Fréchet Spaces and Presheave Morphisms.

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A topological vector space V is both a topological space and a vector space such that the vector space operations are continuous. A topological vector space is locally convex if its topology admits a basis consisting of convex sets (a set A is convex if (1 – t) + ty ∈ A ∀ x, y ∈ A and t ∈ [0, 1].

We say that a locally convex topological vector space is a Fréchet space if its topology is induced by a translation-invariant metric d and the space is complete with respect to d, that is, all the Cauchy sequences are convergent.

A seminorm on a vector space V is a real-valued function p such that ∀ x, y ∈ V and scalars a we have:

(1) p(x + y) ≤ p(x) + p(y),

(2) p(ax) = |a|p(x),

(3) p(x) ≥ 0.

The difference between the norm and the seminorm comes from the last property: we do not ask that if x ≠ 0, then p(x) > 0, as we would do for a norm.

If {pi}{i∈N} is a countable family of seminorms on a topological vector space V, separating points, i.e. if x ≠ 0, there is an i with pi(x) ≠ 0, then ∃ a translation-invariant metric d inducing the topology, defined in terms of the {pi}:

d(x, y) = ∑i=1 1/2i pi(x – y)/(1 + pi(x – y))

The following characterizes Fréchet spaces, giving an effective method to construct them using seminorms.

A topological vector space V is a Fréchet space iff it satisfies the following three properties:

  • it is complete as a topological vector space;
  • it is a Hausdorff space;
  • its topology is induced by a countable family of seminorms {pi}{i∈N}, i.e., U ⊂ V is open iff for every u ∈ U ∃ K ≥ 0 and ε > 0 such that {v|pk(u – v) < ε ∀ k ≤ K} ⊂ U.

We say that a sequence (xn) in V converges to x in the Fréchet space topology defined by a family of seminorms iff it converges to x with respect to each of the given seminorms. In other words, xn → x, iff pi(xn – x) → 0 for each i.

Two families of seminorms defined on the locally convex vector space V are said to be equivalent if they induce the same topology on V.

To construct a Fréchet space, one typically starts with a locally convex topological vector space V and defines a countable family of seminorms pk on V inducing its topology and such that:

  1. if x ∈ V and pk(x) = 0 ∀ k ≥ 0, then x = 0 (separation property);
  2. if (xn) is a sequence in V which is Cauchy with respect to each seminorm, then ∃ x ∈ V such that (xn) converges to x with respect to each seminorm (completeness property).

The topology induced by these seminorms turns V into a Fréchet space; property (1) ensures that it is Hausdorff, while the property (2) guarantees that it is complete. A translation-invariant complete metric inducing the topology on V can then be defined as above.

The most important example of Fréchet space, is the vector space C(U), the space of smooth functions on the open set U ⊆ Rn or more generally the vector space C(M), where M is a differentiable manifold.

For each open set U ⊆ Rn (or U ⊂ M), for each K ⊂ U compact and for each multi-index I , we define

||ƒ||K,I := supx∈K |(∂|I|/∂xI (ƒ)) (x)|, ƒ ∈ C(U)

Each ||.||K,I defines a seminorm. The family of seminorms obtained by considering all of the multi-indices I and the (countable number of) compact subsets K covering U satisfies the properties (1) and (1) detailed above, hence makes C(U) into a Fréchet space. The sets of the form

|ƒ ∈ C(U)| ||ƒ – g||K,I < ε

with fixed g ∈ C(U), K ⊆ U compact, and multi-index I are open sets and together with their finite intersections form a basis for the topology.

All these constructions and results can be generalized to smooth manifolds. Let M be a smooth manifold and let U be an open subset of M. If K is a compact subset of U and D is a differential operator over U, then

pK,D(ƒ) := supx∈K|D(ƒ)|

is a seminorm. The family of all the seminorms  pK,D with K and D varying among all compact subsets and differential operators respectively is a separating family of seminorms endowing CM(U) with the structure of a complete locally convex vector space. Moreover there exists an equivalent countable family of seminorms, hence CM(U) is a Fréchet space. Let indeed {Vj} be a countable open cover of U by open coordinate subsets, and let, for each j, {Kj,i} be a countable family of compact subsets of Vj such that ∪i Kj,i = Vj. We have the countable family of seminorms

pK,I := supx∈K |(∂|I|/∂xI (ƒ)) (x)|, K ∈  {Kj,i}

inducing the topology. CM(U) is also an algebra: the product of two smooth functions being a smooth function.

