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Andrius Kulikauskas

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Tensor Hom adjunction

  • (AxB->C) left adjoint to (A->(B->C))
  • general function left adjoint to values on elements
  • free construction left adjoint to forgetful

4 logics for 4 geometries

  • no simplification - no distance between metalevel and level - affine - contradiction
  • simplify by one perspective relative to the center - get model
  • simplify by two perspectives - get directions, forward and backward
  • simplify by three perspectives - get variables, defined from by the side view

Yoshimasa Nakamura, Alexei Zhedanov. Toda Chain, Sheffer Class of Orthogonal Polynomials and Combinatorial Numbers

S. J. Rapeli, Pratik Shah and A. K. Shukla. Remark on Sheffer Polynomials explains J(D), relates it to A(t)

House rules:

  • The policy of the many
  • Leave things the way you found them
  • Respect constancy

Jesus Antonio: Key to physics is information and symmetry

What is the combinatorics of convex spaces and how does that relate to orthogonal polynomials, which give different ways of looking at the geometry?

5 notions of independency

What are the transition matrices between orthogonal polynomials?

https://www.nature.com/articles/s41534-018-0106-y#:~:text=Surface%20codes%20are%20building%20blocks,models%20describing%20random%20Pauli%20errors

https://en.wikipedia.org/wiki/Probabilistic_programming

https://mathoverflow.net/questions/118857/forcing-in-homotopy-type-theory

probabilistic programming paradigm (quantum computing)

Measurement based quantum computer vs gate based quantum computer

lattice surgery

https://en.wikipedia.org/wiki/Toric_code

topological quantum computer

https://en.wikipedia.org/wiki/One-way_quantum_computer

https://www.aimath.org/WWN/convexalggeom/AIM.pdf

Naturality in homotopy type theory breaks down when we try to do type theory in type theory.

A special case of the universal property of identity types is related to the Yoneda lemma.

Amelia: [The axiom of function extensionality is] inconsistent with many axioms of a more "computational" nature. For example, "formal Church's thesis" says that for any function N→N, there is a "program" (we call it a realizer) that realizes it. You can kinda see what goes wrong: this would be able to tell e.g. "λ x → x" and "λ x → x + 0" apart. You could imagine an assignment of realizers that sidesteps this, though, so to see that it's actually inconsistent takes slightly more work.

What is the relationship between universal properties as proved by the function extensionality principle, and universal properties as given by Kan extensions?

https://github.com/FrozenWinters/stlc SLTC project where Astra formalises the categorical semantics of function types in Agda.

A063573 Counts the number S(n) of lambda terms at level n, in the case of a single variable.

  • Let V be the number of variables.
  • {$S(n+1) = VS(n) + 2S(n)\sum_{i=0}^{n-1}S(i) + S(n)S(n)$}
  • This comes from two steps.
    • Add {$\lambda x.\_ $} in front of a lambda term from level n.
    • Combine two lambda terms {$( \_\;\_ )$} at least one of which comes from {$S(n)$}.
    • When V=1 we get 1,2,10,170,33490...
    • When V=2 we get 2,8,112,15008...

Calculate the combinatorics of the lambda-calculus on a single variable, and if possible, on two or more variables. Is the lambda-calculus equivalent to the recursion relation for orthogonal polynomials?

one-projectionall-constantmany-successor
recursion - all & manycomposition - many & oneminimization - one & all

Have all finite limits is equivalent to

  • Having terminal objects
  • Having a product for any pair of objects
  • Having an equalizer for any pair of parallel arrows

These are the building blocks for limits

Sean Carroll or me? Quantum field theory. Instead of space and time, consider in terms of particles and their interactions. Particle clock steps take us from possible interaction to possible interaction. Problem: field theory is based on Minkowski spacetime rather than on particles.

One-all-many relates questions (selection) and answers (judgement). Many is the regularity that every question is answered relevantly. Zero is "no" as a positive answer.

Induction argument on truncation levels uses the level below (for identities) and the level above (which we're trying to reach). Similarly, the recurrence relation relates the level xP_n(x) with the level below and the level above.

