# Dependent epimorphisms with respect to truncated types ```agda module foundation.dependent-epimorphisms-with-respect-to-truncated-types where ``` <details><summary>Imports</summary> ```agda open import foundation.epimorphisms-with-respect-to-truncated-types open import foundation.universe-levels open import foundation-core.embeddings open import foundation-core.precomposition-dependent-functions open import foundation-core.truncated-types open import foundation-core.truncation-levels ``` </details> ## Idea A **dependent `k`-epimorphism** is a map `f : A → B` such that the [precomposition function](foundation.precomposition-dependent-functions.md) ```text - ∘ f : ((b : B) → C b) → ((a : A) → C (f a)) ``` is an [embedding](foundation-core.embeddings.md) for every family `C` of [`k`-types](foundation.truncated-types.md) over `B`. Clearly, every dependent `k`-epimorphism is a [`k`-epimorphism](foundation.epimorphisms-with-respect-to-truncated-types.md). The converse is also true, i.e., every `k`-epimorphism is a dependent `k`-epimorphism. Therefore it follows that a map `f : A → B` is [`k`-acyclic](synthetic-homotopy-theory.truncated-acyclic-maps.md) if and only if it is a `k`-epimorphism, if and only if it is a dependent `k`-epimorphism. ## Definitions ### The predicate of being a dependent `k`-epimorphism ```agda module _ {l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} where is-dependent-epimorphism-Truncated-Type : (A → B) → UUω is-dependent-epimorphism-Truncated-Type f = {l : Level} (C : B → Truncated-Type l k) → is-emb (precomp-Π f (λ b → type-Truncated-Type (C b))) ``` ## Properties ### Every dependent `k`-epimorphism is a `k`-epimorphism ```agda module _ {l1 l2 : Level} {k : 𝕋} {A : UU l1} {B : UU l2} (f : A → B) where is-epimorphism-is-dependent-epimorphism-Truncated-Type : is-dependent-epimorphism-Truncated-Type k f → is-epimorphism-Truncated-Type k f is-epimorphism-is-dependent-epimorphism-Truncated-Type e X = e (λ _ → X) ``` The converse of the above, that every `k`-epimorphism is a dependent `k`-epimorphism, can be found in the file on [`k`-acyclic maps](synthetic-homotopy-theory.truncated-acyclic-maps.md). ## See also - [`k`-acyclic maps](synthetic-homotopy-theory.truncated-acyclic-maps.md) - [Epimorphisms](foundation.epimorphisms.md) - [Epimorphisms with respect to sets](foundation.epimorphisms-with-respect-to-sets.md) - [Epimorphisms with respect to truncated types](foundation.epimorphisms-with-respect-to-truncated-types.md)