refactor(category): add new folder functor, split adjoint file into separate files
This commit is contained in:
Floris van Doorn
Leonardo de Moura
parent
21723a82b6
commit
0e7b7af1da
17 changed files with 283 additions and 280 deletions
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@ -7,15 +7,10 @@ Development of Category Theory. The following files are in this folder (sorted s
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* [iso](iso.hlean) : iso, mono, epi, split mono, split epi
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* [category](category.hlean) : Categories (i.e. univalent or Rezk-complete precategories)
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* [groupoid](groupoid.hlean)
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* [functor](functor.hlean)
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* [nat_trans](nat_trans.hlean) : Natural transformations
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* [functor](functor/functor.md) (subfolder) : definition and properties of functors
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* [strict](strict.hlean) : Strict categories
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* [nat_trans](nat_trans.hlean) : Natural transformations
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* [constructions](constructions/constructions.md) (subfolder) : basic constructions on categories and examples of categories
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The following files depend on some of the files in the folder [constructions](constructions/constructions.md)
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* [curry](curry.hlean) : Define currying and uncurrying of functors
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* [limits](limits.hlean) : Limits in a category, defined as terminal object in the cone category
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* [colimits](colimits.hlean) : Colimits in a category, defined as the limit of the opposite functor
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* [adjoint](adjoint.hlean) : Adjoint functors and Equivalences (WIP)
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* [yoneda](yoneda.hlean) : Yoneda Embedding (WIP)
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@ -7,7 +7,7 @@ Authors: Floris van Doorn
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Comma category
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-/
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import ..functor ..strict ..category
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import ..functor.functor ..strict ..category
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open eq functor equiv sigma sigma.ops is_trunc iso is_equiv
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@ -7,7 +7,7 @@ Authors: Floris van Doorn
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Discrete category
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-/
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import ..groupoid types.bool ..functor
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import ..groupoid types.bool ..functor.functor
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open eq is_trunc iso bool functor
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@ -7,7 +7,7 @@ Authors: Floris van Doorn
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Some finite categories which are neither discrete nor indiscrete
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-/
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import ..functor types.sum
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import ..functor.functor types.sum
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open bool unit is_trunc sum eq functor equiv
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@ -6,7 +6,7 @@ Authors: Floris van Doorn, Jakob von Raumer
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Category of hsets
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-/
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import ..category types.equiv ..functor types.lift ..limits ..colimits hit.set_quotient hit.trunc
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import ..category types.equiv types.lift ..colimits hit.set_quotient
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open eq category equiv iso is_equiv is_trunc function sigma set_quotient trunc
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@ -6,7 +6,7 @@ Authors: Floris van Doorn, Jakob von Raumer
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Opposite precategory and (TODO) category
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-/
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import ..functor ..category
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import ..functor.functor ..category
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open eq functor iso equiv is_equiv
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@ -6,7 +6,7 @@ Authors: Floris van Doorn, Jakob von Raumer
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Product precategory and (TODO) category
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-/
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import ..category ..functor hit.trunc
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import ..category ..functor.functor hit.trunc
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open eq prod is_trunc functor sigma trunc
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@ -6,7 +6,7 @@ Authors: Floris van Doorn, Jakob von Raumer
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Sum precategory and (TODO) category
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-/
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import ..category ..functor types.sum
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import ..category ..functor.functor types.sum
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open eq sum is_trunc functor lift
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103
hott/algebra/category/functor/adjoint.hlean
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103
hott/algebra/category/functor/adjoint.hlean
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@ -0,0 +1,103 @@
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/-
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Copyright (c) 2015 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn
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Adjoint functors
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-/
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import .attributes
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open category functor nat_trans eq is_trunc iso equiv prod trunc function pi is_equiv
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namespace category
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-- TODO(?): define a structure "adjoint" and then define
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-- structure is_left_adjoint (F : C ⇒ D) :=
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-- (G : D ⇒ C) -- G
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-- (is_adjoint : adjoint F G)
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structure is_left_adjoint [class] {C D : Precategory} (F : C ⇒ D) :=
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(G : D ⇒ C)
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(η : 1 ⟹ G ∘f F)
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(ε : F ∘f G ⟹ 1)
