1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
|
{-# OPTIONS --without-K --rewriting #-}
module Univalence-to-funext where
open import lib.Base
open import lib.Equivalence
open import lib.Function
open import lib.PathGroupoid
open import lib.PathFunctor
open import lib.NType
--============================================================================--
-- Generic paths in Σ types
module _ {i j} {A : Type i} {B : A → Type j} where
Σ-path-pair : (v w : Σ A B) → Type (lmax i j)
Σ-path-pair v w = Σ (fst v == fst w)
(λ p → transport B p (snd v) == snd w)
pair→Σ= : {v w : Σ A B} → Σ-path-pair v w → v == w
pair→Σ= (idp , idp) = idp
Σ=→pair : {v w : Σ A B} → v == w → Σ-path-pair v w
Σ=→pair idp = (idp , idp)
Σ=→pair→Σ= : {v w : Σ A B} (p : v == w) → pair→Σ= (Σ=→pair p) == p
Σ=→pair→Σ= idp = idp
pair→Σ=→pair : {v w : Σ A B} (z : Σ-path-pair v w) →
Σ=→pair (pair→Σ= z) == z
pair→Σ=→pair (idp , idp) = idp
Σ=≃pair : {v w : Σ A B} → (v == w) ≃ (Σ-path-pair v w)
Σ=≃pair = equiv Σ=→pair pair→Σ= pair→Σ=→pair Σ=→pair→Σ=
--============================================================================--
-- Univalence
idtoeqv : ∀ {i} {A B : Type i} → (A == B) → (A ≃ B)
idtoeqv {A = A} idp = ide A
postulate
ua : ∀ {i} {A B : Type i} → (A ≃ B) → (A == B)
idtoeqv-ua-β : ∀ {i} {A B : Type i} (e : A ≃ B) → idtoeqv (ua e) == e
eqv-post∘ : ∀ {i} {A B : Type i} {C : Type i} (p : A == B) →
fst (idtoeqv (ap (λ T → (C → T)) p)) == _∘_ (fst (idtoeqv p))
eqv-post∘ idp = idp
--============================================================================--
-- Happly and lemmas
module _ {i j} {A : Type i} {B : A → Type j} {f g : Π A B} where
idh : f ∼ f
idh a = idp
happly : f == g → f ∼ g
happly p a = ap (λ k → k a) p
tr-happly-lemma : (p : f == g) →
transport (λ k → f ∼ k) p (idh) == happly p
tr-happly-lemma idp = idp
fib-happly-lemma : (h : f ∼ g) → ((f , idh) == (g , h)) ≃ hfiber happly h
fib-happly-lemma h = (equiv from to α β) ∘e Σ=≃pair
where
to : hfiber happly h → Σ-path-pair (f , idh) (g , h)
to (p , q) = (p , tr-happly-lemma p ∙ q)
from : Σ-path-pair (f , idh) (g , h) → hfiber happly h
from (p , q) = (p , ! (tr-happly-lemma p) ∙ q)
β : (b : Σ-path-pair (f , idh) (g , h)) → to (from b) == b
β (p , q) = pair→Σ= (idp , ! (∙-assoc (tr-happly-lemma p)
(! (tr-happly-lemma p)) q) ∙
!-inv-r (tr-happly-lemma p) ∙2 idp)
α : (a : hfiber happly h) → from (to a) == a
α (p , q) = pair→Σ= (idp ,
! (∙-assoc (! (tr-happly-lemma p))
(tr-happly-lemma p) q) ∙
!-inv-l (tr-happly-lemma p) ∙2 idp)
--============================================================================--
-- Equivalences and contractibility
contr-retract : ∀ {i} {A B : Type i}
(r : A → B) (s : B → A) →
(r ∘ s ∼ (idf B)) →
is-contr A → is-contr B
contr-retract r s h p = has-level-in
((r (fst c)) , λ y → ap r (snd c (s y)) ∙ h y)
where c = has-level-apply p
module _ {i j} {A : Type i} {B : Type j} where
is-contr-map : (f : A → B) → Type (lmax i j)
is-contr-map f = (y : B) → is-contr (hfiber f y)
tr-ap-lemma : (f : A → B) {a' a : A} {b : B}
(p : a' == a) (q : f a' == b) →
transport (λ x → f x == b) p q == ! (ap f p) ∙ q
tr-ap-lemma f idp idp = idp
nat-lemma : {k l : B → B} {a b : B} (p : a == b )(h : k ∼ l) →
ap k p ∙ h b == h a ∙ ap l p
nat-lemma {a = a} idp h = ! (∙-unit-r (h a))
contr-map-is-equiv : {f : A → B} → (is-contr-map f → is-equiv f)
contr-map-is-equiv {f = f} c = snd (equiv f g α β)
where
g : B → A
g b = fst (fst (has-level-apply (c b)))
α : f ∘ g ∼ (idf B)
α b = snd (fst (has-level-apply (c b)))
