Schematic Grothendieck Representation

A spectral Grothendieck representation Rep is said to be schematic if for every triple γ ≤ τ ≤ δ in Top(A), for every A in R^{^}(Ring) we have a commutative diagram in R^{^}:

 

If Rep is schematic, then, P : Top(A) → R^{^} is a presheaf with values in R^{^} over the lattice Top(A)^o, for every A in R.

The modality is to restrict attention to Tors(Rep(A)); that is, a lattice in the usual sense; and hence this should be viewed as the commutative shadow of a suitable noncommutative theory.

For obtaining the complete lattice Q(A), a duality is expressed by an order-reversing bijection: (−)⁻¹ : Q(A) → Q((Rep(A))^o). (Rep(A))^o is not a Grothendieck category. It is additive and has a projective generator; moreover, it is known to be a varietal category (also called triplable) in the sense that it has a projective regular generator P, it is co-complete and has kernel pairs with respect to the functor Hom(P, −), and moreover every equivalence relation in the category is a kernel pair. If a comparison functor is constructed via Hom(P, −) as a functor to the category of sets, it works well for the category of set-valued sheaves over a Grothendieck topology.

Now (−)⁻¹ is defined as an order-reversing bijection between idempotent radicals on Rep(A) and (Rep(A))^o, implying we write (Top(A))⁻¹ for the image of Top(A) in Q((Rep( A))^o). This is encoded in the exact sequence in Rep(A):

0 → ρ(M) → M → ρ⁻¹(M) → 0

(reversed in (Rep(A))^o). By restricting attention to hereditary torsion theories (kernel functors) when defining Tors(−), we introduce an asymmetry that breaks the duality because Top(A)⁻¹ is not in Tors((Rep(A))^op). If notationally, TT(G) is the complete lattice of torsion theories (not necessarily hereditary) of the category G; then (TT(G))⁻¹ ≅ TT(G^op). Hence we may view Tors(G)⁻¹ as a complete sublattice of TT(G^op).

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Derivability from Relational Logic of Charles Sanders Peirce to Essential Laws of Quantum Mechanics

Charles Sanders Peirce made important contributions in logic, where he invented and elaborated novel system of logical syntax and fundamental logical concepts. The starting point is the binary relation S_iRS_j between the two ‘individual terms’ (subjects) S_j and S_i. In a short hand notation we represent this relation by R_ij. Relations may be composed: whenever we have relations of the form R_ij, R_jl, a third transitive relation R_il emerges following the rule

R_ijR_kl = δ_jkR_il —– (1)

In ordinary logic the individual subject is the starting point and it is defined as a member of a set. Peirce considered the individual as the aggregate of all its relations

S_i = ∑_j R_ij —– (2)

The individual S_i thus defined is an eigenstate of the R_ii relation

R_iiS_i = S_i —– (3)

The relations R_ii are idempotent

R²_ii = R_ii —– (4)

and they span the identity

∑_i R_ii = 1 —– (5)

The Peircean logical structure bears resemblance to category theory. In categories the concept of transformation (transition, map, morphism or arrow) enjoys an autonomous, primary and irreducible role. A category consists of objects A, B, C,… and arrows (morphisms) f, g, h,… . Each arrow f is assigned an object A as domain and an object B as codomain, indicated by writing f : A → B. If g is an arrow g : B → C with domain B, the codomain of f, then f and g can be “composed” to give an arrow gof : A → C. The composition obeys the associative law ho(gof) = (hog)of. For each object A there is an arrow 1_A : A → A called the identity arrow of A. The analogy with the relational logic of Peirce is evident, R_ij stands as an arrow, the composition rule is manifested in equation (1) and the identity arrow for A ≡ S_i is R_ii.