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(H : Π(c : C), ε (F c) ∘ F (η c) = ID (F c))
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(K : Π(d : D), G (ε d) ∘ η (G d) = ID (G d))
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abbreviation right_adjoint [unfold 4] := @is_left_adjoint.G
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abbreviation unit [unfold 4] := @is_left_adjoint.η
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abbreviation counit [unfold 4] := @is_left_adjoint.ε
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abbreviation counit_unit_eq [unfold 4] := @is_left_adjoint.H
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abbreviation unit_counit_eq [unfold 4] := @is_left_adjoint.K
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theorem is_hprop_is_left_adjoint [instance] {C : Category} {D : Precategory} (F : C ⇒ D)
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: is_hprop (is_left_adjoint F) :=
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begin
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apply is_hprop.mk,
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intro G G', cases G with G η ε H K, cases G' with G' η' ε' H' K',
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assert lem₁ : Π(p : G = G'), p ▸ η = η' → p ▸ ε = ε'
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→ is_left_adjoint.mk G η ε H K = is_left_adjoint.mk G' η' ε' H' K',
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{ intros p q r, induction p, induction q, induction r, esimp,
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apply apd011 (is_left_adjoint.mk G η ε) !is_hprop.elim !is_hprop.elim},
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assert lem₂ : Π (d : carrier D),
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(to_fun_hom G (natural_map ε' d) ∘
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natural_map η (to_fun_ob G' d)) ∘
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to_fun_hom G' (natural_map ε d) ∘
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natural_map η' (to_fun_ob G d) = id,
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{ intro d, esimp,
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rewrite [assoc],
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rewrite [-assoc (G (ε' d))],
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esimp, rewrite [nf_fn_eq_fn_nf_pt' G' ε η d],
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esimp, rewrite [assoc],
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esimp, rewrite [-assoc],
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rewrite [↑functor.compose, -respect_comp G],
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rewrite [nf_fn_eq_fn_nf_pt ε ε' d,nf_fn_eq_fn_nf_pt η' η (G d),▸*],
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rewrite [respect_comp G],
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rewrite [assoc,▸*,-assoc (G (ε d))],
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rewrite [↑functor.compose, -respect_comp G],
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rewrite [H' (G d)],
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rewrite [respect_id,▸*,id_right],
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apply K},
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assert lem₃ : Π (d : carrier D),
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(to_fun_hom G' (natural_map ε d) ∘
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natural_map η' (to_fun_ob G d)) ∘
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to_fun_hom G (natural_map ε' d) ∘
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natural_map η (to_fun_ob G' d) = id,
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{ intro d, esimp,
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rewrite [assoc, -assoc (G' (ε d))],
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esimp, rewrite [nf_fn_eq_fn_nf_pt' G ε' η' d],
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esimp, rewrite [assoc], esimp, rewrite [-assoc],
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rewrite [↑functor.compose, -respect_comp G'],
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rewrite [nf_fn_eq_fn_nf_pt ε' ε d,nf_fn_eq_fn_nf_pt η η' (G' d)],
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esimp,
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rewrite [respect_comp G'],
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rewrite [assoc,▸*,-assoc (G' (ε' d))],
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rewrite [↑functor.compose, -respect_comp G'],
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rewrite [H (G' d)],
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rewrite [respect_id,▸*,id_right],
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apply K'},
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fapply lem₁,
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{ fapply functor.eq_of_pointwise_iso,
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{ fapply change_natural_map,
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{ exact (G' ∘fn1 ε) ∘n !assoc_natural_rev ∘n (η' ∘1nf G)},
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{ intro d, exact (G' (ε d) ∘ η' (G d))},
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{ intro d, exact ap (λx, _ ∘ x) !id_left}},
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{ intro d, fconstructor,
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{ exact (G (ε' d) ∘ η (G' d))},
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{ exact lem₂ d },
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{ exact lem₃ d }}},
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{ clear lem₁, refine transport_hom_of_eq_right _ η ⬝ _,
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krewrite hom_of_eq_compose_right,
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rewrite functor.hom_of_eq_eq_of_pointwise_iso,
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apply nat_trans_eq, intro c, esimp,
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refine !assoc⁻¹ ⬝ ap (λx, _ ∘ x) (nf_fn_eq_fn_nf_pt η η' c) ⬝ !assoc ⬝ _,
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esimp, rewrite [-respect_comp G',H c,respect_id G',▸*,id_left]},
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{ clear lem₁, refine transport_hom_of_eq_left _ ε ⬝ _,
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krewrite inv_of_eq_compose_left,
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rewrite functor.inv_of_eq_eq_of_pointwise_iso,
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apply nat_trans_eq, intro d, esimp,
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krewrite [respect_comp],
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rewrite [assoc,nf_fn_eq_fn_nf_pt ε' ε d,-assoc,▸*,H (G' d),id_right]}
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end
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end category
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147
hott/algebra/category/functor/attributes.hlean
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147
hott/algebra/category/functor/attributes.hlean
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@ -0,0 +1,147 @@
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/-
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Copyright (c) 2015 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn
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Attributes of functors (full, faithful, split essentially surjective, ...)
