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Lie groups are interesting physically because they occur naturally as transformation groups, i.e. as representations—rotations of Euclidean 3-space, Lorentz transformations, unitary transforms of Hilbert space, and so on.
Liam O'Raifeartaigh, group structure of gauge theories
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Direct sum is not the coproduct in the category of quadratic vector spaces. The absence of terminal objects and (co)products in QVec is due to the fact that projections do not generally preserve norms.
José Figueroa O'Farrill, Spin Geometry
The concept of (Grothendieck) topos was introduced by Alexandre Grothendieck in the early 1960s, in order to provide a mathematical underpinning for the 'exotic' cohomology theories needed in algebraic geometry. Every topological space 𝑿 gives rise to a topos (the category of sheaves of sets on 𝑿) and every topos in Grothendieck’s sense can be considered as a 'generalized space'. At the end of the same decade, William Lawvere and Myles Tierney realized that the concept of Grothendieck topos also yielded an abstract notion of mathematical universe within which one could carry out most familiar set-theoretic constructions, but which also, thanks to the inherent 'flexibility' of the notion of topos, could be profitably exploited to construct ‘new mathematical worlds’ having particular properties. A few years later, the theory of classifying toposes added a further fundamental viewpoint to the above-mentioned ones: a topos can be seen not only as a generalized space or as a mathematical universe, but also as a suitable kind of first-order theory (considered up to a general notion of equivalence of theories). In fact, every first-order... geometric theory) admits a canonical topos-theoretic model lying in the classifying topos of the theory, satisfying the universal property that every topos-theoretic model of the theory can be obtained as a pullback of this canonical model along a unique morphism of toposes. Such canonical models of mathematical theories do not exist in the classical set-theoretic setting, but any classical (i.e., set-based) model of the theory can be seen as a point of its classifying topos. Unfortunately, apart from a few isolated exceptions, the study of Grothendieck toposes in the context of Logic was essentially not pursued afterwards. In fact, a great part of the efforts of the categorical community in the past years had been focused on the study of elementary toposes, a more general (elementarily axiomatizable) class of categories in which one can consider models of arbitrary finitary first-order theories, but in which one cannot do geometry in a Grothendieckian sense due to the fact that elementary toposes do not in general admit sites of definition. Even though the study of these more general concepts has shed important light on the original concept of topos (in fact, it was seen that several concepts in Grothendieck's topos theory admitted an 'elementary' formulation), the concept of elementary topos has not been fruitful as a source of 'concrete' applications in different fields of Mathematics, essentially because of the lack of a non-trivial representation theory 'from below' analogous to that provided by sites (which in the context of Grothendieck toposes play the role of 'carriers of concrete mathematical information').
Olivia Caramello
Topological Quantum Field Theories: New TQO Class Of Models
Topically Quantified Field Theories
New Quantum Realm Order: Innovative Models Promise Stronger Quantum Computing
A recent finding in theoretical physics has revealed a new class of models that display behaviours previously thought to be outside the typical descriptions of topological quantum field theories (TQFTs). This is challenging the conventional understanding of topological quantum order (TQO). This innovative discovery offers a compelling vision for quantum computing's future with quantum memories that are more thermally robust than surface codes.
Understanding TQFTs
Topological Quantum Field Theories (TQFTs) have long been thought to describe topological quantum order (TQO). Their properties, especially anyon excitations, are similar to TQO-exhibiting systems. It is generally known that a microscopic model with TQO renormalises to a specific TQFT at the low-energy limit, allowing ground-state degeneracies to be calculated. Essentially metric-independent, TQFTs describe a zero-energy subspace.
Category Theory nicely describes Michael Atiyah's rigorous axiomatisation of TQFTs, inspired by Edward Witten. They are functors that keep their monoidal structure from bordisms to vector spaces.