  • Andrius: I wonder if there are any connections with the arithmetical hierarchy in computability theory. In that hierarchy, the sigmas and the pis are intermixed. So I wonder if there are any ways that pis (products) get interspersed between the truncation levels?
  • Astra: If you have a Pi-type, then the truncatedness level is that of the codomain, so this behavious is a bit different from what you would see in that hierarchy

Homotopy Type Theory

  • Substitution for variables - binding and scope
  • Types are specifications are programs
  • Communicating by algorithm and certain shared assumptions. Discover those assumptions.
  • Term M (program - that when it runs) in type A (program - it runs the way A says it runs)

An empty type has no evidence for it, is not true. A nonempty type, as a proposition, is true. The notion of empty or nonempty is relevant for the sevensome, for describing {$\forall \wedge \exists$}.

How is Yoneda lemma related to matrix row manipulation? And how might that help relate Cramer's rule to Kan extensions and the Yoneda lemma?

Young-Il Choo - MeetUp

The inclusion of Field in CRing has no left adjoint because it would carry Z to an initial field, which does not exist. How might an initial field relate to the field with one element?

Riehl: 2.4. The category of elements A universal property for an object c ∈ C is expressed either by a contravariant functor F together with a representation C (−, c)  F or by a covariant functor F together with a representation C (c, −)  F. The representations define a natural characterization of the maps into (in the contravariant case) or out of (in the covariant case) the object c. Proposition 2.3.1 implies that a universal property characterizes the object c ∈ C up to isomorphism. More precisely, there is a unique isomorphism between c and any other object representing F that commutes with the chosen representations. In such contexts, the phrase “c is the universal object in C with an x” assets that x ∈ Fc is a universal element in the sense of Definition 2.3.3, i.e., x is the element of Fc that classifies the natural isomorphism that defines the representation by the Yoneda lemma. In this section, we prove that the term “universal” is being used in the precise sense alluded to at the beginning of this chapter: the universal element is either initial or terminal in an appropriate category. The category in question, called the category of elements, can be constructed in a canonical way from the data of the representable functor F. The main result of this section, Proposition 2.4.8, proves that any universal property can be understood as defining an initial or terminal object, as variance dictates, completing the promise made in §2.1.

Riehl, page 50: The Yoneda lemma is arguably the most important result in category theory, although it takes some time to explore the depths of the consequences of this simple statement. In §2.3, we define the notion of universal element that witnesses a universal property of some object in a locally small category. The universal element witnessing the universal property of the complete graph is an n-coloring of K n , an element of the set n-Color(K n ). In §2.4, we use the Yoneda lemma to show that the pair comprised of an object characterized by a universal property and its universal element defines either an initial or a terminal object in the category of elements of the functor that it represents. This gives precise meaning to the term universal: it is a synonym for either “initial” or “terminal,” with context disambiguating between the two cases. For instance, K n is the terminal n-colored graph: the terminal object in the category of n-colored graphs and graph homomorphisms that preserve the coloring of vertices.

Comma category Lawvere showed that the functors F : C → D {\displaystyle F:{\mathcal {C}}\rightarrow {\mathcal {D}}} F:{\mathcal {C}}\rightarrow {\mathcal {D}} and G : D → C {\displaystyle G:{\mathcal {D}}\rightarrow {\mathcal {C}}} G:{\mathcal {D}}\rightarrow {\mathcal {C}} are adjoint if and only if the comma categories ( F ↓ i d D ) {\displaystyle (F\downarrow id_{\mathcal {D}})} (F\downarrow id_{{\mathcal {D}}}) and ( i d C ↓ G ) {\displaystyle (id_{\mathcal {C}}\downarrow G)} (id_{{\mathcal {C}}}\downarrow G), with i d D {\displaystyle id_{\mathcal {D}}} id_{{\mathcal {D}}} and i d C {\displaystyle id_{\mathcal {C}}} id_{{\mathcal {C}}} the identity functors on D {\displaystyle {\mathcal {D}}} {\mathcal {D}} and C {\displaystyle {\mathcal {C}}} {\mathcal {C}} respectively, are isomorphic, and equivalent elements in the comma category can be projected onto the same element of C × D {\displaystyle {\mathcal {C}}\times {\mathcal {D}}} {\mathcal {C}}\times {\mathcal {D}}. This allows adjunctions to be described without involving sets, and was in fact the original motivation for introducing comma categories.