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Adjoint functors, isomorphisms and equivalences have their own file
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-/
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import ..constructions.functor function arity
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open category functor nat_trans eq is_trunc iso equiv prod trunc function pi is_equiv
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namespace category
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variables {C D E : Precategory} {F : C ⇒ D} {G : D ⇒ C}
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definition faithful [class] (F : C ⇒ D) := Π⦃c c' : C⦄ ⦃f f' : c ⟶ c'⦄, F f = F f' → f = f'
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definition full [class] (F : C ⇒ D) := Π⦃c c' : C⦄, is_surjective (@(to_fun_hom F) c c')
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definition fully_faithful [class] (F : C ⇒ D) := Π(c c' : C), is_equiv (@(to_fun_hom F) c c')
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definition split_essentially_surjective [class] (F : C ⇒ D) := Π(d : D), Σ(c : C), F c ≅ d
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definition essentially_surjective [class] (F : C ⇒ D) := Π(d : D), ∃(c : C), F c ≅ d
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definition is_weak_equivalence [class] (F : C ⇒ D) := fully_faithful F × essentially_surjective F
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definition is_equiv_of_fully_faithful [instance] [reducible] (F : C ⇒ D) [H : fully_faithful F]
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(c c' : C) : is_equiv (@(to_fun_hom F) c c') :=
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!H
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definition hom_inv [reducible] (F : C ⇒ D) [H : fully_faithful F] (c c' : C) (f : F c ⟶ F c')
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: c ⟶ c' :=
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(to_fun_hom F)⁻¹ᶠ f
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definition reflect_is_iso [constructor] (F : C ⇒ D) [H : fully_faithful F] {c c' : C} (f : c ⟶ c')
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[H : is_iso (F f)] : is_iso f :=
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begin
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fconstructor,
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{ exact (to_fun_hom F)⁻¹ᶠ (F f)⁻¹},
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{ apply eq_of_fn_eq_fn' (to_fun_hom F),
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rewrite [respect_comp,right_inv (to_fun_hom F),respect_id,left_inverse]},
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{ apply eq_of_fn_eq_fn' (to_fun_hom F),
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rewrite [respect_comp,right_inv (to_fun_hom F),respect_id,right_inverse]},
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end
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definition reflect_iso [constructor] (F : C ⇒ D) [H : fully_faithful F] {c c' : C}
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(f : F c ≅ F c') : c ≅ c' :=
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begin
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fconstructor,
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{ exact (to_fun_hom F)⁻¹ᶠ f},
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{ assert H : is_iso (F ((to_fun_hom F)⁻¹ᶠ f)),
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{ have H' : is_iso (to_hom f), from _, exact (right_inv (to_fun_hom F) (to_hom f))⁻¹ ▸ H'},
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exact reflect_is_iso F _},
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end
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theorem reflect_inverse (F : C ⇒ D) [H : fully_faithful F] {c c' : C} (f : c ⟶ c')
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[H : is_iso f] : (to_fun_hom F)⁻¹ᶠ (F f)⁻¹ = f⁻¹ :=
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inverse_eq_inverse (idp : to_hom (@(iso.mk f) (reflect_is_iso F f)) = f)
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definition hom_equiv_F_hom_F [constructor] (F : C ⇒ D)
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[H : fully_faithful F] (c c' : C) : (c ⟶ c') ≃ (F c ⟶ F c') :=
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equiv.mk _ !H
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definition iso_of_F_iso_F (F : C ⇒ D)
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[H : fully_faithful F] (c c' : C) (g : F c ≅ F c') : c ≅ c' :=
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begin
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induction g with g G, induction G with h p q, fapply iso.MK,
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{ rexact (to_fun_hom F)⁻¹ᶠ g},
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{ rexact (to_fun_hom F)⁻¹ᶠ h},
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{ exact abstract begin
