Metric spaces, Pospaces and Local pospaces

Metric Spaces

We want to construct a functor from $Graph$ to the category of metric spaces.

Category of metric spaces:

• morphisms:

• isometries/embeddings:

$\begin{cases} (X, d_X) &⟶ (Y, d_Y) \\ f:X → Y &⟼ \text{ a map st } d_Y(f(x), f(y)) = d_Y(x, y) \end{cases}$

they are monos ⟶ most rigid definition

• $\vert\vert\vert T \vert\vert\vert \; ≝ \; \sup \lbrace \Vert T(x) \Vert \; \mid \; \Vert x \Vert ∈ \overline{B}(0, 1) \rbrace$
Contractive morphism:

if $d_Y(f(x), f(y)) ≤ d_Y(x, y)$, or more generally: $r$-Lipschitz maps: $d_Y(f(x), f(y)) ≤ r_f \cdot d_Y(x, y)$ with $r_f ≥ 0$ (where $r_f$ is the operator norm)

• any distance on a set $X$ induces a topology on $X$ (basis of open sets: open balls):

$U \text{ open iff } ∀x ∈ U, ∃r > 0 \text{ st } B(x, r) ⊆ U$

NB: if $B(x, r_i) ⊆ U_i$, then $B(x, \min_i r_i) ⊆ \bigcap\limits_{i} U_i$ for a finite intersection

most loose definition, because many different metrics may induce the same topology

Length spaces

Distance between $A$ and $B$: length of the shortest path from $A$ to $B$ (may not be the straight line)

Length $l(γ)$ of a path $γ:[0, r] ⟶ (X, d)$: lub of the sums

$\sum\limits_{ i=1 }^n d(γ(t_{i-1}), γ(t_i))$

for all $n ∈ ℕ$, and $0 = t_0 ≤ ⋯ ≤ t_n = r$

NB: it may happen that it is infinite (e.g. if your path is a fractal)

Length space:

distance between $x$ an $x’$: $\inf \lbrace l(γ) \; \mid \; γ \text{ is a path from } x \text{ to } x'\rbrace$

Geodesic path $γ$ between $x$ an $x’$:

if $l(γ) = d(x, x’)$

Geodesic space:

when every two points are related by a geodesic path

Cauchy sequence:

$(x_n) ∈ (X, d)$ st for all $ε > 0$, there exists $N_ε ∈ ℕ$ st for all $n, m ≥ N_ε$, $d_x(x_n, x_m) < ε$

Complete:

all Cauchy sequences converge

Hopf-Rinow theorem

$X$ complete and locally compact iff every closed bounded subset is compact and $X$ is geodesic

NB: recall that a Hausdorff space is locally compact iff each of its points admits a compact neighbourhood.

Isometric embeddings in $ℝ^n$

It is:

• geodesic
• length space
• complete
• locally compact

Now, what about $ℝ^n \backslash \lbrace 0\rbrace$?

• NOT geodesic
• length space
• NOT complete
• locally compact

Now, what about $ℝ^n \backslash [0, 1]^n$?

• NOT geodesic
• NOT length space
• NOT complete
• locally compact

You can always turn a length space into a metric space by setting:

$d_l(x,x') = \inf \lbrace l(γ) \; | \; γ \text{ is a path from } x \text{ to } x' \rbrace$

cf. picture

Each arrow provided with the same length because we want the shortest path on the graph to correspond to the shortest path in the continuous model (and we want the metric graph construction to be functorial)

• geodesic
• length space
• complete
• locally compact

Metric graph construction:

$Grph ⟶ Met_{emb}, Met_{ctr}, Met_{Lip}, Met_{Top} ?$

Certainly not $Met_{emb}$, because there can always be shortcuts in the target metric graph (you can send two points to one point ⟶ not a mono)

Functoriality:

$Grph ⟶ Met_{ctr}$

Directed graphs ⟹ order needed

Partially ordered spaces

Partially ordered space:

$X$ equipped with an order $≤$ whose graph is closed.

If $(X, ≤)$ is partially ordered, then it’s Haussdorff. Indeed:

$≤ ∩ ≤^{op} = \lbrace (x, y) \; \mid \; x=y \rbrace$

Intersection of closed sets ⟹ the diagonal $\lbrace (x, y) \; \mid \; x=y \rbrace$ is closed, which is sufficient for $X$ to be Hausdorff.

