SLIDE 1 Pierre Cagne
joint work with Paul-André Melliès
Université Paris Diderot – Paris 7
Identity types as equality predicates
Reconciling hyperdoctrines with MLTT
HoTT 2019 – Carnegie Mellon University August 12, 2019
SLIDE 2
- 1. Lawvere’s hyperdoctrines
- 2. Reconcile hyperdoctrines with intensional equalities
SLIDE 3
- 1. Lawvere’s hyperdoctrines
SLIDE 4 Lawvere’s hyperdoctrines
An hyperdoctrine is a pseudofunctor 𝑄 ∶ Cop → Cat such that:
- C has fjnite products,
- each 𝑄(𝑔 ) has both a left adjoint ∃𝑔 and a right adjoint ∀𝑔,
- each 𝑄(𝑑) is a cartesian closed category.
What does it have to do with logic?
SLIDE 5 Lawvere’s hyperdoctrines
An hyperdoctrine is a pseudofunctor 𝑄 ∶ Cop → Cat such that:
- C has fjnite products,
- each 𝑄(𝑔 ) has both a left adjoint ∃𝑔 and a right adjoint ∀𝑔,
- each 𝑄(𝑑) is a cartesian closed category.
What does it have to do with logic?
SLIDE 6 Seely’s semantics is an hyperdoctrine
𝐵 𝐶 𝑔 𝑌 𝐵 𝑔 ×𝑞 𝑌 𝑍 𝑟 𝑍 𝑟 𝑍 ∃𝑔𝑟 𝑔 𝑟
Π𝑔𝑟
⌟
𝑞
In particular for 𝑔 = 𝜀𝐵 ∶ 𝐵 → 𝐵 × 𝐵,
∃𝜀𝐵 ∶ id𝐵 ↦ 𝜀𝐵
SLIDE 7 Seely’s semantics is an hyperdoctrine
𝐵 𝐶 𝑔 𝑌 𝐵 𝑔 ×𝑞 𝑌 𝑍 𝑟 𝑍 𝑟 𝑍 ∃𝑔𝑟 𝑔 𝑟
Π𝑔𝑟
⌟
𝑞
In particular for 𝑔 = 𝜀𝐵 ∶ 𝐵 → 𝐵 × 𝐵,
∃𝜀𝐵 ∶ id𝐵 ↦ 𝜀𝐵
SLIDE 8 Seely’s semantics is an hyperdoctrine
𝐵 𝐶 𝑔 𝑌 𝐵 𝑔 ×𝑞 𝑌 𝑍 𝑟 𝑍 𝑟 𝑍 ∃𝑔𝑟 𝑔 𝑟
Π𝑔𝑟
⌟
𝑞
In particular for 𝑔 = 𝜀𝐵 ∶ 𝐵 → 𝐵 × 𝐵,
∃𝜀𝐵 ∶ id𝐵 ↦ 𝜀𝐵
SLIDE 9 Seely’s semantics is an hyperdoctrine
𝐵 𝐶 𝑔 𝑌 𝐵 𝑔 ×𝑞 𝑌 𝑍 𝑟 𝑍 𝑟 𝑍 ∃𝑔𝑟 𝑔 𝑟
Π𝑔𝑟
⌟
𝑞
In particular for 𝑔 = 𝜀𝐵 ∶ 𝐵 → 𝐵 × 𝐵,
∃𝜀𝐵 ∶ id𝐵 ↦ 𝜀𝐵
SLIDE 10 Seely’s semantics is an hyperdoctrine
𝐵 𝐶 𝑔 𝑌 𝐵 𝑔 ×𝑞 𝑌 𝑍 𝑟 𝑍 𝑟 𝑍 ∃𝑔𝑟 𝑔 𝑟
Π𝑔𝑟
⌟
𝑞
In particular for 𝑔 = 𝜀𝐵 ∶ 𝐵 → 𝐵 × 𝐵,
∃𝜀𝐵 ∶ id𝐵 ↦ 𝜀𝐵
SLIDE 11 Seely’s semantics is an hyperdoctrine
𝐵 𝐶 𝑔 𝑌 𝐵 𝑔 ×𝑞 𝑌 𝑍 𝑟 𝑍 𝑟 𝑍 ∃𝑔𝑟 𝑔 𝑟
Π𝑔𝑟
⌟
𝑞
In particular for 𝑔 = 𝜀𝐵 ∶ 𝐵 → 𝐵 × 𝐵,
∃𝜀𝐵 ∶ id𝐵 ↦ 𝜀𝐵
SLIDE 12 Seely’s semantics is an hyperdoctrine
𝐵 𝐶 𝑔 𝑌 𝐵 𝑔 ×𝑞 𝑌 𝑍 𝑟 𝑍 𝑟 𝑍 ∃𝑔𝑟 𝑔 𝑟
Π𝑔𝑟
⌟
𝑞
In particular for 𝑔 = 𝜀𝐵 ∶ 𝐵 → 𝐵 × 𝐵,
∃𝜀𝐵 ∶ id𝐵 ↦ 𝜀𝐵
SLIDE 13 Seely’s semantics is an hyperdoctrine
𝐵 𝐶 𝑔 𝑌 𝐵 𝑔 ×𝑞 𝑌 𝑍 𝑟 𝑍 𝑟 𝑍 ∃𝑔𝑟 𝑔 𝑟
Π𝑔𝑟
⌟
𝑞
In particular for 𝑔 = 𝜀𝐵 ∶ 𝐵 → 𝐵 × 𝐵,
∃𝜀𝐵 ∶ id𝐵 ↦ 𝜀𝐵
SLIDE 14 Seely’s semantics is an hyperdoctrine
𝐵 𝐶 𝑔 𝑌 𝐵 𝑔 ×𝑞 𝑌 𝑍 𝑟 𝑍 𝑟 𝑍 ∃𝑔𝑟 𝑔 𝑟
Π𝑔𝑟
⌟
𝑞
In particular for 𝑔 = 𝜀𝐵 ∶ 𝐵 → 𝐵 × 𝐵,
∃𝜀𝐵 ∶ id𝐵 ↦ 𝜀𝐵
SLIDE 15 Seely’s semantics is an hyperdoctrine
𝐵 𝐶 𝑔 𝑌 𝐵 𝑔 ×𝑞 𝑌 𝑍 𝑟 𝑍 𝑟 𝑍 ∃𝑔𝑟 𝑔 𝑟
Π𝑔𝑟
⌟
𝑞
In particular for 𝑔 = 𝜀𝐵 ∶ 𝐵 → 𝐵 × 𝐵,
∃𝜀𝐵 ∶ id𝐵 ↦ 𝜀𝐵
SLIDE 16 Subsets form an hyperdoctrine
