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values | symbolic_name stringlengths 1 131 | library stringclasses 417
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unop_id (F : Cᵒᵖ ⥤ Dᵒᵖ) : nat_trans.unop (𝟙 F) = 𝟙 (F.unop) | rfl | lemma | category_theory.nat_trans.unop_id | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
remove_op (α : F.op ⟶ G.op) : G ⟶ F | { app := λ X, (α.app (op X)).unop,
naturality' := λ X Y f, quiver.hom.op_inj $
by simpa only [functor.op_map] using (α.naturality f.op).symm } | def | category_theory.nat_trans.remove_op | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [
"quiver.hom.op_inj"
] | Given a natural transformation `α : F.op ⟶ G.op`,
we can take the "unopposite" of each component obtaining a natural transformation `G ⟶ F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
remove_op_id (F : C ⥤ D) : nat_trans.remove_op (𝟙 F.op) = 𝟙 F | rfl | lemma | category_theory.nat_trans.remove_op_id | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
remove_unop {F G : Cᵒᵖ ⥤ Dᵒᵖ} (α : F.unop ⟶ G.unop) : G ⟶ F | { app := λ X, (α.app (unop X)).op,
naturality' := λ X Y f, quiver.hom.unop_inj $
by simpa only [functor.unop_map] using (α.naturality f.unop).symm } | def | category_theory.nat_trans.remove_unop | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [
"quiver.hom.unop_inj"
] | Given a natural transformation `α : F.unop ⟶ G.unop`, we can take the opposite of each
component obtaining a natural transformation `G ⟶ F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
remove_unop_id (F : Cᵒᵖ ⥤ Dᵒᵖ) : nat_trans.remove_unop (𝟙 F.unop) = 𝟙 F | rfl | lemma | category_theory.nat_trans.remove_unop_id | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_op (α : F ⟶ G) : G.left_op ⟶ F.left_op | { app := λ X, (α.app (unop X)).unop,
naturality' := λ X Y f, quiver.hom.op_inj (by simp) } | def | category_theory.nat_trans.left_op | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [
"quiver.hom.op_inj"
] | Given a natural transformation `α : F ⟶ G`, for `F G : C ⥤ Dᵒᵖ`,
taking `unop` of each component gives a natural transformation `G.left_op ⟶ F.left_op`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_op_id : (𝟙 F : F ⟶ F).left_op = 𝟙 F.left_op | rfl | lemma | category_theory.nat_trans.left_op_id | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_op_comp (α : F ⟶ G) (β : G ⟶ H) :
(α ≫ β).left_op = β.left_op ≫ α.left_op | rfl | lemma | category_theory.nat_trans.left_op_comp | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
remove_left_op (α : F.left_op ⟶ G.left_op) : G ⟶ F | { app := λ X, (α.app (op X)).op,
naturality' := λ X Y f, quiver.hom.unop_inj $
by simpa only [functor.left_op_map] using (α.naturality f.op).symm } | def | category_theory.nat_trans.remove_left_op | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [
"quiver.hom.unop_inj"
] | Given a natural transformation `α : F.left_op ⟶ G.left_op`, for `F G : C ⥤ Dᵒᵖ`,
taking `op` of each component gives a natural transformation `G ⟶ F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
remove_left_op_id : nat_trans.remove_left_op (𝟙 F.left_op) = 𝟙 F | rfl | lemma | category_theory.nat_trans.remove_left_op_id | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_op (α : F ⟶ G) : G.right_op ⟶ F.right_op | { app := λ X, (α.app _).op,
naturality' := λ X Y f, quiver.hom.unop_inj (by simp) } | def | category_theory.nat_trans.right_op | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [
"quiver.hom.unop_inj"
] | Given a natural transformation `α : F ⟶ G`, for `F G : Cᵒᵖ ⥤ D`,
taking `op` of each component gives a natural transformation `G.right_op ⟶ F.right_op`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_op_id : (𝟙 F : F ⟶ F).right_op = 𝟙 F.right_op | rfl | lemma | category_theory.nat_trans.right_op_id | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_op_comp (α : F ⟶ G) (β : G ⟶ H) :
