概论
数学
发现
Andrius Kulikauskas
 ms@ms.lt
 +370 607 27 665
 My work is in the Public Domain for all to share freely.
用中文
Software

 Relate the axioms of mereology to the divisions of everything.
 Relate divisions, finite exact sequences and adjunct functors. And also relate long exact sequences, derived functors and the three cycle.
 What is the topology of holes? And what does it mean for them to be filled, unfilled or neither? And what are their complements? And how much does this topology depart from dualism? Think of there being two kinds of objects  two topologies  the cycles (shells) and the fillings/holes  and describe the relationship between these two topologies.
 Relate finite exact sequences of length 4 to the Yates index set theorem.
 For simplexes, relate barycentric form and standard form. Calculate the barycentric coordinates. For example, find the center of a tetrahedron in terms of its vertices.
 Study the chaos of watersheds for the divisions of everything  the twosome, threesome, foursome, etc. Note how a "hill" arises (for example, with the fivesome) and how that hill becomes a division into two (with the sevensome).
 Can roots of unity be thought of as divisions of everything?
 What is the difference between an exact and a nonexact relationship?
 Turing machines  inner states are "states of mind" according to Turing. How do they relate to divisions of everything?
 Study the Wolfram Axiom and Nand.
 Function relates many dimensions (the perspectives) to one dimension (the whole), just like a division. The whole is given by the operation +1. And what do +2 and +3 mean?
 0=1 yields the zero ring. In what sense is this the collapse of a system, as with the eightsome? And how is it related to the finite field {$F_1$} ?
 Interpret the boundary of a simplicial complex. Explain how the boundary of a boundary of a simplicial complex is zero. How does the boundary express orientation?
 Homologija bandyti išsakyti persitvarkymų tarpą tarp pirminės ir antrinės tvarkos.
 Study existential and universal quantifiers as adjunctions and as the basis for the arithmetical hierarchy.
 "For all" and "there exists" are adjoints presumably because they are on opposite sides of a negation wall that distinguishes the internal structure and external relationships. (That wall also distinguishes external context and internal structure.) (And algorithms?) So study that wall, for example, with regard to recursion theory.
 Reikėtų gerai išmokti aritmetinę hierarchiją ir bandyti ją taikyti kitur. Kaip jinai rūšiuoja sąvokas? Kaip ji siejasi su pirmos eilės, antros eilės ir kitokiomis logikomis? Kaip ji siejasi su tikrųjų skaičių ir kitokių skaičių tvėrimu? Kaip kvantoriai išsako septynerybę? Ar septynerybė išsako kvantorių ir neigimo derinius? Kaip jie susiję su požiūriais, požiūrių sudūrimu ir požiūrių grandinėmis, tad su kategorijų teorija ir požiūrių algebra?
 Does the Law of Forms define the sevensome?
 In an exact sequence, is a perspective the group or the homomorphism?
 What is the significance of exact functors for my philosophy?
 Finite exact sequences depend on there being a zero object as in abelian categories. The zero object is distinguished by having unique morphism to and from every other object. In what sense can everything be thought of as this zero object? How is it understood to be divided? In what sense can nothing be thought of as this zero object? How are everything and nothing related here?
 How does the snake lemma relate to the sixsome and to the meaning of life (internalization of external perspectives)?
Overview
Divisions of everything seem to appear in various places in mathematics. Most generally:
 Bott periodicity may describe the eightcycle of divisions and the related clock shifts +1, +2, +3.
 Finite exact sequences may express divisions of everything into perspectives.
 Adjoint strings may express divisions of everything into perspectives.
 Šarūnas Raudys's statistical hierarchy may be based on divisions of everything.
 The separation of a finite number of physical states by quanta, as in the case of spin.
 The real forms of a Lie algebra may express the perspectives in a division of everything.
Finite exact sequences
Abelian categories
 Abelian categories are the general framework for studying exact sequences and deviations from them.
 Exact functors are the relevant functors from and to abelian categories.
 Derived functors indicate deviations from exactness. They are the links {$R^iF(X)$} in the sequences defined by taking the kernel of the map from {$F(I^i)$} modulo the image of the map to {$F(I^i)$}. They apply the leftexact functor {$F$} to a long exact sequence of injective objects.
