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started some minimum at exceptional field theory (the formulation of 11d supergravity that makes the exceptional U-duality symmetry manifest)
One more item for the list of Sullivan models – examples
added a bunch of pointers to the literature (with brief comments) at string scattering amplitude.
Also added a corresponding paragraph at effective field theory.
(this is still in reaction to that MO discussion, specifically to the question here)
I edited Trimble n-category:
added table of contents
added hyperlinks
moved the query boxes that seemed to contain closed discussion to the bottom. I kept the query box where I ask for a section about category theory for Trimble n-categories, but maybe we want to remove that, too. Todd has more on this on his personal web.
Added a link to the page on hypersheaves (which was expanded today).
fianlly added the details of Dugger’s description of cofibrant objects in the projective model structure on simplicial presheaves in the section Cofibrant objects.
After Urs’ post at the café about “Tricategory of conformal nets” at Oberwolfach I took a look at the paper Conformal nets and local field theory and noted that I would have to ask some trivial and boring questions about nomenclature before I could even try to get to the content.
One example is about “Haag duality”: It seems to me that we need a generalization of net index sets on the nLab that includes the bounded open sets used for the Haag-Kastler vacuum representation and the index sets used in the mentioned paper. One of the concept needed would be “causal index set”:
A relation ⊥ on an index set (poset) I is called a causal disjointness relation (and a,b∈I are called causally disjoint if a⊥b) if the following properties are satisfied:
(i) ⊥ is symmetric
(ii) a⊥b and c<b implies a⊥c
(iii) if M⊂I is bounded from above, then a⊥b for all a∈M implies supM⊥b.
(iv) for every a∈I there is a b∈I with a⊥b
A poset with such a relation is called a causal index set.
Well, that’s not completly true, because in the literature that I know there is the additionally assumtion that I contains an infinite unbounded sequence and hence is not finite (that whould be a poset that is ? what? unbounded?), that is not a condition imposed on posets on the nLab.
After this definition one can go on and define “causal complement”, the “causality condition” for a net and then several notions of duality with respect to causal complements etc. all without reference to Minkowski space or any Lorentzian manifolds.
Should I create a page causal index set or is there something similar on the nLab already that I overlooked?
category: people page for
Canadian bacon
I created Galois module. I also added further references to p-divisible group; in particular section 4.2 of Lurie’s survey of elliptic cohomology gives some generalization of the classical theory. I started also a page with -the somehow unfortunate- title relations of certain classes of group schemes- I intended it to give an overview and examples of the basic kinds of group schemes occurring in classical (algebraic) number theory (the page contains more or less two specific examples; so there is still development potential).
added pointer to:
I have created the entry recollement. Adjointness, cohesiveness etc. lovers should be interested.
I corrected an apparent typo:
A 2-monad T as above is lax-idempotent if and only if for any T-algebra a:TA→A there is a 2-cell θa:1⇒η∘a
to
A 2-monad T as above is lax-idempotent if and only if for any T-algebra a:TA→A there is a 2-cell θa:1⇒ηA∘a
It might be nice to say ηA is the unit of the algebra….
added pointer to
and rewrote the Idea-section to make it clear that these authors require not just existence of left and right adjoints, but in fact an ambidextrous adjoint and satisfying an extract coherence condition.
Added a recent reference on Peirce’s Gamma graphs for modal logic. This describes his first approach via broken cuts rather than the later tinctured sheet approach. I keep meaning to see if there’s anything in the latter close to LSR 2-category of modes approach.
According to the broken-cut method, possibility is broken cut surrounding solid cut, while necessity is solid cut surrounding broken cut. Since solid cut is negation, broken cut signifies not-necessarily. Easy to see □¬=¬◊ as the same pattern of three cuts, etc.