A Fréchet space V is said to be a Fréchet algebra if its topology can be defined by a countable family of submultiplicative seminorms, i.e., a countable family {qi)i∈N of seminorms satisfying

qi(ƒg) ≤qi (ƒ) qi(g) ∀ i ∈ N

Let F be a sheaf of real vector spaces over a manifold M. F is a Fréchet sheaf if:

(1)  for each open set U ⊆ M, F(U) is a Fréchet space;

(2)  for each open set U ⊆ M and for each open cover {Ui} of U, the topology of F(U) is the initial topology with respect to the restriction maps F(U) → F(Ui), that is, the coarsest topology making the restriction morphisms continuous.

As a consequence, we have the restriction map F(U) → F(V) (V ⊆ U) as continuous. A morphism of sheaves ψ: F → F’ is said to be continuous if the map F(U) → F'(U) is open for each open subset U ⊆ M.

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Badiou’s Vain Platonizing, or How the World is a Topos? Note Quote.

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As regards the ‘logical completeness of the world’, we need to show that Badiou’s world of T-sets does indeed give rise to a topos.

Badiou’s world consisting of T-Sets – in other words pairs (A, Id) where Id : A × A → T satisfies the particular conditions in respect to the complete Heyting algebra structure of T—is ‘logically closed’, that is, it is an elementary topos. It thus encloses not only pull-backs but also the exponential functor. These make it possible for it to internalize a Badiou’s infinity arguments that operate on the power-functor and which can then be expressed from insde the situation despite its existential status.

We need to demonstrate that Badiou’s world is a topos. Rather than beginning from Badiou’s formalism of T -sets, we refer to the standard mathematical literature based on which T-sets can be regarded as sheaves over the particular Grothendieck-topology on the category T: there is a categorical equivalence between T-sets satisfying the ‘postulate of materialism’ and S hvs(T,J). The complications Badiou was caught up with while seeking to ‘Platonize’ the existence of a topos thus largely go in vain. We only need to show that Shvs(T,J) is a topos.

Consider the adjoint sheaf functor that always exists for the category of presheaves

Idα : SetsCop → Shvs(Cop,J)

, where J is the canonical topology. It then amounts to an equivalence of categories. Thus it suffices to replace this category by the one consisting of presheaves SetsTop. This argument works for any category C rather than the specific category related to an external complete Heyting algebra T. In the category of Sets define YX as the set of functions X → Y. Then in the category of presheaves,

SetsCopYX(U) ≅ Hom(hU,YX) ≅ Hom(hU × X,Y)

, where hU is the representable sheaf hU(V) = Hom(V,U). The adjunction on the right side needs to be shown to exist for all sheaves – not just the representable ones. The proof then follows by an argument based on categorically defined limits, which has an existence. It can also be verified directly that the presheaf YX is actually a sheaf. Finally, for the existence of the subobject-classifier ΩSetsCop, it can be defined as

ΩSetsCop(U) ≅ Hom(hU,Ω) ≅ {sub-presheaves of hU} ≅ {sieves on U}, or alternatively, for the category of proper sheaves Shvs(C,J), as

ΩShvs(C,J)(U) = {closed sieves on U}

Here it is worth reminding ourselves that the topology on T is defined by a basis K(p) = {Θ ⊂ T | ΣΘ = p}. Therefore, in the case of T-sets satisfying the strong ‘postulate of materialism’, Ω(p) consists of all sieves S (downward dense subsets) of T bounded by relation ΣS ≤ p. These sieves are further required to be closed. A sieve S with an envelope ΣS = s is closed if for any other r ≤ s, ie. for all r ≤ s, one has the implication

frs(S) ∈ J(r) ⇒ frs ∈ S

, where frs : r → s is the unique arrow in the poset category. In particular, since ΣS = s for the topology whose basis consists of territories on s, we have the equation 1s(S) = fss(S) = S ∈ J(s). Now the condition that the sieve is closed implies 1s ∈ S. This is only possible when S is the maximal sieve on s—namely it consists of all arrows r → s for r ≤ s. In such a case S itself is closed. Therefore, in this particular case

Ω(p)={↓(s)|s ≤ p} = {hs | s ≤ p}

This is indeed a sheaf whose all amalgamations are ‘real’ in the sense of Badiou’s postulate of materialism. Thus it retains a suitable T-structure. Let us assume now that we are given an object A, which is basically a functor and thus a T-graded family of subsets A(p). For there to exist a sub-functor B ֒→ A comes down to stating that B(p) ⊂ A(p) for each p ∈ T. For each q ≤ p, we also have an injection B(q) ֒→ B(p) compatible (through the subset-representation with respect to A) with the injections A(q) ֒→ B(q). For any given x ∈ A(p), we can now consider the set

φp(x) = {q | q ≤ p and x q ∈ B(q)}

This is a sieve on p because of the compatibility condition for injections, and it is furthermore closed since the map x → Σφp(x) is in fact an atom and thus retains a real representative b ∈ B. Then it turns out that φp(x) =↓ (Eb). We now possess a transformation of functors φ : A → Ω which is natural (diagrammatically compatible). But in such a case we know that B ֒→ A is in turn the pull-back along φ of the arrow true, which is equivalent to the category of T-Sets.