 The degree to which a left exact functor fails to be exact can be measured with its right derived functors; the degree to which a right exact functor fails to be exact can be measured with its left derived functors.
 If the functor F is left adjoint to G, then F is right exact and G is left exact.
 Derived category {$D(A)$} of an abelian category {$A$} has for objects the chain complexes in {$A$} and the morphisms include the derived functors.
Exact sequences
 Exact sequences of functions are those where the image of one function equals the kernel of the next function.
 Exact sequences thus consist of bisections of perspectives.
 The concept of scope: Kernel: irrelevant because goes to zero. Cokernel: irrelevant because outside of scope.
 A simplex is imagined to be embedded into 0, Zero, that which is not there. So the holes are equated to Zero if you go around them. If you go around something that is there, A, then the sum is A rather than Zero. But then the cycle around A is a boundary of A. So we mod out by such boundaries so that they don't affect our search for holes. So homology is counting the unfilled holes. "Cycles" are the boundaries of holes (filled or not); "Boundaries" are the boundaries of filled holes; "Cocycles" are... ; "Coboundaries" are...
 Qiaochu Yuan: Exact sequences are just the chain complexes with trivial homology. Chain complexes are a "linearization" of simplicial complexes in a fairly precise sense, the DoldKan correspondence.
 Qiaochu Yuan: Exact sequences are a natural abstraction of the notion of generators and relations. let R be a ring and M a left Rmodule with generating set S. Then there is a canonical surjection RS→fM→0. The kernel of this surjection describes all the possible relations in S and gives rise to a short exact sequence 0→ker(f)→RS→fM→0. If R is a Principal Ideal Domain, then ker(f) is free, so picking a basis for ker(f) gives an irredundant set of relations among the generators. However, if ker(f) is not free, then picking a defining set of relations T (that is, a generating set in ker(f)) instead gives rise to an exact sequence 0→ker(g)→RT→gRS→fM→0. If ker(g) is not free, then... and so on. From this perspective we are thinking of exact sequences as resolutions.
 Jack Schmidt: Exact sequences are basically a way to keep track of syzygies. Roger Wiegand: Given a commutative ring R, a finitely generated Rmodule M with generators z1, ..., zn, then a syzygy of M is an element (a1,...,an) of Rn for which a1z1 + ... + anzn = 0. Given a generating set, the set of all syzygies is a submodule of Rn, the module of syzygies. This module of syzygies of M is the kernel of the map Rn>M that takes the standard basis elements of Rn to the given set of generators.
 Jason Polak: Short and long exact sequences come up in the question: does A⊗R− preserve a certain injective map? Dually, you can ask whether Hom(A,−) preserves a certain surjective map.
 Dan Rust: A chain complex C of maps di is a sequence ⋯→Ai+1→di+1Ai→diAi−1→⋯ such that di∘di+1=0 for all i. We know that imdi+1⊂kerdi and so we can take a quotient. Let Hn(C)=kerdn/imdn+1. We call this the nth homology of the chain complex C. It turns out that the homology of C is trivial in every degree if and only if C is an exact sequence.
Related thoughts about homology
 Homology  holes  what is not there  thus a topic for explicit vs. implicit math
 A multidimensional torus has holes (Betti numbers) given by the binomial theorem.
 Note that a cross polytope has no totality  no volume and hence no "filling" but is always a cycle that is not a boundary.
 Homology and cohomology are like the relation between 0>1>2>3 and 4>5>6>7.
 Odd cohomology works like fermions, even cohomology works like bosons.
 Boris Novikov: Let X be a space and Y its subspace. If a boundary (in Y) of an ndimensional relative cycle c of X∖Y is a boundary of something in Y then one can build a proper ndimensional cycle of X from c, gluing this "something" to c.
Long exact sequences
 Long exact sequences are those which never terminate.
 Long exact sequences can be thought of the way that growth proceeds, extending the whole.
 Zigzag lemma relates to infinite revolutions along the threecycle.
