Causal Isomorphism as a Diffeomorphism. Some further Rumination on Philosophy of Science. Thought of the Day 82.0

Let (M, gab) and (M′, g′ab) be (temporally oriented) relativistic spacetimes that are both future- and past-distinguishing, and let φ : M → M′ be a ≪-causal isomorphism. Then φ is a diffeomorphism and preserves gab up to a conformal factor; i.e. φ⋆(g′ab) is conformally equivalent to gab.

Under the stated assumptions, φ must be a homeomorphism. If a spacetime (M, gab) is not just past and future distinguishing, but strongly causal, then one can explicitly characterize its (manifold) topology in terms of the relation ≪. In this case, a subset O ⊆ M is open iff, ∀ points p in O, ∃ points q and r in O such that q ≪ p ≪ r and I+(q) ∩ I(r) ⊆ O (Hawking and Ellis). So a ≪-causal isomorphism between two strongly causal spacetimes must certainly be a homeomorphism. Then one invokes a result of Hawking, King, and McCarthy that asserts, in effect, that any continuous ≪-causal isomorphism must be smooth and must preserve the metric up to a conformal factor.

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The following example shows that the proposition fails if the initial restriction on causal structure is weakened to past distinguishability or to future distinguishability alone. We give the example in a two-dimensional version to simplify matters. Start with the manifold R2 together with the Lorentzian metric

gab = (d(at)(db)x) − (sinh2t)(dax)(dbx)

where t, x are global projection coordinates on R2. Next, form a vertical cylinder by identifying the point with coordinates (t, x) with the one having coordinates (t, x + 2). Finally, excise two closed half lines – the sets with respective coordinates {(t, x): x = 0 and t ≥ 0} and {(t, x): x = 1 and t ≥ 0}. Figure shows, roughly, what the null cones look like at every point. (The future direction at each point is taken to be the “upward one.”) The exact form of the metric is not important here. All that is important is the indicated qualitative behavior of the null cones. Along the (punctured) circle C where t = 0, the vector fields (∂/∂t)a and (∂/∂x)a both qualify as null. But as one moves upward or downward from there, the cones close. There are no closed timelike (or null) curves in this spacetime. Indeed, it is future distinguishing because of the excisions. But it fails to be past distinguishing because I(p) = I(q) for all points p and q on C. For all points p there, I(p) is the entire region below C. Now let φ be the bijection of the spacetime onto itself that leaves the “lower open half” fixed but reverses the position of the two upper slabs. Though φ is discontinuous along C, it is a ≪-causal isomorphism. This is the case because every point below C has all points in both upper slabs in its ≪-future.

Category Theory of a Sketch. Thought of the Day 50.0

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If a sketch can be thought of as an abstract concept, a model of a sketch is not so much an interpretation of a sketch, but a concrete or particular instantiation or realization of it. It is tempting to adopt a Kantian terminology here and say that a sketch is an abstract concept, a functor between a sketch and a category C a schema and the models of a sketch the constructions in the “intuition” of the concept.

The schema is not unique since a sketch can be realized in many different categories by many different functors. What varies from one category to the other is not the basic structure of the realizations, but the types of morphisms of the underlying category, e.g., arbitrary functions, continuous maps, etc. Thus, even though a sketch captures essential structural ingredients, others are given by the “environment” in which this structure will be realized, which can be thought of as being itself another structure. Hence, the “meaning” of some concepts cannot be uniquely given by a sketch, which is not to say that it cannot be given in a structuralist fashion.

We now distinguish the group as a structure, given by the sketch for the theory of groups, from the structure of groups, given by a category of groups, that is the category of models of the sketch for groups in a given category, be it Set or another category, e.g., the category of topological spaces with continuous maps. In the latter case, the structure is given by the exactness properties of the category, e.g., Cartesian closed, etc. This is an important improvement over the traditional framework in which one was unable to say whether we should talk about the structure common to all groups, usually taken to be given by the group axioms, or the structure generated by “all” groups. Indeed, one can now ask in a precise manner whether a category C of structures, e.g., the category of (small) groups, is sketchable, that is, whether there exists a sketch S such that Mod(S, Set) is equivalent as a category to C.

There is another category associated to a sketch, namely the theory of that sketch. The theory of a sketch S, denoted by Th(S), is in a sense “freely” constructed from S : the arrows of the underlying graph are freely composed and the diagrams are imposed as equations, and so are the cones and the cocones. Th(S) is in fact a model of S in the previous sense with the following universal property: for any other model M of S in a category C there is a unique functor F: Th(S) → C such that FU = M, where U: S → Th(S). Thus, for instance, the theory of groups is a category with a group object, the generic group, “freely” constructed from the sketch for groups. It is in a way the “universal” group in the sense that any other group in any category can be constructed from it. This is possible since it contains all possible arrows, i.e., all definable operations, obtained in a purely internal or abstract manner. It is debatable whether this category should be called the theory of the sketch. But that may be more a matter of terminology than anything else, since it is clear that the “free” category called the theory is there to stay in one way or another.

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 = 2U is a Boolean lattice, and if the closure cl ∶ 2U → 2U 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:

img_20170204_065156

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.

img_20170204_071135

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 1P ≤ P ≤ 1p∗∗ and then from the universal property of the top element we have 1P** = 1P. 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 0P = 1Q. 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 ∈ 2U and T ∈ 2V we define,

S ∶= {v ∈ V ∶ ∀ s ∈ S, s ∼ v} ∈ 2V,

T ∶= {u ∈ U ∶ ∀ t ∈ T , u ∼ t} ∈ 2U

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

To see this, note that ∀ subsets S ∈ 2U and T ∈ 2V 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 2U and 2V 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:

img_20170204_075300

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,…,xn] in n commuting variables. For each polynomial f(x) ∶= f(x1,…,xn) ∈ K[x] and for each n-tuple of field elements α ∶= (α1,…,αn) ∈ Kn, we denote the evaluation by f(α) ∶= f(α1,…,αn) ∈ 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).