R_ij may receive multiple interpretations: as a transition from the j state to the i state, as a measurement process that rejects all impinging systems except those in the state j and permits only systems in the state i to emerge from the apparatus, as a transformation replacing the j state by the i state. We proceed to a representation of R_ij

R_ij = |r_i⟩⟨r_j| —– (6)

where state ⟨r_i | is the dual of the state |r_i⟩ and they obey the orthonormal condition

⟨r_i |r_j⟩ = δ_ij —– (7)

It is immediately seen that our representation satisfies the composition rule equation (1). The completeness, equation (5), takes the form

∑_n|r_i⟩⟨r_i|=1 —– (8)

All relations remain satisfied if we replace the state |r_i⟩ by |ξ_i⟩ where

|ξ_i⟩ = 1/√N ∑_n |r_i⟩⟨r_n| —– (9)

with N the number of states. Thus we verify Peirce’s suggestion, equation (2), and the state |r_i⟩ is derived as the sum of all its interactions with the other states. R_ij acts as a projection, transferring from one r state to another r state

R_ij |r_k⟩ = δ_jk |r_i⟩ —– (10)

We may think also of another property characterizing our states and define a corresponding operator

Q_ij = |q_i⟩⟨q_j | —– (11)

with

Q_ij |q_k⟩ = δ_jk |q_i⟩ —– (12)

and

∑_n |q_i⟩⟨q_i| = 1 —– (13)

Successive measurements of the q-ness and r-ness of the states is provided by the operator

R_ijQ_kl = |r_i⟩⟨r_j |q_k⟩⟨q_l | = ⟨r_j |q_k⟩ S_il —– (14)

with

S_il = |r_i⟩⟨q_l | —– (15)

Considering the matrix elements of an operator A as Anm = ⟨rn |A |rm⟩ we find for the trace

Tr(S_il) = ∑_n ⟨r_n |S_il |r_n⟩ = ⟨q_l |r_i⟩ —– (16)

From the above relation we deduce

Tr(R_ij) = δ_ij —– (17)

Any operator can be expressed as a linear superposition of the R_ij

A = ∑_i,j A_ijR_ij —– (18)

with

A_ij =Tr(AR_ji) —– (19)

The individual states could be redefined

|r_i⟩ → e^iφ_i |r_i⟩ —– (20)

|q_i⟩ → e^iθ_i |q_i⟩ —– (21)

without affecting the corresponding composition laws. However the overlap number ⟨r_i |q_j⟩ changes and therefore we need an invariant formulation for the transition |r_i⟩ → |q_j⟩. This is provided by the trace of the closed operation R_iiQ_jjR_ii

Tr(R_iiQ_jjR_ii) ≡ p(q_j, r_i) = |⟨r_i |q_j⟩|² —– (22)

The completeness relation, equation (13), guarantees that p(q_j, r_i) may assume the role of a probability since

∑_j p(q_j, r_i) = 1 —– (23)

We discover that starting from the relational logic of Peirce we obtain all the essential laws of Quantum Mechanics. Our derivation underlines the outmost relational nature of Quantum Mechanics and goes in parallel with the analysis of the quantum algebra of microscopic measurement.

Truncation Functors

Let A be an abelian category, and let D = D(A) be the derived category. For any complex A• in A, and n ∈ Z, we let τ_≤nA• be the truncated complex

··· → Aⁿ⁻² → Aⁿ⁻¹ → ker(Aⁿ → Aⁿ⁺¹)→ 0 → 0 → ··· , and dually we let τ_≥nA be the complex

··· → 0 → 0 → coker(Aⁿ⁻¹ → Aⁿ) → Aⁿ⁺¹ → Aⁿ⁺² → ···

Note that

H^m(τ_≤nA•) = H^m(A•) if m ≤ n

= 0 if m > n

and that

H^m(τ_≥nA•) = H^m(A•)  if m ≥ n

= 0 if m < n

One checks that τ_≥n (respectively τ_≤n) extends naturally to an additive functor of complexes which preserves homotopy and takes quasi-isomorphisms to quasi-isomorphisms, and hence induces an additive functor D → D. In fact if D_≤n (respectively D_≥n) is the full subcategory of D whose objects are the complexes A• such that H^m(A•) = 0 for m > n (respectively m < n) then we have additive functors

