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EgbertRijkeEgbertRijke
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Beyond foundation (#751)
In this PR I pulled changes I made in the `foundation` and `foundation-core` folders in the Beyond finite sets PR #623. The goal is to create something mergeable without having to wait for the other unfinished changes in #623. Let's see what monstrosity I created.
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‎src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md

Lines changed: 11 additions & 8 deletions
Original file line numberDiff line numberDiff line change
@@ -17,7 +17,7 @@ open import finite-group-theory.finite-type-groups
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open import finite-group-theory.sign-homomorphism
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open import finite-group-theory.transpositions
1919
20-
open import foundation.action-on-equivalences-families-over-subuniverses
20+
open import foundation.action-on-equivalences-type-families-over-subuniverses
2121
open import foundation.action-on-identifications-functions
2222
open import foundation.contractible-types
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open import foundation.dependent-pair-types
@@ -29,6 +29,7 @@ open import foundation.negation
2929
open import foundation.propositional-truncations
3030
open import foundation.raising-universe-levels
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open import foundation.transport-along-identifications
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open import foundation.type-theoretic-principle-of-choice
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open import foundation.unit-type
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open import foundation.universe-levels
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@@ -69,7 +70,7 @@ module _
6970
( star)
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( transposition Y))
7172
( map-equiv
72-
( action-equiv-family-on-subuniverse
73+
( action-equiv-family-over-subuniverse
7374
( mere-equiv-Prop (Fin (n +ℕ 2)))
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( orientation-Complete-Undirected-Graph (n +ℕ 2))
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( raise l (Fin (n +ℕ 2)) ,
@@ -110,17 +111,19 @@ module _
110111
preserves-id-equiv-orientation-complete-undirected-graph-equiv
111112
( n +ℕ 2)}
112113
{ y =
113-
( action-equiv-family-on-subuniverse
114+
( action-equiv-family-over-subuniverse
114115
( mere-equiv-Prop (Fin (n +ℕ 2)))
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( orientation-Complete-Undirected-Graph (n +ℕ 2))) ,
116-
( preserves-id-equiv-action-equiv-family-on-subuniverse
117+
( compute-id-equiv-action-equiv-family-over-subuniverse
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( mere-equiv-Prop (Fin (n +ℕ 2)))
118119
( orientation-Complete-Undirected-Graph (n +ℕ 2)))}
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( eq-is-contr
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( is-contr-preserves-id-action-equiv-family-on-subuniverse
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( mere-equiv-Prop (Fin (n +ℕ 2)))
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( orientation-Complete-Undirected-Graph (n +ℕ 2))
123-
( is-set-orientation-Complete-Undirected-Graph (n +ℕ 2)))))
121+
( is-contr-equiv' _
122+
( distributive-Π-Σ)
123+
( is-contr-Π
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( unique-action-equiv-family-over-subuniverse
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( mere-equiv-Prop (Fin (n +ℕ 2)))
126+
( orientation-Complete-Undirected-Graph (n +ℕ 2)))))))
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( not-even-difference-orientation-aut-transposition-count
125128
(n +ℕ 2 , (compute-raise l (Fin (n +ℕ 2)))) (star))
126129

