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I added some material on arc-connected spaces to connected space.
I added also a reference to Willard’s General Topology, together with this online link to a Scribd document: Willard. Is this kosher (I am guessing this document is not “pirated”, but I’m not sure)?
Prompted by a question which I received, I went and tried to streamline the old entry Lie infinity-algebroid representation a little:
moved the pevious “Properties”-discussion of complexes of holomorphic bundles to the Examples-section;
added the example of -algebra extensions
added more information to the References-section
cross-linked a bit more with infinity-action and with L-infinity algebra cohomology etc.
I started a separate page for Picard stack (which used to be just a redirect to Picard scheme), stated the general nonsense idea with a pointer to Lurie’s thesis, where this essentially appears.
(BWT, where in the DAG series did this end up? I forget.)
Of course the upshot is that it’s simply the internal hom/mapping stack . I have a question here: it seems clear that the higher versions want to be called the higher intermediate Jacobians (their deformation theory at 0 are the Artin-Mazur formal groups). Why does nobody say this? (Or if they do, where?)
Over in the thread on “Picard infinity-stack” we turned to discussion of Brauer stack. Just for completeness I should probably make this a separate thread here: I had created Brauer stack for the moment only with the following Idea-section
It is traditional to speak, for a suitable scheme , of its Picard group and of its Brauer group. Moreover, it is a classical fact that under suitable conditions the former admits itself a canonical geometric structure that makes it the Picard scheme of . Still well known, if maybe less commonly highlighted, is that this is just the 0-truncation of the Picard stack of , which is simply the mapping stack into the delooping of the multiplicative group. In this form this applies immediately also to more general context such as E-∞ geometry ("spectral geometry") and gives a concept of Picard ∞-stack ("derived Picard stack"). Given this and the relation of the Brauer group to étale cohomology it is clear that the Brauer group similarly arises as the torsion subgroup of the 0-truncation of the ∞-stack which ought to be called the Brauer stack, given as the mapping stack
into the second delooping of the multiplicative group (modulating line 2-bundles). Indeed, just as the Picard stack turns under Lie integration (evaluation on infinitesimally thickened points) and 0-truncation into what is commonly called the formal Picard group, so this Brauer -stack similarly gives what is commonly called the formal Brauer group.
However, while therefore the terminology "Brauer stack" is the evident continuation of a traditional pattern (which in the other direction continues with the group of units and the mapping scheme ), it seems that this terminology has never been introduced in the literature (at time of this writing). (?)
at holomorphic vector bundle I have started a section titled As complex vector bundles with holomorphically flat connections.
This deserves much more discussion (and maybe in a dedicated entry), but for the moment I have there the following paragraphs (with lots of room for further improvement):
+– {: .num_theorem #KoszulMalgrangeTheorem}
Holomorphic vector bundles over a complex manifold are equivalently complex vector bundles which are equipped with a holomorphic flat connection. Under this identification the Dolbeault operator acting on the sections of the holomorphic vector bundle is identified with the holomorphic component of the covariant derivative of the given connection.
The analogous statement is true for generalization of vector bundles to chain complexes of module sheaves with coherent cohomology.
=–
For complex vector bundles over complex varieties this statement is due to Alexander Grothendieck and (Koszul-Malgrange 58), recalled for instance as (Pali 06, theorem 1). It may be understood as a special case of the Newlander-Nirenberg theorem, see (Delzant-Py 10, section 6), which also generalises the proof to infinite-dimensional vector bundles. Over Riemann surfaces, see below, the statement was highlighted in (Atiyah-Bott 83) in the context of the Narasimhan–Seshadri theorem.
The generalization from vector bundles to coherent sheaves is due to (Pali 06). In the genrality of (∞,1)-categories of chain complexes (dg-categories) of holomorphic vector bundles the statement is discussed in (Block 05).
+– {: .num_remark}
The equivalence in theorem \ref{KoszulMalgrangeTheorem} serves to relate a fair bit of differential geometry/differential cohomology with constructions in algebraic geometry. For instance intermediate Jacobians arise in differential geometry and quantum field theory as moduli spaces of flat connections equipped with symplectic structure and Kähler polarization, all of which in terms of algebraic geometry directly comes down moduli spaces of abelian sheaf cohomology with coefficients in the structure sheaf (and/or some variants of that, under the exponential exact sequence).