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apply eq_of_fn_eq_fn' (to_fun_hom F),
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rewrite [respect_comp, respect_id,
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right_inv (to_fun_hom F), right_inv (to_fun_hom F), p],
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end end},
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{ exact abstract begin
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apply eq_of_fn_eq_fn' (to_fun_hom F),
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rewrite [respect_comp, respect_id,
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right_inv (to_fun_hom F), right_inv (@(to_fun_hom F) c' c), q],
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end end}
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end
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definition iso_equiv_F_iso_F [constructor] (F : C ⇒ D)
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[H : fully_faithful F] (c c' : C) : (c ≅ c') ≃ (F c ≅ F c') :=
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begin
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fapply equiv.MK,
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{ exact to_fun_iso F},
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{ apply iso_of_F_iso_F},
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{ exact abstract begin
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intro f, induction f with f F', induction F' with g p q, apply iso_eq,
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esimp [iso_of_F_iso_F], apply right_inv end end},
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{ exact abstract begin
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intro f, induction f with f F', induction F' with g p q, apply iso_eq,
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esimp [iso_of_F_iso_F], apply right_inv end end},
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end
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definition full_of_fully_faithful (H : fully_faithful F) : full F :=
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λc c' g, tr (fiber.mk ((@(to_fun_hom F) c c')⁻¹ᶠ g) !right_inv)
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definition faithful_of_fully_faithful (H : fully_faithful F) : faithful F :=
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λc c' f f' p, is_injective_of_is_embedding p
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definition fully_faithful_of_full_of_faithful (H : faithful F) (K : full F) : fully_faithful F :=
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begin
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intro c c',
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apply is_equiv_of_is_surjective_of_is_embedding,
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{ apply is_embedding_of_is_injective,
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intros f f' p, exact H p},
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{ apply K}
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end
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theorem is_hprop_fully_faithful [instance] (F : C ⇒ D) : is_hprop (fully_faithful F) :=
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by unfold fully_faithful; exact _
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theorem is_hprop_full [instance] (F : C ⇒ D) : is_hprop (full F) :=
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by unfold full; exact _
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theorem is_hprop_faithful [instance] (F : C ⇒ D) : is_hprop (faithful F) :=
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by unfold faithful; exact _
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theorem is_hprop_essentially_surjective [instance] (F : C ⇒ D)
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: is_hprop (essentially_surjective F) :=
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by unfold essentially_surjective; exact _
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theorem is_hprop_is_weak_equivalence [instance] (F : C ⇒ D) : is_hprop (is_weak_equivalence F) :=
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by unfold is_weak_equivalence; exact _
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definition fully_faithful_equiv (F : C ⇒ D) : fully_faithful F ≃ (faithful F × full F) :=
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equiv_of_is_hprop (λH, (faithful_of_fully_faithful H, full_of_fully_faithful H))
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(λH, fully_faithful_of_full_of_faithful (pr1 H) (pr2 H))
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/- alternative proof using direct calculation with equivalences
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definition fully_faithful_equiv (F : C ⇒ D) : fully_faithful F ≃ (faithful F × full F) :=
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calc
