation: does it really mean to say that is an -torsor, as opposed to saying that it is associated to an -torsor?
I have added the relations
coisotropic submanifold Lagrangian submanifold in Poisson Lie algebroid
Dirac structure Lagrangian submanifold in Courant Lie 2-algebroid
to Lagrangian submanifold and cross-linked with various related entries, such as polarization.
There have been two empty pages created lately, both anonymous. They are at Riemann sphere and quasi inverse. It looks as if both were attempts to add something that was aborted.
This page, wall crossing in Aarhus, refers (in the future tense) to a course in 2010. The webpage link is broken as well. Does anyone have a link that could replace that one?
A stub, so that I could link to it from the Café. Redirects from finitely presented object. (Is there any connection with finitely presentable object?) The general definition may not be the best, please check!
Thought I would flag up that there have been two of these lately, ideal in semigroups and liars paradox. I waited to see if their ‘authors’ were going to come back and correct them, but so far they have not.
The entry Mochizuki's proof of abc is non-standard in form but has been updated by someone called Daniel.
created Poisson tensor just for completeness, to be able to point to it from related entries.
created a simple table (co)isotropic subspaces - table for inclusion in other entries, just so as to usefully cross-link the relevant entries
Added a little to the Idea-section of holonomy groupoid. But this deserves to be further expanded upon.
am starting model structure on dg-algebras over an operad
I created a stub for discrete event system as the grey link on tropical semirings was annoying me!
Mentions of the category occur all over the nLab, but with quite a bit of plasticity of meaning. I thought it might be good to have another look at the entry Set and try to describe this plasticity as considered along various axes, to help readers who might be puzzled by “just what does the nLab think the category of sets is?” For example, one reads that the category of sets has marvelous properties such as being a well-pointed topos, and then a little further down one sees that is not a topos according to predicative mathematics. This could be very confusing. Similarly, there are some pages in the nLab that assume satisfies AC without batting an eye, while others discuss arcane weaker choice principles that might satisfy. I think we need to be a just a bit more up-front about this, right on the page Set.
In the definition section on Set, I made a meager start on this by declaring that the nLab adopts a ’pluralist’ position on the matter of sets and , and jotted down a few of the possible axes (“axises”, if I were James Dolan) of meaning and interpretation that guide how one thinks of , e.g., predicative vs. impredicative, classical vs. intuitionist, selection of choice principles, and others. I didn’t think really hard about this, but it might suggest useful ways of organizing the page.
I left out other axes such as “structural vs. material”, and said nothing about type theory. The page set talked more about this; I envision Set as concentrating more on properties of the category of sets.
I got to thinking about this when I began to wonder how Toby thinks about , which is maybe different from how I usually think about it. (Usually it feels slightly alien to me to posit say WISC as a possible choice principle for the category of sets, which for me usually connotes a model of ETCS – normally I’d think of WISC instead as a possible axiom for a topos or a pretopos.) I was wondering whether Toby had a kind of “bottom line” for , say for example “ for me means at least a well-pointed topos with NNO, unless I choose to adopt a predicative mode”, or something like that. Anyway, discussion is invited.
After a few days’ editing, I’m announcing absolute differential form (using my neologism). This is a notion of differential form that can be integrated on a completely unoriented submanifold. Examples from classical differential geometry include the arclength element on a Riemannian manifold and on the complex plane.
Since there are classical examples, people must have thought about these before me, but I have never heard of them. Absolute differential forms are not linear (although they must satisfy a restricted linearity condition), and many typical examples are not smooth (although they are still continuous), so they don’t show up in the usual classification theorems. Has anybody heard of them before?
created brief remark at Kostant-Souriau extension
(beware the hyphen bug, sometimes only Kostant Souriau extension will work, not sure why and when)
(the hyphen bug combined with the cache bug combined with the low responsiveness make for a special experience…)
New stubs N-complex (the homological algebra where with ) and Michel Dubois-Violette. This interested me somewhat over a decade ago. Unfortunately, I missed the seminar talk yesterday in Zagreb by one of my colleagues, Pavle Pandžić, who found with his collaborator, very recently, that a more general and more insightful redefinition of Dirac cohomology, suggested by concrete applications in representation theory, involves the homological algebra of -complexes. I hope there will be some writeup soon available.
I made linearly independent subset to satisfy a link. I put in something about the free-forgetful adjunction and something else about constructive mathematics.
expanded a bit the discussion of morphisms of sites at site
There was a parity error at n-group; I fixed that and put in the low-dimensional examples.
I have created a diambiguation entry at Artin-Mazur codiagonal, as the old links at bisimplicial set lead to the entry on the codiagonal of a coproduct. I have used total simplicial set as the preferred term. Perhaps a more detailed discussion of this might be useful, but I have not got the time at the moment. (I am very slow at doing diagrams, :-( )
Wrote generic proof with some comments about a couple seemingly weaker versions of the axiom of choice that I've never seen mentioned anywhere before (has anyone else?). Toby and I noticed these a little bit ago while thinking about exact completions, but I just now realized that they're actually good for something: proving that the category of anafunctors between two small categories is essentially small (in the "projective" way).
I wrote eventuality filter, although maybe this was unnecessary, and as it was mostly already there at net. Then I took some of the logic from there and adapted it to null set.
created free field theory with the formalization in terms of BV-complexes by Costello-Gwilliam.
Does anyone know if we have a discussion, somewhere, of the theorem of Thomason linking homotopy colimits with Grothendieck constructions. I have looked in places that I thought were likely but found no trace of it, but sometimes things get buried in entries on other topics so are difficult to find.
