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forget_enrichment.of (X : C) : forget_enrichment W C | X | def | category_theory.forget_enrichment.of | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | Typecheck an object of `C` as an object of `forget_enrichment W C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_enrichment.to (X : forget_enrichment W C) : C | X | def | category_theory.forget_enrichment.to | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | Typecheck an object of `forget_enrichment W C` as an object of `C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_enrichment.to_of (X : C) :
forget_enrichment.to W (forget_enrichment.of W X) = X | rfl | lemma | category_theory.forget_enrichment.to_of | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_enrichment.of_to (X : forget_enrichment W C) :
forget_enrichment.of W (forget_enrichment.to W X) = X | rfl | lemma | category_theory.forget_enrichment.of_to | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
category_forget_enrichment : category (forget_enrichment W C) | begin
let I : enriched_category (Type v) (transport_enrichment (coyoneda_tensor_unit W) C) :=
infer_instance,
exact enriched_category_Type_equiv_category C I,
end | instance | category_theory.category_forget_enrichment | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_enrichment.hom_of {X Y : C} (f : (𝟙_ W) ⟶ (X ⟶[W] Y)) :
forget_enrichment.of W X ⟶ forget_enrichment.of W Y | f | def | category_theory.forget_enrichment.hom_of | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | Typecheck a `(𝟙_ W)`-shaped `W`-morphism as a morphism in `forget_enrichment W C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_enrichment.hom_to {X Y : forget_enrichment W C} (f : X ⟶ Y) :
(𝟙_ W) ⟶ (forget_enrichment.to W X ⟶[W] forget_enrichment.to W Y) | f | def | category_theory.forget_enrichment.hom_to | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | Typecheck a morphism in `forget_enrichment W C` as a `(𝟙_ W)`-shaped `W`-morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_enrichment.hom_to_hom_of {X Y : C} (f : (𝟙_ W) ⟶ (X ⟶[W] Y)) :
forget_enrichment.hom_to W (forget_enrichment.hom_of W f) = f | rfl | lemma | category_theory.forget_enrichment.hom_to_hom_of | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_enrichment.hom_of_hom_to {X Y : forget_enrichment W C} (f : X ⟶ Y) :
forget_enrichment.hom_of W (forget_enrichment.hom_to W f) = f | rfl | lemma | category_theory.forget_enrichment.hom_of_hom_to | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_enrichment_id (X : forget_enrichment W C) :
forget_enrichment.hom_to W (𝟙 X) = (e_id W (forget_enrichment.to W X : C)) | category.id_comp _ | lemma | category_theory.forget_enrichment_id | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | The identity in the "underlying" category of an enriched category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_enrichment_id' (X : C) :
forget_enrichment.hom_of W (e_id W X) = (𝟙 (forget_enrichment.of W X : C)) | (forget_enrichment_id W (forget_enrichment.of W X)).symm | lemma | category_theory.forget_enrichment_id' | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_enrichment_comp {X Y Z : forget_enrichment W C} (f : X ⟶ Y) (g : Y ⟶ Z) :
forget_enrichment.hom_to W (f ≫ g) = (((λ_ (𝟙_ W)).inv ≫
(forget_enrichment.hom_to W f ⊗ forget_enrichment.hom_to W g)) ≫ e_comp W _ _ _) | rfl | lemma | category_theory.forget_enrichment_comp | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | Composition in the "underlying" category of an enriched category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
enriched_functor
(C : Type u₁) [enriched_category V C] (D : Type u₂) [enriched_category V D] | (obj : C → D)
(map : Π X Y : C, (X ⟶[V] Y) ⟶ (obj X ⟶[V] obj Y))
(map_id' : ∀ X : C, e_id V X ≫ map X X = e_id V (obj X) . obviously)
(map_comp' : ∀ X Y Z : C,
e_comp V X Y Z ≫ map X Z = (map X Y ⊗ map Y Z) ≫ e_comp V (obj X) (obj Y) (obj Z) . obviously) | structure | category_theory.enriched_functor | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | A `V`-functor `F` between `V`-enriched categories
has a `V`-morphism from `X ⟶[V] Y` to `F.obj X ⟶[V] F.obj Y`,
satisfying the usual axioms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
enriched_functor.id (C : Type u₁) [enriched_category V C] : enriched_functor V C C | { obj := λ X, X,
map := λ X Y, 𝟙 _, } | def | category_theory.enriched_functor.id | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | The identity enriched functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
enriched_functor.comp {C : Type u₁} {D : Type u₂} {E : Type u₃}
[enriched_category V C] [enriched_category V D] [enriched_category V E]
(F : enriched_functor V C D) (G : enriched_functor V D E) :
enriched_functor V C E | { obj := λ X, G.obj (F.obj X),
map := λ X Y, F.map _ _ ≫ G.map _ _, } | def | category_theory.enriched_functor.comp | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | Composition of enriched functors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
enriched_functor.forget {C : Type u₁} {D : Type u₂}
[enriched_category W C] [enriched_category W D]
(F : enriched_functor W C D) : (forget_enrichment W C) ⥤ (forget_enrichment W D) | { obj := λ X, forget_enrichment.of W (F.obj (forget_enrichment.to W X)),
map := λ X Y f, forget_enrichment.hom_of W
(forget_enrichment.hom_to W f ≫ F.map (forget_enrichment.to W X) (forget_enrichment.to W Y)),
map_comp' := λ X Y Z f g, begin
dsimp,
apply_fun forget_enrichment.hom_to W,
{ simp only [... | def | category_theory.enriched_functor.forget | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | An enriched functor induces an honest functor of the underlying categories,
by mapping the `(𝟙_ W)`-shaped morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
graded_nat_trans (A : center V) (F G : enriched_functor V C D) | (app : Π (X : C), A.1 ⟶ (F.obj X ⟶[V] G.obj X))
(naturality :
∀ (X Y : C), (A.2.β (X ⟶[V] Y)).hom ≫ (F.map X Y ⊗ app Y) ≫ e_comp V _ _ _ =
(app X ⊗ G.map X Y) ≫ e_comp V _ _ _) | structure | category_theory.graded_nat_trans | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | The type of `A`-graded natural transformations between `V`-functors `F` and `G`.
