I.B - Vladimir Voevodsky’s new foundational program (2006/2009)

• Mistake in one of his papers

  • Modern mathematical proofs: too complex to be reliably checked by humans
  • ⟶ Since then, has been championing computer proof assistants

  The functional programming language / proof assistant Agda: based on dependent type theory.

Martin-Löf’s Dependent Type Theory (MLTT)… in a nutshell

Contexts

A context $Γ$:

is a list (of assumptions)

$$\, \\ x_1 : A_1, \, x_2:A_2(x_1) \, \ldots, \, x_n:A_n(x_1, \ldots, x_{n-1}) \\$$

of variables, such that each type $A_i(x_1, \ldots, x_{i-1})$ depends on variables $x_1, \ldots, x_{i-1}$ of earlier types $A_1(x_1), \ldots, A_{i-1}(x_1, \ldots, x_{i-2})$.

Types and Terms

Inference rules enable us to derive judgments of the form:

$Σ$-types:

existential quantification represented as an indexed pair, where the type of the second component depends on the first.

$$(a,b):\Sigma _{(x:A)}B(x)$$

⟺

$a:A$ and $b:B(a)$.

Proof assistants make use of the Curry-Howard correspondence.

Substitutions / Context Morphisms

A substitution $σ: Δ ⟶ \overbrace{Γ}^{\rlap{≝\; x_1:A_1,\; x_2:A_2(x_1),\;\ldots,\; x_n:A_n(x_1, \ldots, x_{n-1})}}$:

a list of terms

$$σ ≝ (t_1, \ldots, t_n)$$

such that

Notation: $σ: Δ ⟶ Γ$ applied in $Γ ⊢ t: A$ is written $t[σ]$ (free variables substituted by their corresponding term in $σ$), idem for types.

• $Γ ≝ (n: ℕ, \; p:{\rm Prime}, \; 𝕂: {\rm Field} \, p^n, \; V: 𝕂\text{-}{\rm VectorSpace})$
• $Δ ≝ (q: {\rm PalindromicPrime}, 𝕃: {\rm TopologicalField} \, q^q, \; A : 𝕃\text{-}{\rm Algebra})$

$$\begin{aligned}Δ &\vdash q : ℕ\\Δ &\vdash q :{\rm Prime}\\Δ &\vdash 𝕃 :{\rm Field} \, q^q\\Δ &\vdash A : 𝕃-{\rm Algebra}\end{aligned}$$

Inference rules for Intensional equality

• Formation:

  $$\frac{\Gamma \vdash A\hspace{1em}\Gamma \vdash a {:} A\hspace{1em}\Gamma \vdash a' {:} A}{\Gamma \vdash a ≃_A a'}$$

• Introduction:

  $$\frac{\Gamma \vdash A\hspace{1em}\Gamma \vdash a {:} A}{\Gamma \vdash \refl {:} a ≃_A a}$$

• Elimination:

  $$\frac{\Gamma \vdash x:A\hspace{1em}\Gamma, y: A, p: x ≃_A y \vdash P(y, p)\hspace{1em}\Gamma \vdash d {:} P(x, \refl_x)}{ \Gamma, y: A, p: x ≃_A y \vdash \J(y, p, d) {:} P(y, p)}$$

• Computation:

  $$\J(x, \refl_x, d) ≡ d {:} P(x, \refl_x)$$

I.C. Homotopy Type Theory

• a type $A$ is abstractly seen as a topological space whose points are its objects

  • if $x, y: A$, $x ≃_A y$ represents the paths from $x$ (start point) to $y$ (end point).

  • given two parallel paths $p, q : x ≃_A y$, a path $r : p ≃_{x≃_A y} q$ corresponds to a homotopy/2-path between $p$ and $q$.

  • $r ≃_{p ≃_{x ≃_A y} q} s$ is the type of 3-paths, and so on…

The tower comprised of these paths, paths of paths, paths of paths of paths, … arising from $A$ forms an $ω$-groupoid

The Univalence Axiom

Univalence Axiom in Voevodsky’s univalent program:

equality of types is (weakly) equivalent to equivalence

⟶ This axiom is not constructive per se

Eliminating univalence is a holy grail

… and formalizing a weak $ω$-groupoid model of Type Theory inside Type Theory might pave the way for it!

