refactor(category): move some files to subfolders, and create file with basic functors
This commit is contained in:
Floris van Doorn
Leonardo de Moura
parent
e14754a337
commit
46dba4ee5e
22 changed files with 359 additions and 306 deletions
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@ -11,6 +11,4 @@ Development of Category Theory. The following files are in this folder (sorted s
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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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* [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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* [yoneda](yoneda.hlean) : Yoneda Embedding (WIP)
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* [limits](limits/limits.md) (subfolder) : Limits and colimits in precategories
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@ -5,7 +5,7 @@ Common categories and constructions on categories. The following files are in th
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* [functor](functor.hlean) : Functor category
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* [opposite](opposite.hlean) : Opposite category
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* [hset](hset.hlean) : Category of sets. Includes proof that it is complete and cocomplete
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* [set](set.hlean) : Category of sets
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* [sum](sum.hlean) : Sum category
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* [product](product.hlean) : Product category
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* [comma](comma.hlean) : Comma category
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@ -18,6 +18,3 @@ Discrete, indiscrete or finite categories:
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* [indiscrete](indiscrete.hlean)
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* [terminal](terminal.hlean)
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* [initial](initial.hlean)
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Non-basic topics:
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* [functor2](functor2.hlean) : showing that the functor category has (co)limits if the codomain has them.
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@ -4,4 +4,4 @@ Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn
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-/
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import .functor .hset .opposite .product .comma
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import .functor .set .opposite .product .comma .sum .discrete .indiscrete .terminal .initial
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@ -123,6 +123,8 @@ namespace category
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{ intro c c' f, apply prod_eq: esimp:apply naturality}
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end
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infixr ` ×n `:70 := prod_nat_trans
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definition prod_flip_functor [constructor] (C D : Precategory) : C ×c D ⇒ D ×c C :=
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functor.mk (λp, (p.2, p.1))
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(λp p' h, (h.2, h.1))
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103
hott/algebra/category/constructions/set.hlean
Normal file
103
hott/algebra/category/constructions/set.hlean
Normal file
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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, Jakob von Raumer
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Category of sets
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-/
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import ..functor.basic ..category types.equiv types.lift
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open eq category equiv iso is_equiv is_trunc function sigma
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namespace category
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definition precategory_hset.{u} [reducible] [constructor] : precategory hset.{u} :=
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precategory.mk (λx y : hset, x → y)
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(λx y z g f a, g (f a))
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(λx a, a)
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(λx y z w h g f, eq_of_homotopy (λa, idp))
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(λx y f, eq_of_homotopy (λa, idp))
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(λx y f, eq_of_homotopy (λa, idp))
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definition Precategory_hset [reducible] [constructor] : Precategory :=
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Precategory.mk hset precategory_hset
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abbreviation set [constructor] := Precategory_hset
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namespace set
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local attribute is_equiv_subtype_eq [instance]
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definition iso_of_equiv [constructor] {A B : set} (f : A ≃ B) : A ≅ B :=
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iso.MK (to_fun f)
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(to_inv f)
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(eq_of_homotopy (left_inv (to_fun f)))
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(eq_of_homotopy (right_inv (to_fun f)))
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definition equiv_of_iso [constructor] {A B : set} (f : A ≅ B) : A ≃ B :=
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begin
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apply equiv.MK (to_hom f) (iso.to_inv f),
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exact ap10 (to_right_inverse f),
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exact ap10 (to_left_inverse f)
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end
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definition is_equiv_iso_of_equiv [constructor] (A B : set)
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: is_equiv (@iso_of_equiv A B) :=
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adjointify _ (λf, equiv_of_iso f)
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(λf, proof iso_eq idp qed)
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(λf, equiv_eq idp)
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local attribute is_equiv_iso_of_equiv [instance]
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definition iso_of_eq_eq_compose (A B : hset) : @iso_of_eq _ _ A B =
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@iso_of_equiv A B ∘ @equiv_of_eq A B ∘ subtype_eq_inv _ _ ∘
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@ap _ _ (to_fun (trunctype.sigma_char 0)) A B :=
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eq_of_homotopy (λp, eq.rec_on p idp)
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definition equiv_equiv_iso (A B : set) : (A ≃ B) ≃ (A ≅ B) :=
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equiv.MK (λf, iso_of_equiv f)
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(λf, proof equiv.MK (to_hom f)
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(iso.to_inv f)
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(ap10 (to_right_inverse f))
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(ap10 (to_left_inverse f)) qed)
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(λf, proof iso_eq idp qed)
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(λf, proof equiv_eq idp qed)
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definition equiv_eq_iso (A B : set) : (A ≃ B) = (A ≅ B) :=
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ua !equiv_equiv_iso
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definition is_univalent_hset (A B : set) : is_equiv (iso_of_eq : A = B → A ≅ B) :=