 Let p: E → B be a basepointpreserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is pathconnected. Then there is a long exact sequence of homotopy groups
 Primena trejybę. Wikipedia: Homotopy groups Let p: E → B be a basepointpreserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is pathconnected. Then there is a long exact sequence of homotopy groups: {$\cdots \rightarrow \pi_n(F) \rightarrow \pi_n(E) \rightarrow \pi_n(B) \rightarrow \pi_{n1}(F) \rightarrow \cdots \rightarrow \pi_0(F) \rightarrow 0. $}
 Jack Schmidt: Exact chain complexes that go on forever in both directions are even more loosely described as "We put things in, and we take things out, and we haven't left anything out, but it's pretty hard to say where anything actually went."
Finite exact sequences are divisions of everything
 Finite exact sequences of functions are those that start and end with zero.
 In an exact sequence, the perspective is the group  it is a division of zero  where zero is everything.
 I believe that finite exact sequences can be thought of as divisions of everything.
 Finite exact sequences can be thought of as infinite exact sequences which have turned in upon themselves, giving an autoassociative function. For example, Bott periodicity describes how the infinite exact sequence folds in on itself in an eightcycle as the relationship between a matrix and its entries.
 Matematikos žinojimo rūmuose trejybės ratas sukuria autoasociatyvas sekas  jos iš begalinių "tikslių sekų" padaro baigtines tikslias sekas, tad padalinimus.
 The growth of the finite exact sequence seems to come from the middle, which keeps becoming more refined.
 An exact sequence is a way of intrinsically defining the concept of dimension. Each term in the sequence characterizes elements of a particular dimension. The terms are organized by increasing dimension. Each increase in dimension corresponds to the introduction of a new perspective which expands upon the previous dimension.
 Jack Schmidt: Resolutions are longer sequences that either go off to the left or to the right, and are more loosely "C is B with something like A removed, except the thing removed is only like A with something else removed...".
 Pascal's triangle  the zeros on either end of each row are like Everything at start and finish of an exact sequence.
Exact functors
 If the functor F is left adjoint to G, then F is a right exact functor and G is a left exact functor.
 The degree to which a left exact functor fails to be exact can be measured with its right derived functors; the degree to which a right exact functor fails to be exact can be measured with its left derived functors.
Derived functors
 A short exact sequence often gives rise to a "long exact sequence". The concept of derived functors explains and clarifies many of these observations.
 Suppose we are given a covariant left exact functor F : A → B between two abelian categories A and B. If 0 → A → B → C → 0 is a short exact sequence in A, then applying F yields the exact sequence 0 → F(A) → F(B) → F(C) and one could ask how to continue this sequence to the right to form a long exact sequence. Strictly speaking, this question is illposed, since there are always numerous different ways to continue a given exact sequence to the right. But it turns out that (if A is "nice" enough) there is one canonical way of doing so, given by the right derived functors of F. For every i≥1, there is a functor RiF: A → B, and the above sequence continues like so: 0 → F(A) → F(B) → F(C) → R1F(A) → R1F(B) → R1F(C) → R2F(A) → R2F(B) → ... . From this we see that F is an exact functor if and only if R1F = 0; so in a sense the right derived functors of F measure "how far" F is from being exact.
Interpreting mathematical structures as divisions of everything
 {$ \displaystyle 0\to 0 $} Nullsome: Identifying 0 with itself directly.
 {$ \displaystyle 0\to A\to 0 $} Onesome: Setting A equal to 0.
 {$ \displaystyle 0\to A\to B\to 0 $} Twosome: Setting A isomorphic to B.
 {$ \displaystyle 0\to A\to B\to C\to 0 $} Threesome: Breaking up B into A and C.
 {$ \displaystyle 0\to A\to B\to C\to D\to 0 $} Foursome: Kernel and cokernel. DivGradCurl. Yoneda lemma.
 {$ \displaystyle 0\to A\to B\to C\to D\to E\to 0 $} Fivesome: Euler's formula.
 {$ \displaystyle 0\to A\to B\to C\to D\to E\to F\to 0 $} Sixsome: Related to threecycles ("triangles"). A Characterization of Long Exact Sequences Coming from the Snake Lemma, Jan Stovicek.
 {$ \displaystyle 0\to A\to B\to C\to D\to E\to F\to G\to 0 $}
 {$ \displaystyle 0\to A\to B\to C\to D\to E\to F\to G\to H\to 0 $}
Think of an exact sequence as starting from everything and ending with everything. Everything from above and from below, with the two identified.
