Identity types as equality predicates Reconciling hyperdoctrines - PowerPoint PPT Presentation

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

1. Lawvere’s hyperdoctrines 2. Reconcile hyperdoctrines with intensional equalities

1. Lawvere’s hyperdoctrines

What does it have to do with logic? Lawvere’s hyperdoctrines An hyperdoctrine is a pseudofunctor 𝑄 ∶ C op → 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.

Lawvere’s hyperdoctrines An hyperdoctrine is a pseudofunctor 𝑄 ∶ C op → 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?

In particular for 𝑔 = 𝜀 𝐵 ∶ 𝐵 → 𝐵 × 𝐵 , ∃ 𝜀 𝐵 ∶ id 𝐵 ↦ 𝜀 𝐵 ∃ 𝑔 𝑟 𝑞 ⌟ Π 𝑔 𝑟 ∀ 𝑔 𝑟 • 𝑔 𝑟 𝑍 𝐵 𝑟 𝑍 𝑟 𝑍 𝑌 𝑔 𝐶 Seely’s semantics is an hyperdoctrine 𝐵 𝑔 × 𝑞 𝑌

In particular for 𝑔 = 𝜀 𝐵 ∶ 𝐵 → 𝐵 × 𝐵 , ∃ 𝜀 𝐵 ∶ id 𝐵 ↦ 𝜀 𝐵 ∃ 𝑔 𝑟 𝑞 ⌟ Π 𝑔 𝑟 ∀ 𝑔 𝑟 • 𝑔 𝑟 𝑍 𝐵 𝑟 𝑍 𝑟 𝑍 𝑌 𝑔 𝐶 Seely’s semantics is an hyperdoctrine 𝐵 𝑔 × 𝑞 𝑌

In particular for 𝑔 = 𝜀 𝐵 ∶ 𝐵 → 𝐵 × 𝐵 , ∃ 𝜀 𝐵 ∶ id 𝐵 ↦ 𝜀 𝐵 ∃ 𝑔 𝑟 𝑞 ⌟ Π 𝑔 𝑟 ∀ 𝑔 𝑟 • 𝑔 𝑟 𝑍 𝐵 𝑟 𝑍 𝑟 𝑍 𝑌 𝑔 𝐶 Seely’s semantics is an hyperdoctrine 𝐵 𝑔 × 𝑞 𝑌

In particular for 𝑔 = 𝜀 𝐵 ∶ 𝐵 → 𝐵 × 𝐵 , ∃ 𝜀 𝐵 ∶ id 𝐵 ↦ 𝜀 𝐵 ∃ 𝑔 𝑟 𝑞 ⌟ Π 𝑔 𝑟 ∀ 𝑔 𝑟 • 𝑔 𝑟 𝑍 𝐵 𝑟 𝑍 𝑟 𝑍 𝑌 𝑔 𝐶 Seely’s semantics is an hyperdoctrine 𝐵 𝑔 × 𝑞 𝑌

In particular for 𝑔 = 𝜀 𝐵 ∶ 𝐵 → 𝐵 × 𝐵 , ∃ 𝜀 𝐵 ∶ id 𝐵 ↦ 𝜀 𝐵 ∃ 𝑔 𝑟 𝑞 ⌟ Π 𝑔 𝑟 ∀ 𝑔 𝑟 • 𝑔 𝑟 𝑍 𝐵 𝑟 𝑍 𝑟 𝑍 𝑌 𝑔 𝐶 Seely’s semantics is an hyperdoctrine 𝐵 𝑔 × 𝑞 𝑌

In particular for 𝑔 = 𝜀 𝐵 ∶ 𝐵 → 𝐵 × 𝐵 , ∃ 𝜀 𝐵 ∶ id 𝐵 ↦ 𝜀 𝐵 ∃ 𝑔 𝑟 𝑞 ⌟ Π 𝑔 𝑟 ∀ 𝑔 𝑟 • 𝑔 𝑟 𝑍 𝐵 𝑟 𝑍 𝑟 𝑍 𝑌 𝑔 𝐶 Seely’s semantics is an hyperdoctrine 𝐵 𝑔 × 𝑞 𝑌

In particular for 𝑔 = 𝜀 𝐵 ∶ 𝐵 → 𝐵 × 𝐵 , ∃ 𝜀 𝐵 ∶ id 𝐵 ↦ 𝜀 𝐵 ∃ 𝑔 𝑟 𝑞 ⌟ Π 𝑔 𝑟 ∀ 𝑔 𝑟 • 𝑔 𝑟 𝑍 𝐵 𝑟 𝑍 𝑟 𝑍 𝑌 𝑔 𝐶 Seely’s semantics is an hyperdoctrine 𝐵 𝑔 × 𝑞 𝑌

