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src/foundation/transport-along-higher-identifications.lagda.md
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| # Transport along higher identifications | ||
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| ```agda | ||
| module foundation.transport-along-higher-identifications where | ||
| ``` | ||
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| <details><summary>Imports</summary> | ||
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| ```agda | ||
| open import foundation.action-on-identifications-functions | ||
| open import foundation.commuting-squares-of-identifications | ||
| open import foundation.dependent-pair-types | ||
| open import foundation.homotopies | ||
| open import foundation.identity-types | ||
| open import foundation.path-algebra | ||
| open import foundation.transport-along-identifications | ||
| open import foundation.universe-levels | ||
| open import foundation.whiskering-homotopies | ||
| ``` | ||
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| </details> | ||
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| ### The action on identifications of transport | ||
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| ```agda | ||
| module _ | ||
| {l1 l2 : Level} {A : UU l1} {x y : A} {p p' : x ＝ y} | ||
| where | ||
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| tr² : (B : A → UU l2) (α : p ＝ p') (b : B x) → (tr B p b) ＝ (tr B p' b) | ||
| tr² B α b = ap (λ t → tr B t b) α | ||
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| module _ | ||
| {l1 l2 : Level} {A : UU l1} {x y : A} {p p' : x ＝ y} | ||
| {α α' : p ＝ p'} | ||
| where | ||
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| tr³ : (B : A → UU l2) (β : α ＝ α') (b : B x) → (tr² B α b) ＝ (tr² B α' b) | ||
| tr³ B β b = ap (λ t → tr² B t b) β | ||
| ``` | ||
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| ### Computing 2-dimensional transport in a family of identifications with a fixed source | ||
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| ```agda | ||
| module _ | ||
| {l : Level} {A : UU l} {a b c : A} {q q' : b ＝ c} | ||
| where | ||
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| tr²-Id-right : | ||
| (α : q ＝ q') (p : a ＝ b) → | ||
| coherence-square-identifications | ||
| ( tr² (Id a) α p) | ||
| ( tr-Id-right q' p) | ||
| ( tr-Id-right q p) | ||
| ( identification-left-whisk p α) | ||
| tr²-Id-right α p = | ||
| inv-nat-htpy (λ (t : b ＝ c) → tr-Id-right t p) α | ||
| ``` | ||
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| ### Coherences and algebraic identities for `tr²` | ||
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| ```agda | ||
| module _ | ||
| {l1 l2 : Level} {A : UU l1} {x y : A} | ||
| {B : A → UU l2} | ||
| where | ||
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| tr²-concat : | ||
| {p p' p'' : x ＝ y} (α : p ＝ p') (α' : p' ＝ p'') (b : B x) → | ||
| (tr² B (α ∙ α') b) ＝ (tr² B α b ∙ tr² B α' b) | ||
| tr²-concat α α' b = ap-concat (λ t → tr B t b) α α' | ||
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| module _ | ||
| {l1 l2 : Level} {A : UU l1} {x y z : A} | ||
| {B : A → UU l2} | ||
| where | ||
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| tr²-left-whisk : | ||
| (p : x ＝ y) {q q' : y ＝ z} (β : q ＝ q') (b : B x) → | ||
| coherence-square-identifications | ||
| ( tr² B (identification-left-whisk p β) b) | ||
| ( tr-concat p q' b) | ||
| ( tr-concat p q b) | ||
| ( htpy-right-whisk (tr² B β) (tr B p) b) | ||
| tr²-left-whisk refl refl b = refl | ||
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| tr²-right-whisk : | ||
| {p p' : x ＝ y} (α : p ＝ p') (q : y ＝ z) (b : B x) → | ||
| coherence-square-identifications | ||
| ( tr² B (identification-right-whisk α q) b) | ||
| ( tr-concat p' q b) | ||
| ( tr-concat p q b) | ||
| ( htpy-left-whisk (tr B q) (tr² B α) b) | ||
| tr²-right-whisk refl refl b = inv right-unit | ||
| ``` | ||
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| #### Coherences and algebraic identities for `tr³` | ||
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| ```agda | ||
| module _ | ||
| {l1 l2 : Level} {A : UU l1} {x y z : A} | ||
| {B : A → UU l2} | ||
| where | ||
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| tr³-htpy-swap-path-swap : | ||
| {q q' : y ＝ z} (β : q ＝ q') {p p' : x ＝ y} (α : p ＝ p') (b : B x) → | ||
| coherence-square-identifications | ||
| ( identification-right-whisk | ||
| ( tr³ | ||
| ( B) | ||
| ( path-swap-nat-identification-left-whisk β α) | ||
| ( b)) | ||
| ( tr-concat p' q' b)) | ||
| ( ( identification-right-whisk | ||
| ( tr²-concat | ||
| ( identification-right-whisk α q) | ||
| ( identification-left-whisk p' β) b) | ||
| ( tr-concat p' q' b)) ∙ | ||
| ( vertical-concat-square | ||
| ( tr² B (identification-right-whisk α q) b) | ||
| ( tr² B (identification-left-whisk p' β) b) | ||
| ( tr-concat p' q' b) | ||
| ( tr-concat p' q b) | ||
| ( tr-concat p q b) | ||
| ( htpy-left-whisk (tr B q) (tr² B α) b) | ||
| ( htpy-right-whisk (tr² B β) (tr B p') b) | ||
| ( tr²-right-whisk α q b) | ||
| ( tr²-left-whisk p' β b))) | ||
| ( ( identification-right-whisk | ||
| ( tr²-concat (identification-left-whisk p β) | ||
| ( identification-right-whisk α q') b) | ||
| ( tr-concat p' q' b)) ∙ | ||
| ( vertical-concat-square | ||
| ( tr² B (identification-left-whisk p β) b) | ||
| ( tr² B (identification-right-whisk α q') b) | ||
| ( tr-concat p' q' b) | ||
| ( tr-concat p q' b) | ||
| ( tr-concat p q b) | ||
| ( htpy-right-whisk (tr² B β) (tr B p) b) | ||
| ( htpy-left-whisk (tr B q') (tr² B α) b) | ||
| ( tr²-left-whisk p β b) | ||
| ( tr²-right-whisk α q' b))) | ||
| ( identification-left-whisk | ||
| ( tr-concat p q b) | ||
| ( htpy-swap-nat-right-htpy (tr² B β) (tr² B α) b)) | ||
| tr³-htpy-swap-path-swap {q = refl} refl {p = refl} refl b = refl | ||
| ``` |
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