# Equality in the fibers of a map

Content created by Egbert Rijke, Fredrik Bakke, Jonathan Prieto-Cubides and Raymond Baker.

Created on 2022-01-26.

module foundation.equality-fibers-of-maps where

Imports
open import foundation.action-on-identifications-functions
open import foundation.dependent-pair-types
open import foundation.identity-types
open import foundation.transport-along-identifications
open import foundation.universe-levels

open import foundation-core.equality-dependent-pair-types
open import foundation-core.equivalences
open import foundation-core.families-of-equivalences
open import foundation-core.fibers-of-maps
open import foundation-core.function-types
open import foundation-core.functoriality-dependent-pair-types
open import foundation-core.homotopies


## Idea

In the file foundation-core.fibers-of-maps we already gave one characterization of the identity type of fiber f b, for an arbitrary map f : A → B. Here we give a second characterization, using the fibers of the action on identifications of f.

## Theorem

For any map f : A → B any b : B and any x y : fiber f b, there is an equivalence

(x ＝ y) ≃ fiber (ap f) (pr2 x ∙ inv (pr2 y))


### Proof

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

fiber-ap-eq-fiber-fiberwise :
(s t : fiber f b) (p : pr1 s ＝ pr1 t) →
tr (λ (a : A) → f a ＝ b) p (pr2 s) ＝ pr2 t →
ap f p ＝ pr2 s ∙ inv (pr2 t)
fiber-ap-eq-fiber-fiberwise (.x' , p) (x' , refl) refl =
inv ∘ concat right-unit refl

abstract
is-fiberwise-equiv-fiber-ap-eq-fiber-fiberwise :
(s t : fiber f b) → is-fiberwise-equiv (fiber-ap-eq-fiber-fiberwise s t)
is-fiberwise-equiv-fiber-ap-eq-fiber-fiberwise (x , y) (.x , refl) refl =
is-equiv-comp
( inv)
( concat right-unit refl)
( is-equiv-concat right-unit refl)
( is-equiv-inv (y ∙ refl) refl)

fiber-ap-eq-fiber :
(s t : fiber f b) → s ＝ t →
fiber (ap f {x = pr1 s} {y = pr1 t}) (pr2 s ∙ inv (pr2 t))
pr1 (fiber-ap-eq-fiber s .s refl) = refl
pr2 (fiber-ap-eq-fiber s .s refl) = inv (right-inv (pr2 s))

triangle-fiber-ap-eq-fiber :
(s t : fiber f b) →
fiber-ap-eq-fiber s t ~
tot (fiber-ap-eq-fiber-fiberwise s t) ∘ pair-eq-Σ {s = s} {t}
triangle-fiber-ap-eq-fiber (x , refl) .(x , refl) refl = refl

abstract
is-equiv-fiber-ap-eq-fiber :
(s t : fiber f b) → is-equiv (fiber-ap-eq-fiber s t)
is-equiv-fiber-ap-eq-fiber s t =
is-equiv-left-map-triangle
( fiber-ap-eq-fiber s t)
( tot (fiber-ap-eq-fiber-fiberwise s t))
( pair-eq-Σ {s = s} {t})
( triangle-fiber-ap-eq-fiber s t)
( is-equiv-pair-eq-Σ s t)
( is-equiv-tot-is-fiberwise-equiv
( is-fiberwise-equiv-fiber-ap-eq-fiber-fiberwise s t))

equiv-fiber-ap-eq-fiber :
(s t : fiber f b) →
(s ＝ t) ≃ fiber (ap f {x = pr1 s} {y = pr1 t}) (pr2 s ∙ inv (pr2 t))
pr1 (equiv-fiber-ap-eq-fiber s t) = fiber-ap-eq-fiber s t
pr2 (equiv-fiber-ap-eq-fiber s t) = is-equiv-fiber-ap-eq-fiber s t

map-inv-fiber-ap-eq-fiber :
(s t : fiber f b) →
fiber (ap f {x = pr1 s} {y = pr1 t}) (pr2 s ∙ inv (pr2 t)) →
s ＝ t
map-inv-fiber-ap-eq-fiber (x , refl) (.x , p) (refl , u) =
eq-pair-eq-fiber (ap inv u ∙ inv-inv p)

ap-pr1-map-inv-fiber-ap-eq-fiber :
(s t : fiber f b) →
(v : fiber (ap f {x = pr1 s} {y = pr1 t}) (pr2 s ∙ inv (pr2 t))) →
ap pr1 (map-inv-fiber-ap-eq-fiber s t v) ＝ pr1 v
ap-pr1-map-inv-fiber-ap-eq-fiber (x , refl) (.x , p) (refl , u) =
ap-pr1-eq-pair-eq-fiber (ap inv u ∙ inv-inv p)

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

eq-fiber-fiber-ap :
(q : f x ＝ f y) → (x , q) ＝ (y , refl) → fiber (ap f {x} {y}) q
eq-fiber-fiber-ap q =
tr (fiber (ap f)) right-unit ∘ fiber-ap-eq-fiber f (x , q) (y , refl)

abstract
is-equiv-eq-fiber-fiber-ap :
(q : (f x) ＝ f y) → is-equiv (eq-fiber-fiber-ap q)
is-equiv-eq-fiber-fiber-ap q =
is-equiv-comp
( tr (fiber (ap f)) right-unit)
( fiber-ap-eq-fiber f (x , q) (y , refl))
( is-equiv-fiber-ap-eq-fiber f (x , q) (y , refl))
( is-equiv-tr (fiber (ap f)) right-unit)


## Table of files about fibers of maps

The following table lists files that are about fibers of maps as a general concept.

ConceptFile
Fibers of maps (foundation-core)foundation-core.fibers-of-maps
Fibers of maps (foundation)foundation.fibers-of-maps
Equality in the fibers of a mapfoundation.equality-fibers-of-maps
Functoriality of fibers of mapsfoundation.functoriality-fibers-of-maps
Fibers of pointed mapsstructured-types.fibers-of-pointed-maps
Fibers of maps of finite typesunivalent-combinatorics.fibers-of-maps
The universal property of the family of fibers of mapsfoundation.universal-property-family-of-fibers-of-maps