A **contravariant functor*** ρ : C^{op} → 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).

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

*is a set*

**U***such that*

**I ⊂↓ U**

**(∀ V ∈ ↓ U)(∀ W ∈ ↓ U)[W ⊂ V ∈ I ⇒ W ∈ I]*** 1.3* For a sieve

*on*

**I***and*

**U***in*

**V ⊂ U***,*

**χ**

**I ↓ V := I****∩ ↓****V*** 1.4* A family of

*is an element*

**I**

**X ∈ ∏**_{V∈I}L(V), or X = (X_{V})_{V∈I}* 1.5* A family

*is called a*

**X =****(X**_{V})_{V∈I }*if*

**P-martingale**

**(∀ V ∈ I)(∀ W ∈ I)[W ⊂ V ⇒ E**^{P}[X_{V}| W] = XW]A sieve on * U* is considered as a kind of a time domain.

**2.0*** Ξ : χ^{op} → Set* is a contravariant functor such that for

*in*

**i**^{V}_{U}: V → U*,*

**χ***is the set of all sieves on*

**Ξ(U)***, and that*

**U**

**Ξ(i**^{V}_{U})(I) = I ↓ V for I ∈ Ξ(U)* 2.1* A Grothendieck topology on

*is a sub-factor*

**χ***satisfying the following conditions*

**J → Ξ****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

*if*

**U**

**I ∈ J(U)*** U* is considered as a time horizon of a time domain

*if it is covered by*

**I***.*

**I*** 3.0* Theorem Let

*be a collection of Grothendieck topologies on*

**{J**_{a}| a ∈ A}*Then the sub-functor*

**χ.***defined by J*

**J → Ξ***is a Grothendieck topology.*

**(U) :=***∩*_{a∈A}J_{a}(U)* 4.0* Let

*be a monetary value measure, and*

**Ψ ∈ Set**^{χop}*be a sieve on*

**I**

**U ∈ χ*** 4.1* A family

*is called*

**X = (X**_{V})_{V∈I}*-matching if*

**Ψ**

**(∀ V ∈ I)(∀ W ∈ I)Ψ**^{V∧W}_{V}= Ψ^{V∧W}_{W}(X_{W})* 4.2* A random variable

*be a*

**X***̄*∈ L(U)**-amalgamation for a family**

*Ψ**if*

**X = (X**_{V})_{V ∈ I}

**(∀ V ∈ I)Ψ**^{V}_{U}(X ̄) = X_{V}* 5.0* Let

*be a monetary value measure.*

**Ψ ∈ Set**^{χop}*be a sieve on*

**I***and*

**U ∈ χ***be a family that has*

**X = (X**_{V})_{V∈I}*-amalgamation. Then*

**Ψ***is*

**X***-matching.*

**Ψ**

Let * X ̄∈ L(U) *be a

*-amalgamation. Then for every*

**Ψ***Therefore, for every*

**V ∈ I, X**_{V}= Ψ^{V}_{U}(X*̄*).

**V, W ∈ I Ψ**^{V∧W}_{V}(X_{V}) = X_{V∧W}= Ψ^{V∧W}_{W}(X_{W})**6.0*** Ψ ∈ Set^{χop}* be a monetary value measure.

*be a sieve on*

**I***and*

**U ∈ χ***be a*

**X = (X**_{V})_{V∈I}*-matching family.*

**Ψ*** 6.1* For

*, if*

**V, W ∈ I***, we have*

**W ⊂ V**

**Ψ**^{W}_{V}(X_{V}) = X_{W}**Ψ ^{W}_{V}(X_{V}) = Ψ^{V∧W}_{V} (X_{V}) = Ψ^{V∧W}_{W} (X_{W}) = Ψ^{W}_{W}(X_{W}) = (X_{W})**

* 6.2* If

*,*

**U ∈ I***is the unique*

**X**_{U}*-amalgamation for*

**Ψ***. By*

**X***,*

**6.1***is the unique*

**X**_{U}*-amalgamation for*

**Ψ***. Now, let*

**X***be another*

**X ̄****∈ L(U)***-amalgamation for*

**Ψ***. Then for every*

**X***Put*

**V ∈ I, X**_{V}= Ψ^{V}_{U}(**X ̄).***Then we have*

**V := U.**

**X**_{U}= Ψ^{U}_{U}(X ̄) = 1_{U}(X ̄) = X ̄.* 7.0* Let

*be a Grothendieck topology on*

**J***. A monetary value measure*

**Ψ ∈ Set**^{χop}*is called a sheaf if for any*

**Ψ ∈ Set**^{χop}*Any*

**U ∈ χ.***covering sieve*

**J***and any*

**I ∈ J(U)***-matching family*

**Ψ***has a unique*

**X = (X**_{V})_{V∈I}b, X*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.*

**Ψ-**Let us consider a sieve * I* on

*as a subfunctor*

**U ∈ χ***, that is a contravariant functor*

**I → Hom**_{X}(-,U)*defined by*

**I : χ**^{op}→ Set**I(V) := {i ^{V}_{U}} if V ∈ I**

**:= Φ if V ∉ I**

for **V ∈ χ**

* 8.0* Let

*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*

**M ⊂****Set**^{χop}

**J**_{M}Let **J _{M }**

**:= ∩**

_{Ψ∈M}J_{Ψ}Then, it is the largest Grothendieck topology for which every monetary value measure * M* is a sheaf……..

** **

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