τ_≤n : D → D_≤n ⊂ D

τ_≥n : D → D_≥n ⊂ D

together with obvious functorial maps

iⁿ_A : τ_≤n A• → A•

jⁿ_A : A• → τ_≥n A•

The preceding iⁿ_A , jⁿ_A induce functorial isomorphisms

Hom_D≤n (B•,τ_≤nA•) →^~ Hom_D(B•, A•) (B• ∈ D_≤n) —– (1)

Hom_D≥n (τ_≥nA•,C•) →^~ Hom_D(A•,C• ) (C• ∈ D_≥n) —– (2)

Bijectivity of (1) means that any map φ : B• → A• (in D) with B• ∈ D_≤n factors uniquely via i_A := iⁿ_A

Given φ, we have a commutative diagram

and since B• ∈ D_≤n, therefore i_B is an isomorphism in D, so we can write

φ = i ◦ (τ_≤nφ ◦ i⁻¹_B),

and thus (1) is surjective.

To prove that (1) is also injective, we assume that i_A ◦ τ_≤n φ = 0 and deduce that τ_≤n φ = 0. The assumption means that there is a commutative diagram in K(A)

where s and s′′ are quasi-isomorphisms, and f/s = τ_≤nφ

Applying the (idempotent) functor τ_≥n, we get a commutative diagram

Since τ_≤ns and τ_≤ns′′ are quasi-isomorphisms, we have

τ_≤nφ = τ_≤n f/τ_≤ns = 0/τ_≤ns′′ = 0

as desired.

Galois Connections. Part 3.

Let (P,≤P) and (Q,≤Q) be posets, and consider two set functions ∗ ∶ P ⇄ Q ∶ ∗. We will denote these by p ↦ p ∗ and q ↦ q ∗ for all p ∈ P and q ∈ Q. This pair of functions is called a Galois connection if, for all p ∈ P and q ∈ Q, we have

p ≤ P q ∗ ⇐⇒ q ≤ Q p  ∗

Let ∗ ∶ P ⇄ Q ∶ ∗ be a Galois connection. For all elements x of P or Q we will use the notations x ∗ ∗ ∶= (x ∗)∗ and x ∗ ∗ ∗ ∶= (x ∗ ∗)∗.

(1) For all p ∈ P and q ∈ Q we have

p ≤ P p ∗ ∗ and q ≤ Q q ∗ ∗.

(2) For all elements p1, p2 ∈ P and q1, q2 ∈ Q we have

p1 ≤ P p2 ⇒ p ∗ 2 ≤ Q p ∗ 1 and q1 ≤ Q q2 ⇒ q2 ∗ ≤ P q1 ∗.

(3) For all elements p ∈ P and q ∈ Q we have

p ∗ ∗ ∗ = p ∗ and q ∗ ∗ ∗ = q ∗

Proof:

Since the definition of a Galois connection is symmetric in P and Q, we will simplify the proof by using the notation

x ≤ y ∗ ⇐⇒ y ≤ x ∗

for all elements x,y such that the inequalities make sense. To prove (1) note that for any element x we have x ∗ ≤ x ∗ by the reflexivity of partial order. Then from the definition of Galois connection we obtain,

(x ∗) ≤ (x) ∗ ⇒ (x) ≤ (x ∗) ∗ ⇒ x ≤ x ∗ ∗

To prove (2) consider elements x, y such that x ≤ y. From (1) and the transitivity of partial x ≤ y ≤ y ∗ ∗ ⇒ x ≤ y ∗ ∗. Then from the definition of Galois connection we obtain

(x) ≤ (y ∗) ∗ ⇒ (y ∗) ≤ (x) ∗ ⇒ y ∗ ≤ x ∗.

To prove (3) consider any element x. On the one hand, part (1) tells us that

(x ∗) ≤ (x ∗) ∗ ∗ ⇒ x ∗ ≤ x ∗ ∗ ∗.