In particular for 𝑔 = 𝜀 𝐵 ∶ 𝐵 → 𝐵 × 𝐵 , ∃ 𝜀 𝐵 ∶ id 𝐵 ↦ 𝜀 𝐵 ∃ 𝑔 𝑟 𝑞 ⌟ Π 𝑔 𝑟 ∀ 𝑔 𝑟 • 𝑔 𝑟 𝑍 𝐵 𝑟 𝑍 𝑟 𝑍 𝑌 𝑔 𝐶 Seely’s semantics is an hyperdoctrine 𝐵 𝑔 × 𝑞 𝑌

In particular for 𝑔 = 𝜀 𝐵 ∶ 𝐵 → 𝐵 × 𝐵 , ∃ 𝜀 𝐵 ∶ id 𝐵 ↦ 𝜀 𝐵 ∃ 𝑔 𝑟 𝑞 ⌟ Π 𝑔 𝑟 ∀ 𝑔 𝑟 • 𝑔 𝑟 𝑍 𝐵 𝑟 𝑍 𝑟 𝑍 𝑌 𝑔 𝐶 Seely’s semantics is an hyperdoctrine 𝐵 𝑔 × 𝑞 𝑌

𝑍 ⌟ 𝐶 𝑔 𝑌 𝑞 𝑍 𝑟 𝑍 𝑟 𝐵 ∃ 𝑔 𝑟 𝑔 𝑟 • ∀ 𝑔 𝑟 Π 𝑔 𝑟 Seely’s semantics is an hyperdoctrine 𝐵 𝑔 × 𝑞 𝑌 In particular for 𝑔 = 𝜀 𝐵 ∶ 𝐵 → 𝐵 × 𝐵 , ∃ 𝜀 𝐵 ∶ id 𝐵 ↦ 𝜀 𝐵

In particular for 𝑔 = 𝜀 𝐵 ∶ 𝐵 → 𝐵 × 𝐵 , ∃ 𝜀 𝐵 ∶ 𝐵 ↦ {(𝑏, 𝑏 ′ ) ∈ 𝐵 × 𝐵 ∶ 𝑏 = 𝑏 ′ } ⊆ 𝐶 𝑔 𝑊 𝑉 𝑉 ∃ 𝑔 𝑉 𝐵 ∀ 𝑔 𝑉 ⊆ ⊆ ⊆ Subsets form an hyperdoctrine 𝑔 (𝑉) = {𝑐 ∈ 𝐶 ∶ ∃𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ∧ 𝑏 ∈ 𝑉} 𝑔 ∗ (𝑉) = {𝑐 ∈ 𝐶 ∶ ∀𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ⇒ 𝑏 ∈ 𝑉} 𝑔 −1 (𝑊)

In particular for 𝑔 = 𝜀 𝐵 ∶ 𝐵 → 𝐵 × 𝐵 , ∃ 𝜀 𝐵 ∶ 𝐵 ↦ {(𝑏, 𝑏 ′ ) ∈ 𝐵 × 𝐵 ∶ 𝑏 = 𝑏 ′ } ⊆ 𝐶 𝑔 𝑊 𝑉 𝑉 ∃ 𝑔 𝑉 𝐵 ∀ 𝑔 𝑉 ⊆ ⊆ ⊆ Subsets form an hyperdoctrine 𝑔 (𝑉) = {𝑐 ∈ 𝐶 ∶ ∃𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ∧ 𝑏 ∈ 𝑉} 𝑔 ∗ (𝑉) = {𝑐 ∈ 𝐶 ∶ ∀𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ⇒ 𝑏 ∈ 𝑉} 𝑔 −1 (𝑊)

In particular for 𝑔 = 𝜀 𝐵 ∶ 𝐵 → 𝐵 × 𝐵 , ∃ 𝜀 𝐵 ∶ 𝐵 ↦ {(𝑏, 𝑏 ′ ) ∈ 𝐵 × 𝐵 ∶ 𝑏 = 𝑏 ′ } ⊆ 𝐶 𝑔 𝑊 𝑉 𝑉 ∃ 𝑔 𝑉 𝐵 ∀ 𝑔 𝑉 ⊆ ⊆ ⊆ Subsets form an hyperdoctrine 𝑔 (𝑉) = {𝑐 ∈ 𝐶 ∶ ∃𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ∧ 𝑏 ∈ 𝑉} 𝑔 ∗ (𝑉) = {𝑐 ∈ 𝐶 ∶ ∀𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ⇒ 𝑏 ∈ 𝑉} 𝑔 −1 (𝑊)