Every type has a unique name. Every universe is a type with a unique name. Every term in a type should have a unique name. So why can't we have a universe of unique names for all of the terms, types and universes? And if we can, then don't we run into a paradox? Or not?

https://www.youtube.com/watch?v=ylM1bPtVftk The 4th movement of Beethoven's Symphony No. 5. Conducted by Arthur Nikisch. Recorded in 1913.

{$\alpha$} and {$\beta$} count ascents and descents and these are steps forwards or backwards in the unfolding of space (in time?) and so they may relate to John's picture of evolution taking us forward and backward in time.

Roelof Koekoek & René F. Swarttouw. The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue

Space has 3 dimensions external to the fivesome (5+3=0)(outside the division) and time has 1 dimension internal to the fivesome (the slack inside the division).

https://ww3.math.ucla.edu/dls/emily-riehl/ video about contractibility

An isomorphism is a special morphism but truly it is a pair of morphisms that are inverses to each other. There may be many such pairs relating two objects but in each pair the inverses are unique with respect to each other. So it is similar to complex conjugation.

Bell number interpretation of Sheffer polynomials gives a foundation for (finite) (and countable) set theory.

Mariana M. Odashima, Beatriz G. Prado, E. Vernek. Pedagogical introduction to equilibrium Green's functions: condensed matter examples with numerical implementations.

https://math.stackexchange.com/questions/989083/is-composition-of-covering-maps-covering-map

https://52livingideas.com

Charlier polynomials give the trivial space wrapper (the moments are the Bell numbers). In what way are the Hermite polynomials trivial?

Space wrappers reinterpret Bell numbers.

Types indicate comparability which is a condition for equality.

From a dream: I imagined that I was entering a spherical world full of structures, and that my perspective upon those structures was a hyperbolic geometry, expressing the Lorentz contraction, thus special relativity.

Comma category Adjunctions
  • Lawvere showed that the functors F : C → D {\displaystyle F:{\mathcal {C}}\rightarrow {\mathcal {D}}} F:{\mathcal {C}}\rightarrow {\mathcal {D}} and G : D → C {\displaystyle G:{\mathcal {D}}\rightarrow {\mathcal {C}}} G:{\mathcal {D}}\rightarrow {\mathcal {C}} are adjoint if and only if the comma categories ( F ↓ i d D ) {\displaystyle (F\downarrow id_{\mathcal {D}})} (F\downarrow id_{{\mathcal {D}}}) and ( i d C ↓ G ) {\displaystyle (id_{\mathcal {C}}\downarrow G)} (id_{{\mathcal {C}}}\downarrow G), with i d D {\displaystyle id_{\mathcal {D}}} id_{{\mathcal {D}}} and i d C {\displaystyle id_{\mathcal {C}}} id_{{\mathcal {C}}} the identity functors on D {\displaystyle {\mathcal {D}}} {\mathcal {D}} and C {\displaystyle {\mathcal {C}}} {\mathcal {C}} respectively, are isomorphic, and equivalent elements in the comma category can be projected onto the same element of C × D {\displaystyle {\mathcal {C}}\times {\mathcal {D}}} {\mathcal {C}}\times {\mathcal {D}}. This allows adjunctions to be described without involving sets, and was in fact the original motivation for introducing comma categories.
  • Relate to the definition of adjunctions in terms of the universal mapping property.
  • Consider how HomSets come into play.

OpenShot eksportuoti 30 fps nes iPhone filmuoja 30 fps

Open Source Software to Thank

  • Linux, Ubuntu, OpenShot, Dia, GIMP

Shot with an iPhone XS Max.

https://www.youtube.com/watch?v=tOp2rdvOmd0 Schuller on Stone's Theorem

https://www.freelists.org išbandyti?

https://math.stackexchange.com/questions/69698/wedge-sum-of-circles-and-the-hawaiian-earring

https://www.facebook.com/hackersatcambridge contact team @ hackersatcambridge.com

portray mu as measuring tape

portray mu as super hero measuring tape with two hands ready to hold on

  • Composition - roots
  • Primitive recursion - runners
  • Minimization operator - seed

Weed in cracks of cement

Conjugacy ? the values of adjunction

In defining the minimization operator, and in coding a list of natural numbers with a single natural number:

  • It is problematic to code "nonhalting" as an integer because it cannot serve as an input. So it would have to be an integer that cannot be used as an input. In this way it is an input that got deleted. And so this is where the power of computing is increased, through the introduction of deletion of a variable.