On the other hand, part (1) tells us that x ≤ x ∗ ∗ and then part (2) says that

(x) ≤ (x ∗ ∗) ⇒ (x ∗ ∗) ∗ ≤ (x) ∗ ⇒ x ∗ ∗ ∗ ≤ x ∗

Finally, the antisymmetry of partial order says that x∗∗∗ = x∗, which we interpret as isomorphism of objects in the poset category. The following definition captures the essence of these three basic properties.

Definition of Closure in a Poset. Given a poset (P,≤), we say that a function cl ∶ P → P is a closure operator if it satisfies the following three properties:

(i) Extensive: ∀p ∈ P, p ≤ cl(p)

(ii) Monotone: ∀ p,q ∈ P, p ≤ q ⇒ cl(p) ≤ cl(q)

(iii) Idempotent: ∀ p ∈ P, cl(cl(p)) = p.

[Remark: If P = 2^U is a Boolean lattice, and if the closure cl ∶ 2^U → 2^U also preserves finite unions, then we call it a Kuratowski closure. Kuratowski proved that such a closure is equivalent to a topology on the set U.]

If ∗ ∶ P → Q ∶ ∗ is a Galois connection, then the basic properties above immediately imply that the compositions ∗ ∗ ∶ P → P and ∗ ∗ ∶ Q → Q are closure operators.

Proof: Property (ii) follows from applying property (2) twice and property (iii) follows from applying ∗ to property (3).

Fundamental Theorem of Galois Connections: Any Galois connection ∗ ∶ P ⇄ Q ∶ ∗ determines two closure operators ∗ ∗ ∶ P → P and ∗ ∗ ∶ Q → Q. We will say that the element p ∈  P (resp. q ∈  Q) is ∗ ∗-closed if p∗ ∗ = p (resp. q∗ ∗ = q). Then the Galois connection restricts to an order-reversing bijection between the subposets of ∗ ∗-closed elements.

Proof: Let Q ∗ ⊆ P and P ∗ ⊆ Q denote the images of the functions ∗ ∶ Q → P and ∗ ∶ P  → Q, respectively. The restriction of the connection to these subsets defines an order-reversing bijection:

Indeed, consider any p ∈ Q ∗, so that p = q ∗ for some q ∈ Q. Then by properties (1) and (3) of Galois connections we have

(p) ∗ ∗ = (q ∗) ∗ ∗ ⇒ p ∗ ∗ = q ∗ ∗ ∗ ⇒ p ∗ ∗ = q ∗ ⇒ p ∗ ∗ = p

Similarly, for all q ∈ P ∗ we have q ∗ ∗ = q. The bijections reverse order because of property (2).

Finally, note that Q ∗ and P ∗ are exactly the subsets of ∗ ∗-closed elements in P and Q, respectively. Indeed, we have seen above that every element of Q ∗ is ∗ ∗-closed. Conversely, if p ∈ P is ∗ ∗-closed then we have

p = p ∗ ∗ ⇒ p = (p ∗) ∗,

and it follows that p ∈ Q ∗. Similarly, every element of P ∗ is ∗ ∗-closed.

Thus, a Galois connection is something like a “loose bijection”. It’s not necessarily a bijection but it becomes one after we “tighten it up”. Sort of like tightening your shoelaces.

The shaded subposets here consist of the ∗ ∗-closed elements. They are supposed to look (anti-) isomorphic. The unshaded parts of the posets get “tightened up” into the shaded subposets. Note that the top elements are ∗ ∗-closed. Indeed, property (2) tells us that 1_P ≤ P ≤ 1_p^∗∗ and then from the universal property of the top element we have 1_P^** = 1_P. Since the left hand side is always true, so is the right hand side. But then from the universal property of the top element in Q we conclude that 0^∗_P = 1_Q. As a consequence of this, the arbitrary meet of ∗ ∗-closed elements (if it exists) is still ∗ ∗-closed. We will see, however, that the join of ∗ ∗-closed elements is not necessarily ∗ ∗-closed. And hence not all Galois connections induce topologies.

