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    • (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.

    • 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≁xx\nsim x with ¬(x∼x)\neg\, (x\sim x), and

      • I thought the standout-box-route to be the most efficient and silentest.

      Since this appears not to have worked out, this message.

      Again, it seems that ≁\nsim is nothing else than an abbreviation, definitionally-equal to ¬(x∼x)\neg\, (x\sim x). It seems to be that this should be spelled out, the ≁\nsim 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 ∼\sim, i.e., ≁\nsim 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”.

    • Two days ago, I created

      Ernst Steinitz.

      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.

    • [ 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.

    • The only content at the page Friedemann Brandt is a link which does not function any more.

    • 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.

    • 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 FF 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 −∞:=sq_{-\infty}:=s (in Power’s sense), and q ∞=tq_{\infty}=t, and FF 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.

      herein

      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 GG. Vertex q −∞q_{-\infty} is an ∞\infty-coking in GG. Vertex q ∞q_{\infty} is an ∞\infty-king in GG. Connection to A 2-Categorical Pasting Theorem, Journal of Algebra 129 (1990): therein, the author calls q −∞q_{-\infty} a “source”, and q ∞q_{\infty} 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. ]

    • (New thread since, after a semi-cursory search, no LatestChanges thread for [path] was found.)

      Added to [path] a definition of “inverse path”.

      Also tried to make the definition of “Moore path” clearer. Quibblingly speaking, this term used to be defined by saying what it has, without relating it to the initial definition of “path”. I was tempted to change the definition of “path” to the one given by tom Dieck in “Algebraic Topology”, having aa and bb for the endpoints of an artbitrary interval, which in particular would make it possible to simply say “for Moore path take a=0a=0, b=nb=n”, but then refrained, suspecting that whoever wrote it this way set store by having path to be always a space-modelled-on [0,1][0,1], which for several reasons seems more simple and systematic indeed.

      Motivation is that I try to concentrate on writing an exposition of a theorem of J.A. Power, and for this, I have resolved to use a —mildly—topological writing style, and for this I in particular need to get serious about paths, and I need Moore paths.

      [Incidentally, in the nLab there lives Moore path category which has much to do with a “Moore path” of the type that lives on path since its creation on September 16, 2011. Maybe one should harmonize the two “Moore path”s a little more, saying a few situating thins on either page. Yes, path already links to Moore path category, but, it seems, not the other way round. Nothing urgent here, though.]

      [Incidentally, I had recourse to a footnote in path. I did not forget the advice given recently, it just seemed right here to, simultaneously,

      • give a reference

      • warn readers of some notational issues

      • not clutter the main text with this

      and I found my hand forced by this. If this is inacceptable, you might even just say “make it into this or that format” and I’ll hopefully do so soon. Now back to pasting schemes.]

    • I added some discussion to the comment box at the bottom of constructive mathematics. I'd like to work those quotes in to a section called "criticism" or "opposition". Half of the reason I want to do that is so those quotes are on that page. Does anyone oppose me doing this?

    • I removed the footnote at adjunct (as just noted elsewhere, I don’t think footnotes are usually a good choice). I put a brief mention of the musical notation in the main text, put the example of currying in an “Examples” section, and the references in a “References” section. I removed the discussion about pronunciation entirely; I think there is no need to tell the reader how to pronounce mathematical notation, at least when it is fairly obvious (how else would you pronounce f ♯f^\sharp than “f-sharp”?).

    • I did some cleanup at pasting law:

      • I find itex-style commutative diagrams very hard to read if the objects are not present, so I named the objects.
      • I generally find it execrable to name anything with the letter OO, and the notation in the related propositions was unnecessarily heavy, so I changed it to match the notation in the main proposition.
      • The boldfaced “labels” of the propositions were confusing because they were in the proposition environments but were not part of the proposition statements (I think this was an accident due to multiple editors), so I replaced them by non-bolded informal discussion before the propositions.
      • I made the bullet lists into paragraphs, which I think read better when all the items are short. In general, note that bullet lists should not be started with * * as that produces two bullets at the beginning of the line.