Minimization operator: representations of nullsome have us proceed through all levels (from true to direct, from direct to constant, from constant to significant)

  • necessary (0 is constancy), actual (the meaning of 0), possible (if starts, then ends)
  • object, process, subject

Constancy - search for meaning

  • one, all, many

Significance - go beyond

  • being, doing, thinking

In the search for constancy: take a stand (as to one), follow through (across all), reflect (supposing many)

In physics, orthogonal polynomials relate what is necessary (top down) and actual (bottom up) as with string theory, questions and answers.

The original spectral theorem: Look for subrepresentations such that S is a one-dimensional matrix eigenvalue. Induction argument.

Classical (both x, p) and quantum (x).

Bald and bankrupt Eastern Europe

The Screwing of the Average man: How the rich get richer and you get poorer

Special relativity - causal connection - are they time like connected.

Wick's theorem - are operators of the same particles - propagator connects

Evolution is indicated by learnability and also by sparse communication and natural differences between hierarchies, different orders of magnitude, allowing for a natural hierarchy of niches. Not only the laws of physics are sparse but also the states in nature are sparse.

Rules of physics plus configuration space plus location within that space.

Source of contradiction

  • We are finite, our system is finite, but the Spirit is infinite dimensional

Self-adjoint operators

  • Stone's theorem (the dynamical evolution)
  • Spectral theorem (the structure): One-to-one connection between projections (measure valued projections) and self-adjoint operators.

Quantum measurement projects into eigenstate. The projection operator is a mathematical statement of the collapse of the wave function. If you do it twice, then you don't get anything more.

Self-adjoint operators are weighted sums of projection operators. The weights you can find from experiments by applying a projection operator.

Hamiltonian is the sum of all the projections onto the energy eigenstates with the energies being the weights.

Uncertainty principle - has to do with representations - representation adds a perspective - so that interferes with measuring certain things.

Minimization operator mu - superhero - who clings to ledges and other such things and is stretched and blown by the wind. And the shape mu gives the shape of his body clinging to the left.

https://en.wikipedia.org/wiki/%CE%9C_operator {$\mu$}-operator

I had a dream that i was professor anthony zee... But in a quantum superposition. Was i z or not z ? Z or not z? ..... is there a third way? Yes but there is a fourth way .... Nevermind z here is m4w!

A qubit specifies the relation between affirmation and negation of probabilities. In matrix form, it provides a complex number which is the coefficient that gets multiplied to the negation (in calculating the new affirmation) and whose conjugate gets multipled to the affirmation (in calculating the new negation). In classical bits, this coefficient is simply zero.

Bekaert, Boulanger. The unitary representations of the Poincare group in any spacetime dimension

Five zones of scattering can be thought of as

Measurement establishes a quantity with regard to boundaries - it establishes the zone within which it is - identifies with a step in the algebra - whereas analysis demarcates the boundaries.

Algebra is thinking step-by-step and so it exhibits finiteness. Analysis is discovering the boundaries between steps and so exhibits continuity by discovering the critical points, as in Moore's theory. Healthy irony (verbalization) codes the analogue signal (the emotional tension between expression and meaning) into a discrete alphabet (of boxes organized in a cognitive network).

Antonio

Wenbo

  • Semi-join lattice.

Oliver

  • Distributional semantics

https://music.163.com/#/playlist?id=797393474 Wenbo

The Topos of Music: Geometric Logic of Concepts, Theory, and Performance - worse

Cool Math for Hot Music - better

Midori

https://www.youtube.com/watch?v=OOsRMECWKAE

Recuerdos de la alambra

Kojin Karatani, Sabu Kohso - Architecture as Metaphor_ Language, Number, Money (1995) semi-join lattice semilattice

Hatcher exercise

Osborne IV 40:00 what is needed for a relativistic quantum field theory.

Brody. Quantum Mechanics and Riemann Hypothesis.

Brown. Topology and Groupoids.