In particular for 𝑔 = 𝜀 𝐵 ∶ 𝐵 → 𝐵 × 𝐵 , ∃ 𝜀 𝐵 ∶ 𝐵 ↦ {(𝑏, 𝑏 ′ ) ∈ 𝐵 × 𝐵 ∶ 𝑏 = 𝑏 ′ } ⊆ 𝐶 𝑔 𝑊 𝑉 𝑉 ∃ 𝑔 𝑉 𝐵 ∀ 𝑔 𝑉 ⊆ ⊆ ⊆ Subsets form an hyperdoctrine 𝑔 (𝑉) = {𝑐 ∈ 𝐶 ∶ ∃𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ∧ 𝑏 ∈ 𝑉} 𝑔 ∗ (𝑉) = {𝑐 ∈ 𝐶 ∶ ∀𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ⇒ 𝑏 ∈ 𝑉} 𝑔 −1 (𝑊)

In particular for 𝑔 = 𝜀 𝐵 ∶ 𝐵 → 𝐵 × 𝐵 , ∃ 𝜀 𝐵 ∶ 𝐵 ↦ {(𝑏, 𝑏 ′ ) ∈ 𝐵 × 𝐵 ∶ 𝑏 = 𝑏 ′ } ⊆ 𝐶 𝑔 𝑊 𝑉 𝑉 ∃ 𝑔 𝑉 𝐵 ∀ 𝑔 𝑉 ⊆ ⊆ ⊆ Subsets form an hyperdoctrine 𝑔 (𝑉) = {𝑐 ∈ 𝐶 ∶ ∃𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ∧ 𝑏 ∈ 𝑉} 𝑔 ∗ (𝑉) = {𝑐 ∈ 𝐶 ∶ ∀𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ⇒ 𝑏 ∈ 𝑉} 𝑔 −1 (𝑊)

In particular for 𝑔 = 𝜀 𝐵 ∶ 𝐵 → 𝐵 × 𝐵 , ∃ 𝜀 𝐵 ∶ 𝐵 ↦ {(𝑏, 𝑏 ′ ) ∈ 𝐵 × 𝐵 ∶ 𝑏 = 𝑏 ′ } ⊆ 𝐶 𝑔 𝑊 𝑉 𝑉 ∃ 𝑔 𝑉 𝐵 ∀ 𝑔 𝑉 ⊆ ⊆ ⊆ Subsets form an hyperdoctrine 𝑔 (𝑉) = {𝑐 ∈ 𝐶 ∶ ∃𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ∧ 𝑏 ∈ 𝑉} 𝑔 ∗ (𝑉) = {𝑐 ∈ 𝐶 ∶ ∀𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ⇒ 𝑏 ∈ 𝑉} 𝑔 −1 (𝑊)

In particular for 𝑔 = 𝜀 𝐵 ∶ 𝐵 → 𝐵 × 𝐵 , ∃ 𝜀 𝐵 ∶ 𝐵 ↦ {(𝑏, 𝑏 ′ ) ∈ 𝐵 × 𝐵 ∶ 𝑏 = 𝑏 ′ } ⊆ 𝐶 𝑔 𝑊 𝑉 𝑉 ∃ 𝑔 𝑉 𝐵 ∀ 𝑔 𝑉 ⊆ ⊆ ⊆ Subsets form an hyperdoctrine 𝑔 (𝑉) = {𝑐 ∈ 𝐶 ∶ ∃𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ∧ 𝑏 ∈ 𝑉} 𝑔 ∗ (𝑉) = {𝑐 ∈ 𝐶 ∶ ∀𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ⇒ 𝑏 ∈ 𝑉} 𝑔 −1 (𝑊)

In particular for 𝑔 = 𝜀 𝐵 ∶ 𝐵 → 𝐵 × 𝐵 , ∃ 𝜀 𝐵 ∶ 𝐵 ↦ {(𝑏, 𝑏 ′ ) ∈ 𝐵 × 𝐵 ∶ 𝑏 = 𝑏 ′ } ⊆ 𝐶 𝑔 𝑊 𝑉 𝑉 ∃ 𝑔 𝑉 𝐵 ∀ 𝑔 𝑉 ⊆ ⊆ ⊆ Subsets form an hyperdoctrine 𝑔 (𝑉) = {𝑐 ∈ 𝐶 ∶ ∃𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ∧ 𝑏 ∈ 𝑉} 𝑔 ∗ (𝑉) = {𝑐 ∈ 𝐶 ∶ ∀𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ⇒ 𝑏 ∈ 𝑉} 𝑔 −1 (𝑊)