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assert H₁ : is_equiv (@iso_of_equiv A B ∘ @equiv_of_eq A B ∘ subtype_eq_inv _ _ ∘
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@ap _ _ (to_fun (trunctype.sigma_char 0)) A B), from
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@is_equiv_compose _ _ _ _ _
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(@is_equiv_compose _ _ _ _ _
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(@is_equiv_compose _ _ _ _ _
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_
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(@is_equiv_subtype_eq_inv _ _ _ _ _))
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!univalence)
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!is_equiv_iso_of_equiv,
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let H₂ := (iso_of_eq_eq_compose A B)⁻¹ in
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begin
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rewrite H₂ at H₁,
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assumption
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end
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end set
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definition category_hset [instance] [constructor] [reducible] : category hset :=
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category.mk precategory_hset set.is_univalent_hset
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definition Category_hset [reducible] [constructor] : Category :=
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Category.mk hset category_hset
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abbreviation cset [constructor] := Category_hset
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open functor lift
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definition lift_functor.{u v} [constructor] : set.{u} ⇒ set.{max u v} :=
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functor.mk tlift
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(λa b, lift_functor)
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(λa, eq_of_homotopy (λx, by induction x; reflexivity))
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(λa b c g f, eq_of_homotopy (λx, by induction x; reflexivity))
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open pi sigma.ops
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end category
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@ -107,6 +107,6 @@ namespace category
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{ intro a b f, induction a: induction b: esimp at *; induction f with f; esimp;
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try contradiction: apply naturality}
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end
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infixr ` +n `:65 := sum_nat_trans
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end category
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@ -3,12 +3,12 @@ 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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TODO: move
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TODO: merge with adjoint
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-/
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import .adjoint ..yoneda
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import .adjoint .examples
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open eq functor nat_trans yoneda iso prod is_trunc
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open eq functor nat_trans iso prod is_trunc
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namespace category
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@ -5,173 +5,3 @@ 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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open category prod nat_trans eq prod.ops iso equiv
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namespace functor
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variables {C D E : Precategory} (F F' : C ×c D ⇒ E) (G G' : C ⇒ E ^c D)
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definition functor_curry_ob [reducible] [constructor] (c : C) : E ^c D :=
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functor.mk (λd, F (c,d))
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(λd d' g, F (id, g))
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(λd, !respect_id)
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(λd₁ d₂ d₃ g' g, calc
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F (id, g' ∘ g) = F (id ∘ id, g' ∘ g) : by rewrite id_id
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... = F ((id,g') ∘ (id, g)) : by esimp
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... = F (id,g') ∘ F (id, g) : by rewrite respect_comp)
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definition functor_curry_hom [constructor] ⦃c c' : C⦄ (f : c ⟶ c')
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: functor_curry_ob F c ⟹ functor_curry_ob F c' :=
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begin
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fapply nat_trans.mk,
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{intro d, exact F (f, id)},
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{intro d d' g, calc
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F (id, g) ∘ F (f, id) = F (id ∘ f, g ∘ id) : respect_comp F
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... = F (f, g ∘ id) : by rewrite id_left
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... = F (f, g) : by rewrite id_right
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... = F (f ∘ id, g) : by rewrite id_right
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... = F (f ∘ id, id ∘ g) : by rewrite id_left
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... = F (f, id) ∘ F (id, g) : (respect_comp F (f, id) (id, g))⁻¹ᵖ
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}
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end
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local abbreviation Fhom [constructor] := @functor_curry_hom
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theorem functor_curry_hom_def ⦃c c' : C⦄ (f : c ⟶ c') (d : D) :
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(Fhom F f) d = to_fun_hom F (f, id) := idp
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theorem functor_curry_id (c : C) : Fhom F (ID c) = nat_trans.id :=
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nat_trans_eq (λd, respect_id F _)
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theorem functor_curry_comp ⦃c c' c'' : C⦄ (f' : c' ⟶ c'') (f : c ⟶ c')
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: Fhom F (f' ∘ f) = Fhom F f' ∘n Fhom F f :=
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begin
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apply @nat_trans_eq,
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intro d, calc
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natural_map (Fhom F (f' ∘ f)) d = F (f' ∘ f, id) : by rewrite functor_curry_hom_def
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... = F (f' ∘ f, id ∘ id) : by rewrite id_id
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... = F ((f',id) ∘ (f, id)) : by esimp
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... = F (f',id) ∘ F (f, id) : by rewrite [respect_comp F]
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... = natural_map ((Fhom F f') ∘ (Fhom F f)) d : by esimp
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end
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definition functor_curry [reducible] [constructor] : C ⇒ E ^c D :=
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functor.mk (functor_curry_ob F)
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(functor_curry_hom F)
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(functor_curry_id F)