This is the type of morphisms in `V` from `A` to the `V`-object of natural transformations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
enriched_nat_trans_yoneda (F G : enriched_functor V C D) : Vᵒᵖ ⥤ (Type (max u₁ w)) | { obj := λ A, graded_nat_trans ((center.of_braided V).obj (unop A)) F G,
map := λ A A' f σ,
{ app := λ X, f.unop ≫ σ.app X,
naturality := λ X Y, begin
have p := σ.naturality X Y,
dsimp at p ⊢,
rw [←id_tensor_comp_tensor_id (f.unop ≫ σ.app Y) _, id_tensor_comp, category.assoc,
category.... | def | category_theory.enriched_nat_trans_yoneda | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | A presheaf isomorphic to the Yoneda embedding of
the `V`-object of natural transformations from `F` to `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
enriched_functor_Type_equiv_functor
{C : Type u₁} [𝒞 : enriched_category (Type v) C]
{D : Type u₂} [𝒟 : enriched_category (Type v) D] :
enriched_functor (Type v) C D ≃ (C ⥤ D) | { to_fun := λ F,
{ obj := λ X, F.obj X,
map := λ X Y f, F.map X Y f,
map_id' := λ X, congr_fun (F.map_id X) punit.star,
map_comp' := λ X Y Z f g, congr_fun (F.map_comp X Y Z) ⟨f, g⟩, },
inv_fun := λ F,
{ obj := λ X, F.obj X,
map := λ X Y f, F.map f,
map_id' := λ X, by { ext ⟨⟩, exact F.map_id ... | def | category_theory.enriched_functor_Type_equiv_functor | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [
"inv_fun"
] | We verify that an enriched functor between `Type v` enriched categories
is just the same thing as an honest functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
enriched_nat_trans_yoneda_Type_iso_yoneda_nat_trans
{C : Type v} [enriched_category (Type v) C]
{D : Type v} [enriched_category (Type v) D]
(F G : enriched_functor (Type v) C D) :
enriched_nat_trans_yoneda F G ≅
yoneda.obj ((enriched_functor_Type_equiv_functor F) ⟶ (enriched_functor_Type_equiv_functor G)) | nat_iso.of_components (λ α,
{ hom := λ σ x,
{ app := λ X, σ.app X x,
naturality' := λ X Y f, congr_fun (σ.naturality X Y) ⟨x, f⟩, },
inv := λ σ,
{ app := λ X x, (σ x).app X,
naturality := λ X Y, by { ext ⟨x, f⟩, exact ((σ x).naturality f), }, }})
(by tidy) | def | category_theory.enriched_nat_trans_yoneda_Type_iso_yoneda_nat_trans | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | We verify that the presheaf representing natural transformations
between `Type v`-enriched functors is actually represented by
the usual type of natural transformations! | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor (C : Type u₁) [category.{v₁} C] (D : Type u₂) [category.{v₂} D]
extends prefunctor C D : Type (max v₁ v₂ u₁ u₂) | (map_id' : ∀ (X : C), map (𝟙 X) = 𝟙 (obj X) . obviously)
(map_comp' : ∀ {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z), map (f ≫ g) = (map f) ≫ (map g) . obviously) | structure | category_theory.functor | category_theory.functor | src/category_theory/functor/basic.lean | [
"tactic.reassoc_axiom",
"category_theory.category.basic"
] | [
"prefunctor"
] | `functor C D` represents a functor between categories `C` and `D`.
To apply a functor `F` to an object use `F.obj X`, and to a morphism use `F.map f`.
The axiom `map_id` expresses preservation of identities, and
`map_comp` expresses functoriality.
