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added pointer to:
We already have main references at Richard Askey, so why not ?
brief category:people-entry for hyperlinking references at elliptic genus
Not to confuse with Macdonald conjecture on plane partitions.
started a minimum at functor with smash products (the realization of ring spectra in terms of lax monoidal functors)
In the end this is entirely a story about monoids with respect to Day convolution tensor products. I suppose there is room to say this yet a bit more general abstractly than MMSS00 did.
added at dendroidal set
a section on the relation to simplicial sets
a section on the symmetric monoidal structure on the cat of dendroidal sets
(also added a stubby "overview" section to model structure on dendroidal sets)
I have expanded Lawvere-Tierney topology, also reorganized it in the process
some bare minimum on the free coproduct cocompletion.
The term used to redirect to the entry free cartesian category, where however the simple idea of free coproduct completion wasn’t really brought out.
Person entry.
Warning: we have a webpage for another algebraist, the group theorist Robert A. Wilson, and elsewhere in the Lab Robert Wilson shortcut is carelessly used for the latter. People tend to shorten links in Lab and the confusion and illegal links might occur in this case. Maybe we should not use version without middle initial for these two guys in links.
In statu nascendi.
A method for calculating determinants. It is related to cluster algebras and a special case of Sylvester identity.
I have been adding some material to matroid. I haven’t gotten around to defining oriented matroid yet (and of course there’s much besides to add).
I should say – for those watching the logs and wondering – that I started editing the entry global equivariant homotopy theory such as to reflect Charles Rezk’s account in a coherent way.
But I am not done yet. The entry has now some of the key basics, but is still missing the general statement in its relation to orbispaces. Also some harmonizing of the whole entry may be necessary now, as I moved around some stuff.
So better don’t look at it yet. I hope to bring it into shape tomorrow or so.
(In the process I have split off global orbit category now.)
created a minimum at Penrose-Hawking singularity theorem
I have expanded the Idea-section at 3d quantum gravity and reorganized the remaining material slightly.
I feel unsure about the pointer to “group field theory” in the References. Can anyone list results that have come out of group field theory that are relevant here?
I find the following noteworthy, and I am not sure if this is widely appreciated:
the original discussion of the quantization of 3d gravity by Witten in 1988 happens work out to be precisely along the lines that “loop quantum gravity” once set out to get to work in higher dimensions: one realizes
that the configuration space is equivalently a space of connections;
that these can be characterized by their parallel transport along paths in base space;
that therefore observables of the theory are given by evaluating on choices of paths (an idea that goes by the unfortunate name “spin network”).
All this is in Witten’s 1988 article. Of course the point there is that in the case of 3d this can actually be made to work. The reason is that in this case it is sufficient to restrict to flat connections and for these everything drastically simplifies: their parallel transport depends not on the actual paths but just on their homotopy class, rel boundary. Accordingly the “spin networks” reduce to evaluations on generators of the fundamental group, etc.
Notice that in 4d the analog of this step that Witten easily performs in 3d was never carried out: instead, because it seemed to hard, the LQG literature always passes to a different system, where smooth connections are replaced by parallel transport that is required to be neigher smooth nor in fact continuous. These are called “generalized connections” in the LQG literature. Of course these have nothing much to do with Einstein-gravity: because there the configuration space does not contain such “generalized” fields.
For these reasons I feel a bit uneasy when the entry refers to LQG or spin foams as “other approaches” to discuss 3d quantum gravity. First of all, the existing good discussion by Witten did realize the LQG idea already in that dimension, and it did it correctly. So in which sense are there “other approaches”?
Which insights on 3d quantum gravity do “spin foam”s or does “group field theory”add? If anyone could list some results with concrete pointers to the literature, I’d be most grateful.
I have edited group scheme and algebraic group slightly. To the latter I added Example-pointers to multiplicative group and additive group
added pointer to:
Tim van Beek has graced us with these: Haag-Kastler axioms.
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 is represented as , which comes equipped with an operation . 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 , an element of , that is of . It's not literally true that is of type , of course, but that would be unreasonable in a structural theory; what we do have is a fixed such that, if is inhabited, then for some (necessarily unique) of type (where is the natural inclusion ), 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 if (which is a meaningful statement to Martin-Löf, albeit not a proposition exactly), but if and are given as subsets of some , then we may well have as subsets of without in the sense of identity of their underlying (pre)sets. In particular, if is a surjection and and are the preimages of elements and of , then will not imply that , and the proof of the axiom of choice does not go through. It will go through if and are identical, that is if in the underlying preset of , 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 always exists but belongs to if and only if 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 , and I know the result , so I conclude ; but I don't know for certain. I certainly don't have conservative over , nor with any other theory (other than those that already model , obviously).
Mike Shulman: Where should I look for a proof that 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 .
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 . 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.