- 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 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).
Fiber spaces (fiber bundles?) are understood as finite exact sequences and perhaps vice versa. This outlook is key for Bott periodicity.
{$F{\rightarrow}E\overset{\pi}{\rightarrow}B$}
A perspective is a map with a fixed point. Study fixed point theorems, such as Brower's fixed point theorem, and the inverse function theorem and implicit function theorem.
Finite exact sequences are divisions of everything
I believe that finite exact sequences can be thought of as divisions of everything. I am trying to show this and understand what this means.
Infinite exact sequences can be thought of the way that growth proceeds, extending the whole. 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 eight-cycle as the relationship between a matrix and its entries.
The growth of the finite exact sequence seems to come from the middle, which keeps becoming more refined.
Matematikos žinojimo rūmuose trejybės ratas sukuria autoasociatyvas sekas - jos iš begalinių "tikslių sekų" padaro baigtines tikslias sekas, tad padalinimus.
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.
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; "Co-cycles" are... ; "Co-boundaries" are...
A multi-dimensional 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.
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 $} Foursome: Kernel and cokernel. Div-Grad-Curl. Yoneda lemma.
- {$ \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 $} Sixsome: Related to three-cycles ("triangles"). A Characterization of Long Exact Sequences Coming from the Snake Lemma, Jan Stovicek.
- {$ \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 $}
Roots of unity = divisions of everything?
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)
Threesome
- 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
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 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 Theorem, jinai pakankamai trumpa ir turbūt ne tokia sudėtinga. Ir paaiškinti ką jinai turi bendro su sąmoningumu.
- Bosons - "ryšiai" kodėl - Yoneda. Fermions - "ar".
Fivesome
- Five lemma and the two four-lemmas.
- Analysis allows for work with limits.
- Eccentricity of conic sections - there are five eccentricities (for the circle, parabola, ellipse, hyperbola, line).
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 $}
Sevensome and eightsome
- Logic is the end result of structure, see the sevensome and Greimas' semiotic square.
- 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?
- triangle: 1 unknown 3 vertices +3 edges +1 whole
Finite exact sequences
Ideas
- 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 Dold-Kan 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 R-module 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 R-module 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.
- 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).
- 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...". 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."
- 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.
- 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.
- Boris Novikov: Let X be a space and Y its subspace. If a boundary (in Y) of an n-dimensional relative cycle c of X∖Y is a boundary of something in Y then one can build a proper n-dimensional cycle of X from c, gluing this "something" to c.
- 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 an only if C is an exact sequence.
Examples
- {$ \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 $}
- Let I and J be ideals in a ring R. Prove that there is an exact sequence of R-modules (what are the maps): {$ \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
- Inclusion-exclusion Reference
- Short exact sequence of a projective hypersurface: line bundle.
- Short exact sequence of a complete intersection 0→R(-s-t)→R(-s) sum R(-t)→I→0. See also scheme theoretic intersection.
Read about