See <https://stacks.math.columbia.edu/tag/001B>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id : C ⥤ C | { obj := λ X, X,
map := λ _ _ f, f } | def | category_theory.functor.id | category_theory.functor | src/category_theory/functor/basic.lean | [
"tactic.reassoc_axiom",
"category_theory.category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_obj (X : C) : (𝟭 C).obj X = X | rfl | lemma | category_theory.functor.id_obj | category_theory.functor | src/category_theory/functor/basic.lean | [
"tactic.reassoc_axiom",
"category_theory.category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_map {X Y : C} (f : X ⟶ Y) : (𝟭 C).map f = f | rfl | lemma | category_theory.functor.id_map | category_theory.functor | src/category_theory/functor/basic.lean | [
"tactic.reassoc_axiom",
"category_theory.category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (F : C ⥤ D) (G : D ⥤ E) : C ⥤ E | { obj := λ X, G.obj (F.obj X),
map := λ _ _ f, G.map (F.map f) } | def | category_theory.functor.comp | category_theory.functor | src/category_theory/functor/basic.lean | [
"tactic.reassoc_axiom",
"category_theory.category.basic"
] | [] | `F ⋙ G` is the composition of a functor `F` and a functor `G` (`F` first, then `G`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_map (F : C ⥤ D) (G : D ⥤ E) {X Y : C} (f : X ⟶ Y) :
(F ⋙ G).map f = G.map (F.map f) | rfl | lemma | category_theory.functor.comp_map | category_theory.functor | src/category_theory/functor/basic.lean | [
"tactic.reassoc_axiom",
"category_theory.category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (F : C ⥤ D) : F ⋙ (𝟭 D) = F | by cases F; refl | lemma | category_theory.functor.comp_id | category_theory.functor | src/category_theory/functor/basic.lean | [
"tactic.reassoc_axiom",
"category_theory.category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (F : C ⥤ D) : (𝟭 C) ⋙ F = F | by cases F; refl | lemma | category_theory.functor.id_comp | category_theory.functor | src/category_theory/functor/basic.lean | [
"tactic.reassoc_axiom",
"category_theory.category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_dite (F : C ⥤ D) {X Y : C} {P : Prop} [decidable P]
(f : P → (X ⟶ Y)) (g : ¬P → (X ⟶ Y)) :
F.map (if h : P then f h else g h) = if h : P then F.map (f h) else F.map (g h) | by { split_ifs; refl, } | lemma | category_theory.functor.map_dite | category_theory.functor | src/category_theory/functor/basic.lean | [
"tactic.reassoc_axiom",
"category_theory.category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_prefunctor_obj (F : C ⥤ D) (X : C) :
F.to_prefunctor.obj X = F.obj X | rfl | lemma | category_theory.functor.to_prefunctor_obj | category_theory.functor | src/category_theory/functor/basic.lean | [
"tactic.reassoc_axiom",
"category_theory.category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_prefunctor_map (F : C ⥤ D)
{X Y : C} (f : X ⟶ Y) : F.to_prefunctor.map f = F.map f | rfl | lemma | category_theory.functor.to_prefunctor_map | category_theory.functor | src/category_theory/functor/basic.lean | [
"tactic.reassoc_axiom",
"category_theory.category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_prefunctor_comp (F : C ⥤ D) (G : D ⥤ E) :
F.to_prefunctor.comp G.to_prefunctor = (F ⋙ G).to_prefunctor | rfl | lemma | category_theory.functor.to_prefunctor_comp | category_theory.functor | src/category_theory/functor/basic.lean | [
"tactic.reassoc_axiom",
"category_theory.category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor.category : category.{(max u₁ v₂)} (C ⥤ D) | { hom := λ F G, nat_trans F G,
id := λ F, nat_trans.id F,
comp := λ _ _ _ α β, vcomp α β } | instance | category_theory.functor.category | category_theory.functor | src/category_theory/functor/category.lean | [
"category_theory.natural_transformation",
"category_theory.isomorphism"
] | [] | `functor.category C D` gives the category structure on functors and natural transformations
between categories `C` and `D`.
Notice that if `C` and `D` are both small categories at the same universe level,
this is another small category at that level.
However if `C` and `D` are both large categories at the same univers... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vcomp_eq_comp (α : F ⟶ G) (β : G ⟶ H) : vcomp α β = α ≫ β | rfl | lemma | category_theory.nat_trans.vcomp_eq_comp | category_theory.functor | src/category_theory/functor/category.lean | [
"category_theory.natural_transformation",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vcomp_app' (α : F ⟶ G) (β : G ⟶ H) (X : C) :
(α ≫ β).app X = (α.app X) ≫ (β.app X) | rfl | lemma | category_theory.nat_trans.vcomp_app' | category_theory.functor | src/category_theory/functor/category.lean | [
"category_theory.natural_transformation",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_app {α β : F ⟶ G} (h : α = β) (X : C) : α.app X = β.app X | by rw h | lemma | category_theory.nat_trans.congr_app | category_theory.functor | src/category_theory/functor/category.lean | [
"category_theory.natural_transformation",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_app (F : C ⥤ D) (X : C) : (𝟙 F : F ⟶ F).app X = 𝟙 (F.obj X) | rfl | lemma | category_theory.nat_trans.id_app | category_theory.functor | src/category_theory/functor/category.lean | [
