Category of Super Vector Spaces Becomes a Tensor Category

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The theory of manifolds and algebraic geometry are ultimately based on linear algebra. Similarly the theory of supermanifolds needs super linear algebra, which is linear algebra in which vector spaces are replaced by vector spaces with a Z/2Z-grading, namely, super vector spaces.

A super vector space is a Z/2Z-graded vector space

V = V0 ⊕ V1

where the elements of Vare called even and that of Vodd.

The parity of v ∈ V , denoted by p(v) or |v|, is defined only on non-zero homogeneous elements, that is elements of either V0 or V1:

p(v) = |v| = 0 if v ∈ V0

= 1 if v ∈ V1

The superdimension of a super vector space V is the pair (p, q) where dim(V0) = p and dim(V1) = q as ordinary vector spaces. We simply write dim(V) = p|q.

If dim(V) = p|q, then we can find a basis {e1,…., ep} of V0 and a basis {ε1,….., εq} of V1 so that V is canonically isomorphic to the free k-module generated by {e1,…., ep, ε1,….., εq}. We denote this k-module by kp|q and we will call {e1,…., ep, ε1,….., εq} the canonical basis of kp|q. The (ei) form a basis of kp = k0p|q and the (εj) form a basis for kq = k1p|q.

A morphism from a super vector space V to a super vector space W is a linear map from V to W preserving the Z/2Z-grading. Let Hom(V, W) denote the vector space of morphisms V → W. Thus we have formed the category of super vector spaces that we denote by (smod). It is important to note that the category of super vector spaces also admits an “inner Hom”, which we denote by Hom(V, W); for super vector spaces V, W, Hom(V, W) consists of all linear maps from V to W ; it is made into a super vector space itself by:

Hom(V, W)0 = {T : V → W|T preserves parity}  (= Hom(V, W))

Hom(V, W)1 = {T : V → W|T reverses parity}

If V = km|n, W = kp|q we have in the canonical basis (ei, εj):

Hom(V, W)0 = (A 0 0 D) and Hom(V, W)1 = (0 B C 0)

where A, B, C , D are respectively (p x m), (p x n), (q x m), (q x n) – matrices with entries in k.

In the category of super vector spaces we have the parity reversing functor ∏(V → ∏V) defined by

(∏V)0 = V1, (∏V)1 = V0

The category of super vector spaces admits tensor products: for super vector spaces V, W, V ⊗ W is given the Z/2Z-grading as

(V ⊗ W)0 = (V0 ⊗ W0) ⊕ (V1 ⊗ W1),

(V ⊗ W)1 = (V0 ⊗ W1) ⊕ (V1 ⊗ W0)

The assignment V, W ↦ V ⊗ W is additive and exact in each variable as in the ordinary vector space category. The object k functions as a unit element with respect to tensor multiplication ⊗ and tensor multiplication is associative, i.e., the two products U ⊗ (V ⊗ W) and (U ⊗ V) ⊗ W are naturally isomorphic. Moreover, V ⊗ W ≅ W ⊗ V by the commutative map,

cV,W : V ⊗ W → W ⊗ V

where

v ⊗ w ↦ (-1)|v||w|w ⊗ v

If we are working with the category of vector spaces, the commutativity isomorphism takes v ⊗ w to w ⊗ v. In super linear algebra we have to add the sign factor in front. This is a special case of the general principle called the “sign rule”. The principle says that in making definitions and proving theorems, the transition from the usual theory to the super theory is often made by just simply following this principle, which introduces a sign factor whenever one reverses the order of two odd elements. The functoriality underlying the constructions makes sure that the definitions are all consistent.

The commutativity isomorphism satisfies the so-called hexagon diagram:

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where, if we had not suppressed the arrows of the associativity morphisms, the diagram would have the shape of a hexagon.

The definition of the commutativity isomorphism, also informally referred to as the sign rule, has the following very important consequence. If V1, …, Vn are the super vector spaces and σ and τ are two permutations of n-elements, no matter how we compose associativity and commutativity morphisms, we always obtain the same isomorphism from Vσ(1) ⊗ … ⊗ Vσ(n) to Vτ(1) ⊗ … ⊗ Vτ(n) namely:

Vσ(1) ⊗ … ⊗ Vσ(n) → Vτ(1) ⊗ … ⊗ Vτ(n)

vσ(1) ⊗ … ⊗ vσ(n) ↦ (-1)N vτ(1) ⊗ … ⊗ vτ(n)

where N is the number of pair of indices i, j such that vi and vj are odd and σ-1(i) < σ-1(j) with τ-1(i) > τ-1(j).