𝐵 𝐶 𝑔 𝑊 𝑔 −1(𝑊) 𝑉 𝑉 ∃𝑔𝑉 𝑔 (𝑉) = {𝑐 ∈ 𝐶 ∶ ∃𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ∧ 𝑏 ∈ 𝑉} ∀𝑔𝑉 𝑔∗(𝑉) = {𝑐 ∈ 𝐶 ∶ ∀𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ⇒ 𝑏 ∈ 𝑉} ⊆ ⊆ ⊆ ⊆
In particular for 𝑔 = 𝜀𝐵 ∶ 𝐵 → 𝐵 × 𝐵,
∃𝜀𝐵 ∶ 𝐵 ↦ {(𝑏, 𝑏′) ∈ 𝐵 × 𝐵 ∶ 𝑏 = 𝑏′}
SLIDE 17 Subsets form an hyperdoctrine
𝐵 𝐶 𝑔 𝑊 𝑔 −1(𝑊) 𝑉 𝑉 ∃𝑔𝑉 𝑔 (𝑉) = {𝑐 ∈ 𝐶 ∶ ∃𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ∧ 𝑏 ∈ 𝑉} ∀𝑔𝑉 𝑔∗(𝑉) = {𝑐 ∈ 𝐶 ∶ ∀𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ⇒ 𝑏 ∈ 𝑉} ⊆ ⊆ ⊆ ⊆
In particular for 𝑔 = 𝜀𝐵 ∶ 𝐵 → 𝐵 × 𝐵,
∃𝜀𝐵 ∶ 𝐵 ↦ {(𝑏, 𝑏′) ∈ 𝐵 × 𝐵 ∶ 𝑏 = 𝑏′}
SLIDE 18 Subsets form an hyperdoctrine
𝐵 𝐶 𝑔 𝑊 𝑔 −1(𝑊) 𝑉 𝑉 ∃𝑔𝑉 𝑔 (𝑉) = {𝑐 ∈ 𝐶 ∶ ∃𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ∧ 𝑏 ∈ 𝑉} ∀𝑔𝑉 𝑔∗(𝑉) = {𝑐 ∈ 𝐶 ∶ ∀𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ⇒ 𝑏 ∈ 𝑉} ⊆ ⊆ ⊆ ⊆
In particular for 𝑔 = 𝜀𝐵 ∶ 𝐵 → 𝐵 × 𝐵,
∃𝜀𝐵 ∶ 𝐵 ↦ {(𝑏, 𝑏′) ∈ 𝐵 × 𝐵 ∶ 𝑏 = 𝑏′}
SLIDE 19 Subsets form an hyperdoctrine
𝐵 𝐶 𝑔 𝑊 𝑔 −1(𝑊) 𝑉 𝑉 ∃𝑔𝑉 𝑔 (𝑉) = {𝑐 ∈ 𝐶 ∶ ∃𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ∧ 𝑏 ∈ 𝑉} ∀𝑔𝑉 𝑔∗(𝑉) = {𝑐 ∈ 𝐶 ∶ ∀𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ⇒ 𝑏 ∈ 𝑉} ⊆ ⊆ ⊆ ⊆
In particular for 𝑔 = 𝜀𝐵 ∶ 𝐵 → 𝐵 × 𝐵,
∃𝜀𝐵 ∶ 𝐵 ↦ {(𝑏, 𝑏′) ∈ 𝐵 × 𝐵 ∶ 𝑏 = 𝑏′}
SLIDE 20 Subsets form an hyperdoctrine
𝐵 𝐶 𝑔 𝑊 𝑔 −1(𝑊) 𝑉 𝑉 ∃𝑔𝑉 𝑔 (𝑉) = {𝑐 ∈ 𝐶 ∶ ∃𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ∧ 𝑏 ∈ 𝑉} ∀𝑔𝑉 𝑔∗(𝑉) = {𝑐 ∈ 𝐶 ∶ ∀𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ⇒ 𝑏 ∈ 𝑉} ⊆ ⊆ ⊆ ⊆
In particular for 𝑔 = 𝜀𝐵 ∶ 𝐵 → 𝐵 × 𝐵,
∃𝜀𝐵 ∶ 𝐵 ↦ {(𝑏, 𝑏′) ∈ 𝐵 × 𝐵 ∶ 𝑏 = 𝑏′}
SLIDE 21 Subsets form an hyperdoctrine
𝐵 𝐶 𝑔 𝑊 𝑔 −1(𝑊) 𝑉 𝑉 ∃𝑔𝑉 𝑔 (𝑉) = {𝑐 ∈ 𝐶 ∶ ∃𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ∧ 𝑏 ∈ 𝑉} ∀𝑔𝑉 𝑔∗(𝑉) = {𝑐 ∈ 𝐶 ∶ ∀𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ⇒ 𝑏 ∈ 𝑉} ⊆ ⊆ ⊆ ⊆
In particular for 𝑔 = 𝜀𝐵 ∶ 𝐵 → 𝐵 × 𝐵,
∃𝜀𝐵 ∶ 𝐵 ↦ {(𝑏, 𝑏′) ∈ 𝐵 × 𝐵 ∶ 𝑏 = 𝑏′}
SLIDE 22 Subsets form an hyperdoctrine
𝐵 𝐶 𝑔 𝑊 𝑔 −1(𝑊) 𝑉 𝑉 ∃𝑔𝑉 𝑔 (𝑉) = {𝑐 ∈ 𝐶 ∶ ∃𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ∧ 𝑏 ∈ 𝑉} ∀𝑔𝑉 𝑔∗(𝑉) = {𝑐 ∈ 𝐶 ∶ ∀𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ⇒ 𝑏 ∈ 𝑉} ⊆ ⊆ ⊆ ⊆
In particular for 𝑔 = 𝜀𝐵 ∶ 𝐵 → 𝐵 × 𝐵,
∃𝜀𝐵 ∶ 𝐵 ↦ {(𝑏, 𝑏′) ∈ 𝐵 × 𝐵 ∶ 𝑏 = 𝑏′}
SLIDE 23 Subsets form an hyperdoctrine
𝐵 𝐶 𝑔 𝑊 𝑔 −1(𝑊) 𝑉 𝑉 ∃𝑔𝑉 𝑔 (𝑉) = {𝑐 ∈ 𝐶 ∶ ∃𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ∧ 𝑏 ∈ 𝑉} ∀𝑔𝑉 𝑔∗(𝑉) = {𝑐 ∈ 𝐶 ∶ ∀𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ⇒ 𝑏 ∈ 𝑉} ⊆ ⊆ ⊆ ⊆
In particular for 𝑔 = 𝜀𝐵 ∶ 𝐵 → 𝐵 × 𝐵,
∃𝜀𝐵 ∶ 𝐵 ↦ {(𝑏, 𝑏′) ∈ 𝐵 × 𝐵 ∶ 𝑏 = 𝑏′}
