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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 E “with the property that whenever U is an absolutely convex neighbourhood of 0 then it contains another, say V, such that U maps to a precompact set in the normed vector space EV.”
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 EndoRel have slick sub categories?
Does it need a translation such as found at quality type#quality_types_as_localizations example 4.3
Let Bin be the category of sets equipped with a binary relation i.e. objects are pairs (X,ρ) with X a set and ρ a binary relation on X and morphims (X1,ρ1)→(X2,ρ2) are functions f:X1→X2 such that xρ1y implies f(x)ρ2f(y). 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 C is presented by a model category M, then the stable (infinity,1)-category Spt(C) of spectrum objects in it is presented by projective/injective model structures on the category Spt(M) of spectrum objects in M.
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 Inj 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.
There have been some additions at nLab which ought to be reported at the nForum, including a new page equivariantization. I would recommend that changes at the level of coreduced object, rev 3 and module category, rev 6 be noted here too.
created univalence axiom
I have split off formally etale morphism from etale morphism. Then I added the general-abstract topos theoretic characterization, essentially by copy-and-pasting the discussion from formally smooth morphism (and replacing epimorphisms by isomorphisms)
Stub for stochastic Loewner equation relevant to conformal field theory, percolation, self-avoiding walks etc.
Created sampling theory, for the moment just recording some references of my interest.
We should also have Zak transform soon.
On the article augmented simplicial set, an augmented simplicial set was defined as a presheaf on the full subcategory Δ+ of Cat consisting of free categories over finite linear directed graphs. This is fairly convoluted, so I’ve added a simpler description of a skeleton of this category. My main point, though, is that this category is denoted Δ+.
On the article semi-simplicial set, a semi-simplicial set is defined as a presheaf on the category of finite linearly ordered sets and injective order-preserving maps. This is also denoted Δ+.
This is a bit unfortunate. On augmented simplicial set the alternative notation Δa is suggested for the category on which augmented simplicial sets are presheaves. So, one possible solution is to change the notation to this. However, I suspect that augmented simplicial sets are used more commonly than semi-simplicial sets, at least on the nLab, so this might cause more damage. Can someone fix things someday?
The reason I bring this up is that I’d like to write a bit about augmented semi-simplicial sets, but right now I can’t, due to notational conflicts.
On the basis of recent discussions, I started indexed (∞,1)-category. Should there be separate pages for different kinds of fiber: monoidal, symmetric, closed,…?
I thought it was silly to have indexed functor without indexed category, so I remedied the situation.
I started finite ∞-group, and added that same reference to Sylow p-subgroup.
Created extranatural transformation by moving the relevant information from dinatural transformation and adding the definition. Disagreements are welcome, but I feel that since dinaturals that aren't extranatural are so rare and harder to deal with and understand, extranaturals merit their own page.
I have been adding some more (historical) references to the entry quantum electrodynamics (also at quantum field theory, S-matrix and causal propagator)
have been writing some Idea-section at causal perturbation theory
(currently this has much overlap with locally covariant perturbative quantum field theory, eventially the latter will contain more stuff)
Stephan Alexander Spahn has created descent object, with some definitions from Street’s Categorical and combinatorial aspects of descent theory.
If I get the opportunity this weekend I’ll add details from Street’s Correction to ’Fibrations in bicategories’ and Lack’s Codescent objects and coherence. Anyone know of any other references?
Looking at Street’s paper again, what he describes as the ’n=0’ case of codescent objects looks to be just the notion of a coequalizer. I would have expected reflexive coequalizers, though, because the higher-n case uses n+2-truncated simplicial objects. Is there a reason for this?
[New thread because, although it existed since 2012, pasting scheme appears not to have had a LatestChanges thread]
Started to expand pasting schemes. Intend to do more on this soon, in an integrated fashion with digraph and planar graph.
PLEASE note: ACCIDENTALLY a page pasting schemes was created too, as a result of some arcane issues with pluralized names of pages-still-empty. Please delete pasting schemes.
created a stub for almost Kähler geometric quantization, for the moment just so as to collect some pointers.
created a stub for pseudoholomorphic vector bundle, for the moment just to record references
created a minimum at tensor product of distributions
I have given interaction picture genuine content (the entry used to be effectively empty):
gave it one section “In quantum mechanics” with the standard kind of material going from interacting Hamiltonians to the definition of the S-matrix, and then a section “In quantum field theory” with an outline of which steps in the previous discussion require special technical care and how.
In the process I expanded the entry Dyson formula. (In the end I effectively rewrote it, but now with a little broader perspective and more pointers).
at distribution there used to be a mentioning of “Colombeau algebras”. I have now removed that paragraph there, and have given it its stand-alone stub entry Colombeau algebra, expanding it slightly.