The dual V* of V is defined as

V* := Hom (V, k)

If V is even, V = V0, V* is the ordinary dual of V consisting of all even morphisms V → k. If V is odd, V = V1, then V* is also an odd vector space and consists of all odd morphisms V1 → k. This is because any morphism from V1 to k = k1|0 is necessarily odd and sends odd vectors into even ones. The category of super vector spaces thus becomes what is known as a tensor category with inner Hom and dual.

Egyptology

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The ancient Egyptians conceived man and kosmos to be dual: firstly, the High God or Divine Mind arose out of the Primeval Waters of space at the beginning of manifestation; secondly, the material aspect expressing what is in the Divine Mind must be in a process of ever-becoming. In other words, the kosmos consists of body and soul. Man emanated in the image of divinity is similarly dual and his evolutionary goal is a fully conscious return to the Divine Mind.

Space, symbolized by the Primeval Waters, contains the seeds and possibilities of all living things in their quiescent state. At the right moment for awakenment, all will take up forms in accordance with inherent qualities. Or to express it in another way: the Word uttered by the Divine Mind calls manifested life to begin once more.

Growth is effected through a succession of lives, a concept that is found in texts and implied in symbolism. Herodotus, the Greek historian (5th century B.C.), wrote that

the Egyptians were the first to teach that the human soul is immortal, and at the death of the body enters into some other living thing then coming to birth; and after passing through all creatures of land, sea, and air (which cycle it completes in three thousand years) it enters once more into a human body, at birth.

The theory of reincarnation is often ascribed to Pythagoras, since he spent some time in Egypt studying its philosophy and, according to Herodotus, “adopted this opinion as if it were his own.”

Margaret A. Murray, who worked with Flinders Petrie, illustrates the Egyptian belief by referring to the ka-names of three kings (The ka-name relates to the vital essence of an individual); the first two of the twelfth dynasty: that of Amonemhat I means “He who repeats births,” Senusert I: “He whose births live,” and the ka-name of Setekhy I of the nineteenth dynasty was “Repeater of births.” (The Splendour That Was Egypt)

Reincarnation has been connected with the rites of Osiris, one of the Mysteries or cycles of initiation perpetuated in Egypt. The concept of transformation as recorded in the Egyptian texts has been interpreted in various ways. De Briere expresses it in astronomical terms: “The sensitive soul re-entered by the gate of the gods, or the Capricorn, into the Amenthe, the watery heavens, where it dwelt always with pleasure; until, descending by the gate of men, or the Cancer, it came to animate a new body.” Herodotus writes of transmigration, i.e., that the soul passes through various animals before being reborn in human form. This refers not to the human soul but to the molecules, atoms, and other components that clothe it. They gravitate to vehicles similar in qualities to their former host’s, drawn magnetically to the new milieu by the imprint made by the human soul, whether it be fine or gross. It is quite clear from the Book of the Dead and other texts that the soul itself after death undergoes experiences in the Duat (Dwat) or Underworld, the realm and condition between heaven and earth, or beneath the earth, supposedly traversed by the sun from sunset to sunrise.

The evolution of consciousness is symbolized by the Solar Barque moving through the Duat. In this context the “hours” of travel represent stages of development. Bika Reed states that at a certain “hour” the individual meets the “Rebel in the Soul,”  that is, at the “hour of spiritual transformation.” And translating from the scroll Reed gives: “the soul warns, only if a man is allowed to continue evolving, can the intellect reach the heart.”

Not only does the scripture deal with rituals assumed to apply to after-death conditions — in some respects similar to the Book of the Dead — but also it seems quite patently a ritual connected with initiation from one level of self-becoming to another. Nevertheless the picture that emerges is that of the “deceased” or candidate for initiation reaching a fork offering two paths called “The Two Paths of Liberation” and, while each may take the neophyte to the abode of the Akhu (the “Blessed”) — a name for the gods, and also for the successful initiates — they involve different experiences. One path, passing over land and water, is that of Osiris or cyclic nature and involves many incarnations. The other way leads through fire in a direct or shortened passage along the route of Horus who in many texts symbolizes the divine spark in the heart.