Yevgeny B. Karasik. Duality revolution: Discovery of new types and mechanisms of duality that are revolutionizing science and technology as well as our ability to solve problems

Think of -1-cell as the center (of all things), the spirit. And think of 0-cell not simply as a point but as a 0-dimensional open arc (the point shell) with regard to that center (the spirit). The point shells are glued onto the spirit, and similarly, open arcs are glued onto point shells, and so on, inductively.

https://www.thphys.uni-heidelberg.de/~floerchinger/categories/ Quantum Field Theory

https://bookstore.ams.org/surv-149/12

https://physics.stackexchange.com/questions/566963/what-is-the-exact-meaning-of-lorentz-invariance-for-a-quantum-scalar-field

https://ncatlab.org/nlab/show/cellular+approximation+theorem#applications

Any functor {$F$} can be thought of as {$F:J\rightarrow C$} where {$J$} is the shape, the index set (how) and {$C$} is the image (what). And for any functor we can ask if it has a limit (an object {$L$} with maps {$\psi_X:L\rightarrow F(X)$} for all objects {$X$} of {$J$}, such that for all {$f:X\rightarrow Y$} in {$J$} we have the analogous morphisms commute in {$C$}, and that is universal as such). So that limit is Why. And is the colimit Whether? Or is whether simply the object with its identity morphism? Compare with the Yoneda lemma.

Relate walks on trees to covering groups. What do conjugates (paths) mean? What is the homotopy group?

https://en.wikipedia.org/wiki/Koopman%E2%80%93von_Neumann_classical_mechanics

https://www.amazon.com/Quantum-Challenge-Foundations-Mechanics-Astronomy/dp/076372470X

In what sense are Feynman diagrams relativistic given that they have directions for time and for space?

Instead of thinking of speed of light, think of a clock that doesn't tick, so that t=0 always. And this is the case for the quantum harmonic osciallator and for the particle-clocks with no steps.

One {$\exists x$}, all {$\forall x$}, many {$\neg\exists x \wedge \neg\forall x$}.

Masaki Kashiwara, Pierre Schapira. Categories and Sheaves. 2006

Gerald B. Folland

  • Quantum Field Theory: A Tourist Guide for Mathematicians 2021
  • Quantum Field Theory 2008

Bohm Pilot Wave, Thomas Spencer

Relative invariance - more global than another

Relate the three-cycle (taking a stand, following through, reflecting) to Kan extension as defined by filtering objects and morphisms, acting on them, and then taking the colimit or limit on that filter.

Alan Turing, Cybernetics and the Secrets of Life

My approach to special relativity lets me work in units in my own frame.

https://en.wikipedia.org/wiki/Algorithmic_information_theory Gregory Chaitin = Shannon + Turing = Compression-Decompression as understanding.

https://en.wikipedia.org/wiki/Cristian_S._Calude Philosophy of computation

https://www.youtube.com/watch?v=xP5-iIeKXE8 Life in life

Thinking about the expansion of the universe as a reduction of density, by which the mass of particles becomes ever less important, by which we have an increase of entropy (becoming less deliberate). And we can reverse this by starting with an increase in entropy and arriving at the expansion of the universe.

Relate Ellerman's heteromorphism and comma category.

Consider how the understanding of Yoneda lemma in terms of a left Kan extension, and in particular, the factoring, relates to the push down automata.

San Francisco Meet Up interests: Dependently typed programming languages. Language aspects of category theory. Functional programming. Topos, lambda calculus. Is type theory advantageous? Modeling infinitesimals.

Kan extensions are a framework for universality. Consider example 6.1.3, the Yoneda Lemma. Think in which ways the universality of limits, colimits, adjunctions, etc. is captured by Kan extensions. All of these universal properties can be thought of in terms of initial or terminal objects in the appropriate categories, such as the category of cones, or the comma category for the universal mapping property for adjunctions. So consider the relevant categories. How do they relate to the classification of adjoint strings?

Dan Shiebler. Kan Extensions in Data Science and Machine Learning

Matematika išplaukia iš (poreikių tenkinimo) algoritmų taikymo, vedančio iš duotybių į bendrybes. O tos bendrybės įkūnija, išreiškia tam tikrus prieštaravimus, juos paverčia sąvokomis, kurias galima mąstyti toliau. Pavyzdžiui, apskritimas iškyla iš begalinės simetrijos visom kryptim, arba iš virve aprėpto ploto maksimalizavimo.

  • Vaizduotė (24 matai) ir Neįsivaizduojamieji (2 matai) yra iš viso 26 matai. Ar juos išreiškia stygų teorija?