‎src/finite-group-theory/delooping-sign-homomorphism.lagda.md

Lines changed: 50 additions & 26 deletions
Original file line numberDiff line numberDiff line change
@@ -18,8 +18,9 @@ open import finite-group-theory.permutations
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open import finite-group-theory.sign-homomorphism
1919
open import finite-group-theory.transpositions
2020
21-
open import foundation.action-on-equivalences-families-over-subuniverses
21+
open import foundation.action-on-equivalences-type-families-over-subuniverses
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open import foundation.action-on-identifications-functions
23+
open import foundation.binary-transport
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open import foundation.commuting-squares-of-maps
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open import foundation.contractible-types
2526
open import foundation.coproduct-types
@@ -31,6 +32,7 @@ open import foundation.empty-types
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open import foundation.equality-dependent-pair-types
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open import foundation.equivalence-classes
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open import foundation.equivalence-extensionality
35+
open import foundation.equivalence-induction
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open import foundation.equivalence-relations
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open import foundation.equivalences
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open import foundation.function-extensionality
@@ -129,7 +131,7 @@ module _
129131
unit-trunc-Prop (compute-raise-Fin l1 (n +ℕ 2))))
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( quotient-aut-succ-succ-Fin n (transposition Y))
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( map-equiv
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( action-equiv-family-on-subuniverse
134+
( action-equiv-family-over-subuniverse
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( mere-equiv-Prop (Fin (n +ℕ 2)))
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( D (n +ℕ 2))
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( raise l1 (Fin (n +ℕ 2)) ,
@@ -146,9 +148,8 @@ module _
146148
147149
eq-counting-equivalence-class-R :
148150
(n : ℕ) →
149-
Id
150-
( equivalence-class (R (n +ℕ 2) (Fin-UU-Fin l1 (n +ℕ 2))))
151-
( raise (l2 ⊔ lsuc l3) (Fin 2))
151+
equivalence-class (R (n +ℕ 2) (Fin-UU-Fin l1 (n +ℕ 2))) ＝
152+
raise (l2 ⊔ lsuc l3) (Fin 2)
152153
eq-counting-equivalence-class-R n =
153154
eq-equiv
154155
( equivalence-class (R (n +ℕ 2) (Fin-UU-Fin l1 (n +ℕ 2))))
@@ -165,34 +166,57 @@ module _
165166
(n : ℕ) (X X' : UU-Fin l1 n) →
166167
type-UU-Fin n X ≃ type-UU-Fin n X' → D n X ≃ D n X'
167168
invertible-action-D-equiv n =
168-
action-equiv-family-on-subuniverse (mere-equiv-Prop (Fin n)) (D n)
169+
action-equiv-family-over-subuniverse (mere-equiv-Prop (Fin n)) (D n)
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170171
preserves-id-equiv-invertible-action-D-equiv :
171172
(n : ℕ) (X : UU-Fin l1 n) →
172173
Id (invertible-action-D-equiv n X X id-equiv) id-equiv
173174
preserves-id-equiv-invertible-action-D-equiv n =
174-
preserves-id-equiv-action-equiv-family-on-subuniverse
175+
compute-id-equiv-action-equiv-family-over-subuniverse
175176
( mere-equiv-Prop (Fin n))
176177
( D n)
177178
178-
preserves-R-invertible-action-D-equiv :
179-
( n : ℕ) →
180-
( X X' : UU-Fin l1 n)
181-
( e : type-UU-Fin n X ≃ type-UU-Fin n X') →
182-
( a a' : D n X) →
183-
( sim-Equivalence-Relation (R n X) a a' ↔
184-
sim-Equivalence-Relation
185-
( R n X')
186-
( map-equiv (invertible-action-D-equiv n X X' e) a)
187-
( map-equiv (invertible-action-D-equiv n X X' e) a'))
188-
preserves-R-invertible-action-D-equiv n X X' =
189-
Ind-action-equiv-family-on-subuniverse (mere-equiv-Prop (Fin n)) (D n) X
190-
( λ Y f →
191-
( a a' : D n X) →
192-
( sim-Equivalence-Relation (R n X) a a' ↔
193-
sim-Equivalence-Relation (R n Y) (map-equiv f a) (map-equiv f a')))
194-
( λ a a' → id , id)
195-
( X')
179+
abstract
180+
preserves-R-invertible-action-D-equiv :
181+
( n : ℕ) →
182+
( X X' : UU-Fin l1 n)
183+
( e : type-UU-Fin n X ≃ type-UU-Fin n X') →
184+
( a a' : D n X) →
185+
( sim-Equivalence-Relation (R n X) a a' ↔
186+
sim-Equivalence-Relation
187+
( R n X')
188+
( map-equiv (invertible-action-D-equiv n X X' e) a)
189+
( map-equiv (invertible-action-D-equiv n X X' e) a'))
190+
preserves-R-invertible-action-D-equiv n X =
191+
ind-equiv-subuniverse
192+
( mere-equiv-Prop (Fin n))
193+
( X)
194+
( λ Y f →
195+
( a a' : D n X) →
196+
( sim-Equivalence-Relation (R n X) a a' ↔
197+
sim-Equivalence-Relation
198+
( R n Y)
199+
( map-equiv (invertible-action-D-equiv n X Y f) a)
200+
( map-equiv (invertible-action-D-equiv n X Y f) a')))
201+
( λ a a' →
202+
( binary-tr
203+
( sim-Equivalence-Relation (R n X))
204+
( inv
205+
( htpy-eq-equiv
206+
( preserves-id-equiv-invertible-action-D-equiv n X)
207+
( a)))
208+
( inv
209+
( htpy-eq-equiv
210+
( preserves-id-equiv-invertible-action-D-equiv n X)
211+
( a')))) ,
212+
( binary-tr
213+
( sim-Equivalence-Relation (R n X))
214+
( htpy-eq-equiv
215+
( preserves-id-equiv-invertible-action-D-equiv n X)
216+
( a))
217+
( htpy-eq-equiv
218+
( preserves-id-equiv-invertible-action-D-equiv n X)
219+
( a'))))
196220
197221
raise-UU-Fin-Fin : (n : ℕ) → UU-Fin l1 n
198222
pr1 (raise-UU-Fin-Fin n) = raise l1 (Fin n)
@@ -1490,7 +1514,7 @@ module _
14901514
unit-trunc-Prop (compute-raise-Fin l1 (n +ℕ 2))))) →
14911515
¬ ( x ＝
14921516
( map-equiv
1493-
( action-equiv-family-on-subuniverse
1517+
( action-equiv-family-over-subuniverse
14941518
( mere-equiv-Prop (Fin (n +ℕ 2)))
14951519
( type-UU-Fin 2 ∘ Q (n +ℕ 2))
14961520
( raise l1 (Fin (n +ℕ 2)) ,