(α ≫ β).right_op = β.right_op ≫ α.right_op | rfl | lemma | category_theory.nat_trans.right_op_comp | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
remove_right_op (α : F.right_op ⟶ G.right_op) : G ⟶ F | { app := λ X, (α.app X.unop).unop,
naturality' := λ X Y f, quiver.hom.op_inj $
by simpa only [functor.right_op_map] using (α.naturality f.unop).symm } | def | category_theory.nat_trans.remove_right_op | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [
"quiver.hom.op_inj"
] | Given a natural transformation `α : F.right_op ⟶ G.right_op`, for `F G : Cᵒᵖ ⥤ D`,
taking `unop` of each component gives a natural transformation `G ⟶ F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
remove_right_op_id : nat_trans.remove_right_op (𝟙 F.right_op) = 𝟙 F | rfl | lemma | category_theory.nat_trans.remove_right_op_id | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op (α : X ≅ Y) : op Y ≅ op X | { hom := α.hom.op,
inv := α.inv.op,
hom_inv_id' := quiver.hom.unop_inj α.inv_hom_id,
inv_hom_id' := quiver.hom.unop_inj α.hom_inv_id } | def | category_theory.iso.op | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [
"quiver.hom.unop_inj"
] | The opposite isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop {X Y : Cᵒᵖ} (f : X ≅ Y) : Y.unop ≅ X.unop | { hom := f.hom.unop,
inv := f.inv.unop,
hom_inv_id' := by simp only [← unop_comp, f.inv_hom_id, unop_id],
inv_hom_id' := by simp only [← unop_comp, f.hom_inv_id, unop_id] } | def | category_theory.iso.unop | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | The isomorphism obtained from an isomorphism in the opposite category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_op {X Y : Cᵒᵖ} (f : X ≅ Y) : f.unop.op = f | by ext; refl | lemma | category_theory.iso.unop_op | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_unop {X Y : C} (f : X ≅ Y) : f.op.unop = f | by ext; refl | lemma | category_theory.iso.op_unop | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op (α : F ≅ G) : G.op ≅ F.op | { hom := nat_trans.op α.hom,
inv := nat_trans.op α.inv,
hom_inv_id' := begin ext, dsimp, rw ←op_comp, rw α.inv_hom_id_app, refl, end,
inv_hom_id' := begin ext, dsimp, rw ←op_comp, rw α.hom_inv_id_app, refl, end } | def | category_theory.nat_iso.op | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | The natural isomorphism between opposite functors `G.op ≅ F.op` induced by a natural
isomorphism between the original functors `F ≅ G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
remove_op (α : F.op ≅ G.op) : G ≅ F | { hom := nat_trans.remove_op α.hom,
inv := nat_trans.remove_op α.inv,
hom_inv_id' := begin ext, dsimp, rw ←unop_comp, rw α.inv_hom_id_app, refl, end,
inv_hom_id' := begin ext, dsimp, rw ←unop_comp, rw α.hom_inv_id_app, refl, end } | def | category_theory.nat_iso.remove_op | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | The natural isomorphism between functors `G ≅ F` induced by a natural isomorphism
between the opposite functors `F.op ≅ G.op`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop {F G : Cᵒᵖ ⥤ Dᵒᵖ} (α : F ≅ G) : G.unop ≅ F.unop | { hom := nat_trans.unop α.hom,
inv := nat_trans.unop α.inv,
hom_inv_id' := begin ext, dsimp, rw ←unop_comp, rw α.inv_hom_id_app, refl, end,
inv_hom_id' := begin ext, dsimp, rw ←unop_comp, rw α.hom_inv_id_app, refl, end } | def | category_theory.nat_iso.unop | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | The natural isomorphism between functors `G.unop ≅ F.unop` induced by a natural isomorphism
between the original functors `F ≅ G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op (e : C ≌ D) : Cᵒᵖ ≌ Dᵒᵖ | { functor := e.functor.op,
inverse := e.inverse.op,
unit_iso := (nat_iso.op e.unit_iso).symm,
counit_iso := (nat_iso.op e.counit_iso).symm,