Bordisms (sometimes termed cobordisms) are higher-dimensional manifolds that describe “space” at a specific moment and are “morphisms” between lower-dimensional manifolds. Bordisms are orientated manifolds of one higher dimension with the disjoint union of Σ1 and Σ2 as their border. Closed, orientated (n-1)-manifolds are bordisms, and their equivalence classes are morphisms. Bordism's tensor product operation is disjoint union. Categories are built from “objects” (sets, vector spaces, topological manifolds) and “morphisms” (structure-preserving applications). Applications between categories that preserve composition and identity morphisms are factors. In an associative, commutative, unit-element category, a “tensor product” operation is called symmetric monoidal structure. The usual vector space tensor product. The TQFT factor preserves these features, making it “symmetric monoidal”.
Classification and comprehension are easier with TQFTs' geometric and topological manifold invariants. These theories physically tie (d+1)-dimensional manifolds, which describe spacetime processes, to operators between Hilbert spaces and d-dimensional manifolds, which represent space, to Hilbert spaces (quantum states). This idea motivates TQFTs to link general relativity with quantum theory. Z(Σ) has the same dimension as a TQFT Z applied to a product manifold Σ × S1.
Challenges to Common Purview: Topological Order Beyond TQFT
Contrary to common perception, 'Topological Order Beyond Quantum Field Theory' has proven that the lowest-energy excitations of these new theories are not invariably linked to a finite number of localised anyons of a TQFT. These gapped lowest-energy excitations are anyons that densely cover the system, not TQFTs.
These new models are distinguished by their distance-dependent interactions between anyons, which are expected in realistic experimental setups. However, axiomatically defined TQFTs are metrically independent. Even at infrared wavelengths, the new TQO systems' anyon excitations become metric dependent. The system obviously demonstrates TQO, but this metric dependency and the dense occupation of anyons prevent these theories' low-energy region from being related to a regular TQFT.
The researchers examined these models by performing exact dualities on traditional (Landau) ordering systems. This innovative method transfers Landau-type theories to dual models with topological order while maintaining spatial dimension. The star-plaquette product model (SPPM), a Zq (q≥2) extension of the Kitaev toric code (TC) stabiliser Hamiltonian, received special attention.
Dual high-dimensional classical systems with large free-energy barriers are produced by the SPPM and its extensions, which include stabiliser group elements beyond independent generators. This makes them more thermally stable than the standard Zq TC model, which is dual to decoupled 1D chains and thermally fragile Non-Abelian, string-net, and higher-dimensional models can use this approach, not simply SPPMs.
Quantum Computing Implications
The most important physical consequence of this research is that it may lead to heat-resistant quantum memory. Surface codes reduce errors well, but thermal noise and temperature increase error rates. Current research may be able to construct quantum memory structures that are less susceptible to heat disruptions, enabling more reliable and stable quantum information storage at higher temperatures. These models realise unstudied quantum codes.
This departure from typical TQFT explanations shows that future theoretical models will need more components to accurately reflect topological matter's complex dynamics. This opens up exciting new possibilities for improving quantum technologies and basic physics.
Functors
I keep confusing myself with these, and references to List and Maybe as Functors is also confusing so I want to clear this up for myself (and provide a reference for myself in the future in case I forget).
Its not untrue that they’re Functors but they are also Not Functors sort of. They’re more like domains that can be lifted to by Functors if that makes more sense? But since there isn’t a name for these Functors, we call it a List Functor or a Maybe Functor I suppose.
So the Functor Law (from the Haskell Wiki - a Quite Reliable Source) says that Functors must follow two Laws. Identity preservations (fairly trivial), and composition which is a little more complicated. I think Id is fairly trivial so I’m going to ignore Id.
First things first, basics. fmap is defined as fmap :: (a -> b) -> f a -> f b. The f is referring to the (a -> b)
So when we talk about List or Maybe being functors what we mean is that we can take some Function f defined as f :: a -> b and then call fmap f (Maybe a) and we will get Maybe b out of it, which means that we can do a -> b without taking it out of the Maybe “wrapper”. In a purely category theory sense, a functor is the transformation the normal types a and b to the wrapped types Maybe a and Maybe b and since its a functor the morphism (function) is also preserved into the Maybe domain, it turns into a Maybe a -> Maybe b.
