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Created a stub tangent bundle categories as a link target to be disambiguated from tangent categories (with a hatnote at the latter). What I’m calling “tangent bundle categories” here are usually called just “tangent categories”, but that clashes with our page tangent category, so I invented a variation. Better suggestions are welcome.
New stub order category (redirecting also the more general case of preorder category).
just for completeness, some terminology explanation at auxiliary field.
Started bornological set. Some people call it a bornological space, but that conflicts with the terminology in functional analysis which refers to a locally convex TVS with a suitable “bounded = continuous” property. I quickly wrote that uniformly continuous maps between metric spaces induce bounded maps, but I’ll recheck when I have a free moment.
Gave Nakanishi-Lautrup field a brief entry, and touched the corresponding entry antighost field.
Presently the actual technical content that should go into this entry is this example at A first idea of quantum field theory. But before I paste that into the entry, I will expand and polish it a bit more tomorrow, when I am more awake.
Needed to be able to point to contractible chain complex and discovered that we didn’t have an entry for that, so I quickly created one.
I gave Koszul complex and Idea-section and stated two key Properties in citable form (but without proof), one of them the statement that a sufficient condition for the Koszul complex to be a resolution of is that is Noetherian, the are in the Jacobson radical, and the cohomology in degree -1 vanishes.
Finally I stated the special case of this (here) where is a formal power series algebra over a field and the elements are formal power series with vanishing constant term.
(I have added the relevant facts as citable numbered examples at Noetherian ring and at Jacobson radical.)
This happens to be the case that one need in BV-formalism in field theory. I am writing this out now at A first idea of quantum field theory (here).
balanced monoidal category used to redirect to twist, which started off with a length disambiguation list before moving on to talk about balanced monoidal categories. I thought this was ugly and unhelpful, so I renamed this page to balanced monoidal category and made twist a pure disambiguation page.
I tried to impose some uniformity on the pages about various different kinds of monoidal 2-categories, listing them all at the top of monoidal 2-category and making their Idea sections all parallel. I also added some discussion of the microcosm principle (defining pseudomonoids in monoidal 2-categories of various sorts) to monoidal 2-category as well as compact closed 2-category and microcosm principle.
I tried over category and found the arrows (and a lot else) were not showing up. (On Firefox 56.0, and on Chrome and Safari)
This is after discovering that the accents on Myles Tierney were not appearing. Strangely they do appear at Alain Prouté, (perhaps they are a different font????)
I had occation to link to “formally integrable PDE” from somewhere, and so I created a stub entry just so that the link works. Also created a stub for integrable PDE and cross-linked with integrable system.
Arnold Neumaier contacted me about the previous stubbiness of the entry resurgence and pointed me to his PO comment on the topic (here). I have put that into the Idea-section of the entry, equipped with some links.
Started something at local BRST complex.
on my personal web I am starting a page derived critical locus (schreiber) with some notes.
I think so far I can convince myself of the claim that the page currently ends with (without proof). My next goal is to show that the homotopy fibers discussed there are given by BV-BRST complexes. But I have to interrupt now.
Since somebody put some nonsense at floor (as reported in Forum discussion), I put some content there.
in the course of writing those notes on A first idea of quantum field theory I occasionally find the need to reference more field theory jargon. In this vein I just created a simple entry “relativistic quantum field theory” just with some highlighting of terminology.
(Similarly I had also created locally variational quantum field theory a while back, but did not find the leisure yet to give it substantial content.)
Finally I gave Bogoliubov’s formula its own entry (with material taken from S-matrix)
I have seen filtered object also used in settings where there may not be an (or initial object) and where the filtration can be doubly infinite. In the current entry both seem not to be allowed. Is this deliberate with another term being used for the more general concept?
New entry (improvized, check co- etc.) resolution.
I couldn’t find an existing discussion for this page – I hope I’m not duplicating such an existing discussion.
I just added a bit to the introduction to clarify something that I was confused about – a cartesian bicategory basically abstracts the properties of V-Prof, but only for cartesian V. To me, this feels like an interesting intermediate position between abstracting the properties of V-Prof for general V, and abstracting the properties of a bicategory like Cat.
Of course, feel free to correct, rework, or roll back entirely my additions! Todd in particular has clearly put a lot of work into this page already.
stub for free loop space object
this is what should be mentioned in that section I added to smooth loop space, instead of the based loop space object notion. Maybe later.
I had need to point specifically to ideals in Lie algebras, so I gave them a little entry Lie ideal.
have created an entry for the new book: Higher Algebra
tried to polish and improve the floating TOC
The entry higher algebra itself I am not happy with (well, it’s mostly just a link list). But no time for that right now.
