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A map is an embedding if and only if it has contractible fibers at values #858

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Oct 18, 2023
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A map is an embedding if and only if it has contractible fibers at values #858

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A map is an embedding if and only if it has contractible fibers at va…
tomdjong Oct 18, 2023
891f12d
Minor improvements
tomdjong Oct 18, 2023
d3125b2
Update src/foundation/embeddings.lagda.md
EgbertRijke Oct 18, 2023
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37 changes: 37 additions & 0 deletions src/foundation/embeddings.lagda.md
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Expand Up @@ -14,13 +14,15 @@ open import foundation.commuting-squares-of-maps
open import foundation.cones-over-cospans
open import foundation.dependent-pair-types
open import foundation.equivalences
open import foundation.fibers-of-maps
open import foundation.functoriality-cartesian-product-types
open import foundation.fundamental-theorem-of-identity-types
open import foundation.identity-types
open import foundation.truncated-maps
open import foundation.universe-levels

open import foundation-core.cartesian-product-types
open import foundation-core.contractible-types
open import foundation-core.function-types
open import foundation-core.functoriality-dependent-pair-types
open import foundation-core.homotopies
Expand Down Expand Up @@ -351,3 +353,38 @@ module _
( is-equiv-map-inv-is-equiv L)
( M)
```

### A map is an embedding if and only if it has contractible fibers at values

```agda
module _ {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B)
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where

is-emb-is-contr-fibers-values' :
((a : A) → is-contr (fiber' f (f a))) → is-emb f
is-emb-is-contr-fibers-values' c a =
fundamental-theorem-id (c a) (λ x → ap f {a} {x})

is-emb-is-contr-fibers-values :
((a : A) → is-contr (fiber f (f a))) → is-emb f
is-emb-is-contr-fibers-values c =
is-emb-is-contr-fibers-values'
( λ a →
is-contr-equiv'
( fiber f (f a))
( equiv-fiber f (f a))
( c a))

is-contr-fibers-values-is-emb' :
is-emb f → ((a : A) → is-contr (fiber' f (f a)))
is-contr-fibers-values-is-emb' e a =
fundamental-theorem-id' (λ x → ap f {a} {x}) (e a)

is-contr-fibers-values-is-emb :
is-emb f → ((a : A) → is-contr (fiber f (f a)))
is-contr-fibers-values-is-emb e a =
is-contr-equiv
( fiber' f (f a))
( equiv-fiber f (f a))
( is-contr-fibers-values-is-emb' e a)
```