functor_unit_iso_comp' := λ X, by { apply quiver.hom.unop_inj, dsimp, simp, }, } | def | category_theory.equivalence.op | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [
"quiver.hom.unop_inj"
] | An equivalence between categories gives an equivalence between the opposite categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop (e : Cᵒᵖ ≌ Dᵒᵖ) : C ≌ D | { functor := e.functor.unop,
inverse := e.inverse.unop,
unit_iso := (nat_iso.unop e.unit_iso).symm,
counit_iso := (nat_iso.unop e.counit_iso).symm,
functor_unit_iso_comp' := λ X, by { apply quiver.hom.op_inj, dsimp, simp, }, } | def | category_theory.equivalence.unop | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [
"quiver.hom.op_inj"
] | An equivalence between opposite categories gives an equivalence between the original categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_equiv (A B : Cᵒᵖ) : (A ⟶ B) ≃ (B.unop ⟶ A.unop) | { to_fun := λ f, f.unop,
inv_fun := λ g, g.op,
left_inv := λ _, rfl,
right_inv := λ _, rfl } | def | category_theory.op_equiv | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [
"inv_fun"
] | The equivalence between arrows of the form `A ⟶ B` and `B.unop ⟶ A.unop`. Useful for building
adjunctions.
Note that this (definitionally) gives variants
```
def op_equiv' (A : C) (B : Cᵒᵖ) : (opposite.op A ⟶ B) ≃ (B.unop ⟶ A) :=
op_equiv _ _
def op_equiv'' (A : Cᵒᵖ) (B : C) : (A ⟶ opposite.op B) ≃ (B ⟶ A.unop) :=
op_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subsingleton_of_unop (A B : Cᵒᵖ) [subsingleton (unop B ⟶ unop A)] : subsingleton (A ⟶ B) | (op_equiv A B).subsingleton | instance | category_theory.subsingleton_of_unop | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decidable_eq_of_unop (A B : Cᵒᵖ) [decidable_eq (unop B ⟶ unop A)] : decidable_eq (A ⟶ B) | (op_equiv A B).decidable_eq | instance | category_theory.decidable_eq_of_unop | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_op_equiv (A B : Cᵒᵖ) : (A ≅ B) ≃ (B.unop ≅ A.unop) | { to_fun := λ f, f.unop,
inv_fun := λ g, g.op,
left_inv := λ _, by { ext, refl, },
right_inv := λ _, by { ext, refl, } } | def | category_theory.iso_op_equiv | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [
"inv_fun"
] | The equivalence between isomorphisms of the form `A ≅ B` and `B.unop ≅ A.unop`.
Note this is definitionally the same as the other three variants:
* `(opposite.op A ≅ B) ≃ (B.unop ≅ A)`
* `(A ≅ opposite.op B) ≃ (B ≅ A.unop)`
* `(opposite.op A ≅ opposite.op B) ≃ (B ≅ A)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_unop_equiv : (C ⥤ D)ᵒᵖ ≌ Cᵒᵖ ⥤ Dᵒᵖ | { functor := op_hom _ _,
inverse := op_inv _ _,
unit_iso := nat_iso.of_components (λ F, F.unop.op_unop_iso.op) begin
intros F G f,
dsimp [op_unop_iso],
rw [(show f = f.unop.op, by simp), ← op_comp, ← op_comp],
congr' 1,
tidy,
end,
counit_iso := nat_iso.of_components (λ F, F.unop_op_iso) (by ... | def | category_theory.functor.op_unop_equiv | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | The equivalence of functor categories induced by `op` and `unop`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_op_right_op_equiv : (Cᵒᵖ ⥤ D)ᵒᵖ ≌ (C ⥤ Dᵒᵖ) | { functor :=
{ obj := λ F, F.unop.right_op,
map := λ F G η, η.unop.right_op },
inverse :=
{ obj := λ F, op F.left_op,
map := λ F G η, η.left_op.op },
unit_iso := nat_iso.of_components (λ F, F.unop.right_op_left_op_iso.op) begin
intros F G η,
dsimp,
rw [(show η = η.unop.op, by simp), ← op_com... | def | category_theory.functor.left_op_right_op_equiv | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | The equivalence of functor categories induced by `left_op` and `right_op`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
over (X : T) | costructured_arrow (𝟭 T) X | def | category_theory.over | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | The over category has as objects arrows in `T` with codomain `X` and as morphisms commutative
triangles.
