# Cartesian product

• $Graph$ is cartesian, as it is the presheaf category $Set^{𝒞^{op}}$ where $𝒞$ is the category having two objects and two (non trivial) morphisms in between.

• In $Top$, the topology of $X × Y$ is the coarsest (with the fewer open sets) topology such that the projections $π_X$ and $π_Y$ are continuous (i.e. the topology whose basis are products of open sets).

• In the category of pospaces $PoSp$: the order on $X × Y$ is the largest one (equality = smallest) making the projections morphisms.

\sqsubseteq_{X×Y} \; ≝ \; \sqsubseteq_{X} × \sqsubseteq_{Y}

is closed since it’s a product of closed sets (and the swapping of two coordinates is continuous).

• What about local pospaces $LPo$? The order on $U × V$ with $U ∈ 𝒰, V ∈ 𝒱$ is

(U × V, \sqsubseteq_U × \sqsubseteq_V)

the product of the pospaces $(U, ≤_U) × (V, ≤_V)$ provides an ordered atlas on $X × Y$

• Category of metric spaces and isometries: $Met_{emb}$? We have a problem! The projections must be embeddings, hence injective! This category is not cartesian.

• Category of metric spaces and contractions $Met_{ctr}$? It is cartesian, with

d((x,x'), (y, y')) = \max \lbrace d_X(x, x'), d_Y(y, y')\rbrace
• Category of metric spaces and continuous maps $Met_{top}$? You can take any distance giving you the product toplogy (e.g. the Euclidean/$L_2$ distance).

Infinite products of directed circles does not exists in $Lpo$. If it existed, the underlying topological space should be the infinite torus.

$𝕊^1$: circle

\prod\limits_{ n ∈ ℕ } 𝕊^1: \text{ the coarsest topology st all } π_k: (x_n)_{n ∈ ℕ} ↦ x_k \text{ are continuous}

Basis of this topology:

\prod\limits_{ n ∈ ℕ } V_n \text{ where } V_n \text{ is an open arc in } 𝕊^1 \text{ and } V_n ≠ 𝕊^1 \text{ for finitely many } n∈ ℕ

but when you have the full circle: impossible to put an order! Other way to see it: in the infinite torus, every point is a vortex ⟹ cannot be a local pospace.

So $Lpo$ is too restricted to do algebraic topology.

# Geometric model

For a process

\Pi = (G_1, …, G_n)

one constructs a local pospace (geometric realization)

\vert G_1 \vert × ⋯ × \vert G_n \vert \; ≝ \; \bigsqcup\limits_{p \text{ point of } G_1, …, G_n} \underbrace{\lbrace p \rbrace × ]0, 1[^{\dim p}}_{\text{block}}

One gets the geometric model by removing the blocks $B_p$ such that $p$ is forbidden.

Reminder: Discrete path: a sequence of points $p_1, …, p_q$ st to go from $p_i$ to $p_{i+1}$, either you leave points to go on arrows, or you leave arrows to go on their target point.

# Directed homotopy

h: [0, r] × [0, q] ⟶ \underbrace{X}_{\text{Local pospace}}
1. $h$ is a homotopy
2. for all $s ∈ [0, r]$, $h(s, -): [0, q] ⟶ X$ is a local pospace morphism
3. Stronger than the previous point: $h$ induces a local pospace morphism

NB:

• Just the 1. and 2.: it’s called a weakly directed homotopy
• if $h$ is a directed homotopy, the reverse $(t, s) ⟼ h(t,q-s)$ is not, a priori, a directed homotopy

• Counter-example: there exist weakly directed homotopies that are not directed homotopies: cf. picture

The relation

γ \preccurlyeq δ ⟺ \text{ there exists a directed homotopy from } γ \text{ to } δ

is a preorder. Associated congruence:

γ \sim δ ⟺ \text{ there exists an elementary directed homotopy from } γ \text{ to } δ
Elementary directed homotopy:

Finite concatenation of directed homotopies going back and forth. Two paths that have a elementary directed homotopy between them are said to be dihomotopic.