II. Weak $ω$-groupoids

Comparison of structures and laws of different algebraic structures

A globular set:

is a family of sets \((G_n)_{n∈ℕ}\) and functions \(s_n, t_n: G_n ⟶ G_{n-1}\) (which stand for source and target respectively) such that, for all $n∈ℕ$:

Intuition

• the points \(\bullet\) are members of \(G_0\)
• \(a, b∈ G_1\)
• \(θ ∈ G_2\)
• \(s(θ) ≝ a\)
• \(t(θ) ≝ b\)

$\begin{cases} s(a) = s(b) \\ t(a) = t(b)\end{cases}\quad$ i.e.: $\quad \begin{cases} s(s (𝜃)) = s(t (𝜃)) \\ t(s (𝜃)) = t(t (𝜃))\end{cases}$

All sources and their corresponding targets are parallel.

For all \(n∈ℕ^\ast, \; a, b:G_{n-1}\), $G_n(a, b) ≝ \lbrace f∈G_n \mid s (f) = a \text{ and } t (f) = b\rbrace$

The elements of \(G_n\) are called \(n\)-arrows.

Laws and Structure of $ω$-groupoids

Strict \(𝜔\)-groupoids

Let \(G ≝ \bigsqcup\limits_{n≥0} G_n\) be a strict \(𝜔\)-groupoid.

Dimension skips:

If \(i > j > k ∈ ℕ, \; \; x, x' : G_k, \; f, f' : G_j(x, x')\) and \(𝛼 : G_i(f, f')\), then \(𝛼\) can be seen as \(k\)-arrow of \(G_k(x, x')\)

Indeed: for all \(n∈ℕ^\ast\) and \(f_1, \ldots, f_n ∈ \lbrace s, t \rbrace\):

\[\begin{cases} s \; f_1 \; \ldots \; f_n = s \\ t \; f_1 \; \ldots \; f_n = t \end{cases}\]

⟶ only the leftmost composition matters.

Structure of \(G_n\)

There are $n$ different types of compositions in \(G_n\): $\scol_1^n, \, \ldots, \, \scol_n^n$.

Let \(1 ≤ i < n\).

Laws of \(G_n\)

Identity (\(i\)): \(x:G_i\)
Composition (\(i < j\)): the points \(x, x', x'' : G_i\), the simple arrows \(f, g : G_j\)
Identity Laws (\(i<j\)): the points \(x, x' : G_i\), the simple arrow \(f : G_j\)	Left: Right: idem

Associativity law (\(i < j\)): the points \(x, x', x'', x''' : G_i\), the simple arrows \(f, g, h : G_j\)

Interchange law (\(i < j < k\)): the points \(x, x', x'' : G_i\), the simple arrows \(f, f', f'', g, g', g'' : G_j\), the double arrows \(𝛼, 𝛼', 𝛽, 𝛽' : G_k\)

Weak \(𝜔\)-groupoids: handmade partial construction

\(G_1\)

• Identity:
  \[{\rm id}^1 (\_ ) : \prod\limits_{a: G_0} G_1(a, a)\]
• Inverse:
  \[{\rm inv}^1 (\_ , \_ ) : \prod\limits_{a, b: G_0} G_1(a, b) ⟶ G_1(b, a)\]
• Composition:
  \[\scol^1_1 (\_ , \_ , \_ ) : \prod\limits_{a, b, c: G_0} G_1(a, b) × G_1(b, c) ⟶ G_1(a, c)\]

\(G_2\)

• Identity:
  \[{\rm id}^2 (\_ ) : \prod\limits_{f: G_1} G_2(f, f)\]

• Inverse:
  \[{\rm inv}^2 (\_ , \_ ) : \prod\limits_{f, g: G_1} G_2(f, g) ⟶ G_2(g, f)\]

• Horizontal Composition (along the \(1\)-arrows):

• Vertical Composition (along the \(2\)-arrows):

• Associativity along the \(1\)-arrows}:
  \[𝛼^1_1 (\_ , \_ , \_ ) : \prod\limits_{x, x', x'', x''': G_0} \prod\limits_{\substack{f: G_1(x, x') \\ g: G_1(x', x'') \\ h: G_1(x'', x''')}} G_2((f \scol_1^1 g) \scol_1^1 h, \; f \scol_1^1 (g \scol_1^1 h))\]

“Laws” that have become structure

• Left identity “law”:

  \[𝜆^1 (\_ ) : \prod\limits_{f: G_1} G_2(({\rm id}^1_{s f} \scol_1^1 f), \; f)\]

• Right identity “law”:

  \[𝜌^1(\_ ) : \prod\limits_{f: G_1} G_2((f \scol_1^1 {\rm id}^1_{t f}), \; f)\]

• Left inverse “law”:

  \[l^1(\_ ) : \prod\limits_{f: G_1} G_2(({\rm inv}^1_{sf, tf}(f) \scol_1^1 f), \; {\rm id}^1_{t f})\]

