Directed Interval

Directed Interval Skill

"The directed interval 2 is the walking arrow: a single morphism 0 → 1." — Riehl-Shulman

Overview

The directed interval 2 replaces the undirected interval 𝕀 of cubical type theory with a directed version. This axiomatizes the notion of "time flows forward" essential for modeling reactions.

Core Definitions (Rzk)

#lang rzk-1

-- CUBES: The category of directed cubes
-- 2 is the basic directed interval [0 → 1]

-- The directed interval (primitive)
#define 2 : CUBE

-- Endpoints
#define 0₂ : 2
#define 1₂ : 2

-- The unique arrow (built-in)
-- There is a morphism 0₂ → 1₂ but NOT 1₂ → 0₂

-- Higher cubes built from 2
#define 2×2 : CUBE := 2 × 2

-- Directed square (all arrows point same way)
#define □ : CUBE := 2 × 2

-- Simplex shapes
#define Δ¹ : CUBE := 2
#define Δ² : CUBE := { (t₁, t₂) : 2 × 2 | t₁ ≤ t₂ }
#define Δ³ : CUBE := { (t₁, t₂, t₃) : 2 × 2 × 2 | t₁ ≤ t₂ ∧ t₂ ≤ t₃ }

-- Hom type as extension type
#define hom (A : U) (x y : A) : U
  := { f : 2 → A | f 0₂ = x ∧ f 1₂ = y }
  -- equivalently: (t : 2) → A [t ≡ 0₂ ↦ x, t ≡ 1₂ ↦ y]

Chemputer Semantics

GF(3) Triad

segal-types (-1) ⊗ directed-interval (0) ⊗ rezk-types (+1) = 0 ✓

As a Coordinator (0), directed-interval:

• Mediates between validators and generators
• Provides the "time axis" for computation
• Enables transport along directed paths

Extension Types

The key innovation is extension types for partial elements:

-- Extension type: functions with prescribed boundary
#define extension-type 
  (I : CUBE) (ψ : I → TOPE) (A : I → U) (a : (t : ψ) → A t) : U
  := { f : (t : I) → A t | (t : ψ) → f t = a t }

-- This generalizes path types of cubical TT to directed setting

SplitMix64 Time Axis

# The directed interval in our system
module DirectedInterval
  # Map SplitMix64 index to directed interval position
  def self.to_interval(index, max_index)
    # 0₂ = index 0, 1₂ = index max_index
    index.to_f / max_index.to_f
  end
  
  # Check if one interaction is "after" another in directed time
  def self.after?(i1, i2)
    i1.epoch > i2.epoch  # Epoch = position on directed interval
  end
  
  # Directed hom: all interactions from x to y
  def self.hom(manager, x_epoch, y_epoch)
    manager.interactions.select do |i|
      i.epoch > x_epoch && i.epoch <= y_epoch
    end
  end
end

Julia ACSet Integration

# Directed interval as ACSet
@present SchDirectedInterval(FreeSchema) begin
  Point::Ob
  Arrow::Ob
  
  src::Hom(Arrow, Point)
  tgt::Hom(Arrow, Point)
  
  # The unique arrow 0 → 1
  # No arrow 1 → 0 (directedness)
end

@acset_type DirectedIntervalGraph(SchDirectedInterval)

function walking_arrow()
  g = DirectedIntervalGraph()
  add_parts!(g, :Point, 2)  # 0₂ and 1₂
  add_part!(g, :Arrow, src=1, tgt=2)  # The unique arrow
  g
end

Key Properties

1. No loops: There is no morphism 1₂ → 0₂ (time irreversibility)

2. Higher simplices: Δⁿ built from directed cubes model n-step processes

3. Extension types: Generalize path types to directed setting

4. Cubical structure: Compatible with cubical type theory machinery

References

• Riehl, E. & Shulman, M. (2017). "A type theory for synthetic ∞-categories." §3.
• Rzk documentation
• Licata, D. & Harper, R. (2011). "2-Dimensional Directed Type Theory."