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I notice that Zoran started a page on New Math. Perhaps we should refer to the song of that title. (and if you don’t know of it, look on Google with Lehrer new Math.)
Someone styling themselves ’the corrector’ has made quite major edits to the local-global principle as well as altering FRS-theorem on rational 2d CFT slightly. That latter entry has also been edited by ’the riddler’. My guess is that the changes need checking out, but I am not competent to do so. I checked the first persons IP number and it looked slightly suspicious. It is 115.178.250.123. The other number (185.56.137.14) is listed as being active in forum spam. The change made by the latter would be easy to correct but I will leave it for the moment as evidence of the spam.
I am starting mapping telescope. So far it has the definition and then statement and proof of the fact that the mapping telescope over the stages of a CW-complex is weakly homotopy equivalent to .
More after lunch.
I started an article bijective proof.
In a section on polynomial identities, I give a proof that polynomials in several variables are uniquely determined by their values at natural number arguments, an intuitively obvious statement if there ever was one. Without bothering to look up whether there are standard nice proofs, I cooked up a proof myself. Please let me know if you know of nicer proofs.
Edit: Having written this, it’s painfully obvious how to prove more general statements even more simply. Ah well. I still invite comments.
jotted down quickly the statement of the Schwede-Schipley classification theorem at stable model category.
I have begun making quite extensive changes to the pages concerning cubical sets.
Perhaps most significantly, I have begun trying out a somewhat different style to the usual one at the nLab. I would like the entries concerning cubical sets to be concise, ideally short, and just contain mathematics. However, I of course do not wish to remove previous work. What I have done therefore is to shunt what was before at cube category and cubical set to new pages: cube category - exposition and cubical set - exposition. The name for the latter two is maybe not the best, but I’m not sure what would be better. Any suggestions? I have then re-written cube category, now called category of cubes with a redirect, and cubical set, in the style I have in mind.
I have also created cubical truncation, skeleton, and co-skeleton and cubical Kan complex. I have also edited homotopy hypothesis for 1-types to remove material which is now present at one of the afore-mentioned entries.
Many things could be added. Here is a TODO list for the moment (excluding homotopy hypothesis for 1-types).
1) Expand upon the monoidal structure on cubical sets at cubical set. Explicit description and construction. On own page. [High priority, but will take some time, so may not get done for a while.]
2) Draw what a horn looks like in dimensions 1 and 2 at cubical set, and indicate the same for a boundary. [High priority, and quick to do.]
3) Give an explicit generators and relations description of , but on a separate page. [Lower priority. Does not take much time. I do not think we should give a proof; the only way to convince oneself that the relations are correct is to write out a proof oneself.]
I have defined a cubical horn at cubical set in a way which is slightly novel, I suppose. Because I like everything to be constructively valid, I prefer to avoid a ’removing a face from the boundary’-like definition. But this could be added as a remark to give intuition, once the pictures of a horn are added.
I am working towards making further progress on homotopy hypothesis for 1-types, but needed the notion of a cubical Kan complex.
To keep a consistent style, it is likely that I will keep a close eye on the pages concerning cubical sets, and heavily edit any deviations from the style I am beginning to put in place. If there are objections to this, let me know. I guess this is maybe the first significant example of ’re-factoring’ on the nLab, so it will be interesting to see what people think of it!
I will eventually add one of those panels with links on the right hand side which includes all pages in the new style on cubical sets, called something like ’cubical sets’.
From our discussion on sugaring, I started Einstein summation convention. That revealed that we don’t have a page for ’tensor calculus’.
The difference between the Riemann integral and the Henstock integral is analogous to the difference between a uniformly continuous function and a continuous function. I made some remarks about this at Henstock integral, along with a comment about what's needed to make the definition constructive (which is related).
started something at Bousfield-Friedlander model structure
Created club in a 2-category.
created a brief entry reduced cylinder, just for completeness.
In the course of this I also touched cylinder object and reduced suspension.
I have started adding basic comments on the cofibrant generation of the classical model structure on pointed topological spaces (also a section at classical model structure on topological spaces) and expanded slightly the relevent paragraph at coslice model structure.
To be expanded. But I need to pause now.
At enriched category it uses to say that
A Top-enriched category is a topological category.
But then at topological category, which redirects to topological concrete category it says that it
does not mean Top-enriched category.
Of course for many people it does. But to get the Lab entries straight, and to go along with the entry simplicially enriched category, I started an entry
just for completeness (and since I need the material elsewhere).
Created smothering functor, smothering 2-functor, and surjective on objects functor.
created an entry braid lemma with the statement and the application to the long exact sequence of a triple in (generalized) homology.
At 2-category equipped with proarrows in the section As a double category I have made a little change in the labelling:
there used to be a horizontal arrow labeled “”, but also the ambient 2-category (the one being equipped) is denoted “” and sometimes both symbols, or rather the same symbol with its two different meanings, appeared right next to each other.
So I have relabeled the horizontal arrow now to “”. I tried to take care to do so consistently throughout the paragraph… Hopefully you can agree with this change.
One question: a few months back we chatted vaguely about how equipment data is equivalent to the structure of an internal category in Cat in the sense at internal (infinity,1)-category. Back then I had written a quick note on this at Segal space - Examples - in 1Grpd.
I’d like to expand on that. Is there meanwhile anything in this direction in the literature?
Someone anonymously started stereotype space. How well-used is this notion?
I gave the scan that Colin MacLarty just shared on the mailing list a home on the nLab:
Saunders MacLane,
Bowdoin Summer School 1969
Notes taken by Ellis Cooper (pdf)
Presently the pdf-link points to my Dropbox folder, as I keep forgetting the system password necessary to upload a file of this size to the nLab server. Maybe Mike or Adeel have the energy to upload it.