SLIDE 24 Subsets form an hyperdoctrine
𝐵 𝐶 𝑔 𝑊 𝑔 −1(𝑊) 𝑉 𝑉 ∃𝑔𝑉 𝑔 (𝑉) = {𝑐 ∈ 𝐶 ∶ ∃𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ∧ 𝑏 ∈ 𝑉} ∀𝑔𝑉 𝑔∗(𝑉) = {𝑐 ∈ 𝐶 ∶ ∀𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ⇒ 𝑏 ∈ 𝑉} ⊆ ⊆ ⊆ ⊆
In particular for 𝑔 = 𝜀𝐵 ∶ 𝐵 → 𝐵 × 𝐵,
∃𝜀𝐵 ∶ 𝐵 ↦ {(𝑏, 𝑏′) ∈ 𝐵 × 𝐵 ∶ 𝑏 = 𝑏′}
SLIDE 25 Subsets form an hyperdoctrine
𝐵 𝐶 𝑔 𝑊 𝑔 −1(𝑊) 𝑉 𝑉 ∃𝑔𝑉 𝑔 (𝑉) = {𝑐 ∈ 𝐶 ∶ ∃𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ∧ 𝑏 ∈ 𝑉} ∀𝑔𝑉 𝑔∗(𝑉) = {𝑐 ∈ 𝐶 ∶ ∀𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ⇒ 𝑏 ∈ 𝑉} ⊆ ⊆ ⊆ ⊆
In particular for 𝑔 = 𝜀𝐵 ∶ 𝐵 → 𝐵 × 𝐵,
∃𝜀𝐵 ∶ 𝐵 ↦ {(𝑏, 𝑏′) ∈ 𝐵 × 𝐵 ∶ 𝑏 = 𝑏′}
SLIDE 26 Predicates form an hyperdoctrine
⃗ 𝑦 ⃗ 𝑧 ⃗ 𝑢(⃗ 𝑦) 𝜔(⃗ 𝑧) 𝜔(⃗ 𝑢(⃗ 𝑦)) 𝜒(⃗ 𝑦) 𝜒(⃗ 𝑦) ∃⃗ 𝑦, (⋀𝑗 𝑢𝑗(⃗ 𝑦) = 𝑧𝑗) ∧ 𝜒(⃗ 𝑦) ∀⃗ 𝑦, (⋀𝑗 𝑢𝑗(⃗ 𝑦) = 𝑧𝑗) ⇒ 𝜒(⃗ 𝑦) ⊧ ⊧ ⊧ ⊧
In particular for ⃗
𝑢(⃗ 𝑦) = (⃗ 𝑦, ⃗ 𝑦) ∶ (𝑦1, … , 𝑦𝑜) → (𝑦1, … , 𝑦2𝑜), ∃(⃗
𝑦,⃗ 𝑦) ∶ ⊤ ↦ ⋀ 𝑗
𝑦𝑗 = 𝑦𝑜+𝑗
SLIDE 27 Predicates form an hyperdoctrine
⃗ 𝑦 ⃗ 𝑧 ⃗ 𝑢(⃗ 𝑦) 𝜔(⃗ 𝑧) 𝜔(⃗ 𝑢(⃗ 𝑦)) 𝜒(⃗ 𝑦) 𝜒(⃗ 𝑦) ∃⃗ 𝑦, (⋀𝑗 𝑢𝑗(⃗ 𝑦) = 𝑧𝑗) ∧ 𝜒(⃗ 𝑦) ∀⃗ 𝑦, (⋀𝑗 𝑢𝑗(⃗ 𝑦) = 𝑧𝑗) ⇒ 𝜒(⃗ 𝑦) ⊧ ⊧ ⊧ ⊧
In particular for ⃗
𝑢(⃗ 𝑦) = (⃗ 𝑦, ⃗ 𝑦) ∶ (𝑦1, … , 𝑦𝑜) → (𝑦1, … , 𝑦2𝑜), ∃(⃗
𝑦,⃗ 𝑦) ∶ ⊤ ↦ ⋀ 𝑗
𝑦𝑗 = 𝑦𝑜+𝑗
SLIDE 28 Predicates form an hyperdoctrine
⃗ 𝑦 ⃗ 𝑧 ⃗ 𝑢(⃗ 𝑦) 𝜔(⃗ 𝑧) 𝜔(⃗ 𝑢(⃗ 𝑦)) 𝜒(⃗ 𝑦) 𝜒(⃗ 𝑦) ∃⃗ 𝑦, (⋀𝑗 𝑢𝑗(⃗ 𝑦) = 𝑧𝑗) ∧ 𝜒(⃗ 𝑦) ∀⃗ 𝑦, (⋀𝑗 𝑢𝑗(⃗ 𝑦) = 𝑧𝑗) ⇒ 𝜒(⃗ 𝑦) ⊧ ⊧ ⊧ ⊧
In particular for ⃗
𝑢(⃗ 𝑦) = (⃗ 𝑦, ⃗ 𝑦) ∶ (𝑦1, … , 𝑦𝑜) → (𝑦1, … , 𝑦2𝑜), ∃(⃗
𝑦,⃗ 𝑦) ∶ ⊤ ↦ ⋀ 𝑗
𝑦𝑗 = 𝑦𝑜+𝑗
SLIDE 29 Predicates form an hyperdoctrine
⃗ 𝑦 ⃗ 𝑧 ⃗ 𝑢(⃗ 𝑦) 𝜔(⃗ 𝑧) 𝜔(⃗ 𝑢(⃗ 𝑦)) 𝜒(⃗ 𝑦) 𝜒(⃗ 𝑦) ∃⃗ 𝑦, (⋀𝑗 𝑢𝑗(⃗ 𝑦) = 𝑧𝑗) ∧ 𝜒(⃗ 𝑦) ∀⃗ 𝑦, (⋀𝑗 𝑢𝑗(⃗ 𝑦) = 𝑧𝑗) ⇒ 𝜒(⃗ 𝑦) ⊧ ⊧ ⊧ ⊧
In particular for ⃗
𝑢(⃗ 𝑦) = (⃗ 𝑦, ⃗ 𝑦) ∶ (𝑦1, … , 𝑦𝑜) → (𝑦1, … , 𝑦2𝑜), ∃(⃗
𝑦,⃗ 𝑦) ∶ ⊤ ↦ ⋀ 𝑗
𝑦𝑗 = 𝑦𝑜+𝑗
SLIDE 30 Predicates form an hyperdoctrine
⃗ 𝑦 ⃗ 𝑧 ⃗ 𝑢(⃗ 𝑦) 𝜔(⃗ 𝑧) 𝜔(⃗ 𝑢(⃗ 𝑦)) 𝜒(⃗ 𝑦) 𝜒(⃗ 𝑦) ∃⃗ 𝑦, (⋀𝑗 𝑢𝑗(⃗ 𝑦) = 𝑧𝑗) ∧ 𝜒(⃗ 𝑦) ∀⃗ 𝑦, (⋀𝑗 𝑢𝑗(⃗ 𝑦) = 𝑧𝑗) ⇒ 𝜒(⃗ 𝑦) ⊧ ⊧ ⊧ ⊧
In particular for ⃗
𝑢(⃗ 𝑦) = (⃗ 𝑦, ⃗ 𝑦) ∶ (𝑦1, … , 𝑦𝑜) → (𝑦1, … , 𝑦2𝑜), ∃(⃗
𝑦,⃗ 𝑦) ∶ ⊤ ↦ ⋀ 𝑗
𝑦𝑗 = 𝑦𝑜+𝑗
SLIDE 31 Predicates form an hyperdoctrine
⃗ 𝑦 ⃗ 𝑧 ⃗ 𝑢(⃗ 𝑦) 𝜔(⃗ 𝑧) 𝜔(⃗ 𝑢(⃗ 𝑦)) 𝜒(⃗ 𝑦) 𝜒(⃗ 𝑦) ∃⃗ 𝑦, (⋀𝑗 𝑢𝑗(⃗ 𝑦) = 𝑧𝑗) ∧ 𝜒(⃗ 𝑦) ∀⃗ 𝑦, (⋀𝑗 𝑢𝑗(⃗ 𝑦) = 𝑧𝑗) ⇒ 𝜒(⃗ 𝑦) ⊧ ⊧ ⊧ ⊧
In particular for ⃗
𝑢(⃗ 𝑦) = (⃗ 𝑦, ⃗ 𝑦) ∶ (𝑦1, … , 𝑦𝑜) → (𝑦1, … , 𝑦2𝑜), ∃(⃗
𝑦,⃗ 𝑦) ∶ ⊤ ↦ ⋀ 𝑗
𝑦𝑗 = 𝑦𝑜+𝑗
SLIDE 32 Predicates form an hyperdoctrine
⃗ 𝑦 ⃗ 𝑧 ⃗ 𝑢(⃗ 𝑦) 𝜔(⃗ 𝑧) 𝜔(⃗ 𝑢(⃗ 𝑦)) 𝜒(⃗ 𝑦) 𝜒(⃗ 𝑦) ∃⃗ 𝑦, (⋀𝑗 𝑢𝑗(⃗ 𝑦) = 𝑧𝑗) ∧ 𝜒(⃗ 𝑦) ∀⃗ 𝑦, (⋀𝑗 𝑢𝑗(⃗ 𝑦) = 𝑧𝑗) ⇒ 𝜒(⃗ 𝑦) ⊧ ⊧ ⊧ ⊧