• $\sim_h$: homotopic
• $\sim_d$: dihomotopic
• $\sim_w$: weakly dihomotopic

## Reparametrization

Reparametrization:

a increasing and surjective map $θ: [0, r] ⟶[0, r]$

NB: Increasing and surjective ⟹ continuous

Any directed path is dihomotopic to any reparametrization thereof: if $γ: [0, r] ⟶ X$,

γ \circ θ \sim γ \circ_h \max (id_{[0,r]}, θ)

More generally: if $h: [0, r] × [0, q] ⟶ [0, r]$ is a directed homotopy from $θ: [0, r] ⟶ [0, r]$ to $θ’: [0, r] ⟶ [0, r]$, then $γ \circ θ$ and $γ \circ θ’$ are homotopic.

(cf. picture ⟶ 2-categorical)

### Peano curve

$γ: [0, 1] ⟶ [0, 1]^2$ continuous and surjective (onto). It is not bijective (otherwise $[0,1]$ and $[0, 1]^2$ would be homeomorphic)

If $X$ is a geometric model of a conservative program with mutex (mutual exclusion: semaphor of arity $1$) only (e.g.: not a cube (semaphors of arity 2), for example a cross (three instances of a mutex)).

Given two directed paths on $X$, say $γ, δ$ such that $γ(0) = δ(0)$ and $γ(r) = δ(r)$:

$γ$ and $δ$ are dihomotopic iff $γ$ and $δ$ are homotopic.

(cf. pictures)

## Compatible programs

P = (P_1,…, P_n) \qquad Q = (Q_1, …, Q_m)

Parallel composition:

P \mid Q = (P_1,…, P_n, Q_1, …, Q_m)

Defined only if their initial valuations and their arity maps coincide ⟺ compatible.

### Syntactical Independence

Syntactically independent:

no variables, semaphores, barriers in common

NB:

• Syntactically independence (s.i.):
• implies compatibility
• can be checked statistically
• compositional: pairwise s.i. ⟹ mutually s.i.

Syntactically independence: too strong ⟶ $P$ and $Q$ could share a resource with no problem in some cases (ex: a semaphor of arity $2$).

### Model Independence

Ex:

sem 1 a,b
sem 2 c

∏_1 = Pa Pc Vc Va | Pa Pc Vc Va
∏_2 = Pb Pc Vc Vb | Pb Pc Vc Vb

One resource of $c$ held by both program at a given time (since sem 1 a,b and Pa, Pb just before Pc), so $c$ should not appear in the geometric model.

Model indpendence:

⟦P_1 \; \mid \; ⋯ \; \mid \;P_N⟧ = ⟦P_1⟧ × ⋯ × ⟦P_1⟧

NB: It can be checked statistically.

Syntactic independence ⟹ Model independence ⟹ Observational independence

# Isothetic regions

Let $\Pi = (P_1, …, P_n)$ made up of P,V processes.

The set

\lbrace \text{finite union of intervals/connected subsets of } ℝ \rbrace

is a sub Boolean algebra (stable under ) of $𝒫(ℝ)$.

And let $(C_n)$ be an increasing sequence of cubes:

C_n \; ≝ \; I_1 × ⋯ × I_d

The increasing union remains a cube, the limit is defined component-wise:

\bigcup\limits_{n}^↑ C_n = \Big( \bigcup\limits_{n}^↑ proj_1 C_n \Big) × ⋯ × \Big( \bigcup\limits_{n}^↑ proj_d C_n \Big)

Because connected subsets are stable under product and increasing union.

We will do the same by replacing $ℝ$ by the geometric realization $\vert G \vert$, where $G$ is finite.

The fact that $G$ is finite is crucial: consider a star with infinitely many branches and remove the center point: the complement is no longer a finite union (not stable under complement).

But finiteness is not necessary for all that: consider the infinite chain (it works, similarly to $ℝ$).

## Galois connection

Ex:

\lbrace \text{Intervals of } ℝ \rbrace \overset{α}{\hookrightarrow} \lbrace \text{Subsets of } ℝ \rbrace

And conversely, by taking the smallest interval containing it:

\lbrace \text{Subsets of } ℝ \rbrace \overset{γ}{⟶} \lbrace \text{Intervals of } ℝ \rbrace

Besides:

γ \circ α = id\\ id ⊆ α \circ γ

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