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{-# OPTIONS --without-K #-}
open import Base
module Homotopy.ReflectiveSubCategory {m} where
record rsc : Set (suc m) where
constructor Rsc
field
P : Set m Set m
P-is-prop : (A : Set m) is-prop (P A)
reflect : Set m Set m
to-reflect : (A : Set m) A reflect A
reflect-in-P : (A : Set m) P (reflect A)
adjoint : (A B : Set m) ⦃ B-in-P : P B ⦄
is-equiv (λ (f : reflect A B) f ◯ to-reflect A)
open rsc ⦃...⦄
adjoint-equiv : ⦃ rs : rsc ⦄ (A B : Set m) (B-in-P : P B)
(reflect A B) ≃ (A B)
adjoint-equiv A B B-in-P = (_ , adjoint A B)
map-reflect : ⦃ rs : rsc ⦄ {X Y : Set m} (f : X Y) (reflect X reflect Y)
map-reflect {X} {Y} f =
(adjoint-equiv X (reflect Y) (reflect-in-P Y)) ⁻¹ $ (to-reflect Y ◯ f)
reduce-map-reflect : ⦃ rs : rsc ⦄ {X Y : Set m} (f : X Y)
(map-reflect f ◯ to-reflect X ≡ to-reflect Y ◯ f)
reduce-map-reflect {X} {Y} f =
inverse-right-inverse (adjoint-equiv X (reflect Y) (reflect-in-P Y))
(to-reflect Y ◯ f)
compose-map-reflect : ⦃ rs : rsc ⦄ {X Y Z : Set m} (g : Y Z) (f : X Y)
map-reflect g ◯ map-reflect f ≡ map-reflect (g ◯ f)
compose-map-reflect {X} {Y} {Z} g f =
equiv-is-inj (adjoint-equiv X (reflect Z) (reflect-in-P Z)) _ _
lemma where
lemma : map-reflect g ◯ (map-reflect f ◯ to-reflect X)
≡ map-reflect (g ◯ f) ◯ to-reflect X
lemma = map (λ u map-reflect g ◯ u) (reduce-map-reflect f)
∘ (map (λ u u ◯ f) (reduce-map-reflect g)
∘ ! (reduce-map-reflect (g ◯ f)))
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