In particular for ⃗
𝑢(⃗ 𝑦) = (⃗ 𝑦, ⃗ 𝑦) ∶ (𝑦1, … , 𝑦𝑜) → (𝑦1, … , 𝑦2𝑜), ∃(⃗
𝑦,⃗ 𝑦) ∶ ⊤ ↦ ⋀ 𝑗
𝑦𝑗 = 𝑦𝑜+𝑗
SLIDE 33 Predicates form an hyperdoctrine
⃗ 𝑦 ⃗ 𝑧 ⃗ 𝑢(⃗ 𝑦) 𝜔(⃗ 𝑧) 𝜔(⃗ 𝑢(⃗ 𝑦)) 𝜒(⃗ 𝑦) 𝜒(⃗ 𝑦) ∃⃗ 𝑦, (⋀𝑗 𝑢𝑗(⃗ 𝑦) = 𝑧𝑗) ∧ 𝜒(⃗ 𝑦) ∀⃗ 𝑦, (⋀𝑗 𝑢𝑗(⃗ 𝑦) = 𝑧𝑗) ⇒ 𝜒(⃗ 𝑦) ⊧ ⊧ ⊧ ⊧
In particular for ⃗
𝑢(⃗ 𝑦) = (⃗ 𝑦, ⃗ 𝑦) ∶ (𝑦1, … , 𝑦𝑜) → (𝑦1, … , 𝑦2𝑜), ∃(⃗
𝑦,⃗ 𝑦) ∶ ⊤ ↦ ⋀ 𝑗
𝑦𝑗 = 𝑦𝑜+𝑗
SLIDE 34 Elementary existential doctrines
An hyperdoctrine is a pseudofunctor 𝑄 ∶ Cop → Cat such that:
- C has fjnite products,
- each 𝑄(𝑔 ) has both a left adjoint ∃𝑔 and a right adjoint ∀𝑔,
- each 𝑄(𝑑) is a cartesian closed category.
Defjne the equality predicate over 𝑑 ∈ as the direct image of 1𝑑 along the diagonal
1𝑑 ∃Δ(1𝑑) 𝑑 𝑑 × 𝑑
∃Δ Δ
SLIDE 35 Elementary existential doctrines
An hyperdoctrine is a pseudofunctor 𝑄 ∶ Cop → Cat such that:
- C has fjnite products,
- each 𝑄(𝑔 ) has both a left adjoint ∃𝑔 and a right adjoint ∀𝑔,
- each 𝑄(𝑑) is a Heyting algebra.
Defjne the equality predicate over 𝑑 ∈ as the direct image of 1𝑑 along the diagonal
1𝑑 ∃Δ(1𝑑) 𝑑 𝑑 × 𝑑
∃Δ Δ
SLIDE 36 Elementary existential doctrines
An hyperdoctrine is a pseudofunctor 𝑄 ∶ Cop → Cat such that:
- C has fjnite products,
- each 𝑄(𝑔 ) has both a left adjoint ∃𝑔 and a right adjoint ∀𝑔,
- each 𝑄(𝑑) is a Boolean algebra.
Defjne the equality predicate over 𝑑 ∈ as the direct image of 1𝑑 along the diagonal
1𝑑 ∃Δ(1𝑑) 𝑑 𝑑 × 𝑑
∃Δ Δ
SLIDE 37 Elementary existential doctrines
An elementary existential doctrine is a pseudofunctor 𝑄 ∶ Cop → Cat such that:
- C has fjnite products,
- each 𝑄(𝑔 ) has both a left adjoint ∃𝑔,
- each 𝑄(𝑑) is a category with fjnal object 1𝑑.
Defjne the equality predicate over 𝑑 ∈ as the direct image of 1𝑑 along the diagonal
1𝑑 ∃Δ(1𝑑) 𝑑 𝑑 × 𝑑
∃Δ Δ
SLIDE 38 Elementary existential doctrines
An elementary existential doctrine is a pseudofunctor 𝑄 ∶ Cop → Cat such that:
- C has fjnite products,
- each 𝑄(𝑔 ) has both a left adjoint ∃𝑔,
- each 𝑄(𝑑) is a category with fjnal object 1𝑑.
Defjne the equality predicate over 𝑑 ∈ C as the direct image of 1𝑑 along the diagonal
1𝑑 ∃Δ(1𝑑) 𝑑 𝑑 × 𝑑
∃Δ Δ
SLIDE 39 Grothendieck bifjbrations
A Grothendieck fjbration is a functor 𝑞 ∶ E → B such that
𝐵 𝐶 𝑣 𝑍 𝑣∗𝑍
cart.
𝑌 𝐵′ ∃!
and
𝐵 𝐶 𝑣 𝑌 ∃𝑣𝑌
cocart.
𝑍 𝐶′ ∃!
SLIDE 40 Grothendieck bifjbrations
A Grothendieck fjbration is a functor 𝑞 ∶ E → B such that
𝐵 𝐶 𝑣 𝑍 𝑣∗𝑍
cart.
𝑌 𝐵′ ∃!
and
𝐵 𝐶 𝑣 𝑌 ∃𝑣𝑌
cocart.
𝑍 𝐶′ ∃!
