Lecture 8: Cubical type theory

Globular Sets

Globular sets:

\[src \circ src = src \circ tgt\\ tgt \circ src = tgt \circ tgt\]

Each shape/cell at level $n+1$ in $A_n$ comes with a source and target at level $A_n$.

Simplicial Sets

Semi-simplicial sets:

family of sets $(A_n){n∈ℕ}$ equipped with $d_i: A{n+1} ⟶ A_n$ (for $0 ≤ i ≤ n+1$) s.t.:

\[d_i(d_{j+1}(α)) = d_j(d_i(α)) \qquad j≥i \quad \text{(1)}\\ d_i(d_j(α)) = d_j(d_{i+1}(α)) \qquad j≤i \quad \text{(2)}\]

Each shape at level $n+1$ in $A_{n+1}$ comes with a $i$-th face ($0≤i≤n+1$) in $A_i$ (the $i$-th face omits the $i$-th point).

Simplicial sets:

semi-simplicial sets equipped with an extra operation called $i$-th “degeneracy” from $A_n$ to $A_{n+1}$ ($0≤i≤n$)

The $i$-th degeneracy is usually called $s_i$, they have to satisfy:

\[d_i \circ s_i = id\\ d_{i+1} \circ s_i = id\\ s_i \circ s_j = s_j \circ s_{i+1} \qquad j<i\\ s_i \circ s_{j+1} = s_j \circ s_i \qquad i<j\\\]

\[d_i \circ s_j = s_j \circ d_i \qquad j>i\\ d_i \circ s_{j+1} = s_j \circ d_i \qquad j <i\]

The $i$-th duplicates the $i$-th point

Cubical Sets

Cubical Set:

a family $(A_n)_{n∈ ℕ}$ of sets equipped with the following operations: \(\partial_i^-: A_{n+1} ⟶ A_n \qquad 0 ≤ i ≤ n\\ \partial_i^+: A_{n+1} ⟶ A_n \qquad 0 ≤ i ≤ n\)

and the following properties, for $j≥i$:

\[\partial_i^- (\partial_{j+1}^-(α)) = \partial_{j}^- (\partial_i^-(α)) \\ \partial_i^- (\partial_{j+1}^+(α)) = \partial_{j}^+ (\partial_i^-(α)) \\ \partial_i^+ (\partial_{j+1}^-(α)) = \partial_{j}^- (\partial_i^+(α)) \\ \partial_i^+ (\partial_{j+1}^+(α)) = \partial_{j}^+ (\partial_i^+(α))\]

That is:

\[\partial_i^α \circ \partial_j^β = \partial_j^β \circ \partial_{i+1}^α \qquad j ≤ i\]

+ Degeneracies:

\[ε_i: A_n ⟶ A_{n+1} \quad 0 ≤ i ≤ n\]

with properties:

\[\partial^+_i \circ ε_i = id\\ \partial^-_i \circ ε_i = id\\ \partial^α_{j+1} \circ ε_i = ε_i \circ \partial_j^α \qquad j ≥ i\\ \partial^α_j \circ ε_{i+1} = ε_i \circ \partial_j^α \qquad j ≤ i\\ ε_{i+1} \circ ε_j = ε_j \circ ε_i \qquad i≥j\\ ε_i \circ ε_j = ε_{j+1} \circ ε_i \qquad i<j\]

NB: Can be defined as presheaves over finite ordinals.

Dependent equality

\[\cfrac{Γ ⊢ t: A \quad Γ ⊢ u: B \quad Γ ⊢ p: A=B }{Γ ⊢ t=_p u: 𝒰}\]

This new equlity fits the cubical structure:

• a term $t:A$ can be seen as a point in a cubical set
• a path $p: t =_A u$ can be seen as a line in a cubical set
• a path $α: \underbrace{p}{: A=B} ={ap_2 \; = \; \underbrace{r}{: A=C} \; \underbrace{s}} \underbrace{q}_{: C=D}$ can be seen as a square cubical set