New entry critics of string theory to collect the references on controversies. I think they are often rambling and vague, not technically useful s the main references we want to collect under string theory and books in string theory. I have changed the sentence in string theory about mathematical definition of parts to somewhat more precise
But every now and then some aspect of string theory, some mathematical gadget or consequence found there is isolated and redefined independently and mathematically rigorously, retaining many features originally predicted.
The point is that most often one does not make rigorous the way some thing is defined via string theory, but one isolates an invariant of manifolds for example and defines a similar one via completely different foundations. The typical example is quantum cohomology which is defined in geometric terms and not in terms of field theory any more.
I have one disagreement with the entry: it says that topological quantum field theory has been discovered as part of string theory research, This is not true, TQFTs were found in 1977, 1978, 1980 articles of Albert Schwartz which had nothing to do with string theory. Only much later Atiyah’s formulation is influenced by string theory.
I finally gave The convenient setting of global analysis a category: reference-entry. Started adding pointers to it from the References-section of some relevant entries. But there will be many more left.
I moved some discussion from bicategory to weak enrichment, a new page. (Possibly it was already moved somewhere else, since Mike had already deleted it, but I couldn't find it.)
some basic definitions at
operad for modules over an algebra and operad for bimodules over algebras
created
(but did we have this already as an entry under some other name?)
Hm, the apostrophe in the page title comes out in unicode, I didn’t create it that way…. And strange things happen now when linking to it. There is a link to this page at simplex category for instance which works just fine. But something tends to go wrong…
I have started to extract some of the relevant key steps from Higher Algebra into tensor product of infinity-modules.
For me that currently mainly serves as an index for how to find those needles in the haystack. But eventually I should turn it into a more comprehensive discussion.
(Some of this used to be over at bilinear map, but I have now moved it).
Just some definitions from Higher Algebra:
with some pointers to infinity-algebra over an (infinity,1)-operad, etc. Also to microcosm principle (more on that in a moment).
added to the Properties-section of reflective (infinity,1)-subcategory the statement and detailed proof of the fact that reflective (oo,1)-subcategories are precisely the full subcategories on local objects.
This proof is really not specific to (oo,1)-categories and parallels a corresponding proof for 1-categories essentially verbatim. A similar 1-categorical proof I had once typed into geometric embedding. I should really copy either one of these versions to reflective subcategory.
I just learned about rigidification and decided to record it somewhere.
I’m not sure if the title is good, because there is the notion of the rigidification of quasi-categories.
Surely this notion has a higher analogue that maybe someone knows more about. Surely you could take an -stack and consider the -categorical fiber product to make a notion of inertia, and then rigidify with respect to some subgroup object inside…
Added to reduced scheme a characterization of reducedness by the internal language of the corresponding sheaf topos: A scheme is reduced iff its structure ring is a residue field in the internal sense of .
created a table
infinity-CS theory for binary non-degenerate invariant polynomial - table
adapted from
and included it into the relevant entries
I filled a general-abstract definition into Lie differentiation. Mainly I took the key points from the beginning of Formal moduli problems and reviewed them a notation somewhat more streamlined to Lie-theoretic reasoning. Then I added an indicaton of how differential cohesion fits in. More should be added to the entry.
I’ll see how much time and energy I have left.
Just in case you see this in the Recently Revised-announcement and are wondering:
I was beginning to extract the key steps in the construction of the -category of -algebras and -bimodules internal to a suitable monoidal -category that is in section 4.3 of Higher Algebra.
I have strated to make some notes in this direction at bimodule – Properties – (∞,2)-category of bimodules and at bilinear map – For ∞-modules.
But this is taking more work than I thought and I need to postpone this until next week (and change my plans for our seminar tomorrow…). Therefore for the moment this material sits there “under construction”. Please take that into account if you look at it at all.
(On the other hand, if anyone feels like lending a hand in completing this, I’d sure be happy about it. I’ll come back to this later this week).
I am constructing a table
structure on algebras and their module categories - table
and am including it into the relevant entries. This is a bit experimental for the moment. More details and variants should be added and maybe some of the relations stated in a better way. Help is appreciated.
I find the concept-formation for 2-rings in
particularly clear-sighted. Among other things it improves on the rationale for considering associative algebras as 2-modules/2-vector spaces and sesquialgebras as 2-rings/3-modules/3-vector spaces.
Where Baez-Dolan defined a “2-rig” to be a compatibly monoidal cocomplete category, theses authors observe that one should require a bit more and define a 2-ring to be a compatibly monoidal presentable category. (This follows Jacob Lurie’s discussion, some of which is alluded to at Pr(infinity,1)Cat).
I have now written out some of the basic definitions and statements at 2-ring in a new subsection Compatibly monoidal presentable categories. I also re-organized the full Definition section a bit, adding a lead-in discussion.
I added some material to Puiseux series, notable the proof that for algebraically closed of characteristic zero, they form the algebraic closure of the field of Laurent series . This is to be connected with a number of unwritten topics like Hensel’s lemma, Newton polygon, complete local ring, and others.
Meanwhile, I noticed that the term “local field” has, besides physics meanings, two closely related distinct mathematical meanings. One for which we have a page local field is (non-discrete) “locally compact Hausdorff topological field”, but another is “field of fractions of a complete DVR”. It’s somewhat strange that two such closely related but distinct concepts have the same name – a terrible source of confusion.
I found we needed an entry 2Mod such as to be able to say things like “a sesquialgebra is an algebra internal to ”.
So I started something.
started sesquialgebra
(It’s about time to add some material on how these are 3-modules/3-vector spaces. )
unmotivated stub for Henselian ring
created Gaussian probability distribution, just for completeness