SLIDE 41 Grothendieck bifjbrations
A Grothendieck fjbration is a functor 𝑞 ∶ E → B such that
𝐵 𝐶 𝑣 𝑍 𝑣∗𝑍
cart.
𝑌 𝐵′ ∃!
and
𝐵 𝐶 𝑣 𝑌 ∃𝑣𝑌
cocart.
𝑍 𝐶′ ∃!
SLIDE 42 Grothendieck bifjbrations
A Grothendieck fjbration is a functor 𝑞 ∶ E → B such that
𝐵 𝐶 𝑣 𝑍 𝑣∗𝑍
cart.
𝑌 𝐵′ ∃!
and
𝐵 𝐶 𝑣 𝑌 ∃𝑣𝑌
cocart.
𝑍 𝐶′ ∃!
SLIDE 43 Grothendieck bifjbrations
A Grothendieck fjbration is a functor 𝑞 ∶ E → B such that
𝐵 𝐶 𝑣 𝑍 𝑣∗𝑍
cart.
𝑌 𝐵′ ∃!
and
𝐵 𝐶 𝑣 𝑌 ∃𝑣𝑌
cocart.
𝑍 𝐶′ ∃!
SLIDE 44 Grothendieck bifjbrations
A Grothendieck fjbration is a functor 𝑞 ∶ E → B such that
𝐵 𝐶 𝑣 𝑍 𝑣∗𝑍
cart.
𝑌 𝐵′ ∃!
and
𝐵 𝐶 𝑣 𝑌 ∃𝑣𝑌
cocart.
𝑍 𝐶′ ∃!
SLIDE 45 Grothendieck bifjbrations
A Grothendieck opfjbration is a functor 𝑞 ∶ E → B such that
𝐵 𝐶 𝑣 𝑍 𝑣∗𝑍
cart.
𝑌 𝐵′ ∃!
and
𝐵 𝐶 𝑣 𝑌 ∃𝑣𝑌
cocart.
𝑍 𝐶′ ∃!
SLIDE 46 Grothendieck bifjbrations
A Grothendieck opfjbration is a functor 𝑞 ∶ E → B such that
𝐵 𝐶 𝑣 𝑍 𝑣∗𝑍
cart.
𝑌 𝐵′ ∃!
and
𝐵 𝐶 𝑣 𝑌 ∃𝑣𝑌
cocart.
𝑍 𝐶′ ∃!
SLIDE 47 Grothendieck bifjbrations
A Grothendieck opfjbration is a functor 𝑞 ∶ E → B such that
𝐵 𝐶 𝑣 𝑍 𝑣∗𝑍
cart.
𝑌 𝐵′ ∃!
and
𝐵 𝐶 𝑣 𝑌 ∃𝑣𝑌
cocart.
𝑍 𝐶′ ∃!
SLIDE 48 Grothendieck bifjbrations
A Grothendieck opfjbration is a functor 𝑞 ∶ E → B such that
𝐵 𝐶 𝑣 𝑍 𝑣∗𝑍
cart.
𝑌 𝐵′ ∃!
and
𝐵 𝐶 𝑣 𝑌 ∃𝑣𝑌
cocart.
𝑍 𝐶′ ∃!
SLIDE 49 Grothendieck bifjbrations
A Grothendieck opfjbration is a functor 𝑞 ∶ E → B such that
𝐵 𝐶 𝑣 𝑍 𝑣∗𝑍
cart.
𝑌 𝐵′ ∃!
and
𝐵 𝐶 𝑣 𝑌 ∃𝑣𝑌
cocart.
𝑍 𝐶′ ∃!
SLIDE 50 Grothendieck bifjbrations
A Grothendieck bifjbration is a functor 𝑞 ∶ E → B such that
𝐵 𝐶 𝑣 𝑍 𝑣∗𝑍
cart.
𝑌 𝐵′ ∃!
and
𝐵 𝐶 𝑣 𝑌 ∃𝑣𝑌
cocart.
𝑍 𝐶′ ∃!
SLIDE 51 Grothendieck construction
From an EED 𝑄 ∶ Cop → Cat, construct a Grothendieck bifjbration:
∫
C 𝑄
C
- objects: pairs (𝑌, 𝐵) for 𝑌 ∈ 𝑄(𝐵)
- morphisms (𝑌, 𝐵) → (𝑍, 𝐶): pairs (𝑣, 𝑔 )
where 𝑣 ∶ 𝐵 → 𝐶 and 𝑔 ∶ 𝑌 → 𝑄(𝑣)(𝑍). From a Grothendieck bifjbration 𝑞 ∶
→
, construct:
Cat
𝐵
𝐵
𝐶
𝐶 ̃ 𝑞 𝑣 ∃𝑣 𝑣∗ ⊢
SLIDE 52 Grothendieck construction
From an EED 𝑄 ∶ Cop → Cat, construct a Grothendieck bifjbration:
∫
C 𝑄
C
- objects: pairs (𝑌, 𝐵) for 𝑌 ∈ 𝑄(𝐵)
- morphisms (𝑌, 𝐵) → (𝑍, 𝐶): pairs (𝑣, 𝑔 )
where 𝑣 ∶ 𝐵 → 𝐶 and 𝑔 ∶ 𝑌 → 𝑄(𝑣)(𝑍). From a Grothendieck bifjbration 𝑞 ∶ E → B, construct:
Bop
Cat
𝐵 E𝐵 𝐶 E𝐶
̃ 𝑞 𝑣 ∃𝑣 𝑣∗ ⊢
SLIDE 53 EEDs are extensional
Equality in EEDs is intrinsically extensional.
𝑎 1𝐵 ∃Δ(1𝐵) 𝐷 𝐵 𝐵 × 𝐵
𝜇 𝑨 ∃! Δ 𝑑 ℎ
SLIDE 54 EEDs are extensional
Equality in EEDs is intrinsically extensional.
𝑎 1𝐵 ∃Δ(1𝐵) 𝐷 𝐵 𝐵 × 𝐵
𝜇 𝑨 ∃! Δ 𝑑 ℎ
SLIDE 55 EEDs are extensional
Equality in EEDs is intrinsically extensional.
𝑎 1𝐵 ∃Δ(1𝐵) 𝐷 𝐵 𝐵 × 𝐵
𝜇 𝑨 ∃! Δ 𝑑 ℎ
SLIDE 56 EEDs are extensional
Equality in EEDs is intrinsically extensional.
𝑎 1𝐵 ∃Δ(1𝐵) 𝐷 𝐵 𝐵 × 𝐵
𝜇 𝑨 ∃! Δ 𝑑 ℎ
SLIDE 57 EEDs are extensional
Equality in EEDs is intrinsically extensional.
𝑎 1𝐵 ∃Δ(1𝐵) 𝐷 𝐵 𝐵 × 𝐵
𝜇 𝑨 ∃! Δ 𝑑 ℎ
SLIDE 58
- 2. Reconcile hyperdoctrines with intensional equalities
SLIDE 59 Primer on tribes
A tribe is a category C with terminal object 1 and a class of maps 𝔊 such that:
- 𝐵 → 1 is in 𝔊 for every object 𝐵,
- 𝔊 contains every isomorphism,
- 𝔊 is stable under change of base,
- 𝔊 is stable under composition,
- 𝔊 ∘ LLP(𝔊) = C,
- LLP(𝔊) is stable under change of base along elements of 𝔊
Bare minimum to interpret a type theory with Σ, Id-types.