"category_theory.natural_transformation",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_app {F G H : C ⥤ D} (α : F ⟶ G) (β : G ⟶ H) (X : C) :
(α ≫ β).app X = α.app X ≫ β.app X | rfl | lemma | category_theory.nat_trans.comp_app | category_theory.functor | src/category_theory/functor/category.lean | [
"category_theory.natural_transformation",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_app_assoc {F G H : C ⥤ D} (α : F ⟶ G) (β : G ⟶ H) (X : C) {X' : D}
(f : H.obj X ⟶ X') :
(α ≫ β).app X ≫ f = α.app X ≫ β.app X ≫ f | by rw [comp_app, assoc] | lemma | category_theory.nat_trans.comp_app_assoc | category_theory.functor | src/category_theory/functor/category.lean | [
"category_theory.natural_transformation",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
app_naturality {F G : C ⥤ (D ⥤ E)} (T : F ⟶ G) (X : C) {Y Z : D} (f : Y ⟶ Z) :
((F.obj X).map f) ≫ ((T.app X).app Z) = ((T.app X).app Y) ≫ ((G.obj X).map f) | (T.app X).naturality f | lemma | category_theory.nat_trans.app_naturality | category_theory.functor | src/category_theory/functor/category.lean | [
"category_theory.natural_transformation",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
naturality_app {F G : C ⥤ (D ⥤ E)} (T : F ⟶ G) (Z : D) {X Y : C} (f : X ⟶ Y) :
((F.map f).app Z) ≫ ((T.app Y).app Z) = ((T.app X).app Z) ≫ ((G.map f).app Z) | congr_fun (congr_arg app (T.naturality f)) Z | lemma | category_theory.nat_trans.naturality_app | category_theory.functor | src/category_theory/functor/category.lean | [
"category_theory.natural_transformation",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_of_mono_app (α : F ⟶ G) [∀ (X : C), mono (α.app X)] : mono α | ⟨λ H g h eq, by { ext X, rw [←cancel_mono (α.app X), ←comp_app, eq, comp_app] }⟩ | lemma | category_theory.nat_trans.mono_of_mono_app | category_theory.functor | src/category_theory/functor/category.lean | [
"category_theory.natural_transformation",
"category_theory.isomorphism"
] | [] | A natural transformation is a monomorphism if each component is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
epi_of_epi_app (α : F ⟶ G) [∀ (X : C), epi (α.app X)] : epi α | ⟨λ H g h eq, by { ext X, rw [←cancel_epi (α.app X), ←comp_app, eq, comp_app] }⟩ | lemma | category_theory.nat_trans.epi_of_epi_app | category_theory.functor | src/category_theory/functor/category.lean | [
"category_theory.natural_transformation",
"category_theory.isomorphism"
] | [] | A natural transformation is an epimorphism if each component is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hcomp {H I : D ⥤ E} (α : F ⟶ G) (β : H ⟶ I) : (F ⋙ H) ⟶ (G ⋙ I) | { app := λ X : C, (β.app (F.obj X)) ≫ (I.map (α.app X)),
naturality' := λ X Y f,
begin
rw [functor.comp_map, functor.comp_map, ←assoc, naturality, assoc,
←map_comp I, naturality, map_comp, assoc]
end } | def | category_theory.nat_trans.hcomp | category_theory.functor | src/category_theory/functor/category.lean | [
"category_theory.natural_transformation",
"category_theory.isomorphism"
] | [
"map_comp"
] | `hcomp α β` is the horizontal composition of natural transformations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hcomp_id_app {H : D ⥤ E} (α : F ⟶ G) (X : C) : (α ◫ 𝟙 H).app X = H.map (α.app X) | by {dsimp, simp} | lemma | category_theory.nat_trans.hcomp_id_app | category_theory.functor | src/category_theory/functor/category.lean | [
"category_theory.natural_transformation",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_hcomp_app {H : E ⥤ C} (α : F ⟶ G) (X : E) : (𝟙 H ◫ α).app X = α.app _ | by simp | lemma | category_theory.nat_trans.id_hcomp_app | category_theory.functor | src/category_theory/functor/category.lean | [
"category_theory.natural_transformation",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exchange {I J K : D ⥤ E} (α : F ⟶ G) (β : G ⟶ H)
(γ : I ⟶ J) (δ : J ⟶ K) : (α ≫ β) ◫ (γ ≫ δ) = (α ◫ γ) ≫ (β ◫ δ) | by ext; simp | lemma | category_theory.nat_trans.exchange | category_theory.functor | src/category_theory/functor/category.lean | [
"category_theory.natural_transformation",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
flip (F : C ⥤ (D ⥤ E)) : D ⥤ (C ⥤ E) | { obj := λ k,
{ obj := λ j, (F.obj j).obj k,
map := λ j j' f, (F.map f).app k,
map_id' := λ X, begin rw category_theory.functor.map_id, refl end,
map_comp' := λ X Y Z f g, by rw [map_comp, ←comp_app] },
map := λ c c' f,
{ app := λ j, (F.obj j).map f } }. | def | category_theory.functor.flip | category_theory.functor | src/category_theory/functor/category.lean | [
"category_theory.natural_transformation",
"category_theory.isomorphism"
] | [
"map_comp"
] | Flip the arguments of a bifunctor. See also `currying.lean`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_hom_inv_app (F : C ⥤ D ⥤ E) {X Y : C} (e : X ≅ Y) (Z : D) :
(F.map e.hom).app Z ≫ (F.map e.inv).app Z = 𝟙 _ | by simp [← nat_trans.comp_app, ← functor.map_comp] | lemma | category_theory.map_hom_inv_app | category_theory.functor | src/category_theory/functor/category.lean | [