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(functor_curry_comp F)
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definition functor_uncurry_ob [reducible] (p : C ×c D) : E :=
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to_fun_ob (G p.1) p.2
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definition functor_uncurry_hom ⦃p p' : C ×c D⦄ (f : hom p p')
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: functor_uncurry_ob G p ⟶ functor_uncurry_ob G p' :=
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to_fun_hom (to_fun_ob G p'.1) f.2 ∘ natural_map (to_fun_hom G f.1) p.2
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local abbreviation Ghom := @functor_uncurry_hom
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theorem functor_uncurry_id (p : C ×c D) : Ghom G (ID p) = id :=
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calc
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Ghom G (ID p) = to_fun_hom (to_fun_ob G p.1) id ∘ natural_map (to_fun_hom G id) p.2 : by esimp
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... = id ∘ natural_map (to_fun_hom G id) p.2 : by rewrite respect_id
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... = id ∘ natural_map nat_trans.id p.2 : by rewrite respect_id
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... = id : id_id
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theorem functor_uncurry_comp ⦃p p' p'' : C ×c D⦄ (f' : p' ⟶ p'') (f : p ⟶ p')
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: Ghom G (f' ∘ f) = Ghom G f' ∘ Ghom G f :=
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calc
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Ghom G (f' ∘ f)
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= to_fun_hom (to_fun_ob G p''.1) (f'.2 ∘ f.2) ∘ natural_map (to_fun_hom G (f'.1 ∘ f.1)) p.2 : by esimp
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... = (to_fun_hom (to_fun_ob G p''.1) f'.2 ∘ to_fun_hom (to_fun_ob G p''.1) f.2)
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∘ natural_map (to_fun_hom G (f'.1 ∘ f.1)) p.2 : by rewrite respect_comp
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... = (to_fun_hom (to_fun_ob G p''.1) f'.2 ∘ to_fun_hom (to_fun_ob G p''.1) f.2)
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∘ natural_map (to_fun_hom G f'.1 ∘ to_fun_hom G f.1) p.2 : by rewrite respect_comp
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... = (to_fun_hom (to_fun_ob G p''.1) f'.2 ∘ to_fun_hom (to_fun_ob G p''.1) f.2)
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∘ (natural_map (to_fun_hom G f'.1) p.2 ∘ natural_map (to_fun_hom G f.1) p.2) : by esimp
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... = (to_fun_hom (to_fun_ob G p''.1) f'.2 ∘ natural_map (to_fun_hom G f'.1) p'.2)
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∘ (to_fun_hom (to_fun_ob G p'.1) f.2 ∘ natural_map (to_fun_hom G f.1) p.2) :
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by rewrite [square_prepostcompose (!naturality⁻¹ᵖ) _ _]
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... = Ghom G f' ∘ Ghom G f : by esimp
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definition functor_uncurry [reducible] [constructor] : C ×c D ⇒ E :=
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functor.mk (functor_uncurry_ob G)
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(functor_uncurry_hom G)
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(functor_uncurry_id G)
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(functor_uncurry_comp G)
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theorem functor_uncurry_functor_curry : functor_uncurry (functor_curry F) = F :=
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functor_eq (λp, ap (to_fun_ob F) !prod.eta)
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begin
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intro cd cd' fg,
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cases cd with c d, cases cd' with c' d', cases fg with f g,
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transitivity to_fun_hom (functor_uncurry (functor_curry F)) (f, g),
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apply id_leftright,
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show (functor_uncurry (functor_curry F)) (f, g) = F (f,g),
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from calc
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(functor_uncurry (functor_curry F)) (f, g) = to_fun_hom F (id, g) ∘ to_fun_hom F (f, id) : by esimp
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... = F (id ∘ f, g ∘ id) : by krewrite [-respect_comp F (id,g) (f,id)]
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... = F (f, g ∘ id) : by rewrite id_left
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... = F (f,g) : by rewrite id_right,
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end
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definition functor_curry_functor_uncurry_ob (c : C)
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: functor_curry (functor_uncurry G) c = G c :=
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begin
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fapply functor_eq,
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{ intro d, reflexivity},
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{ intro d d' g, refine !id_leftright ⬝ _, esimp,
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rewrite [▸*, ↑functor_uncurry_hom, respect_id, ▸*, id_right]}
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end
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theorem functor_curry_functor_uncurry : functor_curry (functor_uncurry G) = G :=
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begin
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fapply functor_eq, exact (functor_curry_functor_uncurry_ob G),
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intro c c' f,
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fapply nat_trans_eq,
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intro d,
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apply concat,
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{apply (ap (λx, x ∘ _)),
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apply concat, apply natural_map_hom_of_eq, apply (ap hom_of_eq), apply ap010_functor_eq},
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apply concat,
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{apply (ap (λx, _ ∘ x)), apply (ap (λx, _ ∘ x)),
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apply concat, apply natural_map_inv_of_eq,
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apply (ap (λx, hom_of_eq x⁻¹)), apply ap010_functor_eq},
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apply concat, apply id_leftright,
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apply concat, apply (ap (λx, x ∘ _)), apply respect_id,
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apply id_left
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end
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definition prod_functor_equiv_functor_functor [constructor] (C D E : Precategory)
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: (C ×c D ⇒ E) ≃ (C ⇒ E ^c D) :=
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equiv.MK functor_curry
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functor_uncurry
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functor_curry_functor_uncurry
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functor_uncurry_functor_curry
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variables {F F' G G'}
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definition nat_trans_curry_nat [constructor] (η : F ⟹ F') (c : C)
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: functor_curry_ob F c ⟹ functor_curry_ob F' c :=
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begin
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fapply nat_trans.mk: esimp,