Nullsome
Onesome
 Everything may be an identity map IA and as such may be defined with regard to any particular object A, that is, person or vantage point.
Twosome
 Perhaps adjunction is the division of a monad into two perspectives, free and forgetful.
 The Jordan curve theorem defines inside and outside.
 Fixed points (as with Mandelbrot set)
 The 维基百科: BorsukUlam theorem seems to divide the nsphere into two perspectives: those antipodal points that get mapped to different points and those that get mapped to the same point. What is the source of this twosome?
Threesome  Short exact sequence
 threesome Jacobi identity
 Solvable Lie algebra  the threesome ultimately is exact, Godly, doesn't go on forever
 Similarly, the Derived series for groups, the series of commutator subgroups.
 Relate triangulated categories with representations of threesome.
 Short exact sequence. Defining a perspective relative to a base.
 Alex Youcis: Short exact sequences are algebraified versions of fiber bundles. 0→Y→X→Z→0 indicates that X is some kind of "twisted product" of Y and Z. We should be able to tell properties of X from properties of Y and Z. For example, knowing that B is an abelian groups such that 0→A→B→C→0 tells us that rank(B)=rank(A)+rank(C).
 Leewz: 0→Z2→Z2⊕Z2→Z2→0 and 0→Z2→Z4→Z2→0 have different middles but the same components. One is the direct product, and the other is a semidirect product.
 Fiber spaces (fiber bundles?) are understood as finite exact sequences and perhaps vice versa. {$F{\rightarrow}E\overset{\pi}{\rightarrow}B$}
Foursome
 Given map T: Domain T > Codomain T we have 0> ker T > Domain T > Codomain T > coker T > 0. Interpretation: given a linear equation T(v)=w to solve, the kernel is the space of solutions to the homogeneous equation T(v)=0, and its dimension is the number of degrees of freedom in a solution, if it exists; the cokernel is the space of constraints that must be satisfied if the equation is to have a solution, and its dimension is the number of constraints that must be satisfied for the equation to have a solution. The dimension of the cokernel plus the dimension of the image (the rank) add up to the dimension of the target space, as the dimension of the quotient space W/T(V) is simply the dimension of the space minus the dimension of the image. dim(Domain T)  dim(ker T) + dim(coker T) = dim(Codomain T). In other words:  dim(ker T) + dim (Domain T)  dim (Codomain T) + dim (coker T) = 0.
 Note that the foursome comes up repeatedly in the Snake Lemma.
 Relate to Yoneda lemma
 Recursive functions  There is a jump hierarchy of recursive functions that (by the Yates index set theorem) has one level be "conscious" of the level that is three levels below it, which is thus relevant for the foursome's role in consciousness.
 Reikėtų išmokti Yates Index Set Theorem, jinai pakankamai trumpa ir turbūt ne tokia sudėtinga. Ir paaiškinti ką jinai turi bendro su sąmoningumu.
 Bosons  "ryšiai" kodėl  Yoneda. Fermions  "ar".
 For complex numbers, {$0 \neq 2\pi$} and so they are different when we go around the threecycle, so they yield the foursome: 0, 120, 240, 360.
 Partially defined and totally defined functions describe different levels of knowledge as regards the foursome.
 Memory lets us return to concepts whose importance we note but may not yet comprehend. We allow ourselves to return to them later to inspect and understand them. Thus memory lets us separate selection and comprehension as if they were left and right parentheses. Memory thus creates obligations. Syntax deals with selection and semantics with comprehension. Selection (of expression) and comprehension (of content) are perhaps the two parts of understanding. Distinguishing them enriches the notion of understanding established in God's dance.
 von Neumann architecture: memory, control unit, arithmetical unit, input/output
 Consider X=X in category theory. Find how to interpret it in terms of four levels of knowledge. For example, the identity morphism may express Whether.
Fivesome
 Five lemma and the two fourlemmas.
 Analysis allows for work with limits.
 Eccentricity of conic sections  there are five eccentricities (for the circle, parabola, ellipse, hyperbola, line).