SLIDE 60 Primer on tribes
A tribe is a category C with terminal object 1 and a class of maps 𝔊 such that:
- 𝐵 → 1 is in 𝔊 for every object 𝐵,
- 𝔊 contains every isomorphism,
- 𝔊 is stable under change of base,
- 𝔊 is stable under composition,
- 𝔊 ∘ LLP(𝔊) = C,
- LLP(𝔊) is stable under change of base along elements of 𝔊
Bare minimum to interpret a type theory with Σ, Id-types.
SLIDE 61 Towards hypertribes
Provide a generalization of EEDs with
cod ∶ 𝔊 → C
as an instance. Goal
SLIDE 62 Identity types in a tribe
Interpret Id𝐵 by factorizing:
Id𝐵 𝐵 𝐵 × 𝐵
𝑞𝐵 Δ 𝑠𝐵
LLP(𝔊)
The j-rule is satisfjed:
𝐵 𝐷 Id𝐵 Id𝐵
𝑠𝐵 𝑑 𝑘
SLIDE 63 Identity types in a tribe
Interpret Id𝐵 by factorizing:
Id𝐵 𝐵 𝐵 × 𝐵
𝑞𝐵 Δ 𝑠𝐵
LLP(𝔊)
The j-rule is satisfjed:
𝐵 𝐷 Id𝐵 Id𝐵
𝑠𝐵 𝑑 𝑘
SLIDE 64 Identity types in a tribe
Interpret Id𝐵 by factorizing:
Id𝐵 𝐵 𝐵 × 𝐵
𝑞𝐵 Δ 𝑠𝐵
LLP(𝔊)
The j-rule is satisfjed:
𝐵 𝐷 Id𝐵 Id𝐵
𝑠𝐵 𝑑 𝑘
SLIDE 65 Identity types as an equality predicates?
The previous rules induces:
𝑎 1𝐵 ∃Δ(1𝐵) 𝐷 𝐵 𝐵 × 𝐵
𝜇 𝑨 ∃! Δ 𝑑 ℎ
𝑎 𝐵 Id𝐵 𝐷 𝐵 𝐵 × 𝐵
𝑟 𝑠𝐵 𝑨 ∃ Δ 𝑟𝑨 ℎ 𝑞𝐵
SLIDE 66 Identity types as an equality predicates?
The previous rules induces:
𝑎 1𝐵 ∃Δ(1𝐵) 𝐷 𝐵 𝐵 × 𝐵
𝜇 𝑨 ∃! Δ 𝑑 ℎ
𝑎 𝐵 Id𝐵 𝐷 𝐵 𝐵 × 𝐵
𝑟 𝑠𝐵 𝑨 ∃ Δ 𝑟𝑨 ℎ 𝑞𝐵
SLIDE 67 Identity types as an equality predicates?
The previous rules induces:
𝑎 1𝐵 ∃Δ(1𝐵) 𝐷 𝐵 𝐵 × 𝐵
𝜇 𝑨 ∃! Δ 𝑑 ℎ
𝑎 𝐵 Id𝐵 𝐷 𝐵 𝐵 × 𝐵
𝑟 𝑠𝐵 𝑨 ∃ Δ 𝑟𝑨 ℎ 𝑞𝐵
SLIDE 68 Identity types as an equality predicates?
The previous rules induces:
𝑎 1𝐵 ∃Δ(1𝐵) 𝐷 𝐵 𝐵 × 𝐵
𝜇 𝑨 ∃! Δ 𝑑 ℎ
𝑎 𝐵 Id𝐵 𝐷 𝐵 𝐵 × 𝐵
𝑟 𝑠𝐵 𝑨 ∃ Δ 𝑟𝑨 ℎ 𝑞𝐵
SLIDE 69 Identity types as an equality predicates?
The previous rules induces:
𝑎 1𝐵 ∃Δ(1𝐵) 𝐷 𝐵 𝐵 × 𝐵
𝜇 𝑨 ∃! Δ 𝑑 ℎ
𝑎 𝐵 Id𝐵 𝐷 𝐵 𝐵 × 𝐵
𝑟 𝑠𝐵 𝑨 ∃ Δ 𝑟𝑨 ℎ 𝑞𝐵
SLIDE 70 Relative lifting property
Given a functor 𝑞 ∶ E → B, say that a map 𝑔 in E has the weak left lifting property relatively to 𝑞 against when:
𝑌 𝑎 𝑍 𝑈 𝐵 𝐷 𝐶 𝐸
𝑔 𝑨 𝑢 ∃𝑙 𝑣 𝑑 𝑤 𝑒 ℎ
SLIDE 71 Relative lifting property
Given a functor 𝑞 ∶ E → B, say that a map 𝑔 in E has the weak left lifting property relatively to 𝑞 against when:
𝑌 𝑎 𝑍 𝑈 𝐵 𝐷 𝐶 𝐸
𝑔 𝑨 𝑢 ∃𝑙 𝑣 𝑑 𝑤 𝑒 ℎ
SLIDE 72 Relative lifting property
Given a functor 𝑞 ∶ E → B, say that a map 𝑔 in E has the weak left lifting property relatively to 𝑞 against when:
𝑌 𝑎 𝑍 𝑈 𝐵 𝐷 𝐶 𝐸
𝑔 𝑨 𝑢 ∃𝑙 𝑣 𝑑 𝑤 𝑒 ℎ
SLIDE 73 Relative lifting property
Given a functor 𝑞 ∶ E → B, say that a map 𝑔 in E has the strong left lifting property relatively to 𝑞 against when:
𝑌 𝑎 𝑍 𝑈 𝐵 𝐷 𝐶 𝐸
𝑔 𝑨 𝑢 ∃!𝑙 𝑣 𝑑 𝑤 𝑒 ℎ
SLIDE 74 Relative factorization systems
Defjne a right weak factorization system relative to 𝑞 ∶ E → B to consist of
- two classes 𝔐E, ℜE of morphisms of E,
- and two classes 𝔐B, ℜB of morphisms of B,
such that
and 𝑞(ℜ ) ⊆ ℜ ,
= LLP𝑞(ℜ )
:
𝑎 𝑌 𝑍 𝐷 𝐵 𝐶
∈ℜ 𝑔 ∈𝔐 ∈ℜ ∈𝔐 𝑣
SLIDE 75 Relative factorization systems
Defjne a right weak factorization system relative to 𝑞 ∶ E → B to consist of
- two classes 𝔐E, ℜE of morphisms of E,
