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A (mere) `k`-path-cosplit map is defined inductively - A (mere) `-2`-path-cosplit map is a map that is (merely) a retract - A (mere) `k+1`-path-cosplit map is a map whose action on identifications is (merely) `k`-path-cosplit. We show - Mere `k`-path-cosplitting is a property - `k`-truncated maps are `k`-path-cosplit - (merely) `k`-path-cosplit maps are (merely) `k+1`-path-cosplit - types that map into `k`-truncated types via (mere) `k`-path-cosplit maps are `k`-truncated - (mere) `k`-path-cosplit maps into `k`-truncated types are `k`-truncated.
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| # Mere path-cosplit maps | ||
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| ```agda | ||
| module foundation.mere-path-cosplit-maps where | ||
| ``` | ||
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| <details><summary>Imports</summary> | ||
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| ```agda | ||
| open import foundation.action-on-identifications-functions | ||
| open import foundation.dependent-pair-types | ||
| open import foundation.inhabited-types | ||
| open import foundation.iterated-dependent-product-types | ||
| open import foundation.logical-equivalences | ||
| open import foundation.propositional-truncations | ||
| open import foundation.truncated-maps | ||
| open import foundation.truncation-levels | ||
| open import foundation.universe-levels | ||
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| open import foundation-core.contractible-maps | ||
| open import foundation-core.contractible-types | ||
| open import foundation-core.equivalences | ||
| open import foundation-core.propositions | ||
| open import foundation-core.retractions | ||
| open import foundation-core.truncated-types | ||
| ``` | ||
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| </details> | ||
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| ## Idea | ||
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| In Homotopy Type Theory, there are multiple nonequivalent ways to state that a | ||
| map is "injective" that are more or less informed by the homotopy structures of | ||
| its domain and codomain. A | ||
| {{#concept "mere path-cosplit map" Disambiguation="between types" Agda=is-mere-path-cosplit}} | ||
| is one such notion, lying somewhere between | ||
| [embeddings](foundation-core.embeddings.md) and | ||
| [injective maps](foundation-core.injective-maps.md). In fact, given an integer | ||
| `k ≥ -2`, if we understand `k`-injective map to mean the `k+2`-dimensional | ||
| [action on identifications](foundation.action-on-higher-identifications-functions.md) | ||
| has a converse map, then we have proper inclusions | ||
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| ```text | ||
| k-injective maps ⊃ k-path-cosplit maps ⊃ k-truncated maps. | ||
| ``` | ||
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| While `k`-truncatedness answers the question: | ||
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| > At which dimension is the action on higher identifications of a function | ||
| > always an [equivalence](foundation-core.equivalences.md)? | ||
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| Mere `k`-path-cosplitting instead answers the question: | ||
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| > At which dimension is the action merely a | ||
| > [retract](foundation-core.retracts-of-types.md)? | ||
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| Thus a _merely `-2`-path-cosplit map_ is a map that merely is a retract. A | ||
| _merely `k+1`-path-cosplit map_ is a map whose | ||
| [action on identifications](foundation.action-on-identifications-functions.md) | ||
| is merely `k`-path-cosplit. | ||
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| We show that mere `k`-path-cosplittness coincides with `k`-truncatedness when | ||
| the codomain is `k`-truncated, but more generally mere `k`-path-cosplitting may | ||
| only induce mere retracts on higher homotopy groups. | ||
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| ## Definitions | ||
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| ```agda | ||
| is-mere-path-cosplit : | ||
| {l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} → (A → B) → UU (l1 ⊔ l2) | ||
| is-mere-path-cosplit neg-two-𝕋 f = is-inhabited (retraction f) | ||
| is-mere-path-cosplit (succ-𝕋 k) {A} f = | ||
| (x y : A) → is-mere-path-cosplit k (ap f {x} {y}) | ||
| ``` | ||
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| ## Properties | ||
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| ### Being merely path-cosplit is a property | ||
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| ```agda | ||