      I would also like to rename this page to pasting law for pullbacks. I know that it’s about pushouts too, but that’s a simple dualization and we have redirects. The name “pasting law” seems overly general to me; I can imagine many different “laws” regarding many different kinds of “pasting”.

    • The next-to-latest revision of equivalence of categories had a “query” to add an “intuitively clear” example why the notion of strict isomorphism of categories is too strong to be useful. I cannot think of a better example than the category of pointed sets versus the category of partial functions. In particular since even readers who have never learned category theory are likely to have been weaned on partial functions. I have therefore started to anser to this “query” with a condensed exposition of this example. The exposition had to broken off for the time being though. I intend to finish it tomorrow, complete with a proof that the categories are not isomorphic and a brief intuitive argument why they are (to be considered) the same nevertheless.

      Comments on whether you agree to use this example appreciated.

    • Todd has some interesting thoughts on the page non-unital operad, but I couldn’t find a discussion thread for the page, so I’ll start one here.

      I was recently led to what seems to be a related perspective: there is a certain skew-monoidale MM in the monoidal bicategory Prof, with underlying object the groupoid FBFB of finite sets and bijections. MM induces a lax monoidal structure on the category Prof(1,FB)Prof(1,FB), and a monoid therein is precisely a symmetric operad, defined in the ∘ i\circ_i style used for non-unital operads (I have the impression that the ∘ i\circ_i definition should be attributed to Markl, as opposed to the May-style definition which only works in the unital case – right?). I hadn’t made the connection to the Day convolution that Todd uses.

      One thing I find intriguing about this approach is that you don’t need to construct a whole monad (using various infinite colimits in the process) in order to set up the definitions, nor do you need to introduce a substitution tensor product which might seem ad-hoc especially if you want to vary your groupoid of arities. So it’s a kind of “minimalist” approach to generalized operads. You might also be able to use a non-groupoidal category of arities and perhaps recover notions of Lawvere theory this way.

      So I was wondering – Todd, is the material at the page non-unital operad based on the existing literature, or is this something original that you put up there? Because I want to find out as much as I can about this perspective!

    • As you may have seen from watching the logs, I am beginning to write a page Introduction to Topology. This is meant to be in the style as the previous Introduction to Stable homotopy theory, but now for basic point-set topology, starting from scratch, with some motivation taken from analysis, and ending with basic covering space theory.

      I’ll be developing this during the next months. At the moment it is skeletal. Comments are most welcome.

    • I added to symmetric group in the Properties section a remark about conjugacy classes given by cycle structures, here.

      This deserves to be expanded on, but for the moment I just need a minimum to be able to refer to it from elsewhere.

    • Small quibbles at electromagnetic field - seems to be some electric and magnetic being swapped.

      Edit: try now - I accidentally copied the capitalisation from the discussion topic heading and now it is fixed

    • Created unnatural isomorphism, with references.

      A cleaner working out and linking between the concepts of

      • unnatural isomorphism (a structure which can of course also be constructed in situations where there is a natural isomorphism

      and

      • unnaturally isomorphic (a property)

      appears to be a worthwhile thing to do.

    • I redesigned cycle category, as had been requested there for some time. I'm not sure if the discussion decided whether the first definition was even correct; that discussion is now towards the bottom of the page. I also incorporated material from the erstwhile separate category of cycles.

    • I finally created Hensel’s lemma, using the formulation in Bourbaki’s Commutative Algebra III.4.3. I also want to put in the formulation as alluded to in this M.SE answer to an old question of mine, which is more geometric.

      I also want to point out that at Henselian ring it might be worth expanding (and I can do this in a few weeks) to consider Henselian couples, where one no longer considers just the maximal ideal in a local ring, but any ideal contained in the radical. (This is a generalisation of the usual result in a different direction from Bourbaki’s, and no doubt one can form the pushout of these lemmas.) This point of view is very geometric.

      There is no doubt an interesting treatment using the internal language of an appropriate topos of this circle of ideas.

    • added example of uniform Cauchy sequences of (continuous) functions with values in a complete metric space: here

      (possibly this is already, in more generality, in some other entry?)

    • I started a bare minimum at adinkra and cross-linked with dessins d’enfants.