I have created an entry transgression of differential forms that discusses the concept using the topos of smooth sets. Apart from the traditional definition as the entry considers the formulation as
which simply forms the internal hom into the classifying map of a differential form. I have spelled out the proof that the two definitions are equivalent.
Then the entry contains statement and proof of the situation of “relative” transgression over manifolds with boundary. (This is what yields, when applied to Lepage forms, Lagrangian correspondences between the phase spaces with respect to different Cauchy surfaces, which is what I currently need this material for in the exposition at A first idea of quantum fields.)
Finally there are two examples, a simplistic one and an simple but interesting one related to Chern-Simons theory. These two examples I had kept for a long time already at geometry of physics – integration in the section “Transgression”. That section I have now expanded accordingly, its content now coincides with the entry transgression of differential forms.
gave retarded product its own entry, with material split off from S-matrix.
This entry is to be expanded further. A related stub is now at quantum observable on interacting fields, also to be expanded.
I also created a table-for-inclusion-into-related-entries products in pQFT – table and included it into related entries
I have added in some additional comments plus some references. In particular there is a paper by Barbara Osofsky (The subscript of projective dimension, and the vanishing of _, Bull. Amer. Math. Soc. Volume 80, Number 1 (1974), 8-26.) which gives a neat discussion.
I have added to causal complement the actual definition of causal complements of subsets of Lorentzian manifolds.
The entry used to contain only a more abstract concept, now kept as the second subsection of the Definition section here.
It is a classical fact that a formal deformation quantization of a Lie-Poisson structure is provided by the universal enveloping algebra of the corresponding Lie algebra. Remarkably, this statement generalizes to some extent to more general (polynomial) Poisson algebras. In particular it holds for every such up to degree three in ! This is due to Penkava-Vanhaecke 00.
I have added a quick summary of this theorem to deformation quantization in a new subsection: Existence – Deformation by universal enveloping algebras. I also gave this an entry on its own at polynomial Poisson algebra.
This is maybe remarkable, since there is possibly no physical measurement known which could detect contributions of higher than third order in . Though I’d need to check. This is subtle because order in is different from the usual loop order that is commonly stated (which is order in the coupling constant) and the relation between the two is complicated.)
Also (and that’s how I came across this article) at least in special cases this gives a way to quantize just by universal constructions on Lie algebras, hence this might potentially tell us something about the quantization of Poisson bracket Lie n-algebras (for which no analog of the corresponding Poisson algebra, i.e. with an associative product around, is known).
expansion, a stub containing just the basic definition of an expansion in shape theory.
I have started an entry on the Mittag-Leffler condition.
There is a small error in the current proof that the category of endofunctors on a Q-category is a Q-category. I am going to correct it as soon as I find my way through the notation (I used it different on the paper). It now reads
The -unit is the dual of the original counit
and the counit is the dual of the original unit
The wrong thing is that , not and that is why the unit and counit got interchanged; they should not get interchanged, but and should. I am going to sort this out. Thus where is unit goes .
Edit: the correct version is now below.
Added Baez-Hoffnung reference to quasitopos, as well as the result that concrete sheaves on a concrete site for a quasitopos
There used to be a stub entry Lagrangian quantum field theory. I have now given it a bit more of an Idea-section.
I have added a brief entry causal locality in order to have somethign to point to when discussing the corresponding axiom at local nets of observables (the latter entry I am about to brush up and expand)
I have expanded the Idea-section at constructive quantum field theory.
I am in the process of brushing up and expanding the entry local net of observables.
I have entirely reorganized the Definition-section: it starts now with the general basic definition on arbitrary spacetimes, then gives a list of all the add-on axioms that people consider.
I also finally noticed that this page coexisted with a synonymous entry that I have now moved to causal net of algebras > history, making “local nets” redirect to “causal net”. I made sure to move all the information over.
I am not quite done yet, will continue to add stuff. Maybe a little later.
I have given causal additivity a little stand-alone entry. (This used to be hidden at S-matrix, but in there it is hard to point to usefully.)
I gave Factorization algebras in perturbative quantum field theory its own Reference-entry
Our pages on 2-adjunctions are a mess: lax 2-adjunction doesn’t link to biadjunction, and pseudoadjunction is a stub with a few references and no indication of how it’s related to a biadjunction. I’ll fix this if no one beats me to it, but I actually don’t recall; is a pseudoadjunction just another name for a biadjunction?