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{ intro d, exact η (c, d)},
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{ intro d d' f, apply naturality}
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end
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definition nat_trans_curry [constructor] (η : F ⟹ F')
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: functor_curry F ⟹ functor_curry F' :=
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begin
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fapply nat_trans.mk: esimp,
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{ exact nat_trans_curry_nat η},
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{ intro c c' f, apply nat_trans_eq, intro d, esimp, apply naturality}
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end
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definition nat_trans_uncurry [constructor] (η : G ⟹ G')
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: functor_uncurry G ⟹ functor_uncurry G' :=
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begin
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fapply nat_trans.mk: esimp,
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{ intro v, unfold functor_uncurry_ob, exact (η v.1) v.2},
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{ intro v w f, unfold functor_uncurry_hom,
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rewrite [-assoc, ap010 natural_map (naturality η f.1) v.2, assoc, naturality, -assoc]}
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end
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end functor
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208
hott/algebra/category/functor/examples.hlean
Normal file
208
hott/algebra/category/functor/examples.hlean
Normal file
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@ -0,0 +1,208 @@
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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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Definition of functors involving at least two different constructions of categories
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-/
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import ..constructions.functor ..constructions.product ..constructions.opposite
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..constructions.set
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open category nat_trans eq prod prod.ops
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namespace functor
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section
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open iso equiv
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variables {C D E : Precategory} (F F' : C ×c D ⇒ E) (G G' : C ⇒ E ^c D)
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/- currying a functor -/
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definition functor_curry_ob [reducible] [constructor] (c : C) : D ⇒ E :=
|
||||
F ∘f (constant_functor D c ×f 1)
|
||||
|
||||
definition functor_curry_hom [constructor] ⦃c c' : C⦄ (f : c ⟶ c')
|
||||
: functor_curry_ob F c ⟹ functor_curry_ob F c' :=
|
||||
F ∘fn (constant_nat_trans D f ×n 1)
|
||||
|
||||
local abbreviation Fhom [constructor] := @functor_curry_hom
|
||||
|
||||
theorem functor_curry_id (c : C) : Fhom F (ID c) = nat_trans.id :=
|
||||
nat_trans_eq (λd, respect_id F _)
|
||||
|
||||
theorem functor_curry_comp ⦃c c' c'' : C⦄ (f' : c' ⟶ c'') (f : c ⟶ c')
|
||||
: Fhom F (f' ∘ f) = Fhom F f' ∘n Fhom F f :=
|
||||
begin
|
||||
apply @nat_trans_eq,
|
||||
intro d, calc
|
||||
natural_map (Fhom F (f' ∘ f)) d = F (f' ∘ f, id) : by esimp
|
||||
... = F (f' ∘ f, id ∘ id) : by rewrite id_id
|
||||
... = F ((f',id) ∘ (f, id)) : by esimp
|
||||
... = F (f',id) ∘ F (f, id) : by rewrite [respect_comp F]
|
||||
... = natural_map ((Fhom F f') ∘ (Fhom F f)) d : by esimp
|
||||
end
|
||||
|
||||
definition functor_curry [reducible] [constructor] : C ⇒ E ^c D :=
|
||||
functor.mk (functor_curry_ob F)
|
||||
(functor_curry_hom F)
|
||||
(functor_curry_id F)
|
||||
(functor_curry_comp F)
|
||||
|
||||
/- uncurrying a functor -/
|
||||
|
||||
definition functor_uncurry_ob [reducible] (p : C ×c D) : E :=
|
||||
to_fun_ob (G p.1) p.2
|
||||
|
||||
definition functor_uncurry_hom ⦃p p' : C ×c D⦄ (f : hom p p')
|
||||
: functor_uncurry_ob G p ⟶ functor_uncurry_ob G p' :=
|
||||
to_fun_hom (to_fun_ob G p'.1) f.2 ∘ natural_map (to_fun_hom G f.1) p.2
|
||||
local abbreviation Ghom := @functor_uncurry_hom
|
||||
|
||||
theorem functor_uncurry_id (p : C ×c D) : Ghom G (ID p) = id :=
|
||||
calc
|
||||
Ghom G (ID p) = to_fun_hom (to_fun_ob G p.1) id ∘ natural_map (to_fun_hom G id) p.2 : by esimp
|
||||
... = id ∘ natural_map (to_fun_hom G id) p.2 : by rewrite respect_id
|
||||
... = id ∘ natural_map nat_trans.id p.2 : by rewrite respect_id
|
||||
... = id : id_id
|
||||
|
||||
theorem functor_uncurry_comp ⦃p p' p'' : C ×c D⦄ (f' : p' ⟶ p'') (f : p ⟶ p')
|
||||
: Ghom G (f' ∘ f) = Ghom G f' ∘ Ghom G f :=
|
||||
calc
|
||||
Ghom G (f' ∘ f)
|
||||
= to_fun_hom (to_fun_ob G p''.1) (f'.2 ∘ f.2) ∘ natural_map (to_fun_hom G (f'.1 ∘ f.1)) p.2 : by esimp
|
||||
... = (to_fun_hom (to_fun_ob G p''.1) f'.2 ∘ to_fun_hom (to_fun_ob G p''.1) f.2)
|
||||
∘ natural_map (to_fun_hom G (f'.1 ∘ f.1)) p.2 : by rewrite respect_comp
|
||||
... = (to_fun_hom (to_fun_ob G p''.1) f'.2 ∘ to_fun_hom (to_fun_ob G p''.1) f.2)
|
||||
∘ natural_map (to_fun_hom G f'.1 ∘ to_fun_hom G f.1) p.2 : by rewrite respect_comp
|
||||
... = (to_fun_hom (to_fun_ob G p''.1) f'.2 ∘ to_fun_hom (to_fun_ob G p''.1) f.2)
|
||||
∘ (natural_map (to_fun_hom G f'.1) p.2 ∘ natural_map (to_fun_hom G f.1) p.2) : by esimp
|
||||
... = (to_fun_hom (to_fun_ob G p''.1) f'.2 ∘ natural_map (to_fun_hom G f'.1) p'.2)
|
||||
∘ (to_fun_hom (to_fun_ob G p'.1) f.2 ∘ natural_map (to_fun_hom G f.1) p.2) :
|
||||
by rewrite [square_prepostcompose (!naturality⁻¹ᵖ) _ _]
|
||||
... = Ghom G f' ∘ Ghom G f : by esimp
|
||||
|
||||
definition functor_uncurry [reducible] [constructor] : C ×c D ⇒ E :=
|
||||
functor.mk (functor_uncurry_ob G)
|
||||
(functor_uncurry_hom G)
|
||||
(functor_uncurry_id G)
|
||||
(functor_uncurry_comp G)
|
||||
|
||||
theorem functor_uncurry_functor_curry : functor_uncurry (functor_curry F) = F :=
|
||||
functor_eq (λp, ap (to_fun_ob F) !prod.eta)
|
||||
begin
|
||||
intro cd cd' fg,
|
||||
cases cd with c d, cases cd' with c' d', cases fg with f g,
|
||||
transitivity to_fun_hom (functor_uncurry (functor_curry F)) (f, g),
|
||||
apply id_leftright,
|
||||
show (functor_uncurry (functor_curry F)) (f, g) = F (f,g),
|
||||
from calc
|
||||
(functor_uncurry (functor_curry F)) (f, g) = to_fun_hom F (id, g) ∘ to_fun_hom F (f, id) : by esimp
|
||||
... = F (id ∘ f, g ∘ id) : by krewrite [-respect_comp F (id,g) (f,id)]
|
||||
... = F (f, g ∘ id) : by rewrite id_left
|
||||
... = F (f,g) : by rewrite id_right,
|
||||
end
|
||||
|
||||
definition functor_curry_functor_uncurry_ob (c : C)
|
||||
: functor_curry (functor_uncurry G) c = G c :=
|
||||
begin
|
||||
fapply functor_eq,
|
||||
{ intro d, reflexivity},
|
||||
{ intro d d' g, refine !id_leftright ⬝ _, esimp,
|
||||
rewrite [▸*, ↑functor_uncurry_hom, respect_id, ▸*, id_right]}
|
||||
end
|
||||
|
||||
theorem functor_curry_functor_uncurry : functor_curry (functor_uncurry G) = G :=
|
||||
begin
|
||||
fapply functor_eq, exact (functor_curry_functor_uncurry_ob G),
|
||||
intro c c' f,
|
||||
fapply nat_trans_eq,
|
||||
intro d,
|
||||
apply concat,
|
||||
{apply (ap (λx, x ∘ _)),
|
||||
apply concat, apply natural_map_hom_of_eq, apply (ap hom_of_eq), apply ap010_functor_eq},
|
||||
apply concat,
|
||||
{apply (ap (λx, _ ∘ x)), apply (ap (λx, _ ∘ x)),
|
||||
apply concat, apply natural_map_inv_of_eq,
|
||||
apply (ap (λx, hom_of_eq x⁻¹)), apply ap010_functor_eq},
|
||||
apply concat, apply id_leftright,
|
||||
apply concat, apply (ap (λx, x ∘ _)), apply respect_id,
|
||||
apply id_left
|
||||
end
|
||||
|
||||
/-
|
||||
This only states that the carriers of (C ^ D) ^ E and C ^ (E × D) are equivalent.