See <https://stacks.math.columbia.edu/tag/001G>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
over.inhabited [inhabited T] : inhabited (over (default : T)) | { default :=
{ left := default,
right := default,
hom := 𝟙 _ } } | instance | category_theory.over.inhabited | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
over_morphism.ext {X : T} {U V : over X} {f g : U ⟶ V}
(h : f.left = g.left) : f = g | by tidy | lemma | category_theory.over.over_morphism.ext | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
over_right (U : over X) : U.right = ⟨⟨⟩⟩ | by tidy | lemma | category_theory.over.over_right | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_left (U : over X) : comma_morphism.left (𝟙 U) = 𝟙 U.left | rfl | lemma | category_theory.over.id_left | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_left (a b c : over X) (f : a ⟶ b) (g : b ⟶ c) :
(f ≫ g).left = f.left ≫ g.left | rfl | lemma | category_theory.over.comp_left | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w {A B : over X} (f : A ⟶ B) : f.left ≫ B.hom = A.hom | by have := f.w; tidy | lemma | category_theory.over.w | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk {X Y : T} (f : Y ⟶ X) : over X | costructured_arrow.mk f | def | category_theory.over.mk | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | To give an object in the over category, it suffices to give a morphism with codomain `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_from_hom {X Y : T} : has_coe (Y ⟶ X) (over X) | { coe := mk } | def | category_theory.over.coe_from_hom | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | We can set up a coercion from arrows with codomain `X` to `over X`. This most likely should not
be a global instance, but it is sometimes useful. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_hom {X Y : T} (f : Y ⟶ X) : (f : over X).hom = f | rfl | lemma | category_theory.over.coe_hom | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_mk {U V : over X} (f : U.left ⟶ V.left) (w : f ≫ V.hom = U.hom . obviously) :
U ⟶ V | costructured_arrow.hom_mk f w | def | category_theory.over.hom_mk | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | To give a morphism in the over category, it suffices to give an arrow fitting in a commutative
triangle. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_mk {f g : over X} (hl : f.left ≅ g.left) (hw : hl.hom ≫ g.hom = f.hom . obviously) :
f ≅ g | costructured_arrow.iso_mk hl hw | def | category_theory.over.iso_mk | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | Construct an isomorphism in the over category given isomorphisms of the objects whose forward
direction gives a commutative triangle. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget : over X ⥤ T | comma.fst _ _ | def | category_theory.over.forget | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | The forgetful functor mapping an arrow to its domain.
See <https://stacks.math.columbia.edu/tag/001G>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_obj {U : over X} : (forget X).obj U = U.left | rfl | lemma | category_theory.over.forget_obj | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_map {U V : over X} {f : U ⟶ V} : (forget X).map f = f.left | rfl | lemma | category_theory.over.forget_map | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_cocone (X : T) : limits.cocone (forget X) | { X := X, ι := { app := comma.hom } } | def | category_theory.over.forget_cocone | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | The natural cocone over the forgetful functor `over X ⥤ T` with cocone point `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map {Y : T} (f : X ⟶ Y) : over X ⥤ over Y | comma.map_right _ $ discrete.nat_trans (λ _, f) | def | category_theory.over.map | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | A morphism `f : X ⟶ Y` induces a functor `over X ⥤ over Y` in the obvious way.