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Hyperstructures

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In many areas of mathematics there is a need to have methods taking local information and properties to global ones. This is mostly done by gluing techniques using open sets in a topology and associated presheaves. The presheaves form sheaves when local pieces fit together to global ones. This has been generalized to categorical settings based on Grothendieck topologies and sites.

The general problem of going from local to global situations is important also outside of mathematics. Consider collections of objects where we may have information or properties of objects or subcollections, and we want to extract global information.

This is where hyperstructures are very useful. If we are given a collection of objects that we want to investigate, we put a suitable hyperstructure on it. Then we may assign “local” properties at each level and by the generalized Grothendieck topology for hyperstructures we can now glue both within levels and across the levels in order to get global properties. Such an assignment of global properties or states we call a globalizer. 

To illustrate our intuition let us think of a society organized into a hyperstructure. Through levelwise democratic elections leaders are elected and the democratic process will eventually give a “global” leader. In this sense democracy may be thought of as a sociological (or political) globalizer. This applies to decision making as well.

In “frustrated” spin systems in physics one may possibly think of the “frustation” being resolved by creating new levels and a suitable globalizer assigning a global state to the system corresponding to various exotic physical conditions like, for example, a kind of hyperstructured spin glass or magnet. Acting on both classical and quantum fields in physics may be facilitated by putting a hyperstructure on them.

There are also situations where we are given an object or a collection of objects with assignments of properties or states. To achieve a certain goal we need to change, let us say, the state. This may be very difficult and require a lot of resources. The idea is then to put a hyperstructure on the object or collection. By this we create levels of locality that we can glue together by a generalized Grothendieck topology.

It may often be much easier and require less resources to change the state at the lowest level and then use a globalizer to achieve the desired global change. Often it may be important to find a minimal hyperstructure needed to change a global state with minimal resources.

Again, to support our intuition let us think of the democratic society example. To change the global leader directly may be hard, but starting a “political” process at the lower individual levels may not require heavy resources and may propagate through the democratic hyperstructure leading to a change of leader.

Hence, hyperstructures facilitates local to global processes, but also global to local processes. Often these are called bottom up and top down processes. In the global to local or top down process we put a hyperstructure on an object or system in such a way that it is represented by a top level bond in the hyperstructure. This means that to an object or system X we assign a hyperstructure

H = {B0,B1,…,Bn} in such a way that X = bn for some bn ∈ B binding a family {bi1n−1} of Bn−1 bonds, each bi1n−1 binding a family {bi2n−2} of Bn−2 bonds, etc. down to B0 bonds in H. Similarly for a local to global process. To a system, set or collection of objects X, we assign a hyperstructure H such that X = B0. A hyperstructure on a set (space) will create “global” objects, properties and states like what we see in organized societies, organizations, organisms, etc. The hyperstructure is the “glue” or the “law” of the objects. In a way, the globalizer creates a kind of higher order “condensate”. Hyperstructures represent a conceptual tool for translating organizational ideas like for example democracy, political parties, etc. into a mathematical framework where new types of arguments may be carried through.

Smooth Manifolds: Frölicher space as Weil exponentiable

Thus spoke André Weil,

Nothing is more fruitful – all mathematicians know it – than those obscure analogies, those disturbing reflections of one theory in another; those furtive caresses, those inexplicable discords; nothing also gives more pleasure to the researcher. The day comes when the illusion dissolves; the yoked theories reveal their common source before disappearing. As the Gita teaches, one achieves knowledge and indifference at the same time.

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The notion of Weil algebra is ordinarily defined for a Lie algebra g. In mathematics, the Weil algebra of a Lie algebra g, introduced by Cartan based on unpublished work of André Weil, is a differential graded algebra given by the Koszul algebra Λ (g*) ⊗ S(g*) of its dual g*.

