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Overview

Divisions of everything seem to appear in various places in mathematics. Most generally:

• Bott periodicity may describe the eight-cycle 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 chain complexes and 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 left-exact 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

• Ravi Vakil. Puzzling Through 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; "Co-cycles" are... ; "Co-boundaries" 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 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.
• 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 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.
• 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 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.

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.
• Zig-zag lemma relates to infinite revolutions along the three-cycle.
• Let p: E → B be a basepoint-preserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is path-connected. Then there is a long exact sequence of homotopy groups
• Primena trejybę. Wikipedia: Homotopy groups Let p: E → B be a basepoint-preserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is path-connected. 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_{n-1}(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 chain complexes and 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 eight-cycle 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.
• A short exact sequence is a division but it's interpretation is a chain complex.
• In a short exact sequence, how does it grow through the middle? In what sense do the relationships at either end stay fixed? If we have a sequence 0->A->B->...->Y->Z->0, in what sense can A->...->Z (f:A->Z) be thought of as a map? As an isomorphism? In general, it's not an isomorphism, because image f is zero and not equal to Z. In other words, a short exact sequence is a division.
• Division of everything: A chain complex that is "bolted down" by filling in the holes to get an exact sequence. This helps explain how we can interpret 0->A->0 nontrivially.
• 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.

Divisions of everything are modeled by chain complexes which mentally become short exact sequences by "letting the air out".

• The onesome, twosome define the relative space, which objectively is trivial.
• The action of "letting the air out" is given by the three-cycle, which is the first objectively non-trivial space.
• Then we can further model the absolute space beyond into which we "let the air out" by the foursome, fivesome, sixsome.
• This modeling is complete with the sevensome, whereby both chambers are balanced.
• The eightsome collapses the chambers, turns them inside out.

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 ill-posed, 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 $} Foursome: Kernel and cokernel. Div-Grad-Curl. Yoneda lemma.
• {$ \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 $}

Think of an exact sequence as starting from everything and ending with everything. Everything from above and from below, with the two identified.

Nullsome

• The kernel is the zero.

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

• Orientable -> nonorientable
  • Cross cap introduces contradiction, which breaks the segregation between orientations, whether inside and outside, self and world, or true and false.
  • Prieštaravimu panaikinimas išskyrimas išorės ir vidaus, (kaip kad ramybe - lūkesčių nebuvimu), tai sutapatinama, kaip kad "cross-cap".
• 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: Borsuk-Ulam theorem seems to divide the n-sphere 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

• Mayer-Vietoris 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

• Short exact sequence
  • https://en.wikipedia.org/wiki/Euler_sequence
  • https://en.wikipedia.org/wiki/Projective_bundle
  • https://en.wikipedia.org/wiki/Grassmann_bundle
• 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 three-cycle, 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 four-lemmas.
• 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 three-cycle?

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 $}
• Let I and J be ideals in a ring R. Prove that there is an exact sequence of R-modules (what are the maps): {$ \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.

Study materials

• Intuitive meaning of exact sequences
• What is a syzygy?
• Chain complex
• Homology groups
• Eilenberg-Steenrod axioms
• Good pair and Deformation retract
• Mitchell's embedding theorem
• Nine lemma
• A Cohomological Viewpoint on Elementary School Arithmetic About "carrying". Access restricted.

Videos

• V: Ben Mares. Introduction to cohomology.

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 n-dimensions 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 (three-cycle) 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 three-cycle.

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.

Exact sequence - describes the landscape of truth - from God (zero) within to God (zero) beyond

Exact sequence is a tug-of-war between the two zeros.

• Septynerybėje, viduje, kas yra? ir kai tai padalinama, subliukšta
• Ir kaip 5+1=6 prideda 1 iš vieno šono? Kaip pasislenka kiti?
• Kaip tikslioji seka ilgėdama grindžia (erdvės) padalinimo sąvoką?

Exact sequences - Divisions of everything

Division of everything as based on probabilities, choice, relating two probablity densities, the asymptotic (conscious) base and the (unconscious) variation.

• See: Antonio Jesus

Exact sequences. The maps are the perspectives. The objects are just "filler".

• 0 Nullsome: No map.
• 0->0 Onesome: The map from 0 to itself.
• 0->A->0 Twosome: Means A is 0. Distinguishes the injective map from 0 (opposites coexist) and the surjective map to 0 (all things are the same).
• 0->A->B->0 Threesome: Defines an isomorphism between A and B. The isomorphism is "reflection". Going from B to 0 is taking a stand. Going from 0 to A is following through.
• Short exact sequence has four mappings - the foursome. 0->A->B->C->0
• Fivesome: 0->A->B->C->A'->0
• Sixsome: 0->A->B->C->A'->B'->0
• Sevensome: 0->A->B->C->A'->B'->C'->0
• Eightsome collapses with A''. Is this like the boundary of a boundary is zero?

The collapse may be because we switch over from maps to objects. The infinite series has no zero, has no God? Starting the third three cycle yields a new relative zero. Also, it may be that at this point there is no more any constraint on the central function and so the division loses its purpose as a constraint.