Hausdorff distance:

$\inf$ over $ε$ st

$\begin{cases} ∀ x ∈ K, ∃ x' ∈ K'; d_X(x, x') ≤ ε \\ ∀ x ∈ K', ∃ x' ∈ K; d_X(x, x') ≤ ε \end{cases}$
$ℝ^n \overbrace{⟶}^{\text{sheaf}}_{\text{charts and atlases}} n \text{-manifold}$ $\text{Pospaces } \overbrace{⟶}^{\text{sheaf}}_{\text{ordered charts and atlases}} \text{ Local Pospaces}$

$ℝ$ ordered atlases on $ℝ$ such that the “directed paths” are the increasing maps from $ℝ$ to $ℝ$

• $]a, b[$ with $a<b$ and standard order
• $U ⊆ ℝ$ open subset with the order:

$x \sqsubseteq y ⟺ x ≤_ℝ y \text{ and } [x,y] ⊆ U$

For the circle: consider the flat circle $X$ (cf. picture), then $\exp: X ⟶ S^1$ is a homeomorphism

Morphisms between locally ordered spaces:

$f: (X, 𝒰) ⟶ (Y, 𝒱)$
• $f: X ⟶ Y$ is continuous
• $∀ x ∈ X, ∃ (U, ≤U) ∈ 𝒰, ∃ (V, ≤_U) ∈ 𝒱$, then $f{\mid \; U}$ is monotone

This definition must be invariant under atlas equivalence (because local pospaces are defined up to this equivalence).

Local pospaces

We have a functor

$I: \begin{cases} Pospaces &⟶ Local \; Pospaces \\ (X, ≤_X) &⟼ X, \lbrace U \text{ open subset of } X \text{ with } {≤_{X × X}}_{| U × U}\rbrace \end{cases}$

Question: is this functor

• one-to-one on objects? No, consider

$X \; ≝ \; [0, 1] ∪ [2, 3]$

we can give it these two pospace structures:

• $≤_X \; ≝ \;d {≤_ℝ}_{| X × X}$
• $x \sqsubseteq_X y ⟺ x ≤_ℝ y \text{ and } [x, y] ⊆ X$

then

$(X, ≤_X) \not ≅ (X, \sqsubseteq_X)\\ I(X, ≤_X) ≅ I(X, \sqsubseteq_X) \qquad \text{(with the identity)}$
• Faithful? Yes!

• Full? No! It’s easier to be a local pospace morphism than to be a pospace one.

In the previous example:

$\begin{cases} I(X, ≤_X) &⟶ I(X, \sqsubseteq_X) \\ x &⟼ x \end{cases}$

is a local pospace morphism, but not a pospace one!

Directed loops on local pospaces

A directed path $δ$ on a local pospace $X$ is constant iff its extremities are equal and there exists an ordered chart of some atlas of $X$ that contains the image of $δ$ (because of the fact that charts are ordered).

(cf. picture)

Vortex:

a point every neighborhood of which contains a directed loop

⟹ a local pospace has no vortex.

E.g.: in the complex plane where directed paths go counterclockwise, the orgin is a vertex.

Ordered altas on metric graphs

$G \; ≝ \; A \overset{ s, t }{⟶} V$

Metric graph:

$\vert G \vert \; ≝ \; V \; \sqcup \; A × ]0, 1[$

Basis: open balls of radius $≤ 1/3$ and

• centered at vertices
• centerad at points on arrows

so that balls

• centered at two distinct vertices do not intersect
• centered at two vertices and one inner point ar such that the inner point one intersect at most one of the vertex balls
• the intersection of two balls centered at the same vertex remains an open ball

Partial order on these balls:

• inherited from the real line
• the order of the ingoing arrows is lower than the one the of the outgoing arrows

There’s a functor

$I: \begin{cases} Grpah &⟶ Local \; Pospaces \\ (G, A, V) &⟼ V \sqcup A × ]0, 1[ \end{cases}$

such that the map $f_0, f_1: G’ ⟶ G’$ is sent to

$\begin{cases} V \ni v &⟶ f_0(v) \\ A × ]0, 1[ \ni (α, t) &⟼ (f_1(α), t) \end{cases}$

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