β : g ∘ f ∼ (idf A)
β a = fst (Σ=→pair (snd (has-level-apply (c (f a))) (a , idp)))
equiv-is-contr-map : {f : A → B} → (is-equiv f → is-contr-map f)
equiv-is-contr-map {f = f} e b =
has-level-in (hf , λ y → pair→Σ= (fc y , sc y))
where
g : B → A
g = is-equiv.g e
α : g ∘ f ∼ (idf A)
α = is-equiv.g-f e
β : f ∘ g ∼ (idf B)
β = is-equiv.f-g e
hf : hfiber f b
hf = ((g b) , (β b))
fc : (y : hfiber f b) → (g b) == fst y
fc (a , p) = ap (g) (! p) ∙ α a
sc : (y : hfiber f b) →
transport (λ x → f x == b) (fc (fst y , snd y)) (β b) == snd y
sc (a , p) =
transport (λ x → f x == b) (fc (a , p)) (β b)
=⟨ tr-ap-lemma f (fc (a , p)) (β b) ⟩
! (ap f (fc (a , p))) ∙ (β b)
=⟨ ap ! (ap-∙ f (ap g (! p)) (α a)) ∙2 idp ⟩
! (ap f (ap g (! p)) ∙ ap f (α a)) ∙ (β b)
=⟨ ap ! (idp {a = ap f (ap g (! p))} ∙2 is-equiv.adj e a) ∙2
idp {a = β b} ⟩
! (ap f (ap g (! p)) ∙ (β (f a))) ∙ (β b)
=⟨ ap ! (∘-ap f g (! p) ∙2 idp) ∙2 idp ⟩
! (ap (f ∘ g) (! p) ∙ (β (f a))) ∙ (β b)
=⟨ !-∙ (ap (f ∘ g) (! p)) (β (f a)) ∙2 idp ⟩
(! (β (f a)) ∙ ! (ap (λ x → f (g x)) (! p))) ∙ (β b)
=⟨ (idp {a = ! (β (f a))} ∙2 !-ap (f ∘ g) (! p)) ∙2 idp {a = β b} ⟩
(! (β (f a)) ∙ (ap (f ∘ g) (! (! p)))) ∙ (β b)
=⟨ (idp {a = ! (β (f a))} ∙2 ap (ap (f ∘ g)) (!-! p)) ∙2
idp {a = β b} ⟩
(! (β (f a)) ∙ (ap (f ∘ g) p)) ∙ (β b)
=⟨ ∙-assoc (! (β (f a))) (ap (f ∘ g) p) (β b) ⟩
! (β (f a)) ∙ (ap (f ∘ g) p ∙ (β b))
=⟨ idp {a = ! (β (f a))} ∙2
(nat-lemma p β ∙ idp {a = β (f a)} ∙2 ap-idf p) ⟩
! (β (f a)) ∙ (β (f a) ∙ p)
=⟨ ! (∙-assoc (! (β (f a))) (β (f a)) p) ⟩
(! (β (f a)) ∙ β (f a)) ∙ p
=⟨ !-inv-l (β (f a)) ∙2 idp ⟩
p
=∎
--============================================================================--
-- Univalence implies funext
module _ {i} {A : Type i} {B : A → Type i} {f : Π A B} where
image : A → Type i
image a = Σ (B a) (λ b → f a == b)
graphType : Type i
graphType = Σ A image
prA : graphType → A
prA = fst
prA-is-equiv : graphType ≃ A
prA-is-equiv = equiv prA (λ x → (x , f x , idp)) (λ b → idp) β
where
tr-post-concat : ∀ {i} {X : Type i} {x y z : X} {p : x == y}
(q : z == x) → transport (_==_ z) p q == q ∙ p
tr-post-concat {p = idp} idp = idp
β : (a : graphType) → (fst a , f (fst a), idp) == a
β (a , b , p) = pair→Σ= (idp , (pair→Σ= (p , (tr-post-concat idp))))
fibreOverId : Type i
fibreOverId = hfiber (_∘_ prA) (idf A)
fibreOverId-is-contr : is-contr fibreOverId
fibreOverId-is-contr = prA∘-is-equiv (idf A)
where
p : graphType == A
p = ua prA-is-equiv
pβ : fst (idtoeqv p) == prA
pβ = ap fst (idtoeqv-ua-β prA-is-equiv)
prA∘-is-equiv : is-contr-map (λ g → prA ∘ g)
prA∘-is-equiv = transport is-contr-map
(eqv-post∘ p ∙ ap (λ x → _∘_ x) pβ)
(equiv-is-contr-map
(snd (idtoeqv (ap (λ T → (A → T)) p))))
homotopyType : Type i
homotopyType = Σ (Π A B) (λ g → f ∼ g)
-- We crucially make use of the η-rule for functions here
s : homotopyType → fibreOverId
s (g , h) = ((λ a → (a , g a , h a)) , idp)
r : fibreOverId → homotopyType
r (func , p) = fst ∘ img , snd ∘ img
where
img : (a : A) → image a
img a = transport image (happly p a) (snd (func a))
-- and here
var-funext : is-prop homotopyType
var-funext = contr-is-prop (contr-retract r s (λ _ → idp)
fibreOverId-is-contr)
module _ {i} {A : Type i} {B : A → Type i} {f g : Π A B} where
pre-funext : (h : f ∼ g) → is-contr ((f , idh {g = g}) == (g , h))
pre-funext h = has-level-apply var-funext (f , idh {g = g}) (g , h)
funext : is-equiv (happly {f = f}{g = g})
funext = contr-map-is-equiv
λ h → equiv-preserves-level
(fib-happly-lemma h)
⦃ pre-funext h ⦄
|