"category_theory.natural_transformation",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_inv_hom_app (F : C ⥤ D ⥤ E) {X Y : C} (e : X ≅ Y) (Z : D) :
(F.map e.inv).app Z ≫ (F.map e.hom).app Z = 𝟙 _ | by simp [← nat_trans.comp_app, ← functor.map_comp] | lemma | category_theory.map_inv_hom_app | category_theory.functor | src/category_theory/functor/category.lean | [
"category_theory.natural_transformation",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const : C ⥤ (J ⥤ C) | { obj := λ X,
{ obj := λ j, X,
map := λ j j' f, 𝟙 X },
map := λ X Y f, { app := λ j, f } } | def | category_theory.functor.const | category_theory.functor | src/category_theory/functor/const.lean | [
"category_theory.opposites"
] | [] | The functor sending `X : C` to the constant functor `J ⥤ C` sending everything to `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_obj_op (X : C) :
(const Jᵒᵖ).obj (op X) ≅ ((const J).obj X).op | { hom := { app := λ j, 𝟙 _ },
inv := { app := λ j, 𝟙 _ } } | def | category_theory.functor.const.op_obj_op | category_theory.functor | src/category_theory/functor/const.lean | [
"category_theory.opposites"
] | [] | The contant functor `Jᵒᵖ ⥤ Cᵒᵖ` sending everything to `op X`
is (naturally isomorphic to) the opposite of the constant functor `J ⥤ C` sending everything to `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_obj_unop (X : Cᵒᵖ) :
(const Jᵒᵖ).obj (unop X) ≅ ((const J).obj X).left_op | { hom := { app := λ j, 𝟙 _ },
inv := { app := λ j, 𝟙 _ } } | def | category_theory.functor.const.op_obj_unop | category_theory.functor | src/category_theory/functor/const.lean | [
"category_theory.opposites"
] | [] | The contant functor `Jᵒᵖ ⥤ C` sending everything to `unop X`
is (naturally isomorphic to) the opposite of
the constant functor `J ⥤ Cᵒᵖ` sending everything to `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_obj_unop_hom_app (X : Cᵒᵖ) (j : Jᵒᵖ) : (op_obj_unop.{v₁ v₂} X).hom.app j = 𝟙 _ | rfl | lemma | category_theory.functor.const.op_obj_unop_hom_app | category_theory.functor | src/category_theory/functor/const.lean | [
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_obj_unop_inv_app (X : Cᵒᵖ) (j : Jᵒᵖ) : (op_obj_unop.{v₁ v₂} X).inv.app j = 𝟙 _ | rfl | lemma | category_theory.functor.const.op_obj_unop_inv_app | category_theory.functor | src/category_theory/functor/const.lean | [
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_functor_op_obj_map (X : Cᵒᵖ) {j₁ j₂ : J} (f : j₁ ⟶ j₂) :
(unop ((functor.op (const J)).obj X)).map f = 𝟙 (unop X) | rfl | lemma | category_theory.functor.const.unop_functor_op_obj_map | category_theory.functor | src/category_theory/functor/const.lean | [
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_comp (X : C) (F : C ⥤ D) :
(const J).obj X ⋙ F ≅ (const J).obj (F.obj X) | { hom := { app := λ _, 𝟙 _ },
inv := { app := λ _, 𝟙 _ } } | def | category_theory.functor.const_comp | category_theory.functor | src/category_theory/functor/const.lean | [
"category_theory.opposites"
] | [] | These are actually equal, of course, but not definitionally equal
(the equality requires F.map (𝟙 _) = 𝟙 _). A natural isomorphism is
more convenient than an equality between functors (compare id_to_iso). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uncurry : (C ⥤ (D ⥤ E)) ⥤ ((C × D) ⥤ E) | { obj := λ F,
{ obj := λ X, (F.obj X.1).obj X.2,
map := λ X Y f, (F.map f.1).app X.2 ≫ (F.obj Y.1).map f.2,
map_comp' := λ X Y Z f g,
begin
simp only [prod_comp_fst, prod_comp_snd, functor.map_comp,
nat_trans.comp_app, category.assoc],
slice_lhs 2 3 { rw ← nat_trans.naturality... | def | category_theory.uncurry | category_theory.functor | src/category_theory/functor/currying.lean | [
"category_theory.products.bifunctor"
] | [
"functor.map_id"
] | The uncurrying functor, taking a functor `C ⥤ (D ⥤ E)` and producing a functor `(C × D) ⥤ E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
curry_obj (F : (C × D) ⥤ E) : C ⥤ (D ⥤ E) | { obj := λ X,
{ obj := λ Y, F.obj (X, Y),
map := λ Y Y' g, F.map (𝟙 X, g) },
map := λ X X' f, { app := λ Y, F.map (f, 𝟙 Y) } } | def | category_theory.curry_obj | category_theory.functor | src/category_theory/functor/currying.lean | [
"category_theory.products.bifunctor"
] | [] | The object level part of the currying functor. (See `curry` for the functorial version.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
curry : ((C × D) ⥤ E) ⥤ (C ⥤ (D ⥤ E)) | { obj := λ F, curry_obj F,
map := λ F G T,
{ app := λ X,
{ app := λ Y, T.app (X, Y),
naturality' := λ Y Y' g,
begin
dsimp [curry_obj],
rw nat_trans.naturality,
end },
naturality' := λ X X' f,
begin
ext, dsimp [curry_obj],
rw nat_trans.naturality,
end } }... | def | category_theory.curry | category_theory.functor | src/category_theory/functor/currying.lean | [
"category_theory.products.bifunctor"
] | [] | The currying functor, taking a functor `(C × D) ⥤ E` and producing a functor `C ⥤ (D ⥤ E)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
currying : (C ⥤ (D ⥤ E)) ≌ ((C × D) ⥤ E) | equivalence.mk uncurry curry
(nat_iso.of_components (λ F, nat_iso.of_components
