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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
Created Beck module, mentioned it (once) on the tangent category page.
brief category:people
-entry for hyperlinking references at generalized (Eilenberg-Steenrod) cohomology
Hello,
I’m new here and a bit unsure on whether this be the right place to post my question and ask for advice. I have found many useful hints on the nlab so I decided to give a try to the nforum as well.
I have posted a detailed question on math.stackexchange. The URL is https://math.stackexchange.com/q/5062099/350024. There, the question got two edits; originally it was:
Consider a mathematical model for physical space, let’s say a (static) 2-dimensional plane. Such a model may start from a given set (of points) with some algebraic structure over it. Usually, as a minimum, this structure is assumed to be an Affine Space; then, once a point is chosen as the “origin” (which amounts to placing an “observer” somewhere in the physical plane), the points of the affine space are mapped to vectors in the associated Vector Space.
This works if it is assumed that the observer may measure distances between any two points. But let’s assume that it is only possible to know how distant a point is from the origin (and in which direction) - think of the observer as a “radar”. At this point it doesn’t make sense to add vectors (and also, I’d say, to multiply them by negative numbers). Basically we have a vector space structure stripped off of the sum operation, and having ℝ+ as the “field” of scalars.
I know that if we remove the product a vector space becomes a group, but what does it become if we remove the sum? I was thinking about a Cone, but such a concept is defined as a subset of a vector space, so with a concept of sum actually.
This is a tentative starting point for constructing an algebraic model for Special Relativity from elementary “physical” principles. After some discussions with other users and some thinking, I have posted this tentative answer:
The construction of a cone is indeed eye-catching, but the problem there is defining the “base” set. If we start with a metric, such a set may simply be the unit circle; however, as I also explained in my edits to my original question, I want to deal with the scenario before introducing a metric.
I have been elaborating on the first comment by @psl2Z:
You would have an action of the monoid (K,⋅) on a set V, where (K,+,⋅) is the field.
Let us indeed start from a bare set V, whose elements are thought to represent (relative) positions (with respect to a chosen spatial point, i.e. to an “observer”). Instead of a generic field, I want to consider ℝ from the outset; more precisely, let us assume that we can multiply any element of V by any positive real number and obtain an element of V. (The heuristic idea is that space extends indefinitely in each direction.)
We are then given an operation from ℝ+×V into V. In other words, if α∈ℝ+ and x∈V then αx∈V. Of course, we want 1x=x and β(αx)=(βα)x. Since (ℝ+,⋅) is a group (and not only a monoid), this defines a group action. Further, if α≠1 we want αx≠x for all x∈V, so the action is free. It is not transitive (heuristically, positions in different directions are not proportional to each other).
Let us introduce an equivalence relation ∼ in V by y∼x iff y=αx for some α>0. The equivalence classes are the (oriented) directions. We may even use ℝ× (=ℝ−{0}) for the group, so that each direction has its opposite-oriented one.
I would like now to introduce a concept of dimension and a concept of isotropy. For the former, the problem is that I don’t have the sum in V (to form linear combinations); for the latter, I need to introduce rotations, but I don’t have a metric (to define them as isometries).
Does this make any sense?
added to action groupoid a section on action oo-groupoids
a bare list of references, to be !includ
-ed into the list of references of relevant entries, such as at quantum computing and quantum programming, for ease of updating and syncing
added to Noether theorem a brief paragraph on the symplectic/Hamiltonian Noether theorem
also created axiom UIP, just for completeness. But the entry still needs some reference or else some further details.
stub for cosmic inflation (for the moment just to record some references)
added brief definition/characterization to Chern class
Just noticed that we have a duplicate page Jon Sterling.
I have now moved the (little but relevant) content (including redirects) from there to here.
Unfortunately, the page rename mechanism seems to be broken until further notice, therefore I am hesitant to clear the page Jon Sterling completely, for the time being.
I have created a stub for dependent type theory.
This used to redirect to just type theory, but in that entry it is being escaped to Martin-Löf type theory, so clearly either it should redirect there or have a separate entry. I guess a separate entry is better, since there is dependent type theory that is not of Martin-Löf “type”.
a bare list of references, to be !include
-ed into the References-lists of relevant entries (such as at anyon and quantum Hall effect) for ease of updating and synchronizing
Little page to focus on this important notion, as opposed to the general remarks at walking structure.
created computational trinitarianism, combining a pointer to an exposition by Bob Harper (thanks to David Corfield) with my table logic/category-theory/type-theory.
I added to walking structure a 2-categorical theorem that implies that usually “the underlying X of the walking X is the initial X”. This fact seems like it should be well-known, but I don’t offhand know a reference for it, can anyone give a pointer?
polished a bit and expanded a bit at interval category (nothing deep, just so that it looks better)
Quick page, analogous to walking isomorphism.
I just see that in this entry it said
Classically, 1 was also counted as a prime number, …
If this is really true, it would be good to see a historic reference. But I’d rather the entry wouldn’t push this, since it seems misguided and, judging from web discussion one sees, is a tar pit for laymen to fall into.
The sentence continued with
…[ the number 1 is ] too prime to be prime.
and that does seem like a nice point to make. So I have edited the entry to now read as follows, but please everyone feel invited to have a go at it:
A prime number is a natural number which cannot be written as a product of two smaller numbers, hence a natural number greater than 1, which is divisible only by 1 and by itself.
This means that every natural number n∈ℕ is, up to re-ordering of factors, uniquely expressed as a product of a tuple of prime numbers:
n=2n13n25n37n411n5⋯This is called the prime factorization of n.
Notice that while the number 1∈ℕ is, clearly, only divisible by one and by itself, hence might look like it deserves to be counted as a prime number, too, this would break the uniqueness of this prime factorization. In view of the general phenomenon in classifications in mathematics of objects being too simple to be simple one might say that 1 is “too prime to be prime”.
I am back to working on geometry of physics. I’ll be out-sourcing new paragraphs there to their own nLab entries as much as possible (because the length of the page makes saving and hence previewing it take many minutes, so I need to work in smaller sub-entries and then copy-and-paste).
In this context I now started an entry prequantum field theory. To be further expanded.
This comes with a table of related concepts extended prequantum field theory - table:
extended prequantum field theory
0≤k≤n | transgression to dimension k |
---|---|
0 | extended Lagrangian, universal characteristic map |
k | (off-shell) prequantum (n-k)-bundle |
n−1 | (off-shell) prequantum circle bundle |
n | action functional = prequantum 0-bundle |
this page seemed to be missing (among coding theory, linear code and now quantum error correcting code). Just a minimal idea-section for the moment
I’ve added Peter May’s Galois theory example to M-category in a section “Applications”.