An expert might want to check. I haven’t actually looked into Colombeau algebras beyond a scanning of a review, and presently I don’t plan to delve into the topic. In fact their idea looks misguided to me.
All I mean to do here is to clean up the structure of the entry distribution (see also my comments in the thread on products of distibutions, here) while preserving what others had written before.
created point-supported distribution
created derivative of a distribution
started a minimum at main theorem of perturbative renormalization theory
[new thread since vertex seems not to have had one]]
To comply with
With few exceptions, all edits to the nLab (either the creation of a new page or the revision of an extant one) should be announced at the nForum, in the “Latest Changes” category.
and with
The only real exceptions are very minor edits such as correction of spelling mistakes or obvious typos or indisputable grammatical errors. However, because of this rule there can at times be a large volume of Latest Changes posts; thus a corollary is that Latest Changes posts at the forum should generally be kept very short and to the point. They should also include a link to the nLab page in question (links at the nForum are created with the same syntax as on the nLab itself).
in the rather new writing in the nLab I think I have to announce that a few days ago I added terminological comments to the pre-existing vertex.
I am working on Lie infinity-algebroid.
So far I have completely reworked the old Idea- and Definition-section to one new Idea-section. More to come.
(new thread since edge seems not to have had one)
To comply with
With few exceptions, all edits to the nLab (either the creation of a new page or the revision of an extant one) should be announced at the nForum, in the “Latest Changes” category.
and with
The only real exceptions are very minor edits such as correction of spelling mistakes or obvious typos or indisputable grammatical errors. However, because of this rule there can at times be a large volume of Latest Changes posts; thus a corollary is that Latest Changes posts at the forum should generally be kept very short and to the point. They should also include a link to the nLab page in question (links at the nForum are created with the same syntax as on the nLab itself).
in the rather new writing in the nLab I only now realized that it seems I have to announce that a few days ago I added terminological comments to the pre-existing edge.
This word is often discussed, in particular since it is so geometric-sounding, while graphs nowadays are often considered purely combinatorially.
Reason for the additions was something interelated with working on the nLab coverage of directed graphs, and a consequence was creating Ernst Steinitz.
I discovered that we had a one-line stub entry Klein-Gordon equation. Have given it some first minimum of content now.
I gave Fedosov deformation quantization its own entry, so far with an Idea-section putting the construction in perspective, an informal outline of how the method proceeds, and some references.
I put in a proof sketch (cribbed from Wikipedia) for the Picard–Lindelöf theorem at ordinary differential equation.
An article writing in the nLab was recently created, to give some guidelines which may help relative newcomers fit in with some of the unspoken norms that have developed at the nLab.
[new thread since “irreflexive relation” was not found among the LatestChanges threads]
A few day ago I added a standout box to irreflexive relation suggesting clarifying a notation.
I did this since there seems to be something to be clarified, but there is, as far as I can tell, nothing more to do than replace x≁x with ¬(x∼x), and
Since this appears not to have worked out, this message.
Again, it seems that ≁ is nothing else than an abbreviation, definitionally-equal to ¬(x∼x). It seems to be that this should be spelled out, the ≁ not being defined anywhere (definitely not on the page itself, and I looked around a bit), and it is at this point most probably not meant to denote an apartness-relation, distinct from the relation ∼, i.e., ≁ is not a relation symbol, in other words, not part of the syntax, rather part of a meta-syntax.
expanded the Idea- and the Definition section at G2-manifold (also further at G2). (Still not really complete, though.) Highlighted the relation to 2-plectic geometry and cross-linked there.
Created digraph. Some background: this discussion. Created with permission, in the sense of
If you really want to split off material that is pertinent to digraphs in the graph-theorist’s sense, then I myself would have no objection to a new article “digraph”.
created electric-magnetic duality
(and dual heterotic string theory as an example)
Two days ago, I created
One reason was (I keep this notification short) something like
I am working, for pasting schemes, on plane digraphs -> working with embeddings becomes important -> a central theorem about plane graphs is of course Whitney’s theorem about unique embeddability of 3-connected planar graphs -> arguably the most well-known, but heretofore not nLab-documented theorem about planar 3-connected graphs is a theorem of Steinitz’s.
Created double pushout rewriting and pushout complement, and added a relevant lemma to adhesive category.
[ new thread since this page appears not to have had any ]
Added two references, and two quotations, one of which I cannot substantiate except for assuring you that I have a distinct recollection of it from a seminar at Hamburg.
Part of the reason for doing so was that I rather naturally stumbled upon this pre-existing nLab page, and that today I did sort-of-a-memory-exercise, during which I remembered several quotes, among them, the newly added two.