In the Corpus Hermeticum, Thoth — Tehuti — was the Mind of the Deity, whom the Alexandrian Greeks identified with Hermes. For example, one of the chief books in the Hermetica is the Poimandres treatise, or Pymander. The early trinity Atum-Ptah-Thoth was rendered into Greek as theos (god) — demiourgos or demourgos-nous (Demiurge or Demiurgic Mind) — nous and logos (Mind and Word). The text states that Thoth, after planning and engineering the kosmos, unites himself with the Demiurgic Mind. There are other expressions proving that the Poimandres text is a Hellenized version of Egyptian doctrine. An important concept therein is that of “making-new-again.” The treatise claims that all animal and vegetable forms contain in themselves “the seed of again-becoming” — a clear reference to reimbodiment — “every birth of flesh ensouled . . . shall of necessity renew itself.” G. R. S. Mead interprets this as palingenesis or reincarnation — “the renewal on the karmic wheel of birth-and-death.” (Thrice-Greatest Hermes)

The Corpus Hermeticum or Books of Hermes are believed by some scholars to have been borrowed from Christian texts, but their concepts are definitely ancient Egyptian in origin, translated into Alexandrian Greek, and Latin.

Looking at Walter Scott’s translation of Poimandres, it states that “At the dissolution of your material body, you first yield up the body itself to be changed,” and it will be absorbed by nature. The rest of the individual’s components return to “their own sources, becoming parts of the universe, and entering into fresh combinations to do other work.” After this, the real or inner man “mounts upward through the structure of the heavens,” leaving off in each of the seven zones certain energies and related substances. The first zone is that of the Moon; the second, the planet Mercury; the third, Venus; fourth, the Sun; fifth, Mars; sixth, Jupiter; and seventh, Saturn. “Having been stripped of all that was wrought upon him” in his previous descent into incarnation on Earth, he ascends to the highest sphere, “being now possessed of his own proper power.” Finally, he enters into divinity. “This is the Good; this is the consummation, for those who have got gnosis.” (According to Scott, gnosis in this context means not only knowledge of divinity but also the relationship between man’s real self and the godhead.)

Further on, the Poimandres explains that the mind and soul can be conjoined only by means of an earth-body, because the mind by itself cannot do so, and an earthly body would not be able to endure

the presence of that mighty and immortal being, nor could so great a power submit to contact with a body defiled by passion. And so the mind takes to itself the soul for a wrap

In Hermetica, Isis to Horus, there is the statement:

. . . . For there are [in the world above, two gods] who are attendants of the Providence that governs all. One of them is Keeper of souls; the other is Conductor of souls. The Keeper is he that has in his charge the unembodied souls; the Conductor is he that sends down to earth the souls that are from time to time embodied, and assigns to them their several places. And both he that keeps watch over the souls, and he that sends them forth, act in accordance with God’s will.

There are many texts using the term “transformations” and a good commentary on the concept by R. T. Rundle Clark follows:

In order to reach the heights of the sky the soul had to undergo those transformations which the High God had gone through as he developed from a spirit in the Primeval Waters to his final position as Sun God . . .” — Myth-And-Symbol-In-Ancient-Egypt

This would appear to mean that in entering upon physical manifestation human souls follow the path of the divine and spiritual artificers of the universe.

There is reason to believe that the after-death adventures met with by the soul through the Duat or Underworld were also undergone by a neophyte during initiation. If the trial ends in success, the awakened human being thereafter speaks with the authority of direct experience. In the most ancient days of Egypt, such an initiate was called a “Son of the Sun” for he embodied the solar splendour. For the rest of mankind, the way is slower, progressing certainly, but more gradually, through many lives. The ultimate achievement is the same: to radiate the highest qualities of the spiritual element locked within the aspiring soul.

Symmetry: Mirror of a Manifold is the Opposite of its Fundamental Poincaré ∞-groupoid

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Given a set X = {a,b,c,..} such as the natural numbers N = {0,1,…,p,…}, there is a standard procedure that amounts to regard X as a category with only identity morphisms. This is the discrete functor that takes X to the category denoted by Disc(X) where the hom-sets are given by Hom(a,b) = ∅ if a ≠ b, and Hom(a,b) = {Ida} = 1 if a = b. Disc(X) is in fact a groupoid.

