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| {-# OPTIONS --without-K --rewriting #-} | ||
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| open import lib.Basics | ||
| open import lib.types.Pi | ||
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| module lib.types.Cospan where | ||
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| record Cospan {i j k : ULevel} : Type (lsucc (lmax (lmax i j) k)) where | ||
| constructor cospan | ||
| field | ||
| A : Type i | ||
| B : Type j | ||
| C : Type k | ||
| f : A → C | ||
| g : B → C | ||
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| record ⊙Cospan {i j k : ULevel} : Type (lsucc (lmax (lmax i j) k)) where | ||
| constructor ⊙cospan | ||
| field | ||
| X : Ptd i | ||
| Y : Ptd j | ||
| Z : Ptd k | ||
| f : X ⊙→ Z | ||
| g : Y ⊙→ Z | ||
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| ⊙cospan-out : ∀ {i j k} → ⊙Cospan {i} {j} {k} → Cospan {i} {j} {k} | ||
| ⊙cospan-out (⊙cospan X Y Z f g) = | ||
| cospan (de⊙ X) (de⊙ Y) (de⊙ Z) (fst f) (fst g) |
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| {-# OPTIONS --without-K --rewriting --overlapping-instances #-} | ||
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| open import lib.Basics | ||
| open import lib.NType | ||
| open import lib.types.Cospan | ||
| open import lib.types.Pointed | ||
| open import lib.types.Sigma | ||
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| module lib.types.Pullback where | ||
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| module _ {i j k} (D : Cospan {i} {j} {k}) where | ||
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| open Cospan D | ||
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| record Pullback : Type (lmax i (lmax j k)) where | ||
| constructor pullback | ||
| field | ||
| a : A | ||
| b : B | ||
| h : f a == g b | ||
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| pullback= : {a a' : A} (p : a == a') {b b' : B} (q : b == b') | ||
| {h : f a == g b} {h' : f a' == g b'} (r : h ∙ ap g q == ap f p ∙ h') | ||
| → pullback a b h == pullback a' b' h' | ||
| pullback= idp idp r = | ||
| ap (pullback _ _) (! (∙-unit-r _) ∙ r) | ||
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| pullback-aβ : {a a' : A} (p : a == a') {b b' : B} (q : b == b') | ||
| {h : f a == g b} {h' : f a' == g b'} (r : h ∙ ap g q == ap f p ∙ h') | ||
| → ap Pullback.a (pullback= p q {h = h} {h' = h'} r) == p | ||
| pullback-aβ idp idp r = | ||
| ap Pullback.a (ap (pullback _ _) (! (∙-unit-r _) ∙ r)) | ||
| =⟨ ∘-ap Pullback.a (pullback _ _) _ ⟩ | ||
| ap (λ _ → _) (! (∙-unit-r _) ∙ r) | ||
| =⟨ ap-cst _ _ ⟩ | ||
| idp =∎ | ||
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| pullback-bβ : {a a' : A} (p : a == a') {b b' : B} (q : b == b') | ||
| {h : f a == g b} {h' : f a' == g b'} (r : h ∙ ap g q == ap f p ∙ h') | ||
| → ap Pullback.b (pullback= p q {h = h} {h' = h'} r) == q | ||
| pullback-bβ idp idp r = | ||
| ap Pullback.b (ap (pullback _ _) (! (∙-unit-r _) ∙ r)) | ||
| =⟨ ∘-ap Pullback.b (pullback _ _) _ ⟩ | ||
| ap (λ _ → _) (! (∙-unit-r _) ∙ r) | ||
| =⟨ ap-cst _ _ ⟩ | ||
| idp =∎ | ||
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| module _ {i j k} (D : ⊙Cospan {i} {j} {k}) where | ||
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| ⊙Pullback : Ptd (lmax i (lmax j k)) | ||
| ⊙Pullback = | ||
| ⊙[ Pullback (⊙cospan-out D) , | ||
| pullback (pt X) (pt Y) (snd f ∙ ! (snd g)) ] | ||
| where open ⊙Cospan D | ||
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| module _ {i j k} (D : Cospan {i} {j} {k}) where | ||
| open Cospan D | ||
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| pullback-decomp-equiv : Pullback D ≃ Σ (A × B) (λ {(a , b) → f a == g b}) | ||
| pullback-decomp-equiv = equiv | ||
| (λ {(pullback a b h) → ((a , b) , h)}) | ||
| (λ {((a , b) , h) → pullback a b h}) | ||
| (λ _ → idp) | ||
| (λ _ → idp) | ||
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| module _ {i j k} (n : ℕ₋₂) {D : Cospan {i} {j} {k}} where | ||
| open Cospan D | ||
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| pullback-level : has-level n A → has-level n B → has-level n C | ||
| → has-level n (Pullback D) | ||
| pullback-level pA pB pC = | ||
| equiv-preserves-level ((pullback-decomp-equiv D)⁻¹) where instance _ = pA; _ = pB; _ = pC | ||
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| instance | ||
| pullback-level-instance : {i j k : ULevel} {n : ℕ₋₂} {D : Cospan {i} {j} {k}} → | ||
| {{has-level n (Cospan.A D)}} → {{has-level n (Cospan.B D)}} → {{has-level n (Cospan.C D)}} | ||
| → has-level n (Pullback D) | ||
| pullback-level-instance {n = n} {{pA}} {{pB}} {{pC}} = pullback-level n pA pB pC |
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