SCT.VolumeI.Chapter03.Section03.LocalizationGroupoids

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# Maps from a localization into a groupoid

Every functor into a groupoid inverts the collection. Compose the two
proved equivalences, retaining the comparison with ordinary restriction.

```agda
{-# OPTIONS --safe --without-K #-}
open import Agda.Primitive using (Level; _⊔_)
open import SCT.VolumeI.Chapter01.Theory using (Theory)
import SCT.VolumeI.Chapter01.Section04.MappingAnimae as Mapping
import SCT.VolumeI.Chapter01.Section07.FunctorCategories as Categories
import SCT.VolumeI.Chapter01.Section06.PullbackLaws as Pullbacks
import SCT.VolumeI.Chapter02.Section01.WalkingMorphism as Walking
import SCT.VolumeI.Chapter02.Section01.IntervalEndpoints as Endpoints
import SCT.VolumeI.Chapter02.Section02.SegalAxiom as Segal
import SCT.VolumeI.Chapter02.Section01.CommutativeSquareAxiom as Squares
import SCT.VolumeI.Chapter02.Section03.RezkAxiom as Rezk

module SCT.VolumeI.Chapter03.Section03.LocalizationGroupoids
  {c m a : Level} (𝒯 : Theory c m a) (M : Mapping.MappingAnimae 𝒯)
  (ℱ : Categories.FunctorCategories 𝒯 M) (P : Pullbacks.PullbackStructure 𝒯)
  (I : Walking.WalkingMorphism 𝒯) (E : Endpoints.IntervalEndpoints 𝒯 M ℱ P I)
  (S : Segal.SegalAxiom 𝒯 M ℱ P I E)
  (Q : Squares.CommutativeSquareAxiom 𝒯 M ℱ P I E)
  (R : Rezk.RezkAxiom 𝒯 M ℱ P I E) where

open import SCT.VolumeI.Chapter01.Section05.Setup 𝒯 M
open import SCT.VolumeI.Chapter01.Section07.Functoriality 𝒯 M ℱ using (funPre)
open import SCT.VolumeI.Chapter02.Section04.Groupoids 𝒯 M ℱ P I E R using (IsGroupoid)
open import SCT.VolumeI.Chapter03.Section01.MorphismCollections 𝒯 M P I using (MorphismCollection)
open import SCT.VolumeI.Chapter03.Section01.SubcategoryAxiom 𝒯 M ℱ P I E S using (SubcategoryAxiom)
open import SCT.VolumeI.Chapter03.Section03.Localizations 𝒯 M ℱ P I E S Q R using (module WithSubcategories)
open import SCT.VolumeI.Chapter03.Section03.MappingCalculus.LocalizationRestriction 𝒯 M ℱ P I E S Q R using (module RestrictionAlong)
open import SCT.VolumeI.Chapter03.Section03.InvertingFunctors 𝒯 M ℱ P I E S Q using (module Inverting)
open import SCT.VolumeI.Chapter03.Section03.MappingCalculus.InvertingGroupoids 𝒯 M ℱ P I E S Q R using (inverting-functors-into-groupoid)

module IntoGroupoids (L : SubcategoryAxiom) {C T : CAT} (W : MorphismCollection C)
  (l : MAP C T) (localization : WithSubcategories.IsLocalization L W l) where
  open WithSubcategories L
  module At (D : CAT) (groupoid : IsGroupoid D) where
    module Restriction = RestrictionAlong L W l (IsLocalization.inverts localization) D
      using (functor; comparison)
    isEquiv : IsEquiv (funPre {D = D} l)
    isEquiv = equiv-transport Restriction.comparison
      (equiv-compose Restriction.functor (Inverting.Category.functorInclusion W D L)
        (IsLocalization.universal localization D)
        (inverting-functors-into-groupoid L W D groupoid))
```