But in category theory, there is also a procedure called opposite or dual, that takes a general category C to its opposite Cop. Let us call Cop the reflection of C by the mirror functor (−)op.

Now the problem is that if we restrict this procedure to categories such as Disc(X), there is no way to distinguish Disc(X) from Disc(X)op. And this is what we mean by sets don’t show symmetries. In the program of Voevodsky, we can interpret this by saying that:

The identity type is not good for sets, instead we should use the Equivalence type. But to get this, we need to move to from sets to Kan complexes i.e., ∞-groupoids.

The notion of a Kan complex is an abstraction of the combinatorial structure found in the singular simplicial complex of a topological space. There the existence of retractions of any geometric simplex to any of its horns – simplices missing one face and their interior – means that all horns in the singular complex can be filled with genuine simplices, the Kan filler condition.

At the same time, the notion of a Kan complex is an abstraction of the structure found in the nerve of a groupoid, the Duskin nerve of a 2-groupoid and generally the nerves of n-groupoids ∀ n ≤ ∞ n. In other words, Kan complexes constitute a geometric model for ∞-groupoids/homotopy types which is based on the shape given by the simplex category. Thus Kan complexes serve to support homotopy theory.

So far we’ve used set theory with this lack of symmetries, as foundations for mathematics. Grothendieck has seen this when he moved from sheaves of sets, to sheaves of groupoid (stacks), because he wanted to allow objects to have symmetries (automorphisms). If we look at the Giraud-Grothendieck picture on nonabelian cohomology, then what happens is an extension of coefficients U : Set ֒→ Cat. We should consider first the comma category Cat ↓ U, whose objects are functors C → Disc(X). And then we should consider the full subcategory consisting of functors C → Disc(X) that are equivalences of categories. This will force C to be a groupoid, that looks like a set. And we call such C → Disc(X) a Quillen-Segal U-object.

This category of Quillen-Segal objects should be called the category of sets with symmetries. Following Grothendieck’s point of view, we’ve denoted by CatU[Set] the comma category, and think of it as categories with coefficients or coordinates in sets. This terminology is justified by the fact that the functor U : Set ֒→ Cat is a morphism of (higher) topos, that defines a geometric point in Cat. The category of set with symmetries is like the homotopy neighborhood of this point, similar to a one-point going to a disc or any contractible object. The advantage of the Quillen-Segal formalism is the presence of a Quillen model structure on CatU[Set] such that the fibrant objects are Quillen-Segal objects.

In standard terminology this means that if we embed a set X in Cat as Disc(X), and take an ‘projective resolution’ of it, then we get an equivalence of groupoids P → Disc(X), and P has symmetries. Concretely what happens is just a factorization of the identity (type) Id : Disc(X) → Disc(X) as a cofibration followed by a trivial fibration:

Disc(X)  ֒→ P → Disc(X)

This process of embedding Set ֒→ QS{CatU[Set]} is a minimal homotopy enhancement. The idea is that there is no good notion of homotopy (weak equivalence) in Set, but there are at least two notions in Cat: equivalences of categories and the equivalences of classifying spaces. This last class of weak equivalences is what happens with mirror phenomenons. The mirror of a manifold should be the opposite of its fundamental Poincaré ∞-groupoid.

Banach Spaces

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Some things in linear algebra are easier to see in infinite dimensions, i.e. in Banach spaces. Distinctions that seem pedantic in finite dimensions clearly matter in infinite dimensions.

The category of Banach spaces considers linear spaces and continuous linear transformations between them. In a finite dimensional Euclidean space, all linear transformations are continuous, but in infinite dimensions a linear transformation is not necessarily continuous.

The dual of a Banach space V is the space of continuous linear functions on V. Now we can see examples of where not only is V* not naturally isomorphic to V, it’s not isomorphic at all.

For any real p > 1, let q be the number such that 1/p  + 1/q = 1. The Banach space Lp is defined to be the set of (equivalence classes of) Lebesgue integrable functions f such that the integral of ||f||p is finite. The dual space of Lp is Lq. If p does not equal 2, then these two spaces are different. (If p does equal 2, then so does qL2 is a Hilbert space and its dual is indeed the same space.)