| is-prop-is-mere-path-cosplit : | ||
| {l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} (f : A → B) → | ||
| is-prop (is-mere-path-cosplit k f) | ||
| is-prop-is-mere-path-cosplit neg-two-𝕋 f = | ||
| is-property-is-inhabited (retraction f) | ||
| is-prop-is-mere-path-cosplit (succ-𝕋 k) f = | ||
| is-prop-Π (λ x → is-prop-Π λ y → is-prop-is-mere-path-cosplit k (ap f)) | ||
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| is-mere-path-cosplit-Prop : | ||
| {l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} → (A → B) → Prop (l1 ⊔ l2) | ||
| is-mere-path-cosplit-Prop k f = | ||
| (is-mere-path-cosplit k f , is-prop-is-mere-path-cosplit k f) | ||
| ``` | ||
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| ### If a map is `k`-truncated then it is merely `k`-path-cosplit | ||
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| ```agda | ||
| is-mere-path-cosplit-is-trunc : | ||
| {l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} {f : A → B} → | ||
| is-trunc-map k f → is-mere-path-cosplit k f | ||
| is-mere-path-cosplit-is-trunc neg-two-𝕋 is-trunc-f = | ||
| unit-trunc-Prop (retraction-is-contr-map is-trunc-f) | ||
| is-mere-path-cosplit-is-trunc (succ-𝕋 k) {f = f} is-trunc-f x y = | ||
| is-mere-path-cosplit-is-trunc k | ||
| ( is-trunc-map-ap-is-trunc-map k f is-trunc-f x y) | ||
| ``` | ||
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| ### If a map is `k`-path-cosplit then it is merely `k+1`-path-cosplit | ||
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| ```agda | ||
| is-mere-path-cosplit-succ-is-mere-path-cosplit : | ||
| {l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} {f : A → B} → | ||
| is-mere-path-cosplit k f → is-mere-path-cosplit (succ-𝕋 k) f | ||
| is-mere-path-cosplit-succ-is-mere-path-cosplit | ||
| neg-two-𝕋 {f = f} is-cosplit-f x y = | ||
| rec-trunc-Prop | ||
| ( is-mere-path-cosplit-Prop neg-two-𝕋 (ap f)) | ||
| ( λ r → unit-trunc-Prop (retraction-ap f r)) | ||
| ( is-cosplit-f) | ||
| is-mere-path-cosplit-succ-is-mere-path-cosplit (succ-𝕋 k) is-cosplit-f x y = | ||
| is-mere-path-cosplit-succ-is-mere-path-cosplit k (is-cosplit-f x y) | ||
| ``` | ||
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| ### If a type maps into a `k`-truncted type via a merely `k`-path-cosplit map then it is `k`-truncated | ||
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| ```agda | ||
| is-trunc-domain-is-mere-path-cosplit-is-trunc-codomain : | ||
| {l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} {f : A → B} → | ||
| is-trunc k B → is-mere-path-cosplit k f → is-trunc k A | ||
| is-trunc-domain-is-mere-path-cosplit-is-trunc-codomain neg-two-𝕋 | ||
| {A} {B} {f} is-trunc-B = | ||
| rec-trunc-Prop | ||
| ( is-trunc-Prop neg-two-𝕋 A) | ||
| ( λ r → is-trunc-retract-of (f , r) is-trunc-B) | ||
| is-trunc-domain-is-mere-path-cosplit-is-trunc-codomain | ||
| (succ-𝕋 k) {A} {B} {f} is-trunc-B is-cosplit-f x y = | ||
| is-trunc-domain-is-mere-path-cosplit-is-trunc-codomain k | ||
| ( is-trunc-B (f x) (f y)) | ||
| ( is-cosplit-f x y) | ||
| ``` | ||
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| This result generalizes the following statements: | ||
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| - A type that injects into a set is a set. | ||
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| - A type that embeds into a `k+1`-truncated type is `k+1`-truncated. | ||
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| - A type that maps into a `k`-truncated type via a `k`-truncated map is | ||
| `k`-truncated. | ||
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| ### If the codomain of a mere `k`-path-cosplit map is `k`-truncated then the map is `k`-truncated | ||
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| ```agda | ||
| is-trunc-map-is-mere-path-cosplit-is-trunc-codomain : | ||
| {l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} {f : A → B} → | ||
| is-trunc k B → is-mere-path-cosplit k f → is-trunc-map k f | ||
| is-trunc-map-is-mere-path-cosplit-is-trunc-codomain k is-trunc-B is-cosplit-f = | ||
| is-trunc-map-is-trunc-domain-codomain k | ||
| ( is-trunc-domain-is-mere-path-cosplit-is-trunc-codomain k | ||
| ( is-trunc-B) | ||
| ( is-cosplit-f)) | ||
| ( is-trunc-B) | ||
| ``` | ||
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| ## See also | ||
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| - [Path-cosplit maps](foundation.path-cosplit-maps.md) | ||
| - [Path-split maps](foundation.path-cosplit-maps.md) | ||
| - [Injective maps](foundation-core.injective-maps.md) | ||
| - [Truncated maps](foundation-core.truncated-maps.md) | ||
| - [Embeddings](foundation-core.embeddings.md) |
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