Another thing that came to mind: of course, it is not for me to make recommendations on how to write what nLab page, but just as a useful recommendation or guideline-suggestion: one way to prevent this page on Tutte becoming more or less indistinguishable from e.g. his Wikipedia page, and rather make in interestingly focused, would be to see to it that
Tutte’s crypto-work is left out of the page entirely,
Tutte’s graph-theoretic work is not made the main focus of the page,
Tutte’s homotopical and matroid-theoretic work is emphasized and documented.
I gave support of a distribution its own entry
The only content at the page Friedemann Brandt is a link which does not function any more.
(Created Phillipe Flajolet Thanks for the edits you made already.)
Changes-note. Changed the already existing page 201707071634 to now contain a different svg illustration, planned to be used in an integrated way in pasting schemes soon.
Metadata. Like here, except that in 201707071634 symbols (arrows) indicating what is to be interpreted to 2-cells are given, in the same direction as in Power’s paper.
Changes-note. Changed the already existing page 201707071626 to now contain a different svg illustration, planned to be used in pasting schemes soon.
Metadata. What 201707051600 is: relevant material to create an nLab article on pasting schemes. This is (a labelling of) the (plane diagram underlying the) pasting diagram A. J. Power gives as an example in his proof of his pasting theorem herein.
Unlike there, the 2-cells are not indicated in 201707051600.
Related concepts: pasting diagram, pasting scheme, digraph, planar graph, higher category theory.
Changed 201707051620.
Metadata. by-and-large, cf. this thread. A difference to 201707051600 is that here what A 2-Categorical Pasting Theorem, Journal of Algebra 129 (1990) calls a “boundary of the face F” is indicated by bold arrows.
EDIT: (proof of necessity of hypothesis in [A 2-Categorical Pasting Theorem, Journal of Algebra 129 (1990] and relevance to 201707051600 temporarily removed, to make it more uniform)
EDIT:
Changes note. Changed the already existing page 201707040601 to contain an svg illustration relevant to pasting scheme and [this thread]
Meta data. cf. [this thread]; difference is that in 201707040601 a face F of the plane digraph is named and one of the two orientations of the euclidean plane is indicated by a circular gray arrows. A connection to [Power’s proof] can be seen by letting q−∞:=s (in Power’s sense), and q∞=t, and F the “F” in Power’s paper.
OLD, bug-related discussion:
For some reason unknown to me, the “discussion” (actually, it is merely meant to be the obligatory “log what you do” entry), the discussion with name ‘201707040601’ that I started seems to have technical problems: the comment I entered is not displayed (to me). I would delete it, but apparently it is not possible to delete “discussions” one has started. Please do with it whatever seems most appropriate.
Changes-note. Changed the already existing page 201707051600 I created, to now contain another svg illustration, planned to be used in pasting schemes soon. Sort-of-a-permission for this is
Power’s proof of (I guess you mean) his pasting theorem would probably be very handy to have discussed at the nLab. It would seem to fit at one of pasting diagram or pasting scheme, but less well at an article on some notion of graph I think. If you could even just write down the precise definitions of these various notions, that would also be very fine in my opinion.
End of changes-note
Metadata. What 201707051600 is: relevant material to create an nLab article on pasting schemes. More specifically: to document A. J. Power’s proof of one of the rigorous formalizations of the notational practice of pasting diagrams. 201707051600 shows a plane digraph G. Vertex q−∞ is an ∞-coking in G. Vertex q∞ is an ∞-king in G. Connection to A 2-Categorical Pasting Theorem, Journal of Algebra 129 (1990): therein, the author calls q−∞ a “source”, and q∞ a “sink”. This is fine but not in tune with contemporary (digraph-theoretic) terminology, whereas “king” and “coking” are. These technical digraph-theoretic terms will be defined in digraph.
Related concepts: pasting diagram, pasting scheme, digraph, planar graph, higher category theory.
[ Some additional explanation: it was bad practice of me, partly excusable by the apparent LatestChanges-thread-starting-with-a-numeral-make-that-thread-invisible-forum-software-bug, to have created this page without notification and having it left unused for so long. Within reason, every illustration one publishes should be taken seriously, and documented. Much can be read on this of course, one useful reference for mathematicians is the TikZ&PGF manual, Version 3.0.0, Chapter 7, Guidelines on Graphics. My intentions were well-meant, in particular to improve the documentation of monoidal-enriched bicategories on the nLab. This is still work in progress, but to get the digraph/pasting scheme project under way is more urgent. Will re-use the 201707* named pages for this purpose, for tidiness. ]
noticed that neither the entry intuitionistic mathematics nor the entry constructive mathematics even mentioned the entries type theory or setoid, for instance. I have now added these bare terms in the list of “related entries”, but maybe somebody feels like adding a little paragraph to these entries to do the issue justice?
created Tate diagonal