(λ X, nat_iso.of_components (λ Y, iso.refl _) (by tidy)) (by tidy)) (by tidy))
(nat_iso.of_components (λ F, nat_iso.of_components
(λ X, eq_to_iso (by simp)) (by tidy)) (by tidy)) | def | category_theory.currying | category_theory.functor | src/category_theory/functor/currying.lean | [
"category_theory.products.bifunctor"
] | [] | The equivalence of functor categories given by currying/uncurrying. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
flip_iso_curry_swap_uncurry (F : C ⥤ D ⥤ E) :
F.flip ≅ curry.obj (prod.swap _ _ ⋙ uncurry.obj F) | nat_iso.of_components (λ d, nat_iso.of_components (λ c, iso.refl _) (by tidy)) (by tidy) | def | category_theory.flip_iso_curry_swap_uncurry | category_theory.functor | src/category_theory/functor/currying.lean | [
"category_theory.products.bifunctor"
] | [
"prod.swap"
] | `F.flip` is isomorphic to uncurrying `F`, swapping the variables, and currying. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uncurry_obj_flip (F : C ⥤ D ⥤ E) : uncurry.obj F.flip ≅ prod.swap _ _ ⋙ uncurry.obj F | nat_iso.of_components (λ p, iso.refl _) (by tidy) | def | category_theory.uncurry_obj_flip | category_theory.functor | src/category_theory/functor/currying.lean | [
"category_theory.products.bifunctor"
] | [
"prod.swap"
] | The uncurrying of `F.flip` is isomorphic to
swapping the factors followed by the uncurrying of `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whiskering_right₂ : (C ⥤ D ⥤ E) ⥤ ((B ⥤ C) ⥤ (B ⥤ D) ⥤ (B ⥤ E)) | uncurry ⋙ (whiskering_right _ _ _) ⋙
((whiskering_left _ _ _).obj (prod_functor_to_functor_prod _ _ _)) ⋙ curry | def | category_theory.whiskering_right₂ | category_theory.functor | src/category_theory/functor/currying.lean | [
"category_theory.products.bifunctor"
] | [] | A version of `category_theory.whiskering_right` for bifunctors, obtained by uncurrying,
applying `whiskering_right` and currying back | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_monomorphisms (F : C ⥤ D) : Prop | (preserves : ∀ {X Y : C} (f : X ⟶ Y) [mono f], mono (F.map f)) | class | category_theory.functor.preserves_monomorphisms | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | A functor preserves monomorphisms if it maps monomorphisms to monomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_mono (F : C ⥤ D) [preserves_monomorphisms F] {X Y : C} (f : X ⟶ Y) [mono f] :
mono (F.map f) | preserves_monomorphisms.preserves f | instance | category_theory.functor.map_mono | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_epimorphisms (F : C ⥤ D) : Prop | (preserves : ∀ {X Y : C} (f : X ⟶ Y) [epi f], epi (F.map f)) | class | category_theory.functor.preserves_epimorphisms | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | A functor preserves epimorphisms if it maps epimorphisms to epimorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_epi (F : C ⥤ D) [preserves_epimorphisms F] {X Y : C} (f : X ⟶ Y) [epi f] :
epi (F.map f) | preserves_epimorphisms.preserves f | instance | category_theory.functor.map_epi | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_monomorphisms (F : C ⥤ D) : Prop | (reflects : ∀ {X Y : C} (f : X ⟶ Y), mono (F.map f) → mono f) | class | category_theory.functor.reflects_monomorphisms | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | A functor reflects monomorphisms if morphisms that are mapped to monomorphisms are themselves
monomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono_of_mono_map (F : C ⥤ D) [reflects_monomorphisms F] {X Y : C} {f : X ⟶ Y}
(h : mono (F.map f)) : mono f | reflects_monomorphisms.reflects f h | lemma | category_theory.functor.mono_of_mono_map | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_epimorphisms (F : C ⥤ D) : Prop | (reflects : ∀ {X Y : C} (f : X ⟶ Y), epi (F.map f) → epi f) | class | category_theory.functor.reflects_epimorphisms | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | A functor reflects epimorphisms if morphisms that are mapped to epimorphisms are themselves
epimorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
epi_of_epi_map (F : C ⥤ D) [reflects_epimorphisms F] {X Y : C} {f : X ⟶ Y}
(h : epi (F.map f)) : epi f | reflects_epimorphisms.reflects f h | lemma | category_theory.functor.epi_of_epi_map | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_monomorphisms_comp (F : C ⥤ D) (G : D ⥤ E) [preserves_monomorphisms F]
[preserves_monomorphisms G] : preserves_monomorphisms (F ⋙ G) | { preserves := λ X Y f h, by { rw comp_map, exactI infer_instance } } | instance | category_theory.functor.preserves_monomorphisms_comp | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_epimorphisms_comp (F : C ⥤ D) (G : D ⥤ E) [preserves_epimorphisms F]
[preserves_epimorphisms G] : preserves_epimorphisms (F ⋙ G) | { preserves := λ X Y f h, by { rw comp_map, exactI infer_instance } } | instance | category_theory.functor.preserves_epimorphisms_comp | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_monomorphisms_comp (F : C ⥤ D) (G : D ⥤ E) [reflects_monomorphisms F]