Threesome

• Long exact sequence (based on three-cycle) replaces the map from the zero object and to the zero object with the dropping by one dimension.
• Thinking = reflection = a minimal jiggle = the Hamiltonian. As in an isomorphism between A and B in the exact sequence 0->A->B->0.
• Threesome: Mayer-Vietoris sequences
• Threesome. su(2) is the cross product algebra in R3
• SU(2) root system - and the threesome - let go of one dimension and pick up another dimension.

Foursome

• Does the Free energy principle express the foursome?

Fivesome

• What Does the Gradient Vector Mean Intuitively?
• The Bright Side of Mathematics

Del indicates direction

• In a scalar field, at every point, gradient measures change in all directions ("for all"), yielding a vector.
• In a vector field, at every point, curl measures change of a vector in perpendicular directions, yielding another vector.
• In a vector field, at every point, divergence measures change of a vector in its own direction ("there exists"), yielding a scalar.
• Gradient - if derivative is well defined at each direction then just add them. What about the curl? and the divergence?
• Perturbations possible. Divide freedom over nonfreedom. Slow down or speed up in the existing direction. Veer or not in the perpendicular direction. Consider relative direction.
• How to think of the Laplacian? as taking us from the gradient to the divergence? and how to think of the Dirac operator (on spinors) as a square root of the Laplacian?
• A vector space with a nonzero curl must have, at every point where it is nonzero, a quiet axis, normal to the curl, along which things are held fixed. If there is no curl then there is no such distinguished axis.
• Effects of cause = Why. For why is knowing all of the effects of a cause. And that is how quantum mechanics works - the collapse of the wave function defines the arisal of the wave function - the collapse defines what must have been the inputs, the causes. That is how it works at CERN.
• Functional differential equations relate to double causality.
• Helmholtz decomposition (the fundamental theorem of vector calculus) any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into the sum of an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field. An irrotational vector field has a scalar potential and a solenoidal vector field has a vector potential, thus the Helmholtz decomposition states that a vector field (satisfying appropriate smoothness and decay conditions) can be decomposed as the sum of the {$ -\nabla \phi +\nabla \times \mathbf{A} $}, where {$\phi$} is a scalar field called "scalar potential", and {$\mathbf{A}$} is a vector field, called a vector potential.

Vector field that equals its own curl

• Is there a vector field that is equal to its own curl?
• Biao Ou. Examinations on a Three-Dimensional Differentiable Vector Field That Equals its Own Curl.
• Without curl there would be no map between {$\mathbb{R}^3\rightarrow\mathbb{R}^3$} and {$\mathbb{R}^3\rightarrow\mathbb{R}^3$}.

{$$\nabla f(x,y,z)=(\frac{\partial f}{\partial x}(x,y,z),\frac{\partial f}{\partial y}(x,y,z),\frac{\partial f}{\partial z}(x,y,z))$$}

{$$\textrm{curl}\;\textbf{F}=(\frac{\partial F_3}{\partial y}(x,y,z)-\frac{\partial F_2}{\partial z}(x,y,z), \frac{\partial F_3}{\partial y}(x,y,z)-\frac{\partial F_2}{\partial z}(x,y,z)$$}

Sixsome

• Morality has us choose the base world with regard to which we make our predictions and measure our surprise as per Karl Friston and free energy.

Does the exact sequence for the sixsome make the curl chiral?

• The fact that the three-cycle goes in one direction - is that related to the chirality of the weak force - as per emotions 5+3=0 ?

Exact sequences factor objects, thus divide them. So in what sense is it the morphisms that are the perspectives in a division of everything?

Long list of examples of short exact sequences

• https://mathoverflow.net/questions/363720/short-exact-sequences-every-mathematician-should-know

Short exact sequences

• https://math.stackexchange.com/questions/3496098/short-exact-sequences-as-fiber-bundles If you are looking at a fibre bundle as a spectrum, then we call this a Class. Conversely, if we looking at one as a neighbourhood it is a a Character.
• https://math.stackexchange.com/questions/141215/what-are-exact-sequences-metaphysically-speaking exact sequences are a natural abstraction of the notion of generators and relations. exact sequences as resolutions
• https://math.stackexchange.com/questions/419329/intuitive-meaning-of-exact-sequence "Euler's Formula" relating the number of vertices (V), edges (E), and faces (F). An exact sequence gives an ingredients list using inclusion-exclusion. Exact sequences are basically a way to keep track of syzygy.
• Binomial theorem expresses how the whole is divided, the division and its complement, and the relationships with God and human according to Poincare duality because God is the perspectives in the complement beyond the division. We can express that with Pascal's triangle, how to relate x and y, the kinds of choice frameworks possible.
• Think of perspectives, divisions of everything, in math, as being probability distributions, or more generally, models of probability that, by means of a choice, relate two realms, as does a perspective. And think how all of math could be derived from the unfolding of such perspectives, the relations between realms.

What does it mean that 0->A->B->0 defines an isomorphism? That it can repeat forever? That the direction can be turned around? We have isomorphisms between 0->A and 0->B, and likewise between A->0 and B->0.