- and two classes 𝔐B, ℜB of morphisms of B,
such that
- 𝑞(𝔐E) ⊆ 𝔐B and 𝑞(ℜE) ⊆ ℜB,
- 𝔐
= LLP𝑞(ℜ )
:
𝑎 𝑌 𝑍 𝐷 𝐵 𝐶
∈ℜ 𝑔 ∈𝔐 ∈ℜ ∈𝔐 𝑣
SLIDE 76 Relative factorization systems
Defjne a right weak factorization system relative to 𝑞 ∶ E → B to consist of
- two classes 𝔐E, ℜE of morphisms of E,
- and two classes 𝔐B, ℜB of morphisms of B,
such that
- 𝑞(𝔐E) ⊆ 𝔐B and 𝑞(ℜE) ⊆ ℜB,
- 𝔐E = LLP𝑞(ℜE)
- for every 𝑔 in
:
𝑎 𝑌 𝑍 𝐷 𝐵 𝐶
∈ℜ 𝑔 ∈𝔐 ∈ℜ ∈𝔐 𝑣
SLIDE 77 Relative factorization systems
Defjne a right weak factorization system relative to 𝑞 ∶ E → B to consist of
- two classes 𝔐E, ℜE of morphisms of E,
- and two classes 𝔐B, ℜB of morphisms of B,
such that
- 𝑞(𝔐E) ⊆ 𝔐B and 𝑞(ℜE) ⊆ ℜB,
- 𝔐E = LLP𝑞(ℜE)
- for every 𝑔 in E:
𝑎 𝑌 𝑍 𝐷 𝐵 𝐶
∈ℜE 𝑔 ∈𝔐E ∈ℜB ∈𝔐B 𝑣
SLIDE 78 Relative factorization systems
Defjne a right weak factorization system relative to 𝑞 ∶ E → B to consist of
- two classes 𝔐E, ℜE of morphisms of E,
- and two classes 𝔐B, ℜB of morphisms of B,
such that
- 𝑞(𝔐E) ⊆ 𝔐B and 𝑞(ℜE) ⊆ ℜB,
- 𝔐E = LLP𝑞(ℜE)
- for every 𝑔 in E:
𝑎 𝑌 𝑍 𝐷 𝐵 𝐶
∈ℜE 𝑔 ∈𝔐E ∈ℜB ∈𝔐B 𝑣
SLIDE 79 Relative factorization systems
Defjne a right weak factorization system relative to 𝑞 ∶ E → B to consist of
- two classes 𝔐E, ℜE of morphisms of E,
- and two classes 𝔐B, ℜB of morphisms of B,
such that
- 𝑞(𝔐E) ⊆ 𝔐B and 𝑞(ℜE) ⊆ ℜB,
- 𝔐E = LLP𝑞(ℜE)
- for every 𝑔 in E:
𝑎 𝑌 𝑍 𝐷 𝐵 𝐶
∈ℜE 𝑔 ∈𝔐E ∈ℜB ∈𝔐B 𝑣
SLIDE 80 Relative factorization systems
Defjne a right strong factorization system relative to 𝑞 ∶ E → B to consist of
- two classes 𝔐E, ℜE of morphisms of E,
- and two classes 𝔐B, ℜB of morphisms of B,
such that
- 𝑞(𝔐E) ⊆ 𝔐B and 𝑞(ℜE) ⊆ ℜB,
- 𝔐E = LLP⟂
𝑞 (ℜE)
𝑎 𝑌 𝑍 𝐷 𝐵 𝐶
∈ℜE 𝑔 ∈𝔐E ∈ℜB ∈𝔐B 𝑣
SLIDE 81 Cocartesian morphism as lifting problems
𝑔 is cocartesian if and only if 𝑔 ∈ LLP⟂
𝑞 (Mor (E))
𝐵 𝐶 𝑣 𝑌 𝑞 ∃𝑣𝑌
cocart.
𝑍 𝐶′ ∃! 𝐷 1 𝑎 1 𝑞 is a Grothendieck opfjbration if and only if there is a strong RFS relative to 𝑞
with ℜE = Mor (E) and 𝔐B = Mor (B)
SLIDE 82 Cocartesian morphism as lifting problems
𝑔 is cocartesian if and only if 𝑔 ∈ LLP⟂
𝑞 (Mor (E))
𝐵 𝐶 𝑣 𝑌 𝑞 ∃𝑣𝑌
cocart.
𝑍 𝐶′ ∃! 𝐷 1 𝑎 1 𝑞 is a Grothendieck opfjbration if and only if there is a strong RFS relative to 𝑞
with ℜE = Mor (E) and 𝔐B = Mor (B)
SLIDE 83 Cocartesian morphism as lifting problems
𝑔 is cocartesian if and only if 𝑔 ∈ LLP⟂
𝑞 (any → 1)
𝐵 𝐶 𝑣 𝑌 𝑞 ∃𝑣𝑌
cocart.
𝑍 𝐶′ ∃! 𝐷 1 𝑎 1 𝑞 is a Grothendieck opfjbration if and only if there is a strong RFS relative to 𝑞
with ℜE = Mor (E) and 𝔐B = Mor (B)
SLIDE 84 Cocartesian morphism as lifting problems
𝑔 is cocartesian if and only if 𝑔 ∈ LLP⟂
𝑞 (Mor (E))
𝐵 𝐶 𝑣 𝑌 𝑞 ∃𝑣𝑌
cocart.
𝑍 𝐶′ ∃! 𝐷 1 𝑎 1 𝑞 is a Grothendieck opfjbration if and only if there is a strong RFS relative to 𝑞
with ℜE = Mor (E) and 𝔐B = Mor (B)
SLIDE 85 Cocartesian morphism as lifting problems
𝑔 is cocartesian if and only if 𝑔 ∈ LLP⟂
𝑞 (Mor (E))
𝐵 𝐶 𝑣 𝑌 𝑞 ∃𝑣𝑌
cocart.