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fully_faithful F
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≃ (Π(c c' : C), is_embedding (to_fun_hom F) × is_surjective (to_fun_hom F))
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: pi_equiv_pi_id (λc, pi_equiv_pi_id
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(λc', !is_equiv_equiv_is_embedding_times_is_surjective))
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... ≃ (Π(c : C), (Π(c' : C), is_embedding (to_fun_hom F)) ×
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(Π(c' : C), is_surjective (to_fun_hom F)))
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: pi_equiv_pi_id (λc, !equiv_prod_corec)
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... ≃ (Π(c c' : C), is_embedding (to_fun_hom F)) × full F
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: equiv_prod_corec
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... ≃ faithful F × full F
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: prod_equiv_prod_right (pi_equiv_pi_id (λc, pi_equiv_pi_id
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(λc', !is_embedding_equiv_is_injective)))
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-/
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end category
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@ -6,7 +6,7 @@ Authors: Floris van Doorn
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Definition of currying and uncurrying of functors
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-/
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import .constructions.functor .constructions.product
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import ..constructions.functor ..constructions.product
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open category prod nat_trans eq prod.ops iso equiv
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@ -3,36 +3,16 @@ Copyright (c) 2015 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn
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Properties of functors such as adjoint functors, equivalences, faithful or full functors
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TODO: Split this file in different files
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Functors which are equivalences or isomorphisms
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-/
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import .constructions.functor function arity
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import .adjoint
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open category functor nat_trans eq is_trunc iso equiv prod trunc function pi is_equiv
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namespace category
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variables {C D : Precategory} {F : C ⇒ D} {G : D ⇒ C}
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-- TODO: define a structure "adjoint" and then define
|
||||
-- structure is_left_adjoint (F : C ⇒ D) :=
|
||||
-- (G : D ⇒ C) -- G
|
||||
-- (is_adjoint : adjoint F G)
|
||||
|
||||
structure is_left_adjoint [class] (F : C ⇒ D) :=
|
||||
(G : D ⇒ C)
|
||||
(η : 1 ⟹ G ∘f F)
|
||||
(ε : F ∘f G ⟹ 1)
|
||||
(H : Π(c : C), ε (F c) ∘ F (η c) = ID (F c))
|
||||
(K : Π(d : D), G (ε d) ∘ η (G d) = ID (G d))
|
||||
|
||||
abbreviation right_adjoint [unfold 4] := @is_left_adjoint.G
|
||||
abbreviation unit [unfold 4] := @is_left_adjoint.η
|
||||
abbreviation counit [unfold 4] := @is_left_adjoint.ε
|
||||
abbreviation counit_unit_eq [unfold 4] := @is_left_adjoint.H
|
||||
abbreviation unit_counit_eq [unfold 4] := @is_left_adjoint.K
|
||||
|
||||
structure is_equivalence [class] (F : C ⇒ D) extends is_left_adjoint F :=
|
||||
mk' ::
|
||||
(is_iso_unit : is_iso η)
|
||||
|
|
@ -41,14 +21,8 @@ namespace category
|
|||
abbreviation inverse := @is_equivalence.G
|
||||
postfix ⁻¹ := inverse
|
||||
--a second notation for the inverse, which is not overloaded (there is no unicode superscript F)
|
||||
postfix [parsing_only] `⁻¹F`:std.prec.max_plus := inverse
|
||||
postfix [parsing_only] `⁻¹ᴱ`:std.prec.max_plus := inverse
|
||||
|
||||
definition faithful [class] (F : C ⇒ D) := Π⦃c c' : C⦄ ⦃f f' : c ⟶ c'⦄, F f = F f' → f = f'
|
||||
definition full [class] (F : C ⇒ D) := Π⦃c c' : C⦄, is_surjective (@(to_fun_hom F) c c')
|
||||
definition fully_faithful [class] (F : C ⇒ D) := Π(c c' : C), is_equiv (@(to_fun_hom F) c c')
|
||||
definition split_essentially_surjective [class] (F : C ⇒ D) := Π(d : D), Σ(c : C), F c ≅ d
|
||||
definition essentially_surjective [class] (F : C ⇒ D) := Π(d : D), ∃(c : C), F c ≅ d
|
||||
definition is_weak_equivalence [class] (F : C ⇒ D) := fully_faithful F × essentially_surjective F
|
||||
definition is_isomorphism [class] (F : C ⇒ D) := fully_faithful F × is_equiv (to_fun_ob F)
|
||||
|
||||
structure equivalence (C D : Precategory) :=
|
||||
|
|
@ -63,77 +37,18 @@ namespace category
|
|||
infix ` ≃c `:25 := equivalence
|
||||
infix ` ≅c `:25 := isomorphism
|
||||
|
||||