      Adinkras were introduced as a graphical tool for classifying super multiplets. Later they were realized to also classify super Riemann surfaces in a way related to dessins d’enfants.

      I don’t really know much about this yet. Started the entry to collect some first references. Hope to expand on it later.

    • Created 201707040601 for further use in some notes on icons in 𝒱\mathcal{V}-enriched bicategories that I am writing.

    • I found myself editing the “floating table of contents” for the topic cluster of differential geoemtry a fair bit:

      Recall, this is the entry which is being included in the top right of all entries related to differential geometry, as a little pull-down menu

      The analogous floating table of conents for topology is

      The latter was developed a lot as I was writing Introduction to Topology – 1 and now it felt that the corresponding table for differential geometry was lagging behind. It can still do with more improvement, but maybe it’s looking better now.

    • I needed a way to point to the topological interval [0,1][0,1] regarded as an interval object for the use in homotopy theory. Neither the entry interval nor the entry interval object seemed specific enough for this purpose, so I created topological interval.

    • I discovered that there was no content in the entry path space, so I gave it some.

    • I’ve just edited topological concrete category to correct the claim that topological functors create limits, which is not quite true: for instance, the forgetful U:Top→SetU: \mathrm{Top} \to \mathrm{Set} fails to reflect limits because choosing a finer topology on the limit vertex yields a non-limiting cone with the same image in Set\mathrm{Set}. This is correctly reported on wikipedia and in Joy of Cats, p. 227.

      It is true that topological functors allow you to calculate limits using the image of the diagram under the functor, which is quite powerful. In Joy of Cats, a topological functor is said to “uniquely lift limits” (definition p. 227, proven p. 363). There doesn’t seem to be an nlab page for this property – I suppose it’s not much used by most category theorists.

    • added to partition of unity a paragraph on how to build Cech coboundaries using partitions of unity (but have been lazy about getting the relative signs right).

      It would be good (for me) if we could add some more about smooth partitions of unity, too, eventually.

    • The term “flow of a vector field” used to redirect to exponential map, which however is really concerned with a somewhat different concept. So I have created now a separate entry flow of a vector field.

    • at monoidal adjunction the second item says

      while the left adjoint is necessarily strong

      but should it not say

      while the left adjoint is necessarily oplax

      ?

    • created arithmetic jet space, so far only highlighting the statement that at prime pp these are X×Spec(ℤ)Spec(ℤ p)X \underset{Spec(\mathbb{Z})}{\times}Spec(\mathbb{Z}_p) (regarded so in Borger’s absolute geometry by applying the Witt ring construction (W n) *(W_n)_\ast to it).

      This is what I had hoped that the definition/characterization would be, so I am relieved. Because this is of course just the definition of synthetic differential geometry with Spec(ℤ p)Spec(\mathbb{Z}_p) regarded as the ppth abstract formal disk.

      Well, or at least this is what Buium defines. Borger instead calls (W n) *(W_n)_\ast itself already the arithmetic jet space functor. I am not sure yet if I follow that.

      I am hoping to realize the following: in ordinary differential geometry then synthetic differential infinity-groupoids is cohesive over “formal moduli problems” and here the flat modality ♭\flat is exactly the analog of the above “jet space” construction, in that it evaluates everything on formal disks. Moreover, ♭\flat canonically sits in a fracture suare together with the “cohesive rationalization” operation [Π dR(−),−][\Pi_{dR}(-),-] and hence plays exactly the role of the arithmetic fracture square, but in smooth geometry. I am hoping that Borger’s absolute geometry may be massaged into a cohesive structure over the base Et(Spec(𝔽 1))Et(Spec(\mathbb{F}_1)) that makes the cohesive fracture square reproduce the arithmetic one.

      If Borger’s absolute direct image were base change to Spec(ℤ p)Spec(\mathbb{Z}_p) followed by the Witt vector construction, then this would come really close to being true. Not sure what to make of it being just that Witt vector construction. Presently I have no real idea of what good that actually is (apart from giving any base topos for Et(Spec(Z))Et(Spec(Z)), fine, but why this one? Need to further think about it.)

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