|
||||
In [exponential laws] we prove that these are in fact isomorphic categories
|
||||
-/
|
||||
definition prod_functor_equiv_functor_functor [constructor] (C D E : Precategory)
|
||||
: (C ×c D ⇒ E) ≃ (C ⇒ E ^c D) :=
|
||||
equiv.MK functor_curry
|
||||
functor_uncurry
|
||||
functor_curry_functor_uncurry
|
||||
functor_uncurry_functor_curry
|
||||
|
||||
variables {F F' G G'}
|
||||
definition nat_trans_curry_nat [constructor] (η : F ⟹ F') (c : C)
|
||||
: functor_curry_ob F c ⟹ functor_curry_ob F' c :=
|
||||
begin
|
||||
fapply nat_trans.mk: esimp,
|
||||
{ intro d, exact η (c, d)},
|
||||
{ intro d d' f, apply naturality}
|
||||
end
|
||||
|
||||
definition nat_trans_curry [constructor] (η : F ⟹ F')
|
||||
: functor_curry F ⟹ functor_curry F' :=
|
||||
begin
|
||||
fapply nat_trans.mk: esimp,
|
||||
{ exact nat_trans_curry_nat η},
|
||||
{ intro c c' f, apply nat_trans_eq, intro d, esimp, apply naturality}
|
||||
end
|
||||
|
||||
definition nat_trans_uncurry [constructor] (η : G ⟹ G')
|
||||
: functor_uncurry G ⟹ functor_uncurry G' :=
|
||||
begin
|
||||
fapply nat_trans.mk: esimp,
|
||||
{ intro v, unfold functor_uncurry_ob, exact (η v.1) v.2},
|
||||
{ intro v w f, unfold functor_uncurry_hom,
|
||||
rewrite [-assoc, ap010 natural_map (naturality η f.1) v.2, assoc, naturality, -assoc]}
|
||||
end
|
||||
end
|
||||
|
||||
section
|
||||
open is_trunc
|
||||
|
||||
/- hom-functors -/
|
||||
|
||||
definition hom_functor_assoc {C : Precategory} {a1 a2 a3 a4 a5 a6 : C}
|
||||
(f1 : hom a5 a6) (f2 : hom a4 a5) (f3 : hom a3 a4) (f4 : hom a2 a3) (f5 : hom a1 a2)
|
||||
: (f1 ∘ f2) ∘ f3 ∘ (f4 ∘ f5) = f1 ∘ (f2 ∘ f3 ∘ f4) ∘ f5 :=
|
||||
calc
|
||||
_ = f1 ∘ f2 ∘ f3 ∘ f4 ∘ f5 : by rewrite -assoc
|
||||
... = f1 ∘ (f2 ∘ f3) ∘ f4 ∘ f5 : by rewrite -assoc
|
||||
... = f1 ∘ ((f2 ∘ f3) ∘ f4) ∘ f5 : by rewrite -(assoc (f2 ∘ f3) _ _)
|
||||
... = _ : by rewrite (assoc f2 f3 f4)
|
||||
|
||||
-- the functor hom(-,-)
|
||||
definition hom_functor.{u v} [constructor] (C : Precategory.{u v}) : Cᵒᵖ ×c C ⇒ set.{v} :=
|
||||
functor.mk
|
||||
(λ (x : Cᵒᵖ ×c C), @homset (Cᵒᵖ) C x.1 x.2)
|
||||
(λ (x y : Cᵒᵖ ×c C) (f : @category.precategory.hom (Cᵒᵖ ×c C) (Cᵒᵖ ×c C) x y)
|
||||
(h : @homset (Cᵒᵖ) C x.1 x.2), f.2 ∘[C] (h ∘[C] f.1))
|
||||
(λ x, abstract @eq_of_homotopy _ _ _ (ID (@homset Cᵒᵖ C x.1 x.2))
|
||||
(λ h, concat (by apply @id_left) (by apply @id_right)) end)
|
||||
(λ x y z g f, abstract eq_of_homotopy (by intros; apply @hom_functor_assoc) end)
|
||||
|
||||
-- the functor hom(-, c)
|
||||
definition hom_functor_left.{u v} [constructor] (C : Precategory.{u v}) (c : C)
|
||||
: Cᵒᵖ ⇒ set.{v} :=
|
||||
hom_functor C ∘f (1 ×f constant_functor Cᵒᵖ c)
|
||||
|
||||
-- the functor hom(c, -)
|
||||
definition hom_functor_right.{u v} [constructor] (C : Precategory.{u v}) (c : C)
|
||||
: C ⇒ set.{v} :=
|
||||
hom_functor C ∘f (constant_functor C c ×f 1)
|
||||
|
||||
|
||||
end
|
||||
|
||||
|
||||
|
||||
end functor
|
||||
|
|
@ -6,7 +6,7 @@ Authors: Floris van Doorn, Jakob von Raumer
|
|||
Exponential laws
|
||||
-/
|
||||
|
||||
import .equivalence .curry
|
||||
import .equivalence .examples
|
||||
..constructions.terminal ..constructions.initial ..constructions.product ..constructions.sum
|
||||
..constructions.discrete
|
||||
|
||||
|
|
|
|||
|
|
@ -1,11 +1,14 @@
|
|||
algebra.category.functor
|
||||
========================
|
||||
|
||||
Functors, functor attributes, equivalences, isomorphism, adjointness.
