# Path-cosplit maps

Content created by Fredrik Bakke.

Created on 2024-06-05.

module foundation.path-cosplit-maps where

Imports
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.mere-path-cosplit-maps
open import foundation.propositional-truncations
open import foundation.truncated-maps
open import foundation.truncation-levels
open import foundation.universe-levels

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


## Idea

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 path-cosplit map is one such notion, lying somewhere between embeddings and injective maps. In fact, given an integer k ≥ -2, if we understand k-injective map to mean the k+2-dimensional action on identifications has a converse map, then we have proper inclusions

  k-injective maps ⊃ k-path-cosplit maps ⊃ k-truncated maps.


While k-truncatedness answers the question:

At which dimension is the action on higher identifications of a function always an equivalence?

Being k-path-cosplitting instead answers the question:

At which dimension is the action a retract?

Thus a -2-path-cosplit map is a map equipped with a retraction. A k+1-path-cosplit map is a map whose action on identifications is k-path-cosplit.

We show that k-path-cosplittness coincides with k-truncatedness when the codomain is k-truncated, but more generally k-path-cosplitting may only induce retracts on higher homotopy groups.

## Definitions

is-path-cosplit :
{l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} → (A → B) → UU (l1 ⊔ l2)
is-path-cosplit neg-two-𝕋 f = retraction f
is-path-cosplit (succ-𝕋 k) {A} f = (x y : A) → is-path-cosplit k (ap f {x} {y})


## Properties

### If a map is k-path-cosplit it is merely k-path-cosplit

is-mere-path-cosplit-is-path-cosplit :
{l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} {f : A → B} →
is-path-cosplit k f → is-mere-path-cosplit k f
is-mere-path-cosplit-is-path-cosplit neg-two-𝕋 is-cosplit-f =
unit-trunc-Prop is-cosplit-f
is-mere-path-cosplit-is-path-cosplit (succ-𝕋 k) is-cosplit-f x y =
is-mere-path-cosplit-is-path-cosplit k (is-cosplit-f x y)


### If a map is k-truncated then it is k-path-cosplit

is-path-cosplit-is-trunc :
{l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} {f : A → B} →
is-trunc-map k f → is-path-cosplit k f
is-path-cosplit-is-trunc neg-two-𝕋 is-trunc-f =
retraction-is-contr-map is-trunc-f
is-path-cosplit-is-trunc (succ-𝕋 k) {f = f} is-trunc-f x y =
is-path-cosplit-is-trunc k (is-trunc-map-ap-is-trunc-map k f is-trunc-f x y)


### If a map is k-path-cosplit then it is k+1-path-cosplit

is-path-cosplit-succ-is-path-cosplit :
{l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} {f : A → B} →
is-path-cosplit k f → is-path-cosplit (succ-𝕋 k) f
is-path-cosplit-succ-is-path-cosplit neg-two-𝕋 {f = f} is-cosplit-f x y =
retraction-ap f is-cosplit-f
is-path-cosplit-succ-is-path-cosplit (succ-𝕋 k) is-cosplit-f x y =
is-path-cosplit-succ-is-path-cosplit k (is-cosplit-f x y)


### If a type maps into a k-truncted type via a k-path-cosplit map then it is k-truncated

is-trunc-domain-is-path-cosplit-is-trunc-codomain :
{l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} {f : A → B} →
is-trunc k B → is-path-cosplit k f → is-trunc k A
is-trunc-domain-is-path-cosplit-is-trunc-codomain neg-two-𝕋
{A} {B} {f} is-trunc-B is-cosplit-f =
is-trunc-retract-of (f , is-cosplit-f) is-trunc-B
is-trunc-domain-is-path-cosplit-is-trunc-codomain
(succ-𝕋 k) {A} {B} {f} is-trunc-B is-cosplit-f x y =
is-trunc-domain-is-path-cosplit-is-trunc-codomain k
( is-trunc-B (f x) (f y))
( is-cosplit-f x y)


This result generalizes the following statements:

• A type that injects into a set is a set.

• A type that embeds into a k+1-truncated type is k+1-truncated.

• A type that maps into a k-truncated type via a k-truncated map is k-truncated.

### If the codomain of a k-path-cosplit map is k-truncated then the map is k-truncated

is-trunc-map-is-path-cosplit-is-trunc-codomain :
{l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} {f : A → B} →
is-trunc k B → is-path-cosplit k f → is-trunc-map k f
is-trunc-map-is-path-cosplit-is-trunc-codomain k is-trunc-B is-cosplit-f =
is-trunc-map-is-trunc-domain-codomain k
( is-trunc-domain-is-path-cosplit-is-trunc-codomain k
( is-trunc-B)
( is-cosplit-f))
( is-trunc-B)