Galois connections between Boolean lattices have a particularly nice form, which is closely related to the universal quantifier “∀“. Galois Connections of Boolean Lattices. Let U,V be sets and let ∼ ⊆ U × V be any subset (called a relation) between U and V . As usual, we will write “u ∼ v” in place of the statement “(u,v) ∈ ∼“, and we read this as “u is related to v“. Then for all S ∈ 2^U and T ∈ 2^V we define,

S^∼ ∶= {v ∈ V ∶ ∀ s ∈ S, s ∼ v} ∈ 2^V,

T^∼ ∶= {u ∈ U ∶ ∀ t ∈ T , u ∼ t} ∈ 2^U

The pair of functions S ↦ S^∼ and T ↦ T^∼ is a Galois connection, ∼ ∶ 2^U ⇄ 2^V ∶ ∼.

To see this, note that ∀ subsets S ∈ 2^U and T ∈ 2^V we have

S ⊆ T^∼ ⇐⇒ ∀ s ∈ S, s ∈ T^∼

⇐⇒ ∀ s ∈ S,∀ t ∈ T, s ∼ t

⇐⇒ ∀ t ∈ T, ∀ s ∈ S, s ∼ t

⇐⇒ ∀ t ∈ T, t ∈ S^∼

⇐⇒ T ⊆ S^∼.

Moreover, one can prove that any Galois connection between 2^U and 2^V arises in this way from a unique relation.

Orthogonal Complement: Let V be a vector space over field K and let V ∗ be the dual space, consisting of linear functions α ∶ V → K. We define the relation ⊥ ⊆ V ∗ × V by

α ⊥ v ⇐⇒ α(v) = 0.

The resulting ⊥⊥-closed subsets are precisely the linear subspaces on both sides. Thus the Fundamental Theorem of Galois Connections gives us an order-reversing bijection between the subspaces of V ∗ and the subspaces of V.

Convex Complement: Let V be a Euclidean space, i.e., a real vector space with an inner product ⟨-,-⟩ ∶ V ×V → ℜ. We define the relation ∼ ⊆ V ×V by

u ∼ v ⇐⇒ ⟨u,v⟩ ≤ 0.

∀ S ⊆ V the operation S ↦ S ∼ ∼ gives the cone genrated by S, thus the ∼ ∼-closed sets are precisely the cones. Here is a picture:

Original Galois Connection: Let L be a field and let G be a finite group of automorphisms of L, i.e., each g ∈ G is a function g ∶ L → L preserving addition and multiplication. We define a relation ∼ ⊆ G × L by

g ∼ l ⇐⇒ g(l) = l.

Define K ∶= L ∼ to be the “subfield fixed by G“. The original Fundamental Theorem of Galois Theory says that the ∼ ∼-closed subsets of G are precisely the subgroups and the ∼ ∼-closed subsets of L are precisely the subfields containing K.

Hilbert’s Nullstellensatz: Let K be a field and consider the ring of polynomials K[x] ∶= K[x₁,…,x_n] in n commuting variables. For each polynomial f(x) ∶= f(x₁,…,x_n) ∈ K[x] and for each n-tuple of field elements α ∶= (α₁,…,α_n) ∈ Kn, we denote the evaluation by f(α) ∶= f(α₁,…,α_n) ∈ K. Now we define a relation ∼ ⊆ K[x] × Kn by

f(x) ∼ α ⇐⇒ f(α) = 0

By definition, the closure operator ∼ ∼ on subsets of Kn is called the Zariski closure. It is not difficult to prove that it satisfies the additional property of a Kuratowski closure (i.e., finite unions of closed sets are closed) and hence it defines a topology on Kn, called the Zariski topology. Hilbert’s Nullstellensatz says that if K is algebraically closed, then the ∼ ∼-closed subsets of K[x] are precisely the radical ideals (i.e., ideals closed under taking arbitrary roots).