In the finite dimensional case, a vector space V is isomorphic to its second dual V**. In general, V can be embedded into V**, but V** might be a larger space. The embedding of V in V** is natural, both in the intuitive sense and in the formal sense of natural transformations. We can turn an element of V into a linear functional on linear functions on V as follows.

Let v be an element of V and let f be an element of V*. The action of v on f is simply fv. That is, v acts on linear functions by letting them act on it.

This shows that some elements of V** come from evaluation at elements of V, but there could be more. Returning to the example of Lebesgue spaces above, the dual of L1 is L, the space of essentially bounded functions. But the dual of L is larger than L1. That is, one way to construct a continuous linear functional on bounded functions is to multiply them by an absolutely integrable function and integrate. But there are other ways to construct linear functionals on L.

A Banach space V is reflexive if the natural embedding of V in V** is an isomorphism. For p > 1, the spaces Lp are reflexive.

Suppose that X is a Banach space. For simplicity, we assume that X is a real Banach space, though the results can be adapted to the complex case in the straightforward manner. In the following, B(x0,ε) stands for the closed ball of radius ε centered at x0 while B◦(x0,ε) stands for the open ball, and S(x0,ε) stands for the corresponding sphere.

Let Q be a bounded operator on X. Since we will be interested in the hyperinvariant subspaces of Q, we can assume without loss of generality that Q is one-to-one and has dense range, as otherwise ker Q or Range Q would be hyperinvariant for Q. By {Q}′ we denote the commutant of Q.

Fix a point x0 ≠ 0 in X and a positive real ε<∥x0∥. Let K= Q−1B(x0,ε). Clearly, K is a convex closed set. Note that 0 ∉ K and K≠ ∅ because Q has dense range. Let d = infK||z||, then d > 0. If X is reflexive, then there exists z ∈ K with ||z|| = d, such a vector is called a minimal vector for x0, ε and Q. Even without reflexivity condition, however, one can always find y ∈ K with ||y|| ≤ 2d, such a y will be referred to as a 2-minimal vector for x0, ε and Q.

The set K ∩ B(0, d) is the set of all minimal vectors, in general this set may be empty. If z is a minimal vector, since z ∈ K = Q−1B(x0, ε) then Qz ∈ B(x0, ε). As z is an element of minimal norm in K then, in fact, Qz ∈ S(x0, ε). Since Q is one-to-one, we have

QB(0, d) ∩ B(x0, ε) = Q B(0, d) ∩ K) ⊆ S(x0, ε).

It follows that QB(0,d) and B◦(x0,ε) are two disjoint convex sets. Since one of them has non-empty interior, they can be separated by a continuous linear functional. That is, there exists a functional f with ||f|| = 1 and a positive real c such that f|QB(0,d)  ≤ c and f|B◦(x0,ε) ≥ c. By continuity, f|B(x0,ε) ≥ c. We say that f is a minimal functional for x0, ε, and Q.

We claim that f(x0) ≥ ε. Indeed, for every x with ||x|| ≤ 1 we have x0 − εx ∈ B(x0,ε). It follows that f(x0 − εx) ≥ c, so that f(x0) ≥ c + εf(x). Taking sup over all x with ||x|| ≤ 1 we get f(x0) ≥ c + ε||f|| ≥ ε.

Observe that the hyperplane Qf = c separates K and B(0, d). Indeed, if z ∈ B(0,d), then (Qf)(z) = f(Qz) ≤ c, and if z ∈ K then Qz ∈ B(x0,ε) so that (Q∗f)(z) = f(Qz) ≥ c. For every z with ||z|| ≤ 1

we have dz ∈ B(0,d), so that (Qf)(dz) ≤ c, it follows that Qf ≤ c/d

On the other hand, for every δ > 0 there exists z ∈ K with ||z|| ≤ d+δ, then (Qf)(z) ≥ c ≥ c/(d+δ) ||z||, whence ||Qf|| ≥ c/(d+δ) . It follows that

||Q∗f|| = c/d.

For every z ∈ K we have (Qf)(z) ≥ c = d ||Qf||. In particular, if y is a 2-minimal vector then

(Qf)(y) ≥ 1/2 Qf ||y||….