definition is_equiv_of_fully_faithful [instance] [reducible] (F : C ⇒ D) [H : fully_faithful F]
|
||||
(c c' : C) : is_equiv (@(to_fun_hom F) c c') :=
|
||||
!H
|
||||
|
||||
definition hom_inv [reducible] (F : C ⇒ D) [H : fully_faithful F] (c c' : C) (f : F c ⟶ F c')
|
||||
: c ⟶ c' :=
|
||||
(to_fun_hom F)⁻¹ᶠ f
|
||||
|
||||
definition hom_equiv_F_hom_F [constructor] (F : C ⇒ D)
|
||||
[H : fully_faithful F] (c c' : C) : (c ⟶ c') ≃ (F c ⟶ F c') :=
|
||||
equiv.mk _ !H
|
||||
|
||||
definition iso_of_F_iso_F (F : C ⇒ D)
|
||||
[H : fully_faithful F] (c c' : C) (g : F c ≅ F c') : c ≅ c' :=
|
||||
begin
|
||||
induction g with g G, induction G with h p q, fapply iso.MK,
|
||||
{ rexact (to_fun_hom F)⁻¹ᶠ g},
|
||||
{ rexact (to_fun_hom F)⁻¹ᶠ h},
|
||||
{ exact abstract begin
|
||||
apply eq_of_fn_eq_fn' (to_fun_hom F),
|
||||
rewrite [respect_comp, respect_id,
|
||||
right_inv (to_fun_hom F), right_inv (to_fun_hom F), p],
|
||||
end end},
|
||||
{ exact abstract begin
|
||||
apply eq_of_fn_eq_fn' (to_fun_hom F),
|
||||
rewrite [respect_comp, respect_id,
|
||||
right_inv (to_fun_hom F), right_inv (@(to_fun_hom F) c' c), q],
|
||||
end end}
|
||||
end
|
||||
|
||||
definition iso_equiv_F_iso_F [constructor] (F : C ⇒ D)
|
||||
[H : fully_faithful F] (c c' : C) : (c ≅ c') ≃ (F c ≅ F c') :=
|
||||
begin
|
||||
fapply equiv.MK,
|
||||
{ exact to_fun_iso F},
|
||||
{ apply iso_of_F_iso_F},
|
||||
{ exact abstract begin
|
||||
intro f, induction f with f F', induction F' with g p q, apply iso_eq,
|
||||
esimp [iso_of_F_iso_F], apply right_inv end end},
|
||||
{ exact abstract begin
|
||||
intro f, induction f with f F', induction F' with g p q, apply iso_eq,
|
||||
esimp [iso_of_F_iso_F], apply right_inv end end},
|
||||
end
|
||||
|
||||
definition is_iso_unit [instance] (F : C ⇒ D) [H : is_equivalence F] : is_iso (unit F) :=
|
||||
!is_equivalence.is_iso_unit
|
||||
|
||||
definition is_iso_counit [instance] (F : C ⇒ D) [H : is_equivalence F] : is_iso (counit F) :=
|
||||
!is_equivalence.is_iso_counit
|
||||
|
||||
definition iso_unit (F : C ⇒ D) [H : is_equivalence F] : F ⁻¹F ∘f F ≅ 1 :=
|
||||
definition iso_unit (F : C ⇒ D) [H : is_equivalence F] : F ⁻¹ᴱ ∘f F ≅ 1 :=
|
||||
(@(iso.mk _) !is_iso_unit)⁻¹ⁱ
|
||||
|
||||
definition iso_counit (F : C ⇒ D) [H : is_equivalence F] : F ∘f F ⁻¹F ≅ 1 :=
|
||||
definition iso_counit (F : C ⇒ D) [H : is_equivalence F] : F ∘f F ⁻¹ᴱ ≅ 1 :=
|
||||
@(iso.mk _) !is_iso_counit
|
||||
|
||||
definition full_of_fully_faithful (H : fully_faithful F) : full F :=
|
||||
λc c' g, tr (fiber.mk ((@(to_fun_hom F) c c')⁻¹ᶠ g) !right_inv)
|
||||
|
||||
definition faithful_of_fully_faithful (H : fully_faithful F) : faithful F :=
|
||||
λc c' f f' p, is_injective_of_is_embedding p
|
||||
|
||||
definition fully_faithful_of_full_of_faithful (H : faithful F) (K : full F) : fully_faithful F :=
|
||||
begin
|
||||
intro c c',
|
||||
apply is_equiv_of_is_surjective_of_is_embedding,
|
||||
{ apply is_embedding_of_is_injective,
|
||||
intros f f' p, exact H p},
|
||||
{ apply K}
|
||||
end
|
||||
|
||||
definition split_essentially_surjective_of_is_equivalence (F : C ⇒ D) [H : is_equivalence F]
|
||||
: split_essentially_surjective F :=
|
||||
begin
|
||||
|
|
@ -142,40 +57,12 @@ namespace category
|
|||
{ exact componentwise_iso (@(iso.mk (counit F)) !is_iso_counit) d}
|
||||
end
|
||||
|
||||
definition reflect_is_iso [constructor] (F : C ⇒ D) [H : fully_faithful F] {c c' : C} (f : c ⟶ c')
|
||||
[H : is_iso (F f)] : is_iso f :=
|
||||
begin
|
||||
fconstructor,
|
||||
{ exact (to_fun_hom F)⁻¹ᶠ (F f)⁻¹},
|
||||
{ apply eq_of_fn_eq_fn' (to_fun_hom F),
|
||||
rewrite [respect_comp,right_inv (to_fun_hom F),respect_id,left_inverse]},
|
||||
{ apply eq_of_fn_eq_fn' (to_fun_hom F),
|
||||
rewrite [respect_comp,right_inv (to_fun_hom F),respect_id,right_inverse]},
|
||||
end
|
||||
|
||||
definition reflect_iso [constructor] (F : C ⇒ D) [H : fully_faithful F] {c c' : C}
|
||||
(f : F c ≅ F c') : c ≅ c' :=
|
||||
begin
|
||||
fconstructor,
|
||||
{ exact (to_fun_hom F)⁻¹ᶠ f},
|
||||
{ assert H : is_iso (F ((to_fun_hom F)⁻¹ᶠ f)),
|
||||
{ have H' : is_iso (to_hom f), from _, exact (right_inv (to_fun_hom F) (to_hom f))⁻¹ ▸ H'},
|
||||
exact reflect_is_iso F _},
|
||||
end
|
||||
|
||||
theorem reflect_inverse (F : C ⇒ D) [H : fully_faithful F] {c c' : C} (f : c ⟶ c')
|
||||
[H : is_iso f] : (to_fun_hom F)⁻¹ᶠ (F f)⁻¹ = f⁻¹ :=
|
||||
inverse_eq_inverse (idp : to_hom (@(iso.mk f) (reflect_is_iso F f)) = f)
|
||||
end category
|
||||
|
||||
namespace category
|
||||
section
|
||||
parameters {C D : Precategory} {F : C ⇒ D} {G : D ⇒ C} (η : G ∘f F ≅ 1) (ε : F ∘f G ≅ 1)
|
||||
|
||||
-- variables (η : Πc, G (F c) ≅ c) (ε : Πd, F (G d) ≅ d)
|
||||
-- (pη : Π(c c' : C) (f : hom c c'), f ∘ to_hom (η c) = to_hom (η c') ∘ G (F f))
|
||||
-- (pε : Π⦃d d' : D⦄ (f : hom d d'), f ∘ to_hom (ε d) = to_hom (ε d') ∘ F (G f))
|
||||
|
||||
private definition ηn : 1 ⟹ G ∘f F := to_inv η
|
||||
private definition εn : F ∘f G ⟹ 1 := to_hom ε
|
||||
|
||||
|
|
@ -334,76 +221,6 @@ namespace category
|
|||
{ apply iso_of_eq !strict_right_inverse},