[reflects_monomorphisms G] : reflects_monomorphisms (F ⋙ G) | { reflects := λ X Y f h, (F.mono_of_mono_map (G.mono_of_mono_map h)) } | instance | category_theory.functor.reflects_monomorphisms_comp | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_epimorphisms_comp (F : C ⥤ D) (G : D ⥤ E) [reflects_epimorphisms F]
[reflects_epimorphisms G] : reflects_epimorphisms (F ⋙ G) | { reflects := λ X Y f h, (F.epi_of_epi_map (G.epi_of_epi_map h)) } | instance | category_theory.functor.reflects_epimorphisms_comp | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_epimorphisms_of_preserves_of_reflects (F : C ⥤ D) (G : D ⥤ E)
[preserves_epimorphisms (F ⋙ G)] [reflects_epimorphisms G] : preserves_epimorphisms F | ⟨λ X Y f hf, G.epi_of_epi_map $ show epi ((F ⋙ G).map f), by exactI infer_instance⟩ | lemma | category_theory.functor.preserves_epimorphisms_of_preserves_of_reflects | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_monomorphisms_of_preserves_of_reflects (F : C ⥤ D) (G : D ⥤ E)
[preserves_monomorphisms (F ⋙ G)] [reflects_monomorphisms G] : preserves_monomorphisms F | ⟨λ X Y f hf, G.mono_of_mono_map $ show mono ((F ⋙ G).map f), by exactI infer_instance⟩ | lemma | category_theory.functor.preserves_monomorphisms_of_preserves_of_reflects | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_epimorphisms_of_preserves_of_reflects (F : C ⥤ D) (G : D ⥤ E)
[preserves_epimorphisms G] [reflects_epimorphisms (F ⋙ G)] : reflects_epimorphisms F | ⟨λ X Y f hf, (F ⋙ G).epi_of_epi_map $ show epi (G.map (F.map f)), by exactI infer_instance⟩ | lemma | category_theory.functor.reflects_epimorphisms_of_preserves_of_reflects | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_monomorphisms_of_preserves_of_reflects (F : C ⥤ D) (G : D ⥤ E)
[preserves_monomorphisms G] [reflects_monomorphisms (F ⋙ G)] : reflects_monomorphisms F | ⟨λ X Y f hf, (F ⋙ G).mono_of_mono_map $ show mono (G.map (F.map f)), by exactI infer_instance⟩ | lemma | category_theory.functor.reflects_monomorphisms_of_preserves_of_reflects | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_monomorphisms.of_iso {F G : C ⥤ D} [preserves_monomorphisms F] (α : F ≅ G) :
preserves_monomorphisms G | { preserves := λ X Y f h,
begin
haveI : mono (F.map f ≫ (α.app Y).hom) := by exactI mono_comp _ _,
convert (mono_comp _ _ : mono ((α.app X).inv ≫ F.map f ≫ (α.app Y).hom)),
rw [iso.eq_inv_comp, iso.app_hom, iso.app_hom, nat_trans.naturality]
end } | lemma | category_theory.functor.preserves_monomorphisms.of_iso | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_monomorphisms.iso_iff {F G : C ⥤ D} (α : F ≅ G) :
preserves_monomorphisms F ↔ preserves_monomorphisms G | ⟨λ h, by exactI preserves_monomorphisms.of_iso α,
λ h, by exactI preserves_monomorphisms.of_iso α.symm⟩ | lemma | category_theory.functor.preserves_monomorphisms.iso_iff | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_epimorphisms.of_iso {F G : C ⥤ D} [preserves_epimorphisms F] (α : F ≅ G) :
preserves_epimorphisms G | { preserves := λ X Y f h,
begin
haveI : epi (F.map f ≫ (α.app Y).hom) := by exactI epi_comp _ _,
convert (epi_comp _ _ : epi ((α.app X).inv ≫ F.map f ≫ (α.app Y).hom)),
rw [iso.eq_inv_comp, iso.app_hom, iso.app_hom, nat_trans.naturality]
end } | lemma | category_theory.functor.preserves_epimorphisms.of_iso | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_epimorphisms.iso_iff {F G : C ⥤ D} (α : F ≅ G) :
preserves_epimorphisms F ↔ preserves_epimorphisms G | ⟨λ h, by exactI preserves_epimorphisms.of_iso α,
λ h, by exactI preserves_epimorphisms.of_iso α.symm⟩ | lemma | category_theory.functor.preserves_epimorphisms.iso_iff | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_monomorphisms.of_iso {F G : C ⥤ D} [reflects_monomorphisms F] (α : F ≅ G) :
reflects_monomorphisms G | { reflects := λ X Y f h,
begin
apply F.mono_of_mono_map,
haveI : mono (G.map f ≫ (α.app Y).inv) := by exactI mono_comp _ _,
convert (mono_comp _ _ : mono ((α.app X).hom ≫ G.map f ≫ (α.app Y).inv)),
rw [← category.assoc, iso.eq_comp_inv, iso.app_hom, iso.app_hom, nat_trans.naturality]
end } | lemma | category_theory.functor.reflects_monomorphisms.of_iso | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_monomorphisms.iso_iff {F G : C ⥤ D} (α : F ≅ G) :
reflects_monomorphisms F ↔ reflects_monomorphisms G | ⟨λ h, by exactI reflects_monomorphisms.of_iso α,
λ h, by exactI reflects_monomorphisms.of_iso α.symm⟩ | lemma | category_theory.functor.reflects_monomorphisms.iso_iff | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_epimorphisms.of_iso {F G : C ⥤ D} [reflects_epimorphisms F] (α : F ≅ G) :
reflects_epimorphisms G | { reflects := λ X Y f h,
begin
apply F.epi_of_epi_map,
haveI : epi (G.map f ≫ (α.app Y).inv) := by exactI epi_comp _ _,
convert (epi_comp _ _ : epi ((α.app X).hom ≫ G.map f ≫ (α.app Y).inv)),
rw [← category.assoc, iso.eq_comp_inv, iso.app_hom, iso.app_hom, nat_trans.naturality]
end } | lemma | category_theory.functor.reflects_epimorphisms.of_iso | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_epimorphisms.iso_iff {F G : C ⥤ D} (α : F ≅ G) :
reflects_epimorphisms F ↔ reflects_epimorphisms G | ⟨λ h, by exactI reflects_epimorphisms.of_iso α, λ h, by exactI reflects_epimorphisms.of_iso α.symm⟩ | lemma | category_theory.functor.reflects_epimorphisms.iso_iff | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_epimorphsisms_of_adjunction {F : C ⥤ D} {G : D ⥤ C} (adj : F ⊣ G) :