=–
started Narasimhan–Seshadri theorem, for the moment just to collect references.
I have just deleted a large number of dollar \ , dollar from the bottom of Blakers-Massey theorem. The effect of such is to add a large ammount of blank space at the end of the page. Was this intentional extra space for something? If not, what is causing it? I should add that I have found similar blank space before and deleted that as well.
added some minimum to Hodge cohomology
since it turned out to be hard to find, I gave the discussion of filtered homotopy colimits at combinatorial model category its own entry, and cross-linked:
gave 11d Chern-Simons theory its own (brief, for the moment) entry (splitting off some material from self-dual higher gauge theory)
started a minimum at Calabi-Yau cohomology.
This is an obvious idea that must have been studied before (for ) but I have had no luck with finding much detail so far.
;-). I found a typo ‘gorup’ and did a search on the n-Lab…. great fun! It is good to know others have disobient fingers!
am starting an entry F/M-theory on elliptically fibered Calabi-Yau 4-folds
So far there is mainly an Idea-section.
created a survey-table-for-inclusion
and included it into the relevant entries
Zoran,
I wanted to add a reference to holomorphic Chern-Simons theory, only to realize that the entry didn't exist yet. Didn't you recently write something about holomorphic CS? I can't find it right now...
added references to 3d supergravity, with brief comments, and added a paragraph on how maximally supersymmetric 3d supergravity does admit an -gauge field (while fluxed compactification from 11d allows only proper subgroups of the global U-duality to be gauged)
wrote an entry cubical structure in M-theory.
This reviews two stories from the literature, and points out that these two stories may be related.
I am not sure yet exactly how much they are related. I am asking that here on PO
stub for instanton
started some minimum at cubical structure on a line bundle
Currently, an element x in a nonassociative algebra A is nilpotent if there exist a natural number n such that .
I want to say that a nilpotent left ideal of a ring R is a nilpotent element in the set of left ideals of R. To say that, I have to determine the structure of the set of left ideals of a ring under addition and multiplication. Wikipedia says that the set of ideals of a ring is a complete modular lattice. Is a complete modular lattice a nonassociative algebra? If not, do people talk about nilpotent elements in a lattice?
added to Mizar a quote:
One of the biggest problems that worry the developers of automated deduction systems is that their systems are not sufficiently recognized and exploited by working mathematicians. Unlike the computer algebra systems, the use of which has indeed become ubiquitous in the work of mathematicians these days, deduction systems are still seldom used. They are mostly used to formalize proofs of well-established and widely known classical theorems, the Fundamental Theorem of Algebra formalized in the systems Coq and Mizar may serve as a perfect example here. Such work, however, is not always considered to be really challenging from the viewpoint of mathematicians who are concerned with obtaining their own new results. Therefore it has been recognized as a big challenge for the deduction systems community to prove that some of the state-of-the-art systems are developed enough to cope with formalizing recent mathematics.
on the off-chance that there is anyone besides me who checks MathOverflow less frequently than the Forum:
there was a question on forcing in homotopy type theory. I took the liberty of sharing some thoughts.
My comment reflects topics that we have discussed here at some length already. Nevertheless, when sending this I noticed that some of these discussions need to be better reflected in the Lab. And in particular better than I have commented on them for the moment.
I won’t further look into this right now as I am busy with something else. But later I’d like to come back to this.
Let us define a (co-)homology -cobordism, where is a path connected space with basepoint :
Definition: A (co-)homology cobordism is a cobordism such that for cohomology and for homology, where and .
Definition: A valued (co-)homology QFT is a symmetric monoidal functor , where the morphisms in are (co-)homology-isomorphisms of (co-)homology -cobordisms, defined as an isomorphism such that and .
What could possible uses of such a QFT be? Can this be related to Homotopy QFTs by the Hurewicz homomorphism ?
Mike kindly wrote model of type theory in an (infinity,1)-topos (homotopytypetheory) with an explicit statement of how univalent Tarskian types-of-types have semantics by object classifiers in general -toposes.