In the Alpha case, we’re to think of negated propositions as though written elsewhere on another sheet (or the back of the sheet). There seems to be a three-dimensionality to the graphs, e.g., the conditional as like a tube from one sheet to another, Wikipedia. I gather his later ideas on tinctured graphs had this idea of being inscribed on different sheets.
added pointer to:
I added more to idempotent monad, in particular fixing a mistake that had been on there a long time (on the associated idempotent monad). I had wanted to give an example that addresses Mike’s query box at the bottom, but before going further, I wanted to track down the reference of Joyal-Tierney, or perhaps have someone like Zoran fill in some material on classical descent theory for commutative algebras (he wrote an MO answer about this once) to illustrate the associated idempotent monad.
Some of this (condition 2 in the proposition in the section on algebras) was written as a preparatory step for a to-be-written nLab article on Day’s reflection theorem for symmetric monoidal closed categories, which came up in email with Harry and Ross Street.
I gave root of unity its own entry (it used to redirect to root), copied over the paragraph on properties of roots of unities in fields, and added a paragraph on the arithmetic geometry description via μn=Spec(ℤ[t](tn−1)) and across-pointer with Kummer sequence.
the standard bar complex of a bimodule in homological algebra is a special case of the bar construction of an algebra over a monad. I have added that as an example to bar construction.
I also added the crucial remark (taken from Ginzburg’s lecture notes) that this is where the term “bar” originates from in the first place: the original authors used to write the elements in the bar complex using a notaiton with lots of vertical bars (!).
(That’s a bad undescriptive choice of terminoiogy. But still not as bad as calling something a “triple”. So we have no reason to complain. ;-)
starting some minimum, cross-linking with quaternion-Kähler manifold and Sp(n).Sp(1)
Used unicode subscripts for indices of exceptional Lie groups including title and links. When not linked, usual formulas are used. See discussion here. Links will be re-checked after all titles have been changed. (Removed two redirects for “E10” from the top and added one for “E10” at the bottom of the page.)
I worked on brushing up (infinity,1)-category a little
mostly I added in a section on homotopical categories, using some paragraphs from Andre Joyal's message to the CatTheory mailing list.
in this context I also rearranged the order of the subsections
I removed in the introduction the link to the page "Why (oo,1)-categories" and instead expanded the Idea section a bit.
added a paragraph to the beginning of the subsection on model categories
added the new Dugger/Spivak references on the relation between quasi-cats and SSet-cats (added that also to quasi-category and to relation between quasi-categories and simplicial categories)
a bare list of references, to be !include-ed at proof assistant and at machine learning, for ease of synchronizing
this MO comment made me realize that we didn’t have an entry proof assistant, so I started one
Created:
\tableofcontents
Prevalence refers to ideas revolving around associating an enhanced measurable space to a complete space metrizable space topological group.
Suppose G is a complete space metrizable space topological group. A Borel subset S⊂G is shy if there is a compactly supported nonzero Borel measure μ such that μ(xS)=0 for all x∈G.
The triple (G,BG,SG), where BG is the σ-algebra of Borel subsets and SG is the σ-ideal of shy sets is an enhanced measurable space.
We may also want to complete enhanced measurable space (G,BG,SG), extending the notion of shy and prevalent sets to non-Borel sets.
Brian R. Hunt, Tim Sauer, James A. Yorke, Prevalence: a translation-invariant “almost every” on infinite-dimensional spaces, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 2, 217–238. doi.
Brian R. Hunt, Tim Sauer, James A. Yorke, Prevalence. An addendum to: “Prevalence: a translation-invariant ‘almost every’ on infinite-dimensional spaces”, Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 2, 306–307. doi.
Survey:
giving this its own entry, not to bury the material all at braid group
I began to add a definition of conformal field theory using the Wightman resp. Osterwalder-Schrader axiomatic approach. My intention is to define and explain the most common concepts that appear again and again in the physics literature, but are rarely defined, like “primary field” or “operator product expansion”.
(I remember that I asked myself, when I first saw an operator product expansion, if the existence of one is an axiom or a theorem, I don’t remember reading or hearing an answer of that until I looked in the book by Schottenloher).
added pointer to:
added to path space object an Examples-section with some model category-theoretic discussion, leading up to the statement that in a simplicial model category for fibrant X the powering XΔ[1] is always a path space object.