‎src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md

Lines changed: 11 additions & 11 deletions
Original file line numberDiff line numberDiff line change
@@ -21,7 +21,7 @@ open import finite-group-theory.permutations
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open import finite-group-theory.sign-homomorphism
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open import finite-group-theory.transpositions
2323
24-
open import foundation.action-on-equivalences-families-over-subuniverses
24+
open import foundation.action-on-equivalences-type-families-over-subuniverses
2525
open import foundation.action-on-identifications-functions
2626
open import foundation.contractible-types
2727
open import foundation.coproduct-types
@@ -43,6 +43,7 @@ open import foundation.propositional-truncations
4343
open import foundation.raising-universe-levels
4444
open import foundation.sets
4545
open import foundation.transport-along-identifications
46+
open import foundation.type-theoretic-principle-of-choice
4647
open import foundation.unit-type
4748
open import foundation.universe-levels
4849
@@ -683,7 +684,7 @@ module _
683684
unit-trunc-Prop (compute-raise-Fin l (n +ℕ 2))))
684685
( sign-comp-aut-succ-succ-Fin n (transposition Y))
685686
( map-equiv
686-
( action-equiv-family-on-subuniverse
687+
( action-equiv-family-over-subuniverse
687688
( mere-equiv-Prop (Fin (n +ℕ 2)))
688689
( λ X → Fin (n +ℕ 2) ≃ pr1 X)
689690
( raise l (Fin (n +ℕ 2)) ,
@@ -716,20 +717,19 @@ module _
716717
simpson-comp-equiv (n +ℕ 2) ,
717718
preserves-id-equiv-simpson-comp-equiv (n +ℕ 2)}
718719
{ y =
719-
( action-equiv-family-on-subuniverse
720+
( action-equiv-family-over-subuniverse
720721
( mere-equiv-Prop (Fin (n +ℕ 2)))
721722
( λ X → Fin (n +ℕ 2) ≃ type-UU-Fin (n +ℕ 2) X) ,
722-
( preserves-id-equiv-action-equiv-family-on-subuniverse
723+
( compute-id-equiv-action-equiv-family-over-subuniverse
723724
( mere-equiv-Prop (Fin (n +ℕ 2)))
724725
( λ X → Fin (n +ℕ 2) ≃ type-UU-Fin (n +ℕ 2) X)))}
725726
( eq-is-contr
726-
( is-contr-preserves-id-action-equiv-family-on-subuniverse
727-
( mere-equiv-Prop (Fin (n +ℕ 2)))
728-
( λ X → Fin (n +ℕ 2) ≃ type-UU-Fin (n +ℕ 2) X)
729-
( λ X →
730-
is-set-equiv-is-set
731-
( is-set-Fin (n +ℕ 2))
732-
( is-set-type-UU-Fin (n +ℕ 2) X)))))
727+
( is-contr-equiv' _
728+
( distributive-Π-Σ)
729+
( is-contr-Π
730+
( unique-action-equiv-family-over-subuniverse
731+
( mere-equiv-Prop (Fin (n +ℕ 2)))
732+
( λ Y → Fin (n +ℕ 2) ≃ type-UU-Fin (n +ℕ 2) Y))))))
733733
( not-sign-comp-transposition-count
734734
(n +ℕ 2 , (compute-raise l (Fin (n +ℕ 2)))) (star))
735735

‎src/foundation.lagda.md

Lines changed: 5 additions & 1 deletion
Original file line numberDiff line numberDiff line change
@@ -10,8 +10,10 @@ open import foundation.0-images-of-maps public
1010
open import foundation.0-maps public
1111
open import foundation.1-types public
1212
open import foundation.2-types public
13-
open import foundation.action-on-equivalences-families-over-subuniverses public
1413
open import foundation.action-on-equivalences-functions public
14+
open import foundation.action-on-equivalences-functions-out-of-subuniverses public
15+
open import foundation.action-on-equivalences-type-families public
16+
open import foundation.action-on-equivalences-type-families-over-subuniverses public
1517
open import foundation.action-on-identifications-binary-functions public
1618
open import foundation.action-on-identifications-dependent-functions public
1719
open import foundation.action-on-identifications-functions public
@@ -251,6 +253,7 @@ open import foundation.subtype-identity-principle public
251253
open import foundation.subtypes public
252254
open import foundation.subuniverses public
253255
open import foundation.surjective-maps public
256+
open import foundation.symmetric-binary-relations public
254257
open import foundation.symmetric-difference public
255258
open import foundation.symmetric-identity-types public
256259
open import foundation.symmetric-operations public
@@ -295,6 +298,7 @@ open import foundation.universal-property-coproduct-types public
295298
open import foundation.universal-property-dependent-pair-types public
296299
open import foundation.universal-property-empty-type public
297300
open import foundation.universal-property-fiber-products public
301+
open import foundation.universal-property-identity-systems public
298302
open import foundation.universal-property-identity-types public
299303
open import foundation.universal-property-image public
300304
open import foundation.universal-property-maybe public

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