I have added a brief pointer to this to some relevant Lab entries: to homotopy type theory, to relation between category theory and type theory, to (infinity,1)-topos and to elementary (infinity,1)-topos.
New stubs tau-function, Hirota equation and person entry Mattia Cafasso. Corrections/references to some related entries, e.g. Fredholm determinant.
created a brief entry cohomological field theory and cross-linked a good bit.
added at Leech lattice a pointer to
I created a stub at Long March as someone had started an empty entry there. For the moment it directs back to Galois theory where there is mention of the discussion at Long March, doh! I should prepare a longer entry, but do not understand the topic that well.
Wrote continued fraction, emphasizing coalgebraic aspects. More links should be inserted, and some more material needs to be filled in.
This used to be a super-brief paragraph at topological K-theory; and now it is a slightly longer but still stubby entry comparison map between algebraic and topological K-theory
There seem to be some misleading remarks at Čech model structure on simplicial presheaves.
Accordingly, the (∞,1)-topos presented by the Čech model structure has as its cohomology theory Čech cohomology.
Marc Hoyois seems to says the opposite: there is no deep relation between “Čech” in “Čech cohomology” and in “Čech model structure”.
[…] the corresponding Čech cover morphism .
Notice that by the discussion at model structure on simplicial presheaves - fibrant and cofibrant objects this is a morphism between cofibrant objects.
The Čech nerve is projective-cofibrant if we assume the site has pullbacks. I don’t know how to prove it otherwise. Of course, injective-cofibrancy is trivial.
this question is evidently also relevant to what the correct notion of internal ∞-groupoid may be
Based on the discussion here, it seems that the Čech model structure is not site-independent, even though it can be defined on the category of simplicial sheaves. A very strange state of affairs…
am starting an entry smooth spectrum (in the sense of smooth infinity-groupoid). But nothing much there yet.
minimum at spin orientation of Tate K-theory, for the moment just as to record the reference and the proposition number in there (to go with this MO question)
started some minimum at real-oriented cohomology theory
I gave index an Idea-section.
In the course of this I created some stubby auxiliary entries, such as (in rapidly increasing order of stubbieness)
some basics at congruence subgroup
Vladimir Sotirov has asked a question at contravariant functor.
Stated Fermat’s little theorem.
Created binomial theorem, and added a relevant lemma to freshman’s dream.
I started rewriting von Neumann algebra from the nPOV. So far I rewrote the definition and added some remarks about Sakai's theorem and preduals, but you can already see a proposed list of sections to be written.
I also edited the remarks section to stress the nPOV.
created a category:Reference-entry String theory and the real world (a set of lecture notes on string phenomenology)
(This is to go along with this PhysicsOverflow reply)
added to Simpson conjecture a History section with a paragraph on how Carlos Simpson came up with the conjecture based on that claim by Kapranov-Voevodsky’s (the one whose delicacy Voevodsky now says made him formalize mathematics in HoTT…)
I was unsatisfied with the entry Eilenberg-MacLane object. So I changed the wording at the beginning. Maybe it's an improvement, maybe something better needs to be done.
created equivariant homotopy theory – table displaying the various cohesive -toposes and their bases -toposes (for inclusion in “Related entries” at the relevant entries)
created global orbit category and global equivariant indexing category.
Both entries contain almost the same content at the moment. Both could use more editing, too.
created G-space, a glorified disambiguation page.
I have been adding some stuff to j-invariant, but it’s not really good yet (this here just in case you are watching the logs and are wondering what’s happening)
started something at elliptic fibration
created an entry for Tmf(n)
created M5-brane charge
Here is a note to myself or anyone else to add the following new preprint to Galois group when the nLab is back online.
I’ve only just read the introduction, but it looks pretty great…
started model structure on operads
by the way: I noticed that the page operad has not a single reference. Maybe somebody feels like filling in his favorite ones...
some basics at modular curve
New stubs Édouard Goursat and Goursat theorem and some rearrangement of holomorphic function. I hope to put Goursat’s proof at Goursat theorem (but in the meantime you may see it PlanetMath) and consider its constructive content (probably assuming the fan theorem). But it might be a while before I get around to that.