 A Frölicher space is one flavour of a generalized smooth space. Frölicher smooth spaces are determined by a rule for
  • how to map the real line smoothly into it,
  • and how to map out of the space smoothly to the real line.

In the general context of space and quantity, Frölicher spaces take an intermediate symmetric position: they are both presheaves as well as copresheaves on their test domain (which here is the full subcategory of manifolds on the real line) and both of these structures determine each other.

After assigning, to each pair (X, W ) of a Frölicher space X and a Weil algebra W , another Frölicher space X ⊗ W , called the Weil prolongation of X with respect to W, which naturally extends to a bifunctor FS × W → FS, where FS is the category of Frölicher spaces and smooth mappings, and W is the category of Weil algebras. We also know

The functor · ⊗ W : FS → FS is product-preserving for any Weil algebra W.

Weil Exponentiability

A Frölicher space X is called Weil exponentiable if (X ⊗ (W1 W2))Y = (X ⊗ W1)Y ⊗ W2 —– (1)

holds naturally for any Frölicher space Y and any Weil algebras W1 and W2. If Y = 1, then (1) degenerates into

X ⊗ (W1 W2) = (X ⊗ W1) ⊗ W2 —– (2)

If W1 = R, then (1) degenerates into

(X ⊗ W2)Y = XY ⊗ W2 —– (3)

Proposition: Convenient vector spaces are Weil exponentiable.
Corollary: C-manifolds are Weil exponentiable.

Proposition: If X is a Weil exponentiable Frölicher space, then so is X ⊗ W for any Weil algebra W.

Proposition: If X and Y are Weil exponentiable Frölicher spaces, then so is X × Y.

Proposition: If X is a Weil exponentiable Frölicher space, then so is XY for any Frölicher space Y .

Theorem: Weil exponentiable Frölicher spaces, together with smooth mappings among them, form a Cartesian closed subcategory FSWE of the category FS.

Generally speaking, limits in the category FS are bamboozling. The notion of limit in FS should be elaborated geometrically.

A finite cone D in FS is called a transversal limit diagram providing that D ⊗ W is a limit diagram in FS for any Weil algebra W , where the diagram D ⊗ W is obtained from D by putting ⊗ W to the right of every object and every morphism in D. By taking W = R, we see that a transversal limit diagram is always a limit diagram. The limit of a finite diagram of Frölicher spaces is said to be transversal providing that its limit diagram is a transversal limit diagram.

Lemma: If D is a transversal limit diagram whose objects are all Weil exponentiable, then DX is also a transversal limit diagram for any Frölicher space X, where DX is obtained from D by putting X as the exponential over every object and every morphism in D.

Proof: Since the functor ·X : FS → FS preserves limits, we have DX ⊗ W = (D ⊗ W)X

for any Weil algebra W , so that we have the desired result.

Lemma: If D is a transversal limit diagram whose objects are all Weil exponentiable, then D ⊗ W is also a transversal limit diagram for any Weil algebra W.

Proof: Since the functor W ⊗ · : W → W preserves finite limits, we have (D ⊗ W) ⊗ W′ = D ⊗ (W ⊗ W′)

for any Weil algebra W′, so that we have the desired result.

Grothendieck Sheaves and Topologies Within Monetary Value Measures. Part 2

A contravariant functor ρ : Cop → Set is  called a presheaf for a category C. By definition, a monetary value measure is a presheaf. The name presheaf suggests that it is related to another concept sheaves, which is a quite important concept in some classical branches in mathematics such as algebraic topology. For a given set, a topology defined on it provides a criteria to distinguish continuous functions from given functions on the set. In a similar way, there is a concept called a Grothendieck topology defined on a given category that gives a criteria to distinguish good presheaves (=sheaves) from given presheaves on the category. In both cases, a Grothendieck topology can be seen as a vehicle to identify good functions (presheaves) among general functions (presheaves).

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On the other hand, if we have a set of functions that we want to make continuous, we can find the weakest topology that makes the functions continuous. In a similar way, if we have a set of presheaves that we want to make good, it is known that we can pick a Grothendieck topology with which the presheaves become sheaves. Since a monetary value measure is a presheaf, if we have a set of good monetary value measures (= the monetary value measures that satisfy a given set of axioms), we may find a Grothendieck topology with which the monetary value measures become sheaves. Now suppose we have a weak topology that makes given functions continuous. This, however, does not imply the fact that any continuous function w.r.t. the topology is contained in the originally given functions. Similarly, Suppose that we have a Grothendieck topology that makes all monetary value measures satisfying a given set of axioms sheaves. It, however, does not mean that any sheaf w.r.t. the Grothendieck topology satisfies the given set of axioms.

