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brief category:people
-entry for hyperlinking references at
Created inductive family. Requires more work, but it’s a start.
added to bipermutative category a remark on the relation to bimonoidal categories and basic examples. But what is still missing are the interesting examples.
brief category:people
-entry for hyperlinking references at superstring scattering amplitudes, KLT relations, pure spinor
brief category:people
-entry for hyperlinking references at superstring scattering amplitudes, KLT relations, pure spinor
Added something to fill a link. Please check the definition that I have given; it is the one that is most natural to me, but not the most standard! Feel free to add details of equivalent definitions.
The commutative diagram will not render yet, I am working on that now; should be done shortly. [Edit: done now!]
am clearing this page, since I just noticed that it duplicates the entry Robion Kirby. I suppose the latter should be the entry with “Rob Kirby” a redirect(?)
Created page, with definition.
Berger-Mellies-Weber claim that the nerve theorem for monads with arities constructs Eilenberg-Moore and Kleisli objects in the 2-category of categories with arities, but as far as I can see their proof as written only shows that the Eilenberg-Moore adjunction lives in this 2-category, not that it retains its universal property there. Is there a quick way to see that it does? If this is true, then I think it gives an even more “natural” explanation of the nerve construction, along the lines of Tom’s original blog post.
brief category:people-entry for hyperlinking references at table of marks
added pointer to
for computer-checks of the Riemann hypothesis. (there are probably more recent such?)
I gave sheaf with transfer an Idea-section
(the entry used to me named “Nisnevich sheaves with transfer”. I have renamed it to singular to stay with our convention and removed the “Nisnevich” from the title, as the concept of transfer as such is really not specific to the Nisnevich topology).
The idea section now is the following. (Experts please complain, and I will try to fine tune further):
Given some category (site) of test spaces, suppose one fixes some category of correspondences in equipped with certain cohomological data on their correspondence space. Then a sheaf with transfer on is a contravariant functor on such that the restriction along the canonical embedding makes the resulting presheaf a sheaf.
Traditionally this is considered for the Nisnevich site and constructed from correspondences equipped with algebraic cycles as discussed at pure motive, (e.g. Voevodsky, 2.1 and def. 3.1.1).
The idea is that, looking at it the other way around, the extension of a sheaf to a sheaf with transfer defines a kind of Umkehr map/fiber integration by which the sheaf is not only pulled back along maps, but also pushed forward, hence “transferred” (this concept of course makes sense rather generally in cohomology, see e.g. Piacenza 84, 1.1).
The derived categories those abelian sheaves with transfers for the Nisnevich site with are A1-homotopy invariant provides a model for motives known as Voevodsky motives or similar (Voevodsky, p. 20).
brief category:people-entry for hyperlinking references at Burnside ring and representation ring
Do other people see these final symbols as identical at Serre intersection formula?
Given a regular scheme and subschemes with defining ideal sheaves …
I added a note to the article on the subobject classifier: “In type theory, the type corresponding to the subobject classifier is typically called Prop.”
brief category:people-entry for hyperlinking references at string phenomenology, heterotic string and MSSM
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) is called a causal disjointness relation (and are called causally disjoint if ) if the following properties are satisfied:
(i) is symmetric
(ii) and implies
(iii) if is bounded from above, then for all implies .
(iv) for every there is a with
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 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?
brief category:people-entry for hyperlinking references at G2-manifold and elsewhere
I started writing a bit more about FOLDS, and while I was at it I clarified the relationship between FOLDS' "simple categories]] and direct categories.
brief category:people-entry for hyperlinking references at branched cover and at 4-manifold
brief category:people-entry for hyperlinking references at brane intersection, (p,q)5brane, orientifold, orientifold plane and maybe elsewhere
added a proof to Urysohn’s lemma
created stub for étale morphism of E-∞ rings in order to record the theorem of essential uniqueness of lifts of étale morphism from underlying commutative rings to -rings (which is crucial for the characterization of the moduli stack of derived elliptic curves, and I have cross-linked with that). But otherwise no content yet, due to lack of leisure.
quick note on self-dual Yang-Mills theory
I added a proposition to this subsection which seems valid intuitionistically, but I wouldn’t mind a reality check from someone.
As a kind of supplement to Urs’s running topology series, I wrote an article colimits of normal spaces. Mainly I had wanted to write down a reasonably clean proof of the fact that CW-complexes are spaces, in particular Hausdorff, as called for on the page CW-complexes are paracompact Hausdorff spaces, but working in slightly greater generality. There are a whole bunch of links to stick in, which I plan to get to.
This page has taken me longer than I had first anticipated. Only after some struggle and reading around did I discover the power of the Tietze characterization of normality, which can be used to give a simple proof of the following general fact:
If are normal and if is a closed embedding and a continuous map, the attachment space = pushout is also normal.
This doesn’t seem so easy to prove with one’s bare hands (i.e., just using the usual definition of normality and reasoning away)!
Urs, after recent discussion with Richard about paracompactness, where do matters stand on the page CW-complexes are paracompact Hausdorff spaces? It would be nice to tie up whatever loose ends are still left hanging there.
I added some references to continuation-passing style, as well as a big rambling Idea section.