|
||||
|
||||
* [basic](basic.hlean) : Definition and basic properties of functors
|
||||
* [curry](curry.hlean) : Define currying and uncurrying of functors
|
||||
* [examples](examples.hlean) : Constructions of functors between categories, involving more than one category in the [constructions](../constructions/constructions.md) folder (functors which only depend on one constructions are in the corresponding file). This includes the 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
|
||||
* [exponential_laws](exponential_laws.hlean)
|
||||
* [yoneda](yoneda.hlean) : the Yoneda Embedding
|
||||
|
||||
Note: the functor category is defined in [constructions.functor](../constructions/functor.hlean).
|
||||
Note: the functor category is defined in [constructions.functor](../constructions/functor.hlean). Functors preserving limits is in [limits.functor_preserve](../limits/functor_preserve.hlean).
|
||||
|
|
@ -6,30 +6,12 @@ Authors: Floris van Doorn
|
|||
Yoneda embedding and Yoneda lemma
|
||||
-/
|
||||
|
||||
import .functor.curry .constructions.hset .constructions.opposite .functor.attributes
|
||||
import .examples .attributes
|
||||
|
||||
open category eq functor prod.ops is_trunc iso is_equiv equiv category.set nat_trans lift
|
||||
|
||||
namespace yoneda
|
||||
|
||||
definition representable_functor_assoc [C : Precategory] {a1 a2 a3 a4 a5 a6 : C}
|
||||
(f1 : hom a5 a6) (f2 : hom a4 a5) (f3 : hom a3 a4) (f4 : hom a2 a3) (f5 : hom a1 a2)
|
||||
: (f1 ∘ f2) ∘ f3 ∘ (f4 ∘ f5) = f1 ∘ (f2 ∘ f3 ∘ f4) ∘ f5 :=
|
||||
calc
|
||||
_ = f1 ∘ f2 ∘ f3 ∘ f4 ∘ f5 : by rewrite -assoc
|
||||
... = f1 ∘ (f2 ∘ f3) ∘ f4 ∘ f5 : by rewrite -assoc
|
||||
... = f1 ∘ ((f2 ∘ f3) ∘ f4) ∘ f5 : by rewrite -(assoc (f2 ∘ f3) _ _)
|
||||
... = _ : by rewrite (assoc f2 f3 f4)
|
||||
|
||||
definition hom_functor.{u v} [constructor] (C : Precategory.{u v}) : Cᵒᵖ ×c C ⇒ set.{v} :=
|
||||
functor.mk
|
||||
(λ (x : Cᵒᵖ ×c C), @homset (Cᵒᵖ) C x.1 x.2)
|
||||
(λ (x y : Cᵒᵖ ×c C) (f : @category.precategory.hom (Cᵒᵖ ×c C) (Cᵒᵖ ×c C) x y)
|
||||
(h : @homset (Cᵒᵖ) C x.1 x.2), f.2 ∘[C] (h ∘[C] f.1))
|
||||
(λ x, abstract @eq_of_homotopy _ _ _ (ID (@homset Cᵒᵖ C x.1 x.2))
|
||||
(λ h, concat (by apply @id_left) (by apply @id_right)) end)
|
||||
(λ x y z g f, abstract eq_of_homotopy (by intros; apply @representable_functor_assoc) end)
|
||||
|
||||
/-
|
||||
These attributes make sure that the fields of the category "set" reduce to the right things
|
||||
However, we don't want to have them globally, because that will unfold the composition g ∘ f
|
||||
|
|
@ -45,6 +27,7 @@ namespace yoneda
|
|||
definition yoneda_embedding [constructor] (C : Precategory) : C ⇒ cset ^c Cᵒᵖ :=
|
||||
functor_curry (!hom_functor ∘f !prod_flip_functor)
|
||||
|
||||
|
||||
notation `ɏ` := yoneda_embedding _
|
||||
|
||||
definition yoneda_lemma_hom [constructor] {C : Precategory} (c : C) (F : Cᵒᵖ ⇒ cset)
|
||||
|
|
@ -6,9 +6,9 @@ Authors: Floris van Doorn
|
|||
colimit_functor ⊣ Δ ⊣ limit_functor
|
||||
-/
|
||||
|
||||
import ..colimits ..functor.adjoint2
|
||||
import .colimits ..functor.adjoint2
|
||||
|
||||
open functor category is_trunc prod yoneda --remove
|
||||
open functor category is_trunc prod
|
||||
|
||||
namespace category
|
||||
|
||||
|
|
|
|||
|
|
@ -6,7 +6,7 @@ Authors: Floris van Doorn
|
|||
Colimits in a category
|
||||
-/
|
||||
|
||||
import .limits .constructions.opposite
|
||||
import .limits ..constructions.opposite
|
||||
|
||||
open is_trunc functor nat_trans eq
|
||||
|
||||
7
hott/algebra/category/limits/default.hlean
Normal file
7
hott/algebra/category/limits/default.hlean
Normal file
|
|
@ -0,0 +1,7 @@
|
|||
/-
|
||||
Copyright (c) 2015 Floris van Doorn. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Floris van Doorn
|
||||
-/
|
||||
|
||||
import .limits .colimits
|
||||
|
|
@ -6,7 +6,7 @@ Authors: Floris van Doorn, Jakob von Raumer
|
|||
Functor category has (co)limits if the codomain has them
|
||||
-/
|
||||
|
||||
import ..colimits
|
||||
import .colimits
|
||||
|
||||
open functor nat_trans eq is_trunc
|
||||
|
||||
|
|
@ -6,7 +6,7 @@ Authors: Floris van Doorn
|
|||
Functors preserving limits
|
||||
-/
|
||||
|
||||
import ..colimits ..yoneda
|
||||
import .colimits ..functor.yoneda
|
||||
|
||||
open functor yoneda is_trunc nat_trans
|
||||
|
||||
|
|
|
|||
|
|
@ -6,8 +6,8 @@ Authors: Floris van Doorn
|
|||
Limits in a category
|
||||
-/
|
||||
|
||||
import .constructions.cone .constructions.discrete .constructions.product
|
||||
.constructions.finite_cats .category .constructions.functor
|
||||
import ..constructions.cone ..constructions.discrete ..constructions.product
|
||||
..constructions.finite_cats ..category ..constructions.functor
|
||||
|
||||
open is_trunc functor nat_trans eq
|
||||
|
||||
8
hott/algebra/category/limits/limits.md
Normal file
8
hott/algebra/category/limits/limits.md
Normal file
|
|
@ -0,0 +1,8 @@
|
|||
algebra.category.limits
|
||||
=======================
|
||||
|
||||
* [limits](limits.hlean) : Limits in a category, defined as terminal object in the cone category
|
||||
* [colimits](colimits.hlean) : Colimits in a category, defined as the limit of the opposite functor
|
||||
* [complete](complete.hlean) : Categories which are (co)complete or constructions which preserve (co)completeness
|
||||
* [functor_preserve](functor_preserve.hlean) : Functors which preserve limits and colimits
|
||||
* [adjoint](adjoint.hlean) : the (co)limit functor is adjoint to the diagonal map
|
||||
|
|
@ -3,101 +3,14 @@ Copyright (c) 2015 Floris van Doorn. All rights reserved.
