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Some nLab pages had a gray link to descriptive set theory, which now has the following stub:
Descriptive set theory is the study of the structures and hierarchies of subsets of real numbers (or more generally of subsets of Polish spaces) that are definable by formulas with real parameters in second-order arithmetic.
Such subsets include Borel sets and more generally projective sets that are defined by alternating between taking images under projection maps of previously defined sets and taking complements of previously defined sets. Once the domain of topologists of the Polish schools and Russian analysts of the early 20th century, it is now considered a central area of logic in which set theory and computability theory (recursion theory) meet and interact.
I made the former entry "fibered category" instead a redirect to Grothendieck fibration. It didn't contain any addition information and was just mixing up links. I also made category fibered in groupoids redirect to Grothendieck fibration
I also edited the "Idea"-section at Grothendieck fibration slightly.
That big query box there ought to be eventually removed, and the important information established in the discussion filled into a proper subsection in its own right.
removing query box from page
+– {: .query} Mike Shulman: Is there a formal statement in some formal system along the lines of “a non-extensional choice operator does not imply AC”?
Toby: I don't know about a formal statement, but I can give you an example.
Recall: In Per Martin-Löf's Intuitionistic Type Theory (and many other systems along similar lines), the basic notion axiomatised is not really that of a set (even though it might be called ’set’) but instead a preset (or ’type’). Often one hears that the axiom of choice does hold in these systems, which doesn't imply classical logic due to a lack of quotient (pre)sets. However, if we define a set to be a preset equipped with an equivalence predicate, then the axiom of choice fails (although we have COSHEP if presets come with an identity predicate).
A lot of these systems (including Martin-Löf's) use ’propositions as types’, in which ∃x:AP(x) is represented as ∑x:AP(x), which comes equipped with an operation π:∑x:AP(x)→A. That is not going to get us our choice operator, but since a choice operator is constructively questionable anyway, then let's throw in excluded middle. This is known to not imply choice, but we do have, for every preset A, an element εA of A∨¬A, that is of A⊎∅A. It's not literally true that εA is of type A, of course, but that would be unreasonable in a structural theory; what we do have is a fixed εA such that, if A is inhabited, then εA=ιA(e) for some (necessarily unique) e of type A (where ιA is the natural inclusion A→A⊎∅A), which I think should be considered good enough. This is for presets (types), but every set has a type of elements, so that gets us our operator.
How is this nonextensional? We do have εA=εB if A=B (which is a meaningful statement to Martin-Löf, albeit not a proposition exactly), but if A and B are given as subsets of some U, then we may well have A=B as subsets of U without A=B in the sense of identity of their underlying (pre)sets. In particular, if f:U→V is a surjection and A and B are the preimages of elements x and y of V, then x=Vy will not imply that εA=εB, and the proof of the axiom of choice does not go through. It will go through if x and y are identical, that is if x=y in the underlying preset of V, so again we do get choice for presets (again), but not for sets.
I'm not certain that a nonextensional global choice operator won't imply excluded middle in some other way, but I don't see how it would. You'd want to do something with the idea that εA always exists but belongs to A if and only if A is inhabited, but I don't see how to parse it (just by assuming that it exists) to decide the question.
Mike Shulman: That’s a very nice explanation/example, and it did help me to understand better what’s going on; thanks! (Did you mean to say “excluded middle” and not “AC” in your final paragraph?) What I would really like, though, is a statement like “the addition of a nonextensional global choice operator to ____ set theory is conservative” (i.e. doesn’t enable the proving of any new theorems that doen’t refer explicitly to the choice operator). Of course I am coming from this comment, wondering whether what you suggested really is a way to get a choice operator without implying the axiom of choice.
Toby: Yeah, I really did mean to say ’excluded middle’; remembering that comment, I assume that the real question is whether the thing is OK for a constructivist. I just argued ITT+EM⊨CO, and I know the result ITT+EM¬⊨AC, so I conclude ITT+CO¬⊨AC; but I don't know ITT+CO¬⊨EM for certain. I certainly don't have ITT+CO conservative over ITT, nor with any other theory (other than those that already model CO, obviously).
Mike Shulman: Where should I look for a proof that ITT+EM doesn’t imply AC?
Toby: I'm not sure, it's part of my folk knowledge now. Probably Michael J. Beeson's Foundations of Constructive Mathematics is the best bet. I'll try to get a look in there myself next week; I can see that it's not exactly obvious, and perhaps my memory is wrong now that I think about it.
Mike Shulman: I’m trying to prove the sort of statement I want over at SEAR+?.
Toby: No, I can't get anything at all out of Beeson (or other references) about full AC (for types equipped with equivalence relations) in ITT.
Harry Gindi: I have references for this discussion that should settle the issue at hand:
Bell, J. L., 1993a. ’Hilbert’s epsilon-operator and classical logic’, Journal of Philosophical Logic, 22:1-18
Bell, J. L., 1993b. ’Hilbert’s epsilon operator in intuitionistic type theories’, Mathematical Logic Quarterly 39:323-337
Meyer Viol, W., 1995a. ’A proof-theoretic treatment of assignments’, Bulletin of the IGPL, 3:223-243
Toby: Thanks, Harry! Now I just have to find these journals at the library. =–
Anonymous
I toiuched the formatting and the hyperlinking of the paragraphs on compatibility of limits with other universal constructions.