I have created an entry free spectrum with some minimum content. Currently it just points to a more comprehensive discussion of free spectra over at Model categories of diagram spectra – Free spectra. Once the material there has stabilized more, I’ll copy the relevant bits over to the dedicated free-spectrum entry.
I have added to symmetric spectrum (after the definition in components) also discussion of the definition as -module objects with respect to Day convolution over (here).
I am really in the middle of some editing here, but need to call it quits for tonight.
created a brief entry module over a monoidal functor, just so that I would have something to link to from the discussion of module objects with respect to Day convolution.
for ease of reference, I gave retract argument its own little lemma page
I have finally filled content into the entry derived functor in homological algebra.
That entry had existed in template form for years, with the intention to eventually take up that content, but clearly I had forgotten to actually put it there after I had written it out on my own web at HAI (schreiber). Now I have copied it over.
I gave spectrification its own entry, in order to collect in one place various constructions such as 1-excisive reflection, Joyal’s parameterized sequential spectrification, as well as Lewis-May-Steinberger’s original “polemical definition”.
I wrote independent family of sets, mainly as an excuse for recording a proof that the number of ultrafilters on an infinite cardinal is .
Chenchang Zhu had been running a course titled “higher bundle theory” in Göttingen last semester. It ended up being mostly about Lie groupoids and stacks. She and her students used the relevant Lab pages as lectures notes, and they added more stuff to these Lab pages as they saw the need.
I just learned of this from Chenchang.
She had created an Lab page
which lists the Lab entries that were used and edited.
For instance the first one is Lie groupoid and Chenchang Zhu as well as some of her students added some stuff to that entry, such as this section Morphisms of Lie groupoids. Below that they added a section on Morphisms of Lie algebroids. (Maybe some of this could be reorganized a little now.)
I added a brief section on Cantor’s theorem for posets to Cantor’s theorem, which in one form says that for posets there can be no surjective poset map (taking ).
You might find it amusing to try to prove this yourself in a pleasant way. I found one proof (you can find it here), but it’s possible I was working too hard for it. :-)
I have been expanding Idea-section and References-section at smash product of spectra. (I suppose all technical detail should go to the respective entries for the various models of spectra).
Notice that this is distinct from the entry symmetric smash product of spectra. I think, or thought, it makes sense to keep these separate, but I might easily be convinced otherwise.
started some minimum at Kronecker pairing
in another entry I want to be able to point to context extension, so I created a brief entry
collected some references at model structure for n-excisive functors and added cross links.
I had not been aware before that Lydakis also got a symmetric monoidal smash on the model structure for excisive functors.
gave minimal Kan fibration some actual content. Definition, list of basic properties, brief indication of their proofs, and some comments on the broader story in the Idea-section.
Added some comments to combinatorial spectrum about their relationship to other spectra and to modern "brave new algebra" (moved up out of the query-box discussion).
I have cleaned up and then expanded a little the entry simplicial homotopy group.
(There used to be various suggestions in the entry, in the main text and in the query boxes, that the simplicial homotopy groups should be organized into an -groupoid structure. I had complained about that in the query boxes long time back, and they had been sitting there since, now I have removed them all to make the entry not look as awkward anymore. )
I changed the name of discrete space to discrete object such that it is now consistent with codiscrete object.
I wrote Cantor-Schroeder-Bernstein theorem.
created an entry multiplicative spectral sequence with essentially the notes that Sebastian Goette had recently put on MO (here and here)
added to Quillen adjunction the statement how the SSet-enriched version presents adjoint (infinity,1)-functors.
Also indicated how one shows that a left Quillen functor prserves weak equivalences between cofibrant objects, and dually
I created prestable ∞-category.
After having a window open to do edits at shelf for almost the past week, I’ve decided just to hit submit. Would like to get back to this, especially to explain the linear ordering on braids found by Dehornoy.
am beginning to add some genuine content to Ruelle zeta function (which used to just redirect to zeta function of a dynamical system which in turn is no more that a stub)
not done yet
Started a general page about beta reduction, and a stub for eta reduction but I have to run now.
I had expected that we had this entry already, but it seems we did not: Spanier-Whitehead category
Stub for semantics of a programming language.
Began cotopological localization.
I have added Zurek’s reference
which is surveying the decoherence approach to the collapse of the wave function and the einselection to interpretation of quantum mechanics.
The very first reference in the list is at a link at Perimeter institute which is outdated.
I created a stub for object of finite length to get rid of a grey link and then one for composition series.
A stub for Rees matrix semigroup.
I just added to abelian group and to compact closed category the basic but unmentioned fact that an abelian group can be seen as a discrete compact closed category.
recent activity made me feel it is time for a new context-cluster floating-table-of-contents: started:
and included it as a floating TOC into relevant entries.
(The table itself clearly should be expanded and better organized much more. But it’s maybe a start.)
I have added to (infinity,2)-sheaf a section Examples - Codomain fibration / canonical (∞,2)-sheaf with the statement that for an -topos, the -functor
is always an -sheaf with respect to the canonical topology.
(It’s the -sheaf of “unstable quasicoherent -sheaves”!)
Added the statement of Urysohn’s lemma. The stub that had been there listed only a link, which is now rotten (it was to some lecture notes), so I added a planetmath link which ought to be stable.
basic definition at Thom collaps map
have made explicit the proof that reduced excisive functors are equivalently spectra, here.
started a bare minimum at Adams category (in the process of expanding at stable homotopy category).