|
|||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Floris van Doorn, Jakob von Raumer
|
||||
|
||||
Category of hsets
|
||||
The category of sets is complete and cocomplete
|
||||
-/
|
||||
|
||||
import ..category types.equiv types.lift ..colimits hit.set_quotient
|
||||
import .colimits ..constructions.set hit.set_quotient
|
||||
|
||||
open eq category equiv iso is_equiv is_trunc function sigma set_quotient trunc
|
||||
open eq functor is_trunc sigma pi sigma.ops trunc set_quotient
|
||||
|
||||
namespace category
|
||||
|
||||
definition precategory_hset.{u} [reducible] [constructor] : precategory hset.{u} :=
|
||||
precategory.mk (λx y : hset, x → y)
|
||||
(λx y z g f a, g (f a))
|
||||
(λx a, a)
|
||||
(λx y z w h g f, eq_of_homotopy (λa, idp))
|
||||
(λx y f, eq_of_homotopy (λa, idp))
|
||||
(λx y f, eq_of_homotopy (λa, idp))
|
||||
|
||||
definition Precategory_hset [reducible] [constructor] : Precategory :=
|
||||
Precategory.mk hset precategory_hset
|
||||
|
||||
abbreviation set [constructor] := Precategory_hset
|
||||
|
||||
namespace set
|
||||
local attribute is_equiv_subtype_eq [instance]
|
||||
definition iso_of_equiv [constructor] {A B : set} (f : A ≃ B) : A ≅ B :=
|
||||
iso.MK (to_fun f)
|
||||
(to_inv f)
|
||||
(eq_of_homotopy (left_inv (to_fun f)))
|
||||
(eq_of_homotopy (right_inv (to_fun f)))
|
||||
|
||||
definition equiv_of_iso [constructor] {A B : set} (f : A ≅ B) : A ≃ B :=
|
||||
begin
|
||||
apply equiv.MK (to_hom f) (iso.to_inv f),
|
||||
exact ap10 (to_right_inverse f),
|
||||
exact ap10 (to_left_inverse f)
|
||||
end
|
||||
|
||||
definition is_equiv_iso_of_equiv [constructor] (A B : set)
|
||||
: is_equiv (@iso_of_equiv A B) :=
|
||||
adjointify _ (λf, equiv_of_iso f)
|
||||
(λf, proof iso_eq idp qed)
|
||||
(λf, equiv_eq idp)
|
||||
|
||||
local attribute is_equiv_iso_of_equiv [instance]
|
||||
|
||||
definition iso_of_eq_eq_compose (A B : hset) : @iso_of_eq _ _ A B =
|
||||
@iso_of_equiv A B ∘ @equiv_of_eq A B ∘ subtype_eq_inv _ _ ∘
|
||||
@ap _ _ (to_fun (trunctype.sigma_char 0)) A B :=
|
||||
eq_of_homotopy (λp, eq.rec_on p idp)
|
||||
|
||||
definition equiv_equiv_iso (A B : set) : (A ≃ B) ≃ (A ≅ B) :=
|
||||
equiv.MK (λf, iso_of_equiv f)
|
||||
(λf, proof equiv.MK (to_hom f)
|
||||
(iso.to_inv f)
|
||||
(ap10 (to_right_inverse f))
|
||||
(ap10 (to_left_inverse f)) qed)
|
||||
(λf, proof iso_eq idp qed)
|
||||
(λf, proof equiv_eq idp qed)
|
||||
|
||||
definition equiv_eq_iso (A B : set) : (A ≃ B) = (A ≅ B) :=
|
||||
ua !equiv_equiv_iso
|
||||
|
||||
definition is_univalent_hset (A B : set) : is_equiv (iso_of_eq : A = B → A ≅ B) :=
|
||||
assert H₁ : is_equiv (@iso_of_equiv A B ∘ @equiv_of_eq A B ∘ subtype_eq_inv _ _ ∘
|
||||
@ap _ _ (to_fun (trunctype.sigma_char 0)) A B), from
|
||||
@is_equiv_compose _ _ _ _ _
|
||||
(@is_equiv_compose _ _ _ _ _
|
||||
(@is_equiv_compose _ _ _ _ _
|
||||
_
|
||||
(@is_equiv_subtype_eq_inv _ _ _ _ _))
|
||||
!univalence)
|
||||
!is_equiv_iso_of_equiv,
|
||||
let H₂ := (iso_of_eq_eq_compose A B)⁻¹ in
|
||||
begin
|
||||
rewrite H₂ at H₁,
|
||||
assumption
|
||||
end
|
||||
end set
|
||||
|
||||
definition category_hset [instance] [constructor] [reducible] : category hset :=
|
||||
category.mk precategory_hset set.is_univalent_hset
|
||||
|
||||
definition Category_hset [reducible] [constructor] : Category :=
|
||||
Category.mk hset category_hset
|
||||
|
||||
abbreviation cset [constructor] := Category_hset
|
||||
|
||||
open functor lift
|
||||
definition lift_functor.{u v} [constructor] : set.{u} ⇒ set.{max u v} :=
|
||||
functor.mk tlift
|
||||
(λa b, lift_functor)
|
||||
(λa, eq_of_homotopy (λx, by induction x; reflexivity))
|
||||
