Simplicial Type Theory¶

These formalisations correspond in part to Section 3 of the RS17 paper.

This is a literate rzk file:

#lang rzk-1

Simplices and their subshapes¶

The 1-simplex

#def Δ¹
  : 2 → TOPE
  := \ t → TOP

The 2-simplex

#def Δ²
  : ( 2 × 2) → TOPE
  := \ (t , s) → s ≤ t

The 3-simplex

#def Δ³
  : ( 2 × 2 × 2) → TOPE
  := \ ((t1 , t2) , t3) → t3 ≤ t2 ∧ t2 ≤ t1

The boundary of a 1-simplex

#def ∂Δ¹
  : Δ¹ → TOPE
  := \ t → (t ≡ 0₂ ∨ t ≡ 1₂)

The boundary of a 2-simplex

#def ∂Δ²
  : Δ² → TOPE
  :=
    \ (t , s) → (s ≡ 0₂ ∨ t ≡ 1₂ ∨ s ≡ t)

#def Λ
  : ( 2 × 2) → TOPE
  := \ (t , s) → (s ≡ 0₂ ∨ t ≡ 1₂)

The product of topes defines the product of shapes.

#def shape-prod
  ( I J : CUBE)
  ( ψ : I → TOPE)
  ( χ : J → TOPE)
  : ( I × J) → TOPE
  := \ (t , s) → ψ t ∧ χ s

The square as a product

#def Δ¹×Δ¹
  : ( 2 × 2) → TOPE
  := shape-prod 2 2 Δ¹ Δ¹

The total boundary of the square

#def ∂□
  : ( 2 × 2) → TOPE
  := \ (t , s) → ((∂Δ¹ t) ∧ (Δ¹ s)) ∨ ((Δ¹ t) ∧ (∂Δ¹ s))

The vertical boundary of the square

#def ∂Δ¹×Δ¹
  : ( 2 × 2) → TOPE
  := shape-prod 2 2 ∂Δ¹ Δ¹

The horizontal boundary of the square

#def Δ¹×∂Δ¹
  : ( 2 × 2) → TOPE
  := shape-prod 2 2 Δ¹ ∂Δ¹

The prism from a 2-simplex in an arrow type

#def Δ²×Δ¹
  : ( 2 × 2 × 2) → TOPE
  := shape-prod (2 × 2) 2 Δ² Δ¹

The intersection of shapes is defined by conjunction on topes.

#def shape-intersection
  ( I : CUBE)
  ( ψ χ : I → TOPE)
  : I → TOPE
  := \ t → ψ t ∧ χ t

The union of shapes is defined by disjunction on topes.

#def shapeUnion
  ( I : CUBE)
  ( ψ χ : I → TOPE)
  : I → TOPE
  := \ t → ψ t ∨ χ t