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I have started a page localization of abelian groups with a bunch of facts and their proofs.
Expanded G-set to talk about topological groups as well.
Split off p-adic integer from p-adic number.
I’ve started a page category of G sets. I will continue to fill in more details as I have time. Some of the information from ZFA can probably be moved here.
I created the article motivic coarse spectrum.
I created the article motivic coarse space.
I created the article bornological coarse space.
added the statement that every abelian group admits a free resolution of length 2, here.
The page constant morphism says
As with Set, any morphism which factors through a terminal object is constant but although this is an “if and only if” in Set it need not be in a general category.
I think the two are equivalent in any category with a terminal object, as explained here. That is, a morphism is constant if and only if the natural transformation factors through the terminal object in the presheaf category, so that if the latter is representable then so is the factorization of .
Have I missed something? If not I’ll edit the page.
brief paragraph at Dolbeault cohomology
I have added the definition of the incarnation of EM-spectra as symmetric/orthogonal spectra (here)
Added to Eilenberg-MacLane space a section Cohomology – Of EM spaces with a minimum of pointers and the statement of the rational cohomology ring of .
The recently added link to the stack project pdf file does not work it seems.
I have spelled out the derivation of the Gysin sequence from the Serre spectral sequence at Gysin sequence .
I added to principle of equivalence the following quote:
Either a thing has properties that nothing else has, in which case we can immediately use a description to distinguish it from the others and refer to it; or, on the other hand, there are several things that have the whole set of their properties in common, in which case it is quite impossible to indicate one of them. For if there is nothing to distinguish a thing, I cannot distinguish it, since otherwise it would be distinguished after all. (Tractatus §2.02331)
and a suggestive comment after it about considering such a statement for any given language, rather than the global setting stated in §1 (’the world’).
In the language given by the internal logic of a category one can never distinguish objects that share all their properties! My thesis is that ideas such as the internal language of a category and the Yoneda lemma have precursors in Tractatus, but I’ve not had time to sit down and nut out the details. Others have written about identifying the logic of the Tractatus (eg Potter’s The logic of the Tractatus, Weiss’ Logic in the Tractatus I: Definability, but I haven’t done a thorough search). I haven’t added these comments to the nLab anywhere, but I hope to do so when I flesh out my arguments.
In the process, I added to Tractatus Logico-Philosophicus some online sources for the text.
Created duality involution.
Created 2-category with contravariance and 3-category with contravariance, combining ideas from this preprint of mine with ideas that Vladimir Sotirov had written at contravariant functor (which I moved from there to here).
Created 2-dinatural transformation. The extended example at the bottom is taken from what Vladimir Sotorov wrote at contravariant functor; I moved it over because this is what it’s an example of, but I don’t entirely understand the category-of-enriched-categories part of the example.
Apparently we still didn’t have a page for strict 2-natural transformation, so I made a stub.
Stub for eventual image
I have started a category:reference entry for the article Mandell, May, Schwede, Shipley: Model categories of diagram spectra. I feel that, maybe with the aid of hindsight, there is now room for a somewhat more concise and streamlined presentation of the material in there, for instance by separating basic category theory from the actual constructions a bit more, and I am in the process of typing that into the entry.
So far I worked on part I.
I’ll polish this a bit more, then I am going to feel inclined to copy this over to relevant sections in the entries on sequential-, symmetric- and orthogonal spectra, respectively, for completeness.
There is already a page Ulrik Buchholtz and now we have UlrikBuchholtz.
I have created an entry model structure on topological sequential spectra.
In parts this directly parallels the entry Bousfield-Friedlander model structure.
But now I have spelled out full proof of the model structure and its cofibrant generation: here
I did this by taking the more general proof that I had earlier spelled out at Model categories of diagram spectra, and specializing it to the case of sequential spectra.
The effect of that is that those tedious technical lemmas about the maps of free spectra collapse to something simple, with the result that the actual proof may start right away with less preliminaries, which makes the writeup a bit more transparent. On the other hand, the neat thing is that apart from that analysis of the free spectra the proof is verbatim the same now for all cases (sequential, symmetric, orthogonal spectra and pre-excisive functors), so in the other entries it’ll be possible to turn this around and say: “after this analysis of the free symmetric/orthogonal spectra the proof of their model structure now follows verbatim as at model structure for topological sequential spectra”.
As far as exposition and writeup goes, the only remaining “gap” I left is that at one point the proof invokes that and hence is a topological model structure (this is used in the proof of this lemma ). I plan to spell that out, too. But not tonight.
created The Music of the Spheres, following Ravenel.
Tried to clarify the history and the relationship between the different models at symmetric monoidal smash product of spectra.
expanded/polished a little the Definition at exact couple
At monoidal category the pointers to
#Kelly
in the first two lemmas are broken. There is no item that they point to. What’s the reference?
started a minimum at functor with smash products (the realization of ring spectra in terms of lax monoidal functors)
In the end this is entirely a story about monoids with respect to Day convolution tensor products. I suppose there is room to say this yet a bit more general abstractly than MMSS00 did.
I have added to the entry associative unital algebra a section “Over monoids in a monoidal category” with the general definition in monoidal categories. At the end I have added statement and proof that (for any commutative monoid in a symmetric monoidal category ).
I have put a minimum remark on the framed bordism ring, its relation to the stable stems, literature and basic examples into Homotopy groups of spheres – Relation to framed bordism and (the same) into cobordism ring – Examples – Framed cobordism.
Started Schur multiplier since we didn’t have it. There’s no doubt plenty to say about extensions, etc., and people well-placed to say it in an organised fashion.
I have started editing at Thom’s theorem. So far it has just the definition of the bordism ring, the statement of the theorem and some literature.
stub for tubular neighbourhood
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.