What are Grothendieck topologies and sheaves?

1.0 Let U ∈ χ

1.1 ↓ U := {V ∈ χ | V ⊂ U}

1.2 A sieve on U is a set I ⊂↓ U such that (∀ V ∈ ↓ U)(∀ W ∈ ↓ U)[W ⊂ V ∈ I ⇒ W ∈ I]

1.3 For a sieve I on U and V ⊂ U in χ, I ↓ V := I ∩ ↓ V

1.4 A family of I is an element X ∈ ∏V∈I L(V), or X = (XV)V∈I

1.5 A family X = (XV)V∈I is called a P-martingale if (∀ V ∈ I)(∀ W ∈ I)[W ⊂ V ⇒ EP[XV | W] = XW]

A sieve on U is considered as a kind of a time domain.

2.0 Ξ : χop → Set is a contravariant functor such that for iVU : V → U in χΞ(U) is the set of all sieves on U, and that Ξ(iVU)(I) = I ↓ V for I ∈ Ξ(U)

2.1 A Grothendieck topology on χ is a sub-factor J → Ξ satisfying the following conditions

2.2 (∀ U ∈ χ) ↓ (U ∈ J(U))

2.3 (∀ V ∈ χ)(∀I ∈ J(U))(∀K ∈ Ξ(U))[(∀ V ∈ I)K ↓ V ∈ J(V) ⇒ K ∈ J(U)]

This way sieve I J-covers U if I ∈ J(U)

U is considered as a time horizon of a time domain I if it is covered by I.

3.0 Theorem Let {Ja | a ∈ A} be a collection of Grothendieck topologies on χ. Then the sub-functor J → Ξ defined by J(U) := a∈A Ja(U) is a Grothendieck topology.

4.0 Let Ψ ∈ Setχop be a monetary value measure, and I be a sieve on U ∈ χ

4.1 A family X = (XV)V∈I is called Ψ-matching if (∀ V ∈ I)(∀ W ∈ I)ΨV∧WV = ΨV∧WW (XW)

4.2 A random variable X ̄∈ L(U) be a Ψ-amalgamation for a family X = (XV)V ∈ I if (∀ V ∈ I)ΨVU(X ̄) = XV

5.0 Let Ψ ∈ Setχop be a monetary value measure. I be a sieve on U ∈ χ and X = (XV)V∈I be a family that has Ψ-amalgamation. Then X is Ψ-matching. 

Let X ̄∈  L(U) be a Ψ-amalgamation. Then for every V ∈ I, XV = ΨVU(X ̄). Therefore, for every V, W ∈ I ΨV∧WV (XV) = XV∧W = ΨV∧WW (XW)

6.0 Ψ ∈ Setχop be a monetary value measure. I be a sieve on U ∈ χ and X = (XV)V∈I be a Ψ-matching family.

6.1 For V, W ∈ I, if W ⊂ V, we have ΨWV(XV) = XW

ΨWV(XV) = ΨV∧WV (XV) = ΨV∧WW (XW) = ΨWW(XW) = (XW)

6.2 If U ∈ I, XU is the unique Ψ-amalgamation for X. By 6.1, XU is the unique Ψ-amalgamation for X. Now, let X ̄∈ L(U) be another Ψ-amalgamation for X. Then for every V ∈ I, XV = ΨVU(X ̄). Put V := U. Then we have XU = ΨUU(X ̄) = 1U(X ̄) = X ̄.

7.0 Let J be a Grothendieck topology on Ψ ∈ Setχop. A monetary value measure Ψ ∈ Setχop is called a sheaf if for any U ∈ χ. Any covering sieve I ∈ J(U) and any Ψ-matching family X = (XV)V∈I b, X has a unique Ψ-amalgamation. Now, we will try to find a Grothendieck topology for which a given class of monetary value measures specified by a given set of (extra) axioms are sheaves.

Let us consider a sieve I on U ∈ χ as a subfunctor I → HomX(-,U), that is a contravariant functor I : χop → Set defined by

I(V) := {iVU} if V ∈ I

:= Φ if V ∉ I

for V ∈ χ

8.0 Let M ⊂ Setχop be the collection of all monetary value measures satisfying a given set of axioms. Then, there exists a Grothendieck topology for which all monetary value measures in M sheaves, where the topology is largest among topologies rep resenting the axioms. Let this topology be denoted by JM

Let J:= ∩Ψ∈MJΨ

Then, it is the largest Grothendieck topology for which every monetary value measure M is a sheaf……..