Created loop order (in the sense used in pQFT) and added a brief explanation of how the loop number of planar Feynman diagrams translates to the order of their contribution in .
I needed to explicitly point to Euler’s formula for planar graphs. So I gave it a stub entry of its own and added a corresponding remark and pointer below prop. 2.6 at Euler characteristic.
I’ve finally created the page Inj to record some facts and give something to link to, though I don’t fully understand what I’ve written so it needs to be checked.
What is the reflector ?
I have yet to add some tags and links in quasitopos. See Sandbox/1054.
had been discussed in nForum: power set & Inj though I haven’t reread all the comments there yet.
I haven’t yet made Surj which at least appears in partition.
has been somewhat discussed in nForum:constant functor
There were two pages for the same book - EKMM and Rings, modules and algebras in stable homotopy theory. I merged the former to the latter, since the latter has a more descriptive title.
I have started to add some of the basic definitions and facts to Schwartz space, tempered distribution and Fourier transform of distributions.
Notice that we had an entry titled “Schwartz space” already since May 2013 (rev 1 by Andrew Stacey) which considered not spaces of smooth functions with rapidly decreasing derivatives, but locally convex TVSs “with the property that whenever is an absolutely convex neighbourhood of then it contains another, say , such that maps to a precompact set in the normed vector space .”
I had not been aware of this use of “Schwartz space” before, and Andrew gave neither reference nor discussion of the evident question, whether “the” Schwartz space is “a” Schwartz space. In June 2015 somebody saw our entry and shared his confusion about this point on Maths.SE here, with no reply so far.
I see that this other use of “Schwartz space” appears in Terzioglu 69 (web) where it is attributed to Grothendieck.
Max came across some baroque linking to sections.
HowTo#how_to_make_links_to_subsections_of_a_page gives maybe old advice that may be partially responsible. My improvements should probably be checked..
I’ve added the section relation#the_quasitopos_of_endorelations
I was unsure whether to add this to relation or Quiv and somewhere we should explicitly give the subobject classifier of Quiv.
Do people dislike my terminology or approach? Does have slick sub categories?
Does it need a translation such as found at quality type#quality_types_as_localizations example 4.3
Let be the category of sets equipped with a binary relation i.e. objects are pairs with a set and a binary relation on and morphims are functions such that implies . This is the same as the category of simple directed graphs hence a quasitopos since it corresponds to the separated objects for the double negation topology on the directed graphs.
Finally created vacuum state as a glorified redirect to Hadamard state (for reasons explained).
I accordingly expanded the Idea-section at Hadamard state and at vacuum.
The keyword derivative used to redirect to “differentiable map”. I found that less than useful for many purposes of linking to it, and so I have now split it off as a stand-alone entry. Presently this contains nothing than pointers to other entries. But it is already useful to see how many entries on variants of “derivative” we have, and to have a place to collect them all.
At line integral I have added missing pointers to Cauchy integral theorem and Cauchy integral formula.
I wonder if we should better split off an entry contour integral from line integral (presently the former redirects to the latter). That’s what Wikipedia does, too. At least if we ever bring a decent chunk of material on complex contour integrals in.
We had had no entry for closed differential form/exact differential form. I have created one now.
(The next one of us who teaches differential geometry should use that occasion to boost our basic entries on differential forms to something more decent.)
I moved the discussion which I had added under “General context” on spectrum to the page spectrum object under “In an ordinary category”.
After adding something about model structures, I guess one can add a comment like: if an (infinity,1)-category is presented by a model category , then the stable (infinity,1)-category of spectrum objects in it is presented by projective/injective model structures on the category of spectrum objects in .
Also, I guess I should move the stuff about the (Sus, Ev) adjunction to the pages suspension functor and loop functor.
I’ve added the following definition to power set
I’ve seen appear in discussions (as as a simple thing everybody knows) but sometimes there iis confusion about its properties. Could it use its own page?
(as usual I may be confused/misguided here)
I created a new entry scalar field on the notion of physical field with a brief bit of text.
I removed the redirect of that to ground ring. It seems to me one says maybe “field of scalars” for “ground ring”, but not “scalar field”. (Wikipedia agrees with me, for what it’s worth.) But I left disambiguations, so no harm is done either way.
New entry books in algebraic geometry and cleaning up queries (and something more) in algebraic geometry.
After sitting on this for days and hardly doing a thing, I added some applications to distribution and added a bit to the section on synthetic differential geometry. While I was dawdling, Andrew Stacey stepped in and added to some parts that needed expert attention -- thanks, Andrew.