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Giving this its own entry (split off from cosmos) for ease of hyperlinking to this particular notion (as announced in the thread there).
It’s essentially just material copied over for now, but wording adjusted for flow and clarity (I hope) but I have added a more pertinent Idea-section and started a section on Examples (just a beginning, so far).
collecting some references on the argument for
exactly 24 D7-branes in F-theory on K3
exactly 24 NS5-branes in HET-theory on K3
(this used to be a remark at F-theory, now copied over and expanded here)
and taking the liberty of adding a comment relating this, via “Hypothesis H”, to the third stable homotopy group of spheres seen under Pontryagin’s theorem.
now I have finally the time to come back to this, as announced, and so I am now starting an entry:
relation between type theory and category theory .
So far there is just some literature collected. I now plan to extract the essence of Seely’s artice into the entry in some technical detail.
Since the page geometry of physics – categories and toposes did not save anymore, due to rendering timeouts caused by its size, I have to decompose it, hereby, into sub-pages that are saved and then re-!included separately.
With our new announcement system this means, for better or worse, that I will now have to “announce” these subsections separately. Please bear with me.
I was looking to correct “Issac Newton” in the first line, but it send me an error about the table of contents. I don’t know how to fix that.
brief category:people-entry for hyperlinking references at Tate K-theory and at equivariant elliptic cohomology
following public demand, I created an entry titled
(If you don’t like this title, please have a look first to see if it makes sense afterwards. If still not, I won’t be dogamitic about this and am open for suggestions for other titles.)
For completeness I have added pointer to
though there should really be some accompanying discussion of how this form of the statement is related to the usual one in terms of presheaves.
I worked on synthetic differential geometry:
I rearranged slightly and then expanded the "Idea" section, trying to give a more comprehensive discussion and more links to related entries. Also added more (and briefly commented) references. Much more about references can probably be said, I have only a vague idea of the "prehistory" of the subject, before it became enshrined in the textbooks by Kock, Lavendhomme and Moerdijk-Reyes.
Also, does anyone have an electronic copy of that famous 1967 lecture by Lawvere on "categorical dynamics"? It would be nice to have an entry on that, as it seems to be a most visionary and influential text. If I understand right it gave birth to topos theory, to synthetic differential geometry and all that just as a spin-off of a more ambitious program to formalize physics. If I am not mistaken, we are currently at a point where finally also that last bit is finding a full implmenetation as a research program.
added doi:10.1007/BFb0081551 and pdf-link for Tome 1
I have added to monoidal model category statement and proof (here) of the basic statement:
Let be a monoidal model category. Then 1) the left derived functor of the tensor product exsists and makes the homotopy category into a monoidal category . If in in addition satisfies the monoid axiom, then 2) the localization functor carries the structure of a lax monoidal functor
The first part is immediate and is what all authors mention. But this is useful in practice typically only with the second part.
I have added at HomePage in the section Discussion a new sentence with a new link:
If you do contribute to the nLab, you are strongly encouraged to similarly drop a short note there about what you have done – or maybe just about what you plan to do or even what you would like others to do. See Welcome to the nForum (nlabmeta) for more information.
I had completly forgotton about that page Welcome to the nForum (nlabmeta). I re-doscivered it only after my recent related comment here.
created Tannaka duality
with a short proof of the duality for the category of permutation representations of a group, using the Yoneda lemma three or four times in a row and nothing else.
either I am mixed up (in which case we'll roll back), or I guess this is the way that it's usually done in the literature? I haven't really checked. Sorry, I just needed that quickly as a lemma for my discussion at homotopy group of an infinity-stack
created a stub for holographic entanglement entropy in reaction to this MO question.
added References to BDR 2-vector bundle .
Explained at mapping cone how the mapping cone is model for a homotopy cofiber. In fact I used that to define and motivate the mapping cone.
Then I moved the example in Top to the top of the list, as that is the archetypical example.
Started lift.
weak factorization system has redirects from: lifting property, right lifting property, left lifting property, lifting problem, lifting problems.
Would it be better to have these redirect to lift?
I am wondering about the following:
Let denote the global orbit category of finite groups, i.e. simply the full sub--category of all -groupoids on those of the form for a finite group.
Regarded as an -site with trivial coverage, this is a cohesive -site. Therefore, given any -topos we obtain a new -topos
which has the following properties:
for each finite group there is the usual , but in addition there is an object to be denoted – to be thought of as the the “generic -orbi-singularity”
(namely that arising as the image of the corresponding object in under the Yoneda-embedding and passing along the inverse terminal geometric morphism of )
it carries an adjoint triple of modalities
to be read
such that (at least when is itself cohesive):
(“the purely singular aspect of an orbi-singularity is a plain quotient of a point, hence a point”)
(“the purely smooth aspect of an orbi-singularity is a homotopy quotient of a point)
(“an orbi-singularity is purely orbi-singular”)
I am wondering about the converse:
Suppose an -topos is such that these three conditions hold (the first one without its parenthetical remark).
Can we conclude that is of the form ?
If not, which axioms could be added to make it work?
A stub for M-theory. What’s supposed to be so mysterious about it? Is it that people don’t even know what form it would take?
added to the people-entry Edward Witten a paragraph Fields medal work with a commented list of articles that according to Atiyah won Witten the Fields medal in 1990.
brief entry extremally disconnected topological space
a stub entry, for the moment just to unbreak an early link offered at separation axioms in terms of lifting properties
added the previously missing cross-link with extremally disconnected topological space
(hoping that that this is the right A M Gleason, please check.)
I have added the characterization of Quillen equivalences in the case that the right adjoint creates weak equivalences, here.
added pointer to yesterday’s
Added a reference
Expansion of references section at differential topology.
Seeing that “completion of a space” was redirecting both for
I am making a little disambiguation page here, to disentangle the links.
But no text here yet. And we are lacking entries on adic completion of spaces anyways.
Created:
A seminal book by Daniel M. Kan and Aldridge K. Bousfield.
Lecture Notes in Mathematics 304 (1972; 1987), Springer.
ISBN 3-540-06105-3, 0-387-06105-3
Introduces localization of spaces and completion of spaces, as well as homotopy limits and homotopy colimits.
added to shape theory a section on how strong shape equivalence of paracompact spaces is detected by oo-stacks on these spaces
By the way: I have a question on the secion titled "Abstract shape theory". I can't understand the first sentence there. It looks like this might have been broken in some editing process. Can anyone fix this paragraph and maybe expand on it?
As I’ve already said elsewhere, I’ve been working on this entry and trying to give a precise definition based on my hunches of what guys like Steenrod really meant by “a convenient category of topological spaces”. (I must immediately admit that I’ve never read his paper with that title. Of course, he meant specifically compactly generated Hausdorff spaces, but nowadays I think we can argue more generally.)
I also said elsewhere that my proposed axiom on closed and open subspaces might be up for discussion. The other axioms maybe not so much: dropping any of them would seem to be a deal-breaker for what an algebraic topologist might consider “convenient”. Or so I think.