Merged the previous tiny subsections on this to a single one, now Compatibility with universal constructions.
added the hyperlink to the stand-alone entry adjoints preserve (co-)limits.
Will create an analogous stand-alone entry for limits commute with limits.
starting page on ETCS with a choice operator ε. See SEAR plus epsilon for the allegorical set theory analogue.
Anonymouse
I’ll try to start add some actual content to the entries classical mechanics, quantum mechanics, etc. For the time being I added a simple but good definition to classical mechanics. Of course this must eventually go with more discussion to show any value. I hope to be able to use some nice lecture notes from Igir Khavkine for this eventually.
For the time being, notice there was this old discussion box, which I am herby mving to the forum here:
–
+–{.query} Edit: I changed the above text, incorporating a part of the discussion (Zoran).
Zoran: I disagree. Classical mechanics is classical mechanics of anything: point particles, rigid bodies (the latter I already included), infinite systems (mechanics of strings, membranes, springs, elastic media, classical fields). It includes statics, not only dynamics. The standard textbooks like Goldstein take it exactly in that generality.
One could even count the simplified beginning part of the specialized branches like aerodynamics and hydrodynamics (ideal liquids for example), which are usually studied in separate courses and which in full formulation are not just mechanical systems, as the thermodynamics also affects the dynamics. There are also mechanical models of dissipative systems, where the dissipative part is taken only phenomenologically, e.g. as friction terms. Hydrodynamics can also be considered as a part of rheology.
Toby: I take your point that ’dynamics’ was not the right word. But do you draw any distinction between ’classical mechanics’ and ’classical physics’? Conversely, what word would you use to restrict attention to particles instead of fields, if not ’mechanics’? (Incidentally, I would take point particles as possibly spinning, although I agree that I should not assume that the particle are points anyway.)
Zoran: you see, in classical mechanics you express all you have by attaching mass, position, velocity etc. to the parfts of mechanical systems. Not all classical physics belongs to this kind of description. The thermodynamical quantities may influence the motion of the systemm, but their description is out of the frame of classical mechanics. If you study liquids you have to take into account both the classical mechanics of the liquid continuum but also variations of its temperature, entropy and so on, which are not expressable within the variables of mechanics. Formally speaking of course, the thermodynamics has very similar formal structure as mechanics, for example Gibbs and Helmholtz free energies and enthalpy are like Lagrangean, the quantities which are extremized when certain theremodynamical quantities are kept constant. To answer the terminological question, there is a classical mechanics of point particles and it is called classical mechanics of point particles, there is also cm of fields and cm of rigid bodies.
Toby: So ’mechanics’ for you means ‹not taking into account thermal physics›? That's not the way that I learned it! But I admit that I do not have a slick phrase for that (any more than you have a slick phrase for ‹mechanics of point particles›), so I will try to ascertain how the term is usually used and defer to that. =–
I added a couple of comments about topos models to principle of omniscience.
I tried to implement the connection between D-modules and quasicoherent sheaves a bit more
added to D-module the alternative definition in terms of quasicoherent sheaves of the deRham space
added to deRham space accordingly a pointer to D-modules (also fixed wrong notation in the formulas there)
added to quasicoherent sheaf at the very bottom a pointer to D-modules.
This needs improving. Notably good references should be given.
I have expanded the Idea section at state on a star-algebra and added a bunch of references.
The entry used to be called “state on an operator algebra”, but I renamed it (keeping the redirect) because part of the whole point of the definition is that it makes sense without necessarily having represented the “abstract” star-algebra as a C*-algebra of linear operators.
Moved talk by McLarty to the History section of the refernces from functorial geometry
Person entry, required at standard monomial.
Standard monomial theory for flag varieties and Schubert varieties.
I have added to homotopy group a very brief pointer to Mike’s HoTT formalization of π1(S1).
Eventually I would like to have by default our nLab entries be equipped with detailed pointers to which aspects have been formalized in HoTT (if they have), and in which .v-file precisely.
created placeholder for Hurewicz theorem
Added references on Fox derivative.
I have split off an entry homotopy in a model category from homotopy and then spelled out statement and proof of the basic lemmas.
this is a bare list of references, to be !include-ed into relevant entries
(various parts of this list have been contained at Cohomotopy, Cohomotopy charge map, Thom’s theorem, Pontryagin-Thom construction and elsewhere; this here now to ease harmonizing/completing these lists)
brief category:people-entry for hyperlinking references at supergeometry supermanifolds, super Lie groups, functorial geometry
I have spelled out statement and proof of:
every reflection is a reflective localization,
reflective localizations are given by full subcategories of local objects.
I have also tried to produce decent cross-links to various entries that mention something related. But there will be more left to do.
I am starting higher Segal space (while sitting in a talk by Mikhail Kapranov about them…)
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