117 lines
5.3 KiB
Text
117 lines
5.3 KiB
Text
/-
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Copyright (c) 2015 Ulrik Buchholtz. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Ulrik Buchholtz
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Functoriality of quotients and a condition for when an equivalence is induced.
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-/
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import types.sigma .quotient
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open eq is_equiv equiv prod prod.ops sigma sigma.ops
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namespace quotient
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section
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variables {A : Type} (R : A → A → Type)
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{B : Type} (Q : B → B → Type)
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(f : A → B) (k : Πa a' : A, R a a' → Q (f a) (f a'))
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include f k
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protected definition functor [reducible] : quotient R → quotient Q :=
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quotient.elim (λa, class_of Q (f a)) (λa a' r, eq_of_rel Q (k a a' r))
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variables [F : is_equiv f] [K : Πa a', is_equiv (k a a')]
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include F K
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protected definition functor_inv [reducible] : quotient Q → quotient R :=
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quotient.elim (λb, class_of R (f⁻¹ b))
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(λb b' q, eq_of_rel R ((k (f⁻¹ b) (f⁻¹ b'))⁻¹
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((right_inv f b)⁻¹ ▸ (right_inv f b')⁻¹ ▸ q)))
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protected definition is_equiv [instance]
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: is_equiv (quotient.functor R Q f k):=
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begin
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fapply adjointify _ (quotient.functor_inv R Q f k),
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{ intro qb, induction qb with b b b' q,
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{ apply ap (class_of Q), apply right_inv },
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{ apply eq_pathover, rewrite [ap_id,ap_compose' (quotient.elim _ _)],
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do 2 krewrite elim_eq_of_rel, rewrite (right_inv (k (f⁻¹ b) (f⁻¹ b'))),
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have H1 : pathover (λz : B × B, Q z.1 z.2)
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((right_inv f b)⁻¹ ▸ (right_inv f b')⁻¹ ▸ q)
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(prod_eq (right_inv f b) (right_inv f b')) q,
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begin apply pathover_of_eq_tr, krewrite [prod_eq_inv,prod_eq_transport] end,
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have H2 : square
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(ap (λx : (Σz : B × B, Q z.1 z.2), class_of Q x.1.1)
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(sigma_eq (prod_eq (right_inv f b) (right_inv f b')) H1))
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(ap (λx : (Σz : B × B, Q z.1 z.2), class_of Q x.1.2)
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(sigma_eq (prod_eq (right_inv f b) (right_inv f b')) H1))
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(eq_of_rel Q ((right_inv f b)⁻¹ ▸ (right_inv f b')⁻¹ ▸ q))
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(eq_of_rel Q q),
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from
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natural_square (λw : (Σz : B × B, Q z.1 z.2), eq_of_rel Q w.2)
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(sigma_eq (prod_eq (right_inv f b) (right_inv f b')) H1),
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krewrite (ap_compose' (class_of Q)) at H2,
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krewrite (ap_compose' (λz : B × B, z.1)) at H2,
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rewrite sigma.ap_pr1 at H2, rewrite sigma_eq_pr1 at H2,
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krewrite prod.ap_pr1 at H2, krewrite prod_eq_pr1 at H2,
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krewrite (ap_compose' (class_of Q) (λx : (Σz : B × B, Q z.1 z.2), x.1.2)) at H2,
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krewrite (ap_compose' (λz : B × B, z.2)) at H2,
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rewrite sigma.ap_pr1 at H2, rewrite sigma_eq_pr1 at H2,
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krewrite prod.ap_pr2 at H2, krewrite prod_eq_pr2 at H2,
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apply H2 } },
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{ intro qa, induction qa with a a a' r,
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{ apply ap (class_of R), apply left_inv },
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{ apply eq_pathover, rewrite [ap_id,(ap_compose' (quotient.elim _ _))],
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do 2 krewrite elim_eq_of_rel,
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have H1 : pathover (λz : A × A, R z.1 z.2)
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((left_inv f a)⁻¹ ▸ (left_inv f a')⁻¹ ▸ r)
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(prod_eq (left_inv f a) (left_inv f a')) r,
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begin apply pathover_of_eq_tr, krewrite [prod_eq_inv,prod_eq_transport] end,
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have H2 : square
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(ap (λx : (Σz : A × A, R z.1 z.2), class_of R x.1.1)
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(sigma_eq (prod_eq (left_inv f a) (left_inv f a')) H1))
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(ap (λx : (Σz : A × A, R z.1 z.2), class_of R x.1.2)
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(sigma_eq (prod_eq (left_inv f a) (left_inv f a')) H1))
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(eq_of_rel R ((left_inv f a)⁻¹ ▸ (left_inv f a')⁻¹ ▸ r))
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(eq_of_rel R r),
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begin
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exact
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natural_square (λw : (Σz : A × A, R z.1 z.2), eq_of_rel R w.2)
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(sigma_eq (prod_eq (left_inv f a) (left_inv f a')) H1)
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end,
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krewrite (ap_compose' (class_of R)) at H2,
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krewrite (ap_compose' (λz : A × A, z.1)) at H2,
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rewrite sigma.ap_pr1 at H2, rewrite sigma_eq_pr1 at H2,
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krewrite prod.ap_pr1 at H2, krewrite prod_eq_pr1 at H2,
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krewrite (ap_compose' (class_of R) (λx : (Σz : A × A, R z.1 z.2), x.1.2)) at H2,
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krewrite (ap_compose' (λz : A × A, z.2)) at H2,
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rewrite sigma.ap_pr1 at H2, rewrite sigma_eq_pr1 at H2,
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krewrite prod.ap_pr2 at H2, krewrite prod_eq_pr2 at H2,
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have H3 :
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(k (f⁻¹ (f a)) (f⁻¹ (f a')))⁻¹
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((right_inv f (f a))⁻¹ ▸ (right_inv f (f a'))⁻¹ ▸ k a a' r)
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= (left_inv f a)⁻¹ ▸ (left_inv f a')⁻¹ ▸ r,
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begin
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rewrite [adj f a,adj f a',ap_inv',ap_inv'],
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rewrite [-(tr_compose _ f (left_inv f a')⁻¹ (k a a' r)),
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-(tr_compose _ f (left_inv f a)⁻¹)],
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rewrite [-(fn_tr_eq_tr_fn (left_inv f a')⁻¹ (λx, k a x) r),
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-(fn_tr_eq_tr_fn (left_inv f a)⁻¹
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(λx, k x (f⁻¹ (f a')))),
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left_inv (k _ _)]
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end,
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rewrite H3, apply H2 } }
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end
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end
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section
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open equiv.ops
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variables {A : Type} (R : A → A → Type)
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{B : Type} (Q : B → B → Type)
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(f : A ≃ B) (k : Πa a' : A, R a a' ≃ Q (f a) (f a'))
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include f k
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/- This could also be proved using ua, but then it wouldn't compute -/
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protected definition equiv : quotient R ≃ quotient Q :=
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equiv.mk (quotient.functor R Q f k) _
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end
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end quotient
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