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At overt space there was a remark that since the definition quantifies over “spaces”, the overtness of a single space might depend on the general meaning chosen for “space”, but that no example was known to the author. I added an example involving synthetic topology, which may not be quite what the author of that remark was thinking of, but which I think is interesting.
I incorporated some of my spiel from the blog into the page type theory.
There has GOT to be a better photograph than that! Is there anyone here in Oxford? Can they go and get a picture for us?
I made some very minor changes to the introduction at descent. I hesitate to do more but at present the discussion does not seem that readable to me. Can someone look at it to see what they think? The intro seems to plunge in deep very quickly and so the ‘idea’ of descent as that of gluing local information together, does not come across to me. The article is lso quite long and perhaps needs splitting up a bit.
Added some content to display map from Taylor’s book. Not very deep, mostly as a reference to the respective section for me.
Created basic outline with some important connections. Yang-Mills measure, after all the main concept which makes this special case interesting, and references will be added later.
Edit: Crosslinked D=2 Yang-Mills theory on related pages: D=2 QCD, D=4 Yang-Mills theory, D=5 Yang-Mills theory.
(Today’s arXiv) A homotopification
and few more additions.
I added the HoTT introduction rule for ’the’, then added a speculative remark on why say things like
The Duck-billed Platypus is a primitive mammal that lives in Australia.
Created:
An internal category object in the category of smooth manifolds in which the source and target maps are submersions.
Sometimes, the smooth manifold of morphisms is allowed to have a boundary, in which case the restrictions of the source and target maps to the boundary are required to be submersions themselves.
i have split off (copied over) the paragraph on the first uncountable ordinal from countable ordinal to first uncountable ordinal, just in order to make it possible to link to “first uncountable ordinal” more directly. Cross-linked with long line.
brief category:people-entry for hyperlinking references at skyrmion, atomic nucleus
I tried to brush-up the References at period a little.
I have trouble downloading the first one, which is
My system keeps telling me that the pdf behind this link is broken. Can anyone see it? (It may well just be my system misbehaving, wouldn’t be the first time…).
at decidable proposition I found the simple basic idea a bit too deeply hidden in the text. In an attempt to improve on this I have added right before the subsections of the Idea-section this quick preview:
External decidability: either p or ¬p may be deduced in the metalanguage;
Internal decidability: p∨¬p may be deduced, hence “p or not p” holds in the object language.
Okay?
Recorded at subsingleton that a different nomenclature also exists, in which “subterminal” and “subsingleton” are not synonymous (see for instance Anders Kock in page 2 of Algebras for the Partial Map Classifier Monad).
an entry for mere proposition had been missing. Created a minimum, just so as to satisfy links.
moving the following ancient query box out of the entry:
+– {: .query} What about the ’or’ of parental threat? Consider the logician parent who says “Come here or I’ll smack you” to his child and smacks even after obedience as they believe in the inclusive ’or’. -David
That's no different from ’If you don't come here, then I'll smack you.’, which also suggests (but does not state) the converse. And in fact, no parent, logician or otherwise, is actually making the promise implied by the ¬(p∧q) clause; if the child comes to such a parent and then kicks the parent in the shin, then the parent will still smack the child. Instead, if you want to make that promise, then you say ’If you come here, then I won't smack you.’ explicitly. This has a very different tenor (unless you say it in a wink-nudge mafia kind of way), as it's a promise rather than a threat. (I know, it's only a promise, which is still different in tenor than a statement that is both promise and threat, as an exclusive disjunction would be. But I still hold that your statement is only a threat.) Note that a logician child who believes the parent's literal expression would still choose to come if avoiding smacking is the highest priority; but the reason is that refusal guarantees a smack, not that obedience necessarily avoids it. That is why the wise child also throws in a contrite expression and an oral apology, to improve the odds. —Toby
I see there’s a literature on the subject including “The Myth of the Exclusive ’Or’” (Mind, 80 (317), 116–121). —David
Also: I argued above that the meaning of ’Come here or I'll smack you’ must be weaker than exclusive disjunction, since the parent will smack the child anyway under some circumstances. However, I agree that it is stronger than inclusive disjunction, but that is because we may go beyond the literal meaning of the words and apply a Gricean implicature. To be specific, if the parent intends to smack the child regardless, then the parent should say ’I'll smack you’ by the Maxim of Quantity, but the parent in fact said something more wordy. Thus we conclude that the parent does not intend to smack the child if the child comes, without ruling out the possibility that the parent will still smack the child for some other reason, as yet unanticipated. —Toby =–
Created:
The dissolution locale ℭL of a locale L is defined as the poset of its sublocales (equivalently: nuclei on L) equipped with the relation of reverse inclusion.
There is a canonical morphism of locales
ι:ℭL→Lsuch that the map ι* sends an open a∈L to the open in ℭL given by the open sublocale of a.
The map ℭL→L can be considered an analogue of the canonical map Td→T for a topological space T, where Td is the underlying set of T equipped with the discrete topology.
In particular, discontinuous maps L→M could be defined as morphisms of locales ℭL→M, see Picado–Pultr, XIV.7.3.
Original reference:
Expository account:
created shifted tangent bundle because I thought somebody was asking about that on the blog, but now looking more closely I find that maybe nobody asked for that...
I started a stub at affine logic as I saw the link requested in a couple of places.
The cut rule for linear logic used to be stated as
If Γ⊢A and A⊢Δ, then Γ⊢Δ.
I don’t think this is general enough, so I corrected it to
If Γ⊢A,Φ and Ψ,A⊢Δ, then Ψ,Γ⊢Δ,Φ.
the entry Galois theory used to be a stub with only some links. I have now added plenty of details.
I have tried to expand a bit the text at the beginning of the category:people entry Alexander Grothendieck, mention more of what his work was about, add more hyperlinks. It could still be much improved, but right now it reads as follows:
The french mathematician Alexandre Grothendieck, (in English usually Alexander Grothendieck), has created a work whose influence has shown him to be the greatest pure mathematician of the 20th century; and his ideas continue to be developed in this century.
Initially working on topological vector spaces and analysis, Grothendieck then made revolutionary advances in algebraic geometry by developing sheaf and topos theory and abelian sheaf cohomology and formulating algebraic geometry in these terms (locally ringed spaces, schemes). Later topos theory further developed independently and today serves as the foundation also for other kinds of geometry. Notably its homotopy theoretic refinement to higher topos theory serves as the foundation for modern derived algebraic geometry.
Grothendieck’s work is documented in texts known as EGA (with Dieudonné), an early account FGA, and the many volume account SGA of the seminars at l’IHÉS, Bures-sur-Yvette, where he was based at the time. (See the wikipedia article for some indication of the story from there until the early 1980s.)
By the way, in view of the recent objection to referring to people as “famous” in category:people entries: the lead-in sentence here is not due to me, it has been this way all along. One might feel that it should be rephrased, but I leave that to those who feel strongly about it.
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)
Added to Maslov index and to Lagrangian Grassmannian the following quick cohomological definition of the Maslov index:
The first ordinary cohomology of the stable Lagrangian Grassmannian with integer coefficients is isomorphic to the integers
H1(LGrass,ℤ)≃ℤ.The generator of this cohomology group is called the universal Maslov index
u∈H1(LGrass,ℤ).Given a Lagrangian submanifold Y↪X of a symplectic manifold (X,ω), its tangent bundle is classified by a function
i:Y→LGrass.The _Maslov index of Y is the universal Maslov index pulled back along this map
i*u∈H1(Y,ℤ).