Now, lets talk about composition (Id is simple, Id should be the same in all domains, if its a -> a then it certainly should be Maybe a -> Maybe b). Composition says composition of morphisms is preserved so
fmap (f . g) == fmap f . fmap g
I think this is most easy to do by example. Lets even use List as an example so that we can see how functions can effortlessly lift. Composition rule says that if f :: a -> b and g :: b -> c and you fmap f . g which is the same as fmap g ( f a) you will get the same as if you did the two fmaps separately. That means that fmap f . fmap g is the same as fmap g (fmap f a). Ok. Confusing. Lets make a list, lst = [1..26] and lets have some f map integers to letters f :: Int -> Char. Obviously if we fmap f lst we will get a list of of letters back (A-Z if we implement f right) but lets say we also implement g :: Char -> String where each character became a word starting with that letter. So there are two different ways of doing it right? fmap f . g will compose f and g together, which means that it goes Int -> Char -> String which becomes Int -> String and then that function is mapped over the list. Alternatively, we can do fmap f . fmap g - first we want to fmap f lst which gives us a list of Chars and then we map over g mapping Strings from the Chars in the list.
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Morphisms
It's another long, boring math post. If you are afraid of math I suggest you drop everything immediately, run in the opposite direction as fast as you can, cover your ears, and shout "LALALALA" as loud as you can while ignoring the funny looks everyone is giving you as a lot of abstract math is up ahead. Or you could just ignore this post, it's your choice. Disclaimer that I probably should of attached with my other math posts: Ok, I'm really just making up words as I go along to describe concepts for a theory I've thought up. I'm not just sticking to established theory. Chances are I will misuse official terminology, unintentionally describe theory that has already been discovered, or say nonsense. First things first, before I explain what morphisms are I'm going to explain what sets are. A set is, like everything else described in these posts, a mathematical object. They represent a collection of other ( some would say one could include itself, but that brings up so logical problems a la Russell's Paradox, unless that old dead white dude wasn't named Russell) objects where every object in existence either belongs to the set, or it does not. These objects that belong to the set are called the elements of the set. Thor total number of elements of a set is referred to the cardinality of the set, and this number is often infinite. Anyway, the important part about sets is they have certain objects in them, and each object can only be in an object once or not at all as sets simply measure whether or not objects belong or not. Now, if all the objects in set A also belong to set B, than set A is considered to be a subset of set B. Set B may be a subset of set A as well, but only if the two sets are identical to each-other, meaning they are really the same set. That's all there is to a set. Now, there is another type of objects called categories. These are thought of as types of objects, so really a category is simply a glorified set, so I guess technically you could arbitrary define any set as a category, but generally you can reason what sets would logically work as category. The set of all sets, assuming such a thing can even exist (see Russell's paradox), would make a good category. Other categories include numbers, matrices, and morphisms, which I will discuss in a moment. Generally categories are not thought of as being sets after they are declared to be categories. As sets can have subsets, categories can have subcategories, where I guess all categories are a subcategory of objects. Now for the important part. Morphisms are a broad category, a very broad and important category which I consider to be the basis of all mathematics, as I believe all other objects can be created with morphisms. Now what is a morphism? A morphism is an object that transforms a bunch of other objects into another object. The objects beings transformed are the input, or arguments, and the object created is the output. The number of objects a morphism transforms at a time, or the number of arguments of the morphism is called it's artity or something like that. Now this is probably pretty confusing, so I well give an example. + represents addition, which is a binary morphism. Binary means it has an artity of two, which means it takes two objects, which happen to both be numbers, and addition outputs a number which is an object, which also happens to be a number. The first two numbers are addition's arguments. In the expression 2+3 2 is the first argument of addition here, 3 is the second, and 5 is the output. Generally, a morphism is represented with a symbol, for example f, its arguments are represented with other symbols, for example $ and ;, and the morphism transforming the objects is represented by first placing the symbol, then placing an opening parenthese [ ( ], followed by all the arguments in order from first argument to last with commas [ , ] between them, and finished with a closing parenthese [ ) ], though they are often denoted with other notations. If we represent addition as f, two as $, and three as ;, then using this format f($,;) would represent two plus three, or we could use the normal symbols so +(2,3) is two plus three, or we could just say 2+3 like a normal person. When f($,;) is used it no longer just denotes a morphism and some objects, it denotes the output, which here is five. Keep in mind that this is not the only notation, and if you cleverly define objects you can cleverly define morphisms to prove all sorts of things. The simplest type of of morphism is the unary morphism ( well, technically it's the nonary morphism, with artity 0, but those are pointless as they just immediately transform to another object), which has one argument. This means it takes an object, and transforms it to another object. Usually transformations using them/ their output are denoted f(x), the notation mentioned in the last paragraph is used far more often in unary functions than binary operations. Now unary morphisms have some special properties. The first is that they have a set associated with them called it's domain. The domain is a set of all values were the morphism is defined. So f(x) denotes some output if x belongs to f's domain, otherwise the expression is meaningless. Likewise, it has another associated set called it's domain. This is a set of all possible outputs of the morphism. It should be noted when a unary morphism's range is it's domain, and also when it is true that for every object on the morphism's range there is exactly one object in the domain which will transform into the first object when the morphism is applied. If a unary morphism has those noted properties than it will have interesting resulting properties. The other important property of unary morphisms is that every morphism with a finite artity can be constructed from unary morphisms, which I will now prove through mathematical induction. First, we can represent a binary morphism with a pseudo-binary morphism which is actually a unary morphism. The domain of this morphism is all objects which are valid for the first argument for the binary morphism, while it's range is a set of unary morphisms. This pseudo-binary operation is denoted f2, and it's defined so that f2(x2)(x1) = f(x2,x1). The process used to create f2 and reduce f's artity by one after we input a number is called currying Now we can define any fn for any n so fn(xn) = f(n-1) for any j-nary morphism and any f1 so f1(x1) = f(xj,x[j-1],...,x1), proving the validity of the process for any operation with artity j so fj(xj)(x[j-1])...(x1) = f(xj,x[j-1],...,x1). The last thing I'd like to discuss for now is operations. Operations are morphisms that have been bound to a category, and all of it's arguments and it's output are elements of that category. For example, addition and multiplication are both binary operations of numbers. There are a few operations I'd like to discuss. The first is union, and it's an operation of sets. It's output is a set whose elements belong to ANY of the inputs. The second is also an operation of sets, and it's called union. It's output is a set whose elements belong to ALL of the inputs. The last operation I'd like to discuss is composition, and it's a binary operation of unary operations. Since I don't have the symbol here I'll denote it with •, and I'll use the standard notation used with addition. It's defined so (f•g)(x) = f(g(x)). It's extremely important to remember than it does NOT mean that f•g = f(g). TL;DR; Morphisms MORPH stuff, hence the name.
1D, 2D, circle, back
I don't know how one gets these ideas. The Box-Muller algorithm--which quickly fashions a normal distribution out of a few dice rolls--comes from the following unintuitive chain of logic.
Take a problem in a simple domain (1-D).
Put it into a more complicated domain (2-D).
Change from grid coordinates to Navy coordinates (rect → polar).
Do the problem there, then push it back.
It's normal to move problems to other domains to solve them: f⁻¹( g( f(●))). Some might say that's the whole point of math. But this is so unintuitive, to make something more convoluted before making it simpler.
Using the Fourier transform, for example, you take a wave (like music) and turn it into a frequency (like notes). Well both music and notes are understandable domains -- sometimes you want one, sometimes you want the other, so you just transition back and forth between the domains as suits. You use the equalizer to adjust things in the second domain, and then you go back to the music domain to listen to the transformed song.
Not this time, though. You go into a seemingly unrelated domain, solve the problem there, and then come back. Very weird. Or as some people like to say, "clever".