preserves_epimorphisms F | { preserves := λ X Y f hf,
⟨begin
introsI Z g h H,
replace H := congr_arg (adj.hom_equiv X Z) H,
rwa [adj.hom_equiv_naturality_left, adj.hom_equiv_naturality_left, cancel_epi,
equiv.apply_eq_iff_eq] at H
end⟩ } | lemma | category_theory.functor.preserves_epimorphsisms_of_adjunction | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [
"adj",
"equiv.apply_eq_iff_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_epimorphisms_of_is_left_adjoint (F : C ⥤ D) [is_left_adjoint F] :
preserves_epimorphisms F | preserves_epimorphsisms_of_adjunction (adjunction.of_left_adjoint F) | instance | category_theory.functor.preserves_epimorphisms_of_is_left_adjoint | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_monomorphisms_of_adjunction {F : C ⥤ D} {G : D ⥤ C} (adj : F ⊣ G) :
preserves_monomorphisms G | { preserves := λ X Y f hf,
⟨begin
introsI Z g h H,
replace H := congr_arg (adj.hom_equiv Z Y).symm H,
rwa [adj.hom_equiv_naturality_right_symm, adj.hom_equiv_naturality_right_symm,
cancel_mono, equiv.apply_eq_iff_eq] at H
end⟩ } | lemma | category_theory.functor.preserves_monomorphisms_of_adjunction | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [
"adj",
"equiv.apply_eq_iff_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_monomorphisms_of_is_right_adjoint (F : C ⥤ D) [is_right_adjoint F] :
preserves_monomorphisms F | preserves_monomorphisms_of_adjunction (adjunction.of_right_adjoint F) | instance | category_theory.functor.preserves_monomorphisms_of_is_right_adjoint | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_monomorphisms_of_faithful (F : C ⥤ D) [faithful F] : reflects_monomorphisms F | { reflects := λ X Y f hf, ⟨λ Z g h hgh, by exactI F.map_injective ((cancel_mono (F.map f)).1
(by rw [← F.map_comp, hgh, F.map_comp]))⟩ } | instance | category_theory.functor.reflects_monomorphisms_of_faithful | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_epimorphisms_of_faithful (F : C ⥤ D) [faithful F] : reflects_epimorphisms F | { reflects := λ X Y f hf, ⟨λ Z g h hgh, by exactI F.map_injective ((cancel_epi (F.map f)).1
(by rw [← F.map_comp, hgh, F.map_comp]))⟩ } | instance | category_theory.functor.reflects_epimorphisms_of_faithful | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_epi_equiv [full F] [faithful F] : split_epi f ≃ split_epi (F.map f) | { to_fun := λ f, f.map F,
inv_fun := λ s, begin
refine ⟨F.preimage s.section_, _⟩,
apply F.map_injective,
simp only [map_comp, image_preimage, map_id],
apply split_epi.id,
end,
left_inv := by tidy,
right_inv := by tidy, } | def | category_theory.functor.split_epi_equiv | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [
"inv_fun",
"map_comp",
"map_id"
] | If `F` is a fully faithful functor, split epimorphisms are preserved and reflected by `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_split_epi_iff [full F] [faithful F] : is_split_epi (F.map f) ↔ is_split_epi f | begin
split,
{ intro h, exact is_split_epi.mk' ((split_epi_equiv F f).inv_fun h.exists_split_epi.some), },
{ intro h, exact is_split_epi.mk' ((split_epi_equiv F f).to_fun h.exists_split_epi.some), },
end | lemma | category_theory.functor.is_split_epi_iff | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [
"inv_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_mono_equiv [full F] [faithful F] : split_mono f ≃ split_mono (F.map f) | { to_fun := λ f, f.map F,
inv_fun := λ s, begin
refine ⟨F.preimage s.retraction, _⟩,
apply F.map_injective,
simp only [map_comp, image_preimage, map_id],
apply split_mono.id,
end,
left_inv := by tidy,
right_inv := by tidy, } | def | category_theory.functor.split_mono_equiv | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [
"inv_fun",
"map_comp",
"map_id"
] | If `F` is a fully faithful functor, split monomorphisms are preserved and reflected by `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_split_mono_iff [full F] [faithful F] : is_split_mono (F.map f) ↔ is_split_mono f | begin
split,
{ intro h, exact is_split_mono.mk' ((split_mono_equiv F f).inv_fun h.exists_split_mono.some), },
{ intro h, exact is_split_mono.mk' ((split_mono_equiv F f).to_fun h.exists_split_mono.some), },
end | lemma | category_theory.functor.is_split_mono_iff | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [
"inv_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_map_iff_epi [hF₁ : preserves_epimorphisms F] [hF₂ : reflects_epimorphisms F] :
epi (F.map f) ↔ epi f | begin
split,
{ exact F.epi_of_epi_map, },
{ introI h,
exact F.map_epi f, },
end | lemma | category_theory.functor.epi_map_iff_epi | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_map_iff_mono [hF₁ : preserves_monomorphisms F] [hF₂ : reflects_monomorphisms F] :
mono (F.map f) ↔ mono f | begin
split,
{ exact F.mono_of_mono_map, },
{ introI h,
exact F.map_mono f, },
end | lemma | category_theory.functor.mono_map_iff_mono | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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