𝑍 𝐶′ ∃! 𝐷 1 𝑎 1 𝑞 is a Grothendieck opfjbration if and only if there is a strong RFS relative to 𝑞
with ℜE = Mor (E) and 𝔐B = Mor (B)
SLIDE 86 Anodyne maps as lifting problems
𝑔 is anodyne if and only if (𝑔 , 𝑣) ∈ LLPcod(Reedy fjbrations) 𝑌 𝑎 𝑍 𝑈 𝐵 𝐷 𝐶 𝐸
𝑔 𝑨 𝑞
𝑢 ∃𝑙 𝑡 𝑣 𝑑 𝑤 𝑒 𝑟 ℎ
𝔊
cod C C is a tribe if and only if there is a weak RFS relative to cod with
ℜ𝔊 = {Reedy fjbrations} and 𝔐C = Mor (C)
SLIDE 87 Anodyne maps as lifting problems
𝑔 is anodyne if and only if (𝑔 , 𝑣) ∈ LLPcod(Reedy fjbrations) 𝑌 𝑎 𝑍 𝑈 𝐵 𝐷 𝐶 𝐸
𝑔 𝑨 𝑞
𝑢 ∃𝑙 𝑡 𝑣 𝑑 𝑤 𝑒 𝑟 ℎ
𝔊
cod C C is a tribe if and only if there is a weak RFS relative to cod with
ℜ𝔊 = {Reedy fjbrations} and 𝔐C = Mor (C)
SLIDE 88 Anodyne maps as lifting problems
𝑔 is anodyne if and only if (𝑔 , 𝑣) ∈ LLPcod(Reedy fjbrations) 𝑌 𝑎 𝑍 𝑈 𝐵 𝐷 𝐶 𝐸
𝑔 𝑨 𝑞
𝑢 ∃𝑙 𝑡 𝑣 𝑑 𝑤 𝑒 𝑟 ℎ
𝔊
cod C C is a tribe if and only if there is a weak RFS relative to cod with
ℜ𝔊 = {Reedy fjbrations} and 𝔐C = Mor (C)
SLIDE 89 Anodyne maps as lifting problems
𝑔 is anodyne if and only if (𝑔 , 𝑣) ∈ LLPcod(Reedy fjbrations) 𝑌 𝑎 𝑍 𝑈 𝐵 𝐷 𝐶 𝐸
𝑔 𝑨 𝑞
𝑢 ∃𝑙 𝑡 𝑣 𝑑 𝑤 𝑒 𝑟 ℎ
𝔊
cod C C is a tribe if and only if there is a weak RFS relative to cod with
ℜ𝔊 = {Reedy fjbrations} and 𝔐C = Mor (C)
SLIDE 90 Anodyne maps as lifting problems
𝑔 is anodyne if and only if (𝑔 , 𝑣) ∈ LLPcod(Reedy fjbrations) 𝑌 𝑎 𝑍 𝑈 𝐵 𝐷 𝐶 𝐸
𝑔 𝑨 𝑞
𝑢 ∃𝑙 𝑡 𝑣 𝑑 𝑤 𝑒 𝑟 ℎ
𝔊
cod C C is a tribe if and only if there is a weak RFS relative to cod with
ℜ𝔊 = {Reedy fjbrations} and 𝔐C = Mor (C)
SLIDE 91 Reconcile Lawvere’s equality and identity types
1𝐵 𝑎 ∃Δ(1𝐵) 𝑈 𝐵 𝐷 𝐵 × 𝐵 𝐸
cocart.
𝑨 ∃!𝑙 Δ 𝑑 ℎ
𝐵 𝑎 Id𝐵 𝑈 𝐵 𝐷 𝐵 × 𝐵 𝐸
𝑠𝐵 𝑨 𝑟 ∃𝑙 Δ 𝑑 𝑞𝐵 ℎ
SLIDE 92 Reconcile Lawvere’s equality and identity types
1𝐵 𝑎 ∃Δ(1𝐵) 𝑈 𝐵 𝐷 𝐵 × 𝐵 𝐸
cocart.
𝑨 ∃!𝑙 Δ 𝑑 ℎ
𝐵 𝑎 Id𝐵 𝑈 𝐵 𝐷 𝐵 × 𝐵 𝐸
𝑠𝐵 𝑨 𝑟 ∃𝑙 Δ 𝑑 𝑞𝐵 ℎ
SLIDE 93 Reconcile Lawvere’s equality and identity types
1𝐵 𝑎 ∃Δ(1𝐵) 𝑈 𝐵 𝐷 𝐵 × 𝐵 𝐸
cocart.
𝑨 ∃!𝑙 Δ 𝑑 ℎ
𝐵 𝑎 Id𝐵 𝑈 𝐵 𝐷 𝐵 × 𝐵 𝐸
𝑠𝐵 𝑨 𝑟 ∃𝑙 Δ 𝑑 𝑞𝐵 ℎ
SLIDE 94 Thank you.
http://www.normalesup.org/~cagne/
This document is licensed under CC-BY-SA 4.0 International.
SLIDE 95 Lawvere’s insight
Consider the groupoid hyperdoctrine: Grpdop → CAT
G ↦ Psh(G)
This should not to be taken as indicative of a lack of vitality of [the groupoid] hyperdoctrine, or even of a lack of a satisfactory theory
- f equality for it. Rather, it indicates that we have probably been
too naive in defjning equality in a manner too closely suggested by the classical conception. — Lawvere
« »
SLIDE 96 Lawvere’s insight
Consider the groupoid hyperdoctrine: Grpdop → CAT
G ↦ Psh(G)
This should not to be taken as indicative of a lack of vitality of [the groupoid] hyperdoctrine, or even of a lack of a satisfactory theory
- f equality for it. Rather, it indicates that we have probably been
too naive in defjning equality in a manner too closely suggested by the classical conception. — Lawvere
« »
SLIDE 97 What is it good for?
A model 𝑁 of a fjrst-order theory 𝕌 can be interpreted as: ctxop Cat Setop
𝔑
𝕌
Sub(−)
𝜈
Cop
where 𝔑 ∶ ⃗
𝑦 ↦ 𝑁|⃗
𝑦|, and 𝜈⃗ 𝑦 ∶ 𝜒(⃗
𝑦) ↦ { ⃗ 𝑛 ∣ 𝑁 ⊧ 𝜒( ⃗ 𝑛)}
SLIDE 98 What is it good for?
A 𝑄-model 𝑁 of a fjrst-order theory 𝕌 can be defjned as: ctxop Cat Setop
𝔑
𝕌 𝑄 𝜈
Cop
where 𝔑 and 𝜈 have good properties.
SLIDE 99
Type-theoretic equality predicates
𝑦∶𝐷 ⊢ 𝑎(𝑦) type 𝑦, 𝑧∶𝐵 ⊢ 𝑑(𝑦, 𝑧)∶𝐷 𝑦∶𝐵 ⊢ 𝑨(𝑦)∶𝑎(𝑑(𝑦, 𝑦)) 𝑦, 𝑧∶𝐵, 𝑞∶Eq𝐵(𝑦, 𝑧) ⊢ 𝑘𝑨(𝑦, 𝑧, 𝑞)∶𝑎(𝑑(𝑦, 𝑧)) 𝑦∶𝐷 ⊢ 𝑎(𝑦) type 𝑦, 𝑧∶𝐵 ⊢ 𝑑(𝑦, 𝑧)∶𝐷 𝑦∶𝐵 ⊢ 𝑨(𝑦)∶𝑎(𝑑(𝑦, 𝑦)) 𝑦∶𝐵 ⊢ 𝑘𝑨(𝑦, 𝑦, refl𝑦) ≡ 𝑨(𝑦) 𝐷 ⊢ 𝑎 type 𝑦, 𝑧∶𝐵 ⊢ 𝑑(𝑦, 𝑧)∶𝐷 𝑦, 𝑧∶𝐵, 𝑞∶Eq𝐵(𝑦, 𝑧) ⊢ 𝑙(𝑦, 𝑧, 𝑞)∶𝑎(𝑑(𝑦, 𝑧)) 𝑦, 𝑧∶𝐵, 𝑞∶Eq𝐵(𝑦, 𝑧) ⊢ 𝑘𝑙(𝑦,𝑦,refl𝑦)(𝑦, 𝑧, 𝑞) ≡ 𝑙(𝑦, 𝑧, 𝑞)