Mathematics is described in terms of set theory. The category of graphs {$\textrm{Set}^{A}$} where {$A$} is the category with two objects, edges E and vertices V, and two nontrivial morphisms target {$t:E\rightarrow V$} and source {$s:E\rightarrow V$}. Similarly, all mathematical structures and their structure preserving morphisms should have a similar expression in terms of sets and their relationshps. Work out various examples. Then study the role of {$\textrm{Set}^{X}$} in the Yoneda Lemma.

SL(2,C) character variety related to hyperbolic geometry. SL2(C) character varieties

Universal enveloping algebra is an abstraction where the generators are free and thus yield infinite generators. Whereas the Lie algebra may be in terms of concrete matrices and the underlying generators, when understood not in terms of the Lie bracket but in terms of matrix multiplication, may have relations such as {$x^2=0$}, {$h^2=1$}.

Information is what you learn. What you learn grows at the boundary, has the shape of the boundary. A shape can be thought of as being created by integrating over these boundaries as they increase.

Tai-Danae Bradley: Information is on the Boundary

Prove that the matrix made up of eigenvectors diagonalizes a matrix.

In special relativity, think of distance squared over time as surface area per time, the difference beween the surface areas of two spheres, one expanding with velocity v, and the other with velocity c.

For John: How could we get negative energy? Consider how to get imaginary square roots. For example, if a speed is greater than the speed of light, then the relationship between time and position is multiplied by an imaginary number.

Quaternions, Dirac equation: Pauli matrices are the three-cycle for learning and they are extended by a fourth dimension of non-learning (what is absolutely true or false) for the foursome.

https://en.wikipedia.org/wiki/Lebesgue_covering_dimension

Covering spaces with repetition yield the spaces they cover.

Enveloping algebra (important for adjunctions) is related to Hochschild cohomology.

Think of combinatorial quantum physics operator (raising and lowering operator) as a derived functor.

Shmuel Weinberger. Algebraic theory of topology. Manifolds and K-theory: the legacy of Andrew Ranicki

Lifting a path is like inverting a functor. How is that related to adjunctions? Adjunction is conditional inversion.

Unclear whether the empty space is path connected.

Think of a universal covering space as expressing the unfolding of a space, thus expressing eternal life.

Wikipedia Library

Peace conference

Relate triangulated categories (with squiggles {$X\rightsquigarrow W = X\rightarrow TW$}) to monads with likewise squiggles.

Samwel Kongere vaizdo įrašai

Nafsi Afrika Acrobats - Pyramid of Peace

Research/Notes

  • Monads deal with scopes: none, some, and so on. The logic of the sevensome.

https://www.wolframphysics.org/technical-introduction/potential-relation-to-physics/motion-and-special-relativity/

Relate {$F_1$} with the basis element 1 in a Clifford algebra.

The house of knowledge for mathematics describes 4 representations (properties) of everything (onesome, totality), which through their unity establish, define space as algebraic, consisting of enumerated dimensions:

  • center (nullsome)
  • balance
  • set of roots of a polynomial
  • list of basis vectors

(Relate this to the binomial theorem.) And it describes 4 representation of the nullsome (center), which through their unity establish, define a point as analytic. This describes four choices:

  • induction (adding a vertex, converting the center to a vertex, recursively)
  • max or min (adding an axis, as with cross polytopes)
  • least upper or greatest lower bound (making a division, a separation on one side or the other)
  • limit (center?)

Meanings are variously related by adjunctions. They enrich the meaning and extend the context.

  • Think of my understanding of my three grandfathers as changing with context.

4 levels of knowledge is sufficient (in the chain complexes). The house of knowledge describes those 4 levels. It relates the analytical view of a point with the algebraic view of a space. Consider the Zig Zag Lemma as applying the three-cycle to set up four levels of knowledge, 4 x 3 = 12 circumstances.

Counterquestions

  • Consider them as a subset of the utility graph {$K_{3,3}$} which describes the three utilities problem and arises in the proof of Kuratowski's theorem characterizing planar graphs.
  • The utility graph can be drawn as a hexagon, in which case only one graph can cross the center if it is to be a planar graph. In that case the center line goes from God's perspective to the world's situation. And this arrangement makes person-in-general and person-in-particular equal in status. Thus it provides a context for such equality of status. And it defines a division of everything into two: general (not knowing) and particular (knowing). It supports the equality of gender.
  • Consider how the counterquestions define divisions of everything and relate to Bott periodicity.
  • Consider how the counterquestions arise in Jesus's house of knowledge and how that relates to the house of knowledge for mathematics.
  • Consider how the counterquestions express visualization and paradox.