|
||||
end
|
||||
|
||||
theorem is_hprop_is_left_adjoint [instance] {C : Category} {D : Precategory} (F : C ⇒ D)
|
||||
: is_hprop (is_left_adjoint F) :=
|
||||
begin
|
||||
apply is_hprop.mk,
|
||||
intro G G', cases G with G η ε H K, cases G' with G' η' ε' H' K',
|
||||
assert lem₁ : Π(p : G = G'), p ▸ η = η' → p ▸ ε = ε'
|
||||
→ is_left_adjoint.mk G η ε H K = is_left_adjoint.mk G' η' ε' H' K',
|
||||
{ intros p q r, induction p, induction q, induction r, esimp,
|
||||
apply apd011 (is_left_adjoint.mk G η ε) !is_hprop.elim !is_hprop.elim},
|
||||
assert lem₂ : Π (d : carrier D),
|
||||
(to_fun_hom G (natural_map ε' d) ∘
|
||||
natural_map η (to_fun_ob G' d)) ∘
|
||||
to_fun_hom G' (natural_map ε d) ∘
|
||||
natural_map η' (to_fun_ob G d) = id,
|
||||
{ intro d, esimp,
|
||||
rewrite [assoc],
|
||||
rewrite [-assoc (G (ε' d))],
|
||||
esimp, rewrite [nf_fn_eq_fn_nf_pt' G' ε η d],
|
||||
esimp, rewrite [assoc],
|
||||
esimp, rewrite [-assoc],
|
||||
rewrite [↑functor.compose, -respect_comp G],
|
||||
rewrite [nf_fn_eq_fn_nf_pt ε ε' d,nf_fn_eq_fn_nf_pt η' η (G d),▸*],
|
||||
rewrite [respect_comp G],
|
||||
rewrite [assoc,▸*,-assoc (G (ε d))],
|
||||
rewrite [↑functor.compose, -respect_comp G],
|
||||
rewrite [H' (G d)],
|
||||
rewrite [respect_id,▸*,id_right],
|
||||
apply K},
|
||||
assert lem₃ : Π (d : carrier D),
|
||||
(to_fun_hom G' (natural_map ε d) ∘
|
||||
natural_map η' (to_fun_ob G d)) ∘
|
||||
to_fun_hom G (natural_map ε' d) ∘
|
||||
natural_map η (to_fun_ob G' d) = id,
|
||||
{ intro d, esimp,
|
||||
rewrite [assoc, -assoc (G' (ε d))],
|
||||
esimp, rewrite [nf_fn_eq_fn_nf_pt' G ε' η' d],
|
||||
esimp, rewrite [assoc], esimp, rewrite [-assoc],
|
||||
rewrite [↑functor.compose, -respect_comp G'],
|
||||
rewrite [nf_fn_eq_fn_nf_pt ε' ε d,nf_fn_eq_fn_nf_pt η η' (G' d)],
|
||||
esimp,
|
||||
rewrite [respect_comp G'],
|
||||
rewrite [assoc,▸*,-assoc (G' (ε' d))],
|
||||
rewrite [↑functor.compose, -respect_comp G'],
|
||||
rewrite [H (G' d)],
|
||||
rewrite [respect_id,▸*,id_right],
|
||||
apply K'},
|
||||
fapply lem₁,
|
||||
{ fapply functor.eq_of_pointwise_iso,
|
||||
{ fapply change_natural_map,
|
||||
{ exact (G' ∘fn1 ε) ∘n !assoc_natural_rev ∘n (η' ∘1nf G)},
|
||||
{ intro d, exact (G' (ε d) ∘ η' (G d))},
|
||||
{ intro d, exact ap (λx, _ ∘ x) !id_left}},
|
||||
{ intro d, fconstructor,
|
||||
{ exact (G (ε' d) ∘ η (G' d))},
|
||||
{ exact lem₂ d },
|
||||
{ exact lem₃ d }}},
|
||||
{ clear lem₁, refine transport_hom_of_eq_right _ η ⬝ _,
|
||||
krewrite hom_of_eq_compose_right,
|
||||
rewrite functor.hom_of_eq_eq_of_pointwise_iso,
|
||||
apply nat_trans_eq, intro c, esimp,
|
||||
refine !assoc⁻¹ ⬝ ap (λx, _ ∘ x) (nf_fn_eq_fn_nf_pt η η' c) ⬝ !assoc ⬝ _,
|
||||
esimp, rewrite [-respect_comp G',H c,respect_id G',▸*,id_left]},
|
||||
{ clear lem₁, refine transport_hom_of_eq_left _ ε ⬝ _,
|
||||
krewrite inv_of_eq_compose_left,
|
||||
rewrite functor.inv_of_eq_eq_of_pointwise_iso,
|
||||
apply nat_trans_eq, intro d, esimp,
|
||||
krewrite [respect_comp],
|
||||
rewrite [assoc,nf_fn_eq_fn_nf_pt ε' ε d,-assoc,▸*,H (G' d),id_right]}
|
||||
end
|
||||
|
||||
theorem is_hprop_is_equivalence [instance] {C : Category} {D : Precategory} (F : C ⇒ D) : is_hprop (is_equivalence F) :=
|
||||
begin
|
||||
assert f : is_equivalence F ≃ Σ(H : is_left_adjoint F), is_iso (unit F) × is_iso (counit F),
|
||||
|
|
@ -416,47 +233,9 @@ namespace category
|
|||
apply is_trunc_equiv_closed_rev, exact f,
|
||||
end
|
||||
|
||||
theorem is_hprop_fully_faithful [instance] (F : C ⇒ D) : is_hprop (fully_faithful F) :=
|
||||
by unfold fully_faithful; exact _
|
||||
|
||||
theorem is_hprop_full [instance] (F : C ⇒ D) : is_hprop (full F) :=
|
||||
by unfold full; exact _
|
||||
|
||||
theorem is_hprop_faithful [instance] (F : C ⇒ D) : is_hprop (faithful F) :=
|
||||
by unfold faithful; exact _
|
||||
|
||||
theorem is_hprop_essentially_surjective [instance] (F : C ⇒ D)
|
||||
: is_hprop (essentially_surjective F) :=
|
||||
by unfold essentially_surjective; exact _
|
||||
|
||||
theorem is_hprop_is_weak_equivalence [instance] (F : C ⇒ D) : is_hprop (is_weak_equivalence F) :=
|
||||
by unfold is_weak_equivalence; exact _
|
||||
|
||||
theorem is_hprop_is_isomorphism [instance] (F : C ⇒ D) : is_hprop (is_isomorphism F) :=
|
||||
by unfold is_isomorphism; exact _
|
||||
|
||||
definition fully_faithful_equiv (F : C ⇒ D) : fully_faithful F ≃ (faithful F × full F) :=
|
||||
equiv_of_is_hprop (λH, (faithful_of_fully_faithful H, full_of_fully_faithful H))
|
||||
(λH, fully_faithful_of_full_of_faithful (pr1 H) (pr2 H))
|
||||
|
||||
/- alternative proof using direct calculation with equivalences
|
||||
|
||||
definition fully_faithful_equiv (F : C ⇒ D) : fully_faithful F ≃ (faithful F × full F) :=
|
||||
calc
|
||||
fully_faithful F
|
||||
≃ (Π(c c' : C), is_embedding (to_fun_hom F) × is_surjective (to_fun_hom F))
|
||||
: pi_equiv_pi_id (λc, pi_equiv_pi_id
|
||||
(λc', !is_equiv_equiv_is_embedding_times_is_surjective))
|
||||
... ≃ (Π(c : C), (Π(c' : C), is_embedding (to_fun_hom F)) ×
|
||||