• Right inverse “law”:

  \[r^1(\_ ) : \prod\limits_{f: G_1} G_2((f \scol_1^1 {\rm inv}^1_{sf, tf}(f)), \; {\rm id}^1_{s f})\]

\(G_3\), and so on…

At level \(3\), in addition, we have to take into account

• the interchange law between the 2-arrows and the 1-arrows:

  \[𝛼^1_2 : \prod\limits_{x, x': G_0} \prod\limits_{f, f', f'', f''': G_1(x, x')} \prod\limits_{\substack{α: G_2(f, f') \\ β: G_2(f', f'') \\ γ: G_2(f'', f''')}} G_3((α \scol_2^2 β) \scol_2^2 γ, \; α \scol_2^2 (β \scol_2^2 γ))\]

• the different compositions \(\scol_1^3, \scol_2^3, \scol_3^3\)

• the associativity along the \(2\)-arrows:

  \[𝛼^1_2 : \prod\limits_{x, x': G_0} \prod\limits_{f, f', f'', f''': G_1(x, x')} \prod\limits_{\substack{α: G_2(f, f') \\ β: G_2(f', f'') \\ γ: G_2(f'', f''')}} G_3((α \scol_2^2 β) \scol_2^2 γ, \; α \scol_2^2 (β \scol_2^2 γ))\]

• etc…

But that’s not it!

Coherence laws in $G_3$:

Equations between elements of $G_1$ have been replaced by new 2-arrows.

But one also requires that these 2-arrows satisfy some new equations (represented by new 3-arrows) of their own, called coherence laws!

For instance, there are 5 ways to parenthesize the composition of 4 1-arrows in $G_1$, which are related as follows:

We impose a coherence law - the pentagon identity - (which becomes a new 3-arrow), stating that all these ways are identical (up to a 3-arrow).

Analogously, another coherence law says that composing with the left identity then with the right identity and composing with the right identity then with the left one are the same thing.

II. Brunerie Type Theory

Brunerie type theory \(𝔹\):

where \(\isContr\) (which stands for is contractible) is inductively defined as follows:

The intuition behind is that

• the elements of the base type \(\star\) correspond to the elements of the set \(G_0\) of the underlying globular set of the weak \(ω\)-groupoid.

• if \(x, y: \star\) the elements of \(x ≃ y\) are the 1-arrows of \(G_1(x, y)\)

• if \(x, y: \star\) and \(f, g: x ≃ y\), the elements of \(f ≃ g\) are those of \(G_1(f, g)\)

• and so on…

a context
\(Γ ≝ \bullet.(x \; y:\star).(f \; g : x ≃ y).(α : f ≃ g)\) will be depicted as

A context is contractible if it can be “reduced to a point”, by contracting along the arrows.

Example

• the context \(Γ ≝ \bullet.(x \; y \; z \; t:\star).(f : x ≃ y).(g : y ≃ z).(h : y ≃ t)\) is contractible

• but the context \(Γ' ≝ \bullet.(x \; y \; z \; t:\star).(f : x ≃ y).(g : y ≃ z).(h \; i : y ≃ t)\) is not:

because of the hole between \(y\) and \(t\) (we cannot contract \(y\) and \(t\)).

But if the hole between \(y\) and \(t\) is filled by a 2-arrow, like in
\(Γ'' ≝ \bullet.(x \; y \; z \; t:\star).(f : x ≃ y).(g : y ≃ z).(h \; i : y ≃ t).(α ≃ h \; i)\)

then it becomes contractible.

Example: the associativity law along the \(1\)-arrows.

Let \(x, y, z, t : \star, \; f : x≃y, \; g: y≃z, \; h: z≃t\) and
\(Γ_{x, y, z, t} ≝ \bullet.(x \; y \; z \; t : \star).(f : x ≃ y).(g : y ≃ z).(h : z ≃ t)\)

First, how to define composition along the \(1\)-arrows?

As

• the context \(Γ_{x, y, z, t}\) is contractible
• and \(Γ_{x, y, z, t} ⊢ x ≃ z\) (since \(Γ_{x, y, z, t} ⊢ x, z : \star\))

\(f \scol_1^1 g : x ≃ z\) can be defined in \(Γ_{x, y, z, t}\) as
\(\coh_{x ≃ z}^{Γ_{x, y, z, t}}\)

Idem for the other compositions.

Now, there exists a

in \(Γ_{x, y, z, t}\), since \(Γ_{x, y, z, t}\) is contractible and \(Γ_{x, y, z, t} ⊢ (f \scol_1^1 g) \scol_1^1 h ≃ f \scol_1^1 (g \scol_1^1 h)\))!