Enveloping algebra (important for adjunctions) is related to Hochschild cohomology which is a special case of the functor Ext.

  • {$\operatorname{Hom}_{A^e}(A,M)$} (where {$A^e:=A\otimes_k A^{op}$} is the enveloping algebra of A and A is considered an A-bimodule via the usual left and right multiplication)

Think of combinatorial quantum physics operator (raising and lowering operator) as a derived functor.

Counterquestions

Potential energy source {$\frac{y}{2}$} (adding free cell) is in balance with kinetic energy source {$i\frac{\partial}{\partial y}$} (half-link)

A chain complex is loose and has slack, which is the basis for homology. An exact sequence is tight and has no slack. A division of everything is tight and has no slack.

  • Santykis su Dievu yra atgarsis, kaip kad dalelytė turi santykį su savo lauku.
  • Fizikos dėsnių raida yra pavyzdys Dievo įsakymo patobulinimo.

Fivesome

John Baez, Michael Shulman. Lectures on n-Categories and Cohomology.

  • (-1)-categories are hom(x,y) sets where x and y are parallel 0-morphisms in a 0-category, which is to say, a set. But the only 0-morphisms in a set are the identity morphisms. Thus hom(x,y) is either an identity morphism (when x=y) or the empty set (otherwise). These are the two possible (-1)-categories.
  • (-2)-categories are hom(x,y) sets where x and y are -1-morphisms in a -1-category. But there is only one non-empty (-1)-category and it has only one morphism. Thus there is only one (-2)-category and it consists of this unique morphism. This category expresses necessary equality when there is only one choice. That is reminiscent of the choice from a single choice which is modeled by {$F_1$}, the field with one element.

Note that there is only one empty set. But there could also be many empty sets. And all can be thought of as an empty set. Can the search for constancy be considered a search for emptiness?

Emily Riehl, Dominic Verity. Elements of ∞-Category Theory

Foursome

For C and D categories we have

  • f is (essentially) 0-surjective {$⇔$} f is (essentially) surjective on objects;
  • f is (essentially) 1-surjective {$⇔$} f is full;
  • f is (essentially) 2-surjective {$⇔$} f is faithful;
  • f is always 3-surjective.

Foursome

A functor between ordinary categories (1-categories) can be:

  • essentially surjective ≃ essentially 0-surjective
  • full ≃ essentially 1-surjective
  • faithful ≃ essentially 2-surjective
  • Every 1-functor is essentially k-surjective for all k≥3.

A functor {$F:C→D$} is essentially surjective if it is surjective on objects “up to isomorphism”: If for every object {$y$} of {$D$}, there exists an object {$x$} of {$C$} and an isomorphism {$F(x)≅y$} in D.

A functor F:C→D can be:

essentially (k≥0)-surjectiveforgets nothingremembers everything
essentially (k≥1)-surjectiveforgets only propertiesremembers at least stuff and structure
essentially (k≥2)-surjectiveforgets at most structureremembers at least stuff
essentially (k≥3)-surjectivemay forget everythingmay remember nothing

This formalism captures the intuition of how “stuff”, “structure”, and “properties” are expected to be related:

  • stuff may be equipped with structure;
  • structure may have (be equipped with) properties.
  • Install: OBS Studio

Yoneda lemma lets you go from natural isomorphism of homsets to natural isomorphism of functors.

Are the doubts and counterquestions related to electromagnetism, U(1) and the related gauge theory?

Observing symmetry requires breaking symmetry.

https://www.masterclass.com/classes/terence-tao-teaches-mathematical-thinking

Is the associativity diagram for monoidal categories an example of the fivesome?

Involution is square root of permutation. Compare with spin as square root of geometry.

Math Discovery

  • How is gravity related to the argument by continuity?

Local and global quantum are linked by experiments, by "the complicated interplay between infrared and ultraviolet affects", by a conspiracy of IR/UV mixing.

Walks

  • Independent entries vs. Rotational invariance yield {$P[X]\propto e^{-\frac{1}{2}\textrm{Tr}X^2}$}.
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Šis puslapis paskutinį kartą keistas August 10, 2022, at 10:35 AM