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I have created a table
on pairs of entries about physics that are in algebra/geometry duality to each other.
And I have included it into the relevant entries.
Entry collineation dedicated to the notions of collineation and correlation in projective geometry.
New entry combinatorial design
Someone (Alessandra Capotosti) has overwritten the Home Page! I will roll back and create a temporary page for what is there.
Later I have rolled back the Home Page to a version from August 2015. Please check if any changes since then we important. (There was a revision from anonymous coward and one other.)
Recently there’s been a slew of papers constructing new model structures for -presheaves and new Quillen equivalences between them. I added a list of all the ones I can think of to (∞,1)-presheaf, with references. A lot of gray links though.
I have created stub entries for Lambda-algebra and for Curtis algorithm (and have cross-linked with a bunch of related entries), for the moment just so as to record some references.
I created a stub about essential sublocales. I’ll polish the entry a bit more in a few hours and then link to it from other entries.
I’m not sure how to name the left adjoint to the nucleus . Provisionally I named it “”, in allusion to the flat modality. I refrained from naming it “”, since this symbool seems most often to refer to the induced action on subobjects or types.
Unfortunately I don’t have access to Kelly and Lawvere’s article On the complete lattice of essential localizations. It probably contains a few more properties of essential sublocales which I’d like to copy to the nLab entry.
added statement and proof of the (or one version of the) Serre long exact sequence of a Serre fibration with highly connected base and fibers.
Started the stub for semilinear map. More to come.
I have begun the page homotopy hypothesis for 1-types with a view to giving a proof. It will take some time before it is complete, I will be building it up gradually.
I also added a link pointing to this new page from homotopy hypothesis.
The proof that I will give has some novel aspects, such as using cubical sets, and is I guess slightly original, though it is only really a variation on the usual arguments. It has been known to me for many years.
New entry archetypal example. I am very glad to thank Prof. Joel W. Robbin, my thesis advisor, for making me aware of this useful concept in late 1990s.
I have added a few more words to CW approximation (it’s still just a brief informal entry, though)
I created n-connected map. If anybody knows sources for Propositions 2 and 4 which actually prove them it would be nice to add them.
Created display logic. (In another thread I already announced the creation of bunched logic.)
I have added statement of the basic fact that -spheres are cosets of orthogonal groups to coset space (also to n-sphere). Then I added a section “Properties – Sequences of coset spaces” with the basic statement about sequences induced from the consecutive inclusion of two subgroups, and an example involving orthogonal groups. Just basic stuff, for reference.
We have two pages radial and star domain that to me seem to be about almost the same thing, albeit the latter also contains other material on more general star-shaped regions (neighbourhoods etc), and has redirects for that title. The first was made by Andrew in 2010, the second by Todd last year. Given that we have a preference for nouns in titles, I’d rather radial was called radial set, if that is the page that is kept for that concept.
I created the page density of a subset. It will overlap a little with topics like probability measure, measure, but has a different flavour, and could be expanded to consider other families of densities, that are less overtly probabilistic.
Reference
has been added at linguistics and at logic. Among other things it studied the logic in natural languages, which is quite different from classical and mathematical logic. In trying to develop a natural theory of meaning, he is not happy with intensionality being neglected in mathematical logic, and even when intensional aspects are accounted for (like in the notion of possible worlds) he argues that they are not the true intensional aspects as human mind sees them but rather just upgrade of extensional aspects. Quite good critic of Chomskian linguistics (e.g. minimalist program) as well.
As announced in another thread, I created Hilbert system.
However, I am a bit confused about exactly how a Hilbert system formalizes mathematical practice. In particular, how does it formalize hypothetical reasoning? When I want to prove a theorem like “If and then ”, I start out by assuming and and trying to prove . I know how to formalize this in natural deduction: I start a derivation with and at the top, and when I’ve gotten to then I apply implies-intro, cross out the and , and conclude . And in a type theory or sequent calculus, I am trying to prove a hypothetical sequent , after which I apply implies-intro again to get . But in a system where the only rules are about deducing “global” theorems, how do I formalize the hypothetical-reasoning method of proving an implication?
gave Atiyah-Hirzebruch spectral sequence a minimum of an Idea-section and added a minimum paragraph with pointers to applications to D-brane charges in string theory here, also on the D-brane charge page itself here
New stubs New Math and mathematics education.
I have spent few hours to split off the entry elementary mathematics from mathematics education. Books about elementary mathematics (as well as introductions into the foundations of mathematics) are books about a particular mathematical subject (“content”) rather than about education or mathematical didactics. In particular, the books on elementary geometry can be now found in elementary mathematics. I hope this division will be useful for further development, while my choices in the distribution are not meant to be final (in particular, more refinements needed).
Page about the major free software platform for graphing geometrical pictures for usage in mathematics education, geogebra. It has also some packets fro statistics and other fields.
Corrected a large number of typos at fundamental groupoid of a cubical set and the cubical nerve of a groupoid, broke it into sections, and changed notation slightly for the content which was there before (what are now the first five sections). Then added the last three sections.
Would like eventually some more details/links in the final section, but will not have time for this tonight.
If you see any typos or other errors, please let me know, or go ahead and fix them!
Created intercategory.
Created quintet construction.
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