See <https://stacks.math.columbia.edu/tag/001G>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_obj_left : ((map f).obj U).left = U.left | rfl | lemma | category_theory.over.map_obj_left | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_obj_hom : ((map f).obj U).hom = U.hom ≫ f | rfl | lemma | category_theory.over.map_obj_hom | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_map_left : ((map f).map g).left = g.left | rfl | lemma | category_theory.over.map_map_left | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id : map (𝟙 Y) ≅ 𝟭 _ | nat_iso.of_components (λ X, iso_mk (iso.refl _) (by tidy)) (by tidy) | def | category_theory.over.map_id | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [
"map_id"
] | Mapping by the identity morphism is just the identity functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_comp {Y Z : T} (f : X ⟶ Y) (g : Y ⟶ Z) : map (f ≫ g) ≅ map f ⋙ map g | nat_iso.of_components (λ X, iso_mk (iso.refl _) (by tidy)) (by tidy) | def | category_theory.over.map_comp | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [
"map_comp"
] | Mapping by the composite morphism `f ≫ g` is the same as mapping by `f` then by `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_reflects_iso : reflects_isomorphisms (forget X) | { reflects := λ Y Z f t, by exactI
⟨⟨over.hom_mk (inv ((forget X).map f))
((as_iso ((forget X).map f)).inv_comp_eq.2 (over.w f).symm),
by tidy⟩⟩ } | instance | category_theory.over.forget_reflects_iso | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_faithful : faithful (forget X) | {}. | instance | category_theory.over.forget_faithful | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_of_epi_left {f g : over X} (k : f ⟶ g) [hk : epi k.left] : epi k | (forget X).epi_of_epi_map hk | lemma | category_theory.over.epi_of_epi_left | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_of_mono_left {f g : over X} (k : f ⟶ g) [hk : mono k.left] : mono k | (forget X).mono_of_mono_map hk | lemma | category_theory.over.mono_of_mono_left | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | If `k.left` is a monomorphism, then `k` is a monomorphism. In other words, `over.forget X` reflects
monomorphisms.
The converse of `category_theory.over.mono_left_of_mono`.
This lemma is not an instance, to avoid loops in type class inference. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono_left_of_mono {f g : over X} (k : f ⟶ g) [mono k] : mono k.left | begin
refine ⟨λ (Y : T) l m a, _⟩,
let l' : mk (m ≫ f.hom) ⟶ f := hom_mk l (by { dsimp, rw [←over.w k, reassoc_of a] }),
suffices : l' = hom_mk m,
{ apply congr_arg comma_morphism.left this },
rw ← cancel_mono k,
ext,
apply a,
end | instance | category_theory.over.mono_left_of_mono | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | If `k` is a monomorphism, then `k.left` is a monomorphism. In other words, `over.forget X` preserves
monomorphisms.
The converse of `category_theory.over.mono_of_mono_left`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_slice_forward : over f ⥤ over f.left | { obj := λ α, over.mk α.hom.left,
map := λ α β κ, over.hom_mk κ.left.left (by { rw auto_param_eq, rw ← over.w κ, refl }) } | def | category_theory.over.iterated_slice_forward | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | Given f : Y ⟶ X, this is the obvious functor from (T/X)/f to T/Y | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_slice_backward : over f.left ⥤ over f | { obj := λ g, mk (hom_mk g.hom : mk (g.hom ≫ f.hom) ⟶ f),
map := λ g h α, hom_mk (hom_mk α.left (w_assoc α f.hom)) (over_morphism.ext (w α)) } | def | category_theory.over.iterated_slice_backward | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | Given f : Y ⟶ X, this is the obvious functor from T/Y to (T/X)/f | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_slice_equiv : over f ≌ over f.left | { functor := iterated_slice_forward f,
inverse := iterated_slice_backward f,
unit_iso :=
nat_iso.of_components
(λ g, over.iso_mk (over.iso_mk (iso.refl _) (by tidy)) (by tidy))
(λ X Y g, by { ext, dsimp, simp }),
counit_iso :=
nat_iso.of_components
(λ g, over.iso_mk (iso.refl _) (by tidy))
... | def | category_theory.over.iterated_slice_equiv | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | Given f : Y ⟶ X, we have an equivalence between (T/X)/f and T/Y | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_slice_forward_forget :
