Lecture 8: Cubical type theory

Globular Sets

Globular sets:

$$\begin{array}{c}src\circ src=src\circ tgt\\ tgt\circ src=tgt\circ tgt\end{array}$$

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 $(A_n){n∈ℕ}$equippedwith$d_i: A{n+1} ⟶ A_n$(for$0 ≤ i ≤ n+1$) s.t.:

$$\begin{array}{c}{d}_i(d_{j+1}(\alpha ))=d_j(d_i(\alpha ))j\ge i\text{(1)}\\ d_i(d_j(\alpha ))=d_j(d_{i+1}(\alpha ))j\le i\text{(2)}\end{array}$$

Each shape at level $n+1$ in $A_{n+1}$ comes with a $i$-th face ($0\le i\le n+1$) 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 $A_{n+1}$ ($0\le i\le n$)

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

$$\begin{array}{c}{d}_i\circ s_i=id\\ d_{i+1}\circ s_i=id\\ s_i\circ s_j=s_j\circ s_{i+1}j<i\\ s_i\circ s_{j+1}=s_j\circ s_{i}i<j\end{array}$$

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

Cubical Sets

Cubical Set:

a family $(A_n{)}_{n\in \mathbb{N}}$ of sets equipped with the following operations: $$\begin{gathered} {\partial }_i^{-}:A_{n+1}⟶A_{n}0\le i\le n \\ {\partial }_i^{+}:A_{n+1}⟶A_{n}0\le i\le n \end{gathered}$$

and the following properties, for $j\ge i$:

$$\begin{array}{c}{\partial }_i^{-}({\partial }_{j+1}^{-}(\alpha ))={\partial }_j^{-}({\partial }_i^{-}(\alpha ))\\ {\partial }_i^{-}({\partial }_{j+1}^{+}(\alpha ))={\partial }_j^{+}({\partial }_i^{-}(\alpha ))\\ {\partial }_i^{+}({\partial }_{j+1}^{-}(\alpha ))={\partial }_j^{-}({\partial }_i^{+}(\alpha ))\\ {\partial }_i^{+}({\partial }_{j+1}^{+}(\alpha ))={\partial }_j^{+}({\partial }_i^{+}(\alpha ))\end{array}$$

That is:

$${\partial }_i^{\alpha }\circ {\partial }_j^{\beta }={\partial }_j^{\beta }\circ {\partial }_{i+1}^{\alpha }j\le i$$

+ Degeneracies:

$${\epsilon }_i:A_n⟶A_{n+1}0\le i\le n$$

with properties:

$$\begin{array}{c}{\partial }_i^{+}\circ {\epsilon }_i=id\\ {\partial }_i^{-}\circ {\epsilon }_i=id\\ {\partial }_{j+1}^{\alpha }\circ {\epsilon }_i={\epsilon }_i\circ {\partial }_j^{\alpha }j\ge i\\ {\partial }_j^{\alpha }\circ {\epsilon }_{i+1}={\epsilon }_i\circ {\partial }_j^{\alpha }j\le i\\ {\epsilon }_{i+1}\circ {\epsilon }_j={\epsilon }_j\circ {\epsilon }_{i}i\ge j\\ {\epsilon }_i\circ {\epsilon }_j={\epsilon }_{j+1}\circ {\epsilon }_{i}i<j\end{array}$$

NB: Can be defined as presheaves over finite ordinals.

Dependent equality

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