(λa b c g f, eq_of_homotopy (λx, by induction x; reflexivity))
|
||||
|
||||
open pi sigma.ops
|
||||
local attribute Category.to.precategory [unfold 1]
|
||||
local attribute category.to_precategory [unfold 2]
|
||||
|
||||
|
|
@ -154,6 +67,7 @@ namespace category
|
|||
{ exact _}
|
||||
end
|
||||
|
||||
|
||||
definition is_cocomplete_set_cone.{u v w} [constructor]
|
||||
(I : Precategory.{v w}) (F : I ⇒ set.{max u v w}ᵒᵖ) : cone_obj F :=
|
||||
begin
|
||||
|
|
@ -189,6 +103,4 @@ namespace category
|
|||
{ intro v w r, apply is_hprop.elimo}}},
|
||||
end
|
||||
|
||||
|
||||
|
||||
end category
|
||||
|
|
@ -29,7 +29,9 @@ The rows indicate the chapters, the columns the sections.
|
|||
|
||||
Theorems and definitions in the library which are not in the book:
|
||||
|
||||
* One major difference is that in this library we heavily use pathovers, so we need less theorems about transports, but instead corresponding theorems about pathovers. These are in [init.pathover](init/pathover.hlean). For higher paths there are [squares](cubical/square.hlean), [squareovers](cubical/squareover.hlean), and the rudiments of [cubes](cubical/cube.hlean) and [cubeovers](cubical/cubeover.hlean).
|
||||
* A major difference is that in this library we heavily use pathovers [D. Licata, G. Brunerie. A Cubical Approach to Synthetic Homotopy Theory]. This means that we need less theorems about transports, but instead corresponding theorems about pathovers. These are in [init.pathover](init/pathover.hlean). For higher paths there are [squares](cubical/square.hlean), [squareovers](cubical/squareover.hlean), and the rudiments of [cubes](cubical/cube.hlean) and [cubeovers](cubical/cubeover.hlean).
|
||||
|
||||
* The category theory library is more extensive than what is presented in the book. For example, we have [limits](algebra/category/limits/limits.md).
|
||||
|
||||
Chapter 1: Type theory
|
||||
---------
|
||||
|
|
@ -159,7 +161,7 @@ Every file is in the folder [algebra.category](algebra/category/category.md)
|
|||
- 9.2 (Functors and transformations): [functor.basic](algebra/category/functor/basic.hlean), [nat_trans](algebra/category/nat_trans.hlean), [constructions.functor](algebra/category/constructions/functor.hlean)
|
||||
- 9.3 (Adjunctions): [functor.adjoint](algebra/category/functor/adjoint.hlean)
|
||||
- 9.4 (Equivalences): [functor.equivalence](algebra/category/functor/equivalence.hlean) and [functor.attributes](algebra/category/functor/attributes.hlean) (partially)
|
||||
- 9.5 (The Yoneda lemma): [constructions.opposite](algebra/category/constructions/opposite.hlean), [constructions.product](algebra/category/constructions/product.hlean), [yoneda](algebra/category/yoneda.hlean) (up to Theorem 9.5.9)
|
||||
- 9.5 (The Yoneda lemma): [constructions.opposite](algebra/category/constructions/opposite.hlean), [constructions.product](algebra/category/constructions/product.hlean), [functor.yoneda](algebra/category/functor/yoneda.hlean) (up to Theorem 9.5.9)
|
||||
- 9.6 (Strict categories): [strict](algebra/category/strict.hlean) (only definition)
|
||||
- 9.7 (†-categories): not formalized
|
||||
- 9.8 (The structure identity principle): not formalized
|
||||
|
|
|
|||
|
|
@ -41,7 +41,7 @@
|
|||
"⁻¹ᵉ" "⁻¹ᶠ" "⁻¹ᵍ" "⁻¹ʰ" "⁻¹ⁱ" "⁻¹ᵐ" "⁻¹ᵒ" "⁻¹ᵖ" "⁻¹ʳ" "⁻¹ᵛ" "⁻¹ˢ" "⁻²" "⁻²ᵒ"
|
||||
"⬝e" "⬝i" "⬝o" "⬝op" "⬝po" "⬝h" "⬝v" "⬝hp" "⬝vp" "⬝ph" "⬝pv" "⬝r" "◾" "◾o"
|
||||
"∘n" "∘f" "∘fi" "∘nf" "∘fn" "∘n1f" "∘1nf" "∘f1n" "∘fn1"
|
||||
"^c" "≃c" "≅c" "×c" "×f" "+c" "+f")
|
||||
"^c" "≃c" "≅c" "×c" "×f" "×n" "+c" "+f" "+n")
|
||||
"lean constants")
|
||||
(defconst lean-constants-regexp (regexp-opt lean-constants))
|
||||
(defconst lean-numerals-regexp
|
||||
|
|
|
|||
Loading…
Reference in a new issue