 Daniel Murfet. Metric and Hilbert spaces. Videos and lecture notes. What is space?
 Murfet: Space is the blank placed between words, characters, symbols etc.
 Space is the default asymmetry that presumes that nonexistence  emptiness  is more prominent than existence  occupied place in space. Thus we can treat nonexistence and existence differently. Time is the default asymmetry that existence is more prominent than nonexistence.
 Max Jammer books on space and time
Sixsome
 Derived functors manifest the threesome, ever perfecting one's position, increasing the kernel, the zero. {$ \displaystyle 0\to F(C)\to F(B)\to F(A)\to R^{1}F(C)\to R^{1}F(B)\to R^{1}F(A)\to R^{2}F(C)\to \cdots $}
 A circle (through polarity) defines triplets of points, and triplets of lines, thus sixsomes. The center of a circle is perhaps a fourth point (with every triplet) much like the identity is related to the threecycle?
Sevensome and eightsome
 Logic is the end result of structure, see the sevensome and Greimas' semiotic square.
 triangle: 1 unknown 3 vertices +3 edges +1 whole
Examples of exact sequences
 {$ \displaystyle 1\to N \to G \to G/N\to 1 $}
 {$ \displaystyle 1\to C_n \to D_{2n} \to C_2\to 1 $}
 {$ \displaystyle \Bbb{H}_1\ \xrightarrow{\text{grad}}\ \Bbb{H}_\text{curl}\ \xrightarrow{\text{curl}}\ \Bbb{H}_\text{div}\ \xrightarrow{\text{div}}\ \Bbb{L}_2 $}
 Let I and J be ideals in a ring R. Prove that there is an exact sequence of Rmodules (what are the maps): {$ \displaystyle 0\to {I\cap J}\to {I \oplus J} \to {I+J} \to 0 $} Gathman. Exact sequences.
 1→SLn(F)→GLn(F)→F×→1
 0→Z→R→R/Z→0
 A fibre bundle F→E→B induces a long exact sequence. If F→E is the homotopy fibre of E→B, then we get a long exact sequence …→πn(F)→πn(E)→πn(B)→πn−1(F)→πn−1(E)→….
 Binomial theorem: Euler's formula for vertices, faces, edges: 0→Z[S]→Z[F]→Z[E]→Z[V]→Z[e]→0 Reference
 0→im(f)→B→cok(f)→0 is exact, for f:A→B
 0→ker(f)→A→fB→cok(f)→0 is exact
 If A→aB→bC→0 and 0→C→cD→dE are exact, then A→aB−→cbD→dE is exact
 0→ker(f)→A→fim(f)→0 is exact, for f:A→B
 Inclusionexclusion Reference
 Short exact sequence of a projective hypersurface: line bundle.
 Short exact sequence of a complete intersection 0→R(st)→R(s) sum R(t)→I→0. See also scheme theoretic intersection.
Study materials
Videos
Other possibilities
Spin states
Spin 1/2 means there are two states separated by a quanta of energy +/ h.
So this is like divisions of everything:
 Spin 0 total spin: onesome
 Spin 1/2: fermions: twosome
 Spin 1: three states: threesome
 Spin 3/2: composite particles: foursome
 Spin 2: graviton: fivesome (time/space)
The real forms of a Lie algebra
 Real forms are similar to Clifford algebras in that in ndimensions you have n versions, n signatures. Each models the taking up of one of the n perspectives in a division of everything
 Perspective:Division = real form:complex Lie algebra/group = real Clifford algebra / complex Clifford algebra
 Circle (threecycle) vs. Line (link to unconditional)  sixsome  and real forms.
Notes
 Relate adjoint functors and exact functors with divisions of everything.
 A type is inhabited by a term. Or it may have no term and not be inhabited by anything. This logical distinction is vital for the sevensome.
 What does it mean to have no functor? How do we describe the nullsome?
 Long sequence expresses the endless perfection by the threecycle.
Partition of unity is not quite related.
 Shulman: twosome: Equality in mathematics is a proposition, and in particular something that can be hypothesized and proven or disproven
 What are the conditions in a short exact sequence 0  A  B  C  0 that let B be the direct sum of A and C ?
 Ravi Vakil. Puzzling Through Exact Sequences.