(Π(c' : C), is_surjective (to_fun_hom F)))
|
||||
: pi_equiv_pi_id (λc, !equiv_prod_corec)
|
||||
... ≃ (Π(c c' : C), is_embedding (to_fun_hom F)) × full F
|
||||
: equiv_prod_corec
|
||||
... ≃ faithful F × full F
|
||||
: prod_equiv_prod_right (pi_equiv_pi_id (λc, pi_equiv_pi_id
|
||||
(λc', !is_embedding_equiv_is_injective)))
|
||||
-/
|
||||
|
||||
/- closure properties -/
|
||||
|
||||
definition is_isomorphism_id [instance] (C : Precategory) : is_isomorphism (1 : C ⇒ C) :=
|
||||
|
|
@ -482,16 +261,16 @@ namespace category
|
|||
definition is_equivalence_id (C : Precategory) : is_equivalence (1 : C ⇒ C) := _
|
||||
|
||||
definition is_equivalence_strict_inverse (F : C ⇒ D) [K : is_equivalence F]
|
||||
: is_equivalence F ⁻¹F :=
|
||||
: is_equivalence F ⁻¹ᴱ :=
|
||||
is_equivalence.mk F (iso_counit F) (iso_unit F)
|
||||
|
||||
/-
|
||||
definition is_equivalence_compose (G : D ⇒ E) (F : C ⇒ D)
|
||||
[H : is_equivalence G] [K : is_equivalence F] : is_equivalence (G ∘f F) :=
|
||||
is_equivalence.mk
|
||||
(F ⁻¹F ∘f G ⁻¹F)
|
||||
(F ⁻¹ᴱ ∘f G ⁻¹ᴱ)
|
||||
abstract begin
|
||||
rewrite [functor.assoc,-functor.assoc F ⁻¹F],
|
||||
rewrite [functor.assoc,-functor.assoc F ⁻¹ᴱ],
|
||||
-- refine ((_ ∘fi _) ∘if _) ⬝i _,
|
||||
end end
|
||||
abstract begin
|
||||
|
|
@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
|
|||
Authors: Floris van Doorn, Jakob von Raumer
|
||||
-/
|
||||
|
||||
import .iso types.pi
|
||||
import ..iso types.pi
|
||||
|
||||
open function category eq prod prod.ops equiv is_equiv sigma sigma.ops is_trunc funext iso
|
||||
open pi
|
||||
10
hott/algebra/category/functor/functor.md
Normal file
10
hott/algebra/category/functor/functor.md
Normal file
|
|
@ -0,0 +1,10 @@
|
|||
algebra.category.functor
|
||||
========================
|
||||
|
||||
* [functor](functor.hlean) : Definition of functors
|
||||
* [curry](curry.hlean) : Define currying and uncurrying of functors
|
||||
* [attributes](attributes.hlean): Attributes of functors (full, faithful, split essentially surjective, ...)
|
||||
* [adjoint](adjoint.hlean) : Adjoint functors and equivalences
|
||||
* [equivalence](equivalence.hlean) : Equivalences and Isomorphisms
|
||||
|
||||
Note: the functor category is defined in [constructions.functor](../constructions/functor.hlean).
|
||||
|
|
@ -3,7 +3,8 @@ Copyright (c) 2015 Floris van Doorn. All rights reserved.
|
|||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Author: Floris van Doorn, Jakob von Raumer
|
||||
-/
|
||||
import .functor .iso
|
||||
|
||||
import .functor.functor
|
||||
open eq category functor is_trunc equiv sigma.ops sigma is_equiv function pi funext iso
|
||||
|
||||
structure nat_trans {C : Precategory} {D : Precategory} (F G : C ⇒ D)
|
||||
|
|
|
|||
|
|
@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
|
|||
Authors: Floris van Doorn, Jakob von Raumer
|
||||
-/
|
||||
|
||||
import .precategory .functor
|
||||
import .functor.functor
|
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|
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open is_trunc eq
|
||||
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|
|
@ -46,35 +46,3 @@ namespace category
|
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end ops
|
||||
|
||||
end category
|
||||
|
||||
/-section
|
||||
open decidable unit empty
|
||||
variables {A : Type} [H : decidable_eq A]
|
||||
include H
|
||||
definition set_hom (a b : A) := decidable.rec_on (H a b) (λh, unit) (λh, empty)
|
||||
theorem set_hom_subsingleton [instance] (a b : A) : subsingleton (set_hom a b) := _
|
||||
definition set_compose {a b c : A} (g : set_hom b c) (f : set_hom a b) : set_hom a c :=
|
||||
decidable.rec_on
|
||||
(H b c)
|
||||
(λ Hbc g, decidable.rec_on
|
||||
(H a b)
|
||||
(λ Hab f, rec_on_true (trans Hab Hbc) ⋆)
|
||||
(λh f, empty.rec _ f) f)
|
||||
(λh (g : empty), empty.rec _ g) g
|
||||
omit H
|
||||
definition discrete_precategory (A : Type) [H : decidable_eq A] : precategory A :=
|
||||
mk (λa b, set_hom a b)
|
||||
(λ a b c g f, set_compose g f)
|
||||
(λ a, decidable.rec_on_true rfl ⋆)
|
||||
(λ a b c d h g f, @subsingleton.elim (set_hom a d) _ _ _)
|
||||
(λ a b f, @subsingleton.elim (set_hom a b) _ _ _)
|
||||
(λ a b f, @subsingleton.elim (set_hom a b) _ _ _)
|
||||
definition Discrete_category (A : Type) [H : decidable_eq A] := Mk (discrete_category A)
|
||||
end
|
||||
section
|
||||
open unit bool
|
||||
definition category_one := discrete_category unit
|
||||
definition Category_one := Mk category_one
|
||||
definition category_two := discrete_category bool
|
||||
definition Category_two := Mk category_two
|
||||
end-/
|
||||
|
|
|
|||
|
|
@ -6,7 +6,7 @@ Authors: Floris van Doorn
|
|||
Yoneda embedding and Yoneda lemma
|
||||
-/
|
||||
|
||||
import .curry .constructions.hset .constructions.opposite .adjoint
|
||||
import .functor.curry .constructions.hset .constructions.opposite .functor.attributes
|
||||
|
||||
open category eq functor prod.ops is_trunc iso is_equiv equiv category.set nat_trans lift
|
||||
|
||||
|
|
|
|||
Loading…
Reference in a new issue