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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
I wrote something short on Categories, Allegories – hopefully not too subjective.
This used to redirect to lax natural transformation, which doesn’t leave much room to talk about the strong case. So I created pseudonatural transformation and had things redirect there, although I didn’t actually make use yet of that room.
(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 a and b for the endpoints of an artbitrary interval, which in particular would make it possible to simply say “for Moore path take a=0, b=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], 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♯ than “f-sharp”?).
had occasion to record some minimum at Borel-Moore homology
I did some cleanup at pasting law:
* * 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”.
It seems that Peter recently deleted the example
The collection of “admissible” (“open”) morphisms in a geometry (for structured (∞,1)-toposes).
from two-out-of-three. Peter, I think this sort of edit should be announced at the forum. However, just looking at the page geometry (for structured (∞,1)-toposes) it seems to me that the deletion was correct: the definition of “geometry” only asks for 2/3 of 2-out-of-3 (closure under composition and left cancellability). Urs, is that right?
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.
Added some structure and references to Barr embedding theorem.
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 M in the monoidal bicategory Prof, with underlying object the groupoid FB of finite sets and bijections. M induces a lax monoidal structure on the category Prof(1,FB), and a monoid therein is precisely a symmetric operad, defined in the ∘i style used for non-unital operads (I have the impression that the ∘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
I have written out a detailed classical point-set proof that the fundamental group of the circle is the integers: here.
Created unnatural isomorphism, with references.
A cleaner working out and linking between the concepts of
and
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.
Moved the definition of constant functor from cone to a new page constant functor.
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.
Made a start at perfectoid field.
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.
The term “bounded function” used to redirect to “bornological space”. I have given it its own entry.
(Noticed this when beginning to write out a proof at Tietze extension theorem.)
Created 201707040601 for further use in some notes on icons in 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.
started monodromy
I needed a way to point to the topological interval [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.
Created a minimal entry for Haskell Curry.
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→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. 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.
I added a brief description of how the exotic 7-spheres are constructed at exotic smooth structure.
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
?
I created article cotopology including a redirect from cocompact space.
I separated p-derivation from Fermat quotient.
created arithmetic jet space, so far only highlighting the statement that at prime p these are X×Spec(ℤ)Spec(ℤp) (regarded so in Borger’s absolute geometry by applying the Witt ring construction (Wn)* 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(ℤp) regarded as the pth abstract formal disk.
Well, or at least this is what Buium defines. Borger instead calls (Wn)* 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 ♭ is exactly the analog of the above “jet space” construction, in that it evaluates everything on formal disks. Moreover, ♭ canonically sits in a fracture suare together with the “cohesive rationalization” operation [ΠdR(−),−] 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(𝔽1)) that makes the cohesive fracture square reproduce the arithmetic one.
If Borger’s absolute direct image were base change to Spec(ℤp) 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)), fine, but why this one? Need to further think about it.)
while writing out the proof of the fundamental product theorem in K-theory I had occasion to record that
I put a disambiguation block at the top of inverse image to point to preimage.
Created parametric right adjoint.
created diagram chasing lemmas - contents (with some information on what implies what) and included it as a floating TOC into the relevant entries.
Created twosets20170617. Contains an svg illustration of a full subcategory of Set consisting of a terminal object and a two-element set. Uses the convention that an identity arrow is labelled by its object. Intended for use in some graph-theoretical considerations from an nPOV. Sufficiently general to be possibly of use in some other nLab articles too.
Comment/question on terminology in the (?,1)-case at regular epimorphism
Created isotropy group of a topos, now that Simon’s preprint finally explains what it actually is!
I am looking for a decent account of the homotopy ring spectrum structure on KU with KU0=BU×ℤ that would be self-contained for a reader with good point-set topology background, but not involving E∞ or model category theory.
What I find in the literature is all sketchy, but maybe I am looking in the wrong places.
First, a discussion of the H-space structure on BU×ℤ in the first place I find on p. 205 (213 of 251) in A Concise Course in Algebraic Topology. But for the crucial step it there only says:
we merely affirm that, by fairly elaborate arguments, one can pass to colimits to obtain a product
Is there a reference that would spell this out?
Next, for the proof of the homotopy ring spectrum structure on KU, the idea is indicated on the first page of
James McLure, H∞-ring spectra via space-level homotopy theory (pdf)
Is there a place where this would be spelled out in some detail?
I have started at topological K-theory a section “For non-compact spaces” (here).
Created assignment and operation.
I had need to record the traditional point-set construction of cofiber sequences in more detail. Now at topological cofiber sequence.