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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
Added today’s
(Made a trivial edit in order to create the discussion page.)
I don’t like this terminology. In all other contexts I can think of, denseness is a property of a subset or subtopos. This particular subtopos happens to be a dense one, and the dense sieves for this topology are called by set-theorists merely “dense” because this is the only topology on posets they consider. But it’s not the unique dense topology on a category, so we shouldn’t call it “the” dense topology. Why not call it the “double-negation topology” since that’s what it is?
Added the definition of a exterior covariant derivative on a vector bundle, and reformatted the existing content (exterior covariant derivative on principal bundle).
In Definition 1.7, does the induced external covariant derivative coincide with external covariant derivative in the sense of Definition 1.1 under the induced connection on the vector bundle? (EDIT: Yes.)
wrote out statement and proof that locally compact and sigma-compact spaces are paracompact
I created the page
I gave the differential cohomology hexagon its own page (split off from tangent cohesive (infinity,1)-topos), added an Idea-section, added references, and expanded the formal discussion a little bit more.
For those who haven’t seen it yet: there is a ballet choreography dancing an artistinc impression of the differential cohomology diagram here:
started an entry F-theory (the string-theoretic notion)
stub for tachyon, but out of time now
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-entry for hyperlinking references at Connes-Lott model
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-entry for hyperlinking references at Connes-Lott model
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-entry for hyperlinking references at Perry-Schwarz action, self-dual higher gauge fields and M5-brane
Added quadrability of a cospan (and coquadrability of a span) as well as quadrability/coquadrability of a category C. The word "quadrable" means "squarable", but "squarable" isn't a real word. The word "quadrable" is the proper translation of the French "carrable".
Note that while the translation "quadrable" is uncommon, the term "carrable" in French is standard.
added pointer to
I have created an entry spectral symmetric algebra with some basics, and with pointers to Strickland-Turner’s Hopf ring spectra and Charles Rezk’s power operations.
In particular I have added amplification that even the case that comes out fairly trivial in ordinary algebra, namely is interesting here in stable homotopy theory, and similarly .
I am wondering about the following:
In view of the discussion at spectral super scheme, then for an even periodic ring spectrum, the superpoint over has to be
This of course is just the base change/extension of scalars under Spec of the “absolute superpoint”
(which might deserve this notation even though the sphere spectrum is of course not even periodic).
This looks like a plausible answer to the quest that David C. and myself were on in another thread, to find a plausible candidate in spectral geometry of the ordinary superpoint , regarded as the base of the brane bouquet.
brief category:people
-entry for hyperlinking references at first-order formulation of gravity
some basics at spherical fibration
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-entry for hyperlinking references at cobordism cohomology theory, cohomotopy twisted cohomotopy etc.
Over in another thread, David Roberts asks for explanation of a bunch of terms in QFT (here).
In further reaction I have started a minimum of explanation for one more item in the list: split supersymmetry.
brief category:people
-entry for hyperlinking references at Cheeger-Gromoll splitting theorem and at Gromoll-Meyer sphere
some minimum, in order to give a home to today’s
in order to give a home to this article:
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-entry for hyperlinking references at collar neighbourhood theorem (hope I identified the correct Wikipedia page for this author)
Finally added to fracture theorem the basic statement of the “arithmetic fracture square”, hence the following discussion.
The number theoretic statement is the following:
+– {: .num_prop #ArithmeticFractureSquare}
The integers are the fiber product of all the p-adic integers with the rational numbers over the rationalization of the former, hence there is a pullback diagram in CRing of the form
Equivalently this is the fiber product of the rationals with the integral adeles over the ring of adeles
=–
In the context of a modern account of categorical homotopy theory this appears for instance as (Riehl 14, lemma 14.4.2).
+– {: .num_remark}
Under the function field analogy we may think of
as an arithmetic curve over F1;
as the ring of functions on the formal disks around all the points in this curve;
as the ring of functions on the complement of a finite number of points in the curve;
is the ring of functions on punctured formal disks around all points, at most finitely many of which do not extend to the unpunctured disk.
Under this analogy the arithmetic fracture square of prop. \ref{ArithmeticFractureSquare} says that the curve has a cover whose patches are the complement of the curve by some points, and the formal disks around these points.
This kind of cover plays a central role in number theory, see for instance thr following discussions:
=–
I got tired of having to fight my way through Kelly’s monster yet again, and created transfinite construction of free algebras. I couldn’t really think of a good name for this page; suggestions are welcome.
I have spent some minutes starting to put some actual expository content into the Idea-section on higher gauge theory. Needs to be much expanded, still, but that’s it for the moment.
Added (not very elegantly) the relation to Meyer-Vietoris sequence at Dold-Thom theorem.
am finally splitting this off from Hopf invariant
created an entry for the statement that the kernel of integration is the exact differential forms with a pointer to a proof, and cross-linked with Lie integration and de Rham theorem
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-entry for hyperlinking references at L-infinity algebra and perturbative quantum field theory