In particular for 𝑔 = 𝜀 𝐵 ∶ 𝐵 → 𝐵 × 𝐵 , ∃ 𝜀 𝐵 ∶ 𝐵 ↦ {(𝑏, 𝑏 ′ ) ∈ 𝐵 × 𝐵 ∶ 𝑏 = 𝑏 ′ } ⊆ 𝐶 𝑔 𝑊 𝑉 𝑉 ∃ 𝑔 𝑉 𝐵 ∀ 𝑔 𝑉 ⊆ ⊆ ⊆ Subsets form an hyperdoctrine 𝑔 (𝑉) = {𝑐 ∈ 𝐶 ∶ ∃𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ∧ 𝑏 ∈ 𝑉} 𝑔 ∗ (𝑉) = {𝑐 ∈ 𝐶 ∶ ∀𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ⇒ 𝑏 ∈ 𝑉} 𝑔 −1 (𝑊)

⊆ 𝐶 𝑔 𝑊 ⊆ 𝑉 𝑉 ∃ 𝑔 𝑉 𝐵 ∀ 𝑔 𝑉 ⊆ ⊆ Subsets form an hyperdoctrine 𝑔 (𝑉) = {𝑐 ∈ 𝐶 ∶ ∃𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ∧ 𝑏 ∈ 𝑉} 𝑔 ∗ (𝑉) = {𝑐 ∈ 𝐶 ∶ ∀𝑏 ∈ 𝐵, 𝑔 (𝑏) = 𝑐 ⇒ 𝑏 ∈ 𝑉} 𝑔 −1 (𝑊) In particular for 𝑔 = 𝜀 𝐵 ∶ 𝐵 → 𝐵 × 𝐵 , ∃ 𝜀 𝐵 ∶ 𝐵 ↦ {(𝑏, 𝑏 ′ ) ∈ 𝐵 × 𝐵 ∶ 𝑏 = 𝑏 ′ }

In particular for ⃗ 𝑦) ∶ (𝑦 1 , … , 𝑦 𝑜 ) → (𝑦 1 , … , 𝑦 2𝑜 ) , 𝑦) ∶ ⊤ ↦ ⋀ 𝑦 𝑗 = 𝑦 𝑜+𝑗 𝑧 𝑦) 𝑦, ⃗ 𝑦) = (⃗ 𝑢(⃗ 𝑦 ⊧ ⊧ ⊧ ⊧ 𝑦) 𝑦,⃗ ⃗ ∀⃗ ⃗ ⃗ 𝑗 ∃ (⃗ 𝑦) 𝜒(⃗ 𝑦) 𝜒(⃗ 𝑦)) 𝑢(⃗ 𝜔(⃗ 𝑧) 𝜔(⃗ 𝑦) 𝑢(⃗ ∃⃗ Predicates form an hyperdoctrine 𝑦, (⋀ 𝑗 𝑢 𝑗 (⃗ 𝑦) = 𝑧 𝑗 ) ∧ 𝜒(⃗ 𝑦, (⋀ 𝑗 𝑢 𝑗 (⃗ 𝑦) = 𝑧 𝑗 ) ⇒ 𝜒(⃗

In particular for ⃗ 𝑦) ∶ (𝑦 1 , … , 𝑦 𝑜 ) → (𝑦 1 , … , 𝑦 2𝑜 ) , 𝑦) ∶ ⊤ ↦ ⋀ 𝑦 𝑗 = 𝑦 𝑜+𝑗 𝑧 𝑦) 𝑦, ⃗ 𝑦) = (⃗ 𝑢(⃗ 𝑦 ⊧ ⊧ ⊧ ⊧ 𝑦) 𝑦,⃗ ⃗ ∀⃗ ⃗ ⃗ 𝑗 ∃ (⃗ 𝑦) 𝜒(⃗ 𝑦) 𝜒(⃗ 𝑦)) 𝑢(⃗ 𝜔(⃗ 𝑧) 𝜔(⃗ 𝑦) 𝑢(⃗ ∃⃗ Predicates form an hyperdoctrine 𝑦, (⋀ 𝑗 𝑢 𝑗 (⃗ 𝑦) = 𝑧 𝑗 ) ∧ 𝜒(⃗ 𝑦, (⋀ 𝑗 𝑢 𝑗 (⃗ 𝑦) = 𝑧 𝑗 ) ⇒ 𝜒(⃗