iterated_slice_forward f ⋙ forget f.left = forget f ⋙ forget X | rfl | lemma | category_theory.over.iterated_slice_forward_forget | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_slice_backward_forget_forget :
iterated_slice_backward f ⋙ forget f ⋙ forget X = forget f.left | rfl | lemma | category_theory.over.iterated_slice_backward_forget_forget | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
post (F : T ⥤ D) : over X ⥤ over (F.obj X) | { obj := λ Y, mk $ F.map Y.hom,
map := λ Y₁ Y₂ f, over.hom_mk (F.map f.left) (by tidy; erw [← F.map_comp, w]) } | def | category_theory.over.post | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | A functor `F : T ⥤ D` induces a functor `over X ⥤ over (F.obj X)` in the obvious way. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
under (X : T) | structured_arrow X (𝟭 T) | def | category_theory.under | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | The under category has as objects arrows with domain `X` and as morphisms commutative
triangles. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
under.inhabited [inhabited T] : inhabited (under (default : T)) | { default :=
{ left := default,
right := default,
hom := 𝟙 _ } } | instance | category_theory.under.inhabited | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
under_morphism.ext {X : T} {U V : under X} {f g : U ⟶ V}
(h : f.right = g.right) : f = g | by tidy | lemma | category_theory.under.under_morphism.ext | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
under_left (U : under X) : U.left = ⟨⟨⟩⟩ | by tidy | lemma | category_theory.under.under_left | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_right (U : under X) : comma_morphism.right (𝟙 U) = 𝟙 U.right | rfl | lemma | category_theory.under.id_right | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_right (a b c : under X) (f : a ⟶ b) (g : b ⟶ c) :
(f ≫ g).right = f.right ≫ g.right | rfl | lemma | category_theory.under.comp_right | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w {A B : under X} (f : A ⟶ B) : A.hom ≫ f.right = B.hom | by have := f.w; tidy | lemma | category_theory.under.w | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk {X Y : T} (f : X ⟶ Y) : under X | structured_arrow.mk f | def | category_theory.under.mk | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | To give an object in the under category, it suffices to give an arrow with domain `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_mk {U V : under X} (f : U.right ⟶ V.right) (w : U.hom ≫ f = V.hom . obviously) :
U ⟶ V | structured_arrow.hom_mk f w | def | category_theory.under.hom_mk | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | To give a morphism in the under category, it suffices to give a morphism fitting in a
commutative triangle. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_mk {f g : under X} (hr : f.right ≅ g.right) (hw : f.hom ≫ hr.hom = g.hom) : f ≅ g | structured_arrow.iso_mk hr hw | def | category_theory.under.iso_mk | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | Construct an isomorphism in the over category given isomorphisms of the objects whose forward
direction gives a commutative triangle. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_mk_hom_right {f g : under X} (hr : f.right ≅ g.right) (hw : f.hom ≫ hr.hom = g.hom) :
(iso_mk hr hw).hom.right = hr.hom | rfl | lemma | category_theory.under.iso_mk_hom_right | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_mk_inv_right {f g : under X} (hr : f.right ≅ g.right) (hw : f.hom ≫ hr.hom = g.hom) :
(iso_mk hr hw).inv.right = hr.inv | rfl | lemma | category_theory.under.iso_mk_inv_right | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget : under X ⥤ T | comma.snd _ _ | def | category_theory.under.forget | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | The forgetful functor mapping an arrow to its domain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_obj {U : under X} : (forget X).obj U = U.right | rfl | lemma | category_theory.under.forget_obj | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_map {U V : under X} {f : U ⟶ V} : (forget X).map f = f.right | rfl | lemma | category_theory.under.forget_map | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_cone (X : T) : limits.cone (forget X) | { X := X, π := { app := comma.hom } } | def | category_theory.under.forget_cone | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | The natural cone over the forgetful functor `under X ⥤ T` with cone point `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map {Y : T} (f : X ⟶ Y) : under Y ⥤ under X | comma.map_left _ $ discrete.nat_trans (λ _, f) | def | category_theory.under.map | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | A morphism `X ⟶ Y` induces a functor `under Y ⥤ under X` in the obvious way. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_obj_right : ((map f).obj U).right = U.right | rfl | lemma | category_theory.under.map_obj_right | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_obj_hom : ((map f).obj U).hom = f ≫ U.hom | rfl | lemma | category_theory.under.map_obj_hom | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_map_right : ((map f).map g).right = g.right | rfl | lemma | category_theory.under.map_map_right | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp {Y Z : T} (f : X ⟶ Y) (g : Y ⟶ Z) : map (f ≫ g) ≅ map g ⋙ map f | nat_iso.of_components (λ X, iso_mk (iso.refl _) (by tidy)) (by tidy) | def | category_theory.under.map_comp | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [
"map_comp"
] | Mapping by the composite morphism `f ≫ g` is the same as mapping by `f` then by `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_reflects_iso : reflects_isomorphisms (forget X) | { reflects := λ Y Z f t, by exactI
⟨⟨under.hom_mk (inv ((under.forget X).map f)) ((is_iso.comp_inv_eq _).2 (under.w f).symm),
by tidy⟩⟩ } | instance | category_theory.under.forget_reflects_iso | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_of_mono_right {f g : under X} (k : f ⟶ g) [hk : mono k.right] : mono k | (forget X).mono_of_mono_map hk | lemma | category_theory.under.mono_of_mono_right | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_of_epi_right {f g : under X} (k : f ⟶ g) [hk : epi k.right] : epi k | (forget X).epi_of_epi_map hk | lemma | category_theory.under.epi_of_epi_right | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | If `k.right` is a epimorphism, then `k` is a epimorphism. In other words, `under.forget X` reflects
epimorphisms.
The converse of `category_theory.under.epi_right_of_epi`.
This lemma is not an instance, to avoid loops in type class inference. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
epi_right_of_epi {f g : under X} (k : f ⟶ g) [epi k] : epi k.right | begin
refine ⟨λ (Y : T) l m a, _⟩,
let l' : g ⟶ mk (g.hom ≫ m) := hom_mk l
(by { dsimp, rw [←under.w k, category.assoc, a, category.assoc] }),
suffices : l' = hom_mk m,
{ apply congr_arg comma_morphism.right this },
rw ← cancel_epi k,
ext,
apply a,
end | instance | category_theory.under.epi_right_of_epi | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | If `k` is a epimorphism, then `k.right` is a epimorphism. In other words, `under.forget X` preserves
epimorphisms.
The converse of `category_theory.under.epi_of_epi_right`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
post {X : T} (F : T ⥤ D) : under X ⥤ under (F.obj X) | { obj := λ Y, mk $ F.map Y.hom,
map := λ Y₁ Y₂ f, under.hom_mk (F.map f.right) (by tidy; erw [← F.map_comp, w]), } | def | category_theory.under.post | category_theory | src/category_theory/over.lean | [
"category_theory.structured_arrow",
"category_theory.punit",
"category_theory.functor.reflects_isomorphisms",
"category_theory.functor.epi_mono"
] | [] | A functor `F : T ⥤ D` induces a functor `under X ⥤ under (F.obj X)` in the obvious way. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
paths (V : Type u₁) : Type u₁ | V | def | category_theory.paths | category_theory | src/category_theory/path_category.lean | [
"category_theory.eq_to_hom",
"category_theory.quotient",
"combinatorics.quiver.path"
] | [] | A type synonym for the category of paths in a quiver. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
category_paths : category.{max u₁ v₁} (paths V) | { hom := λ (X Y : V), quiver.path X Y,
id := λ X, quiver.path.nil,
comp := λ X Y Z f g, quiver.path.comp f g, } | instance | category_theory.paths.category_paths | category_theory | src/category_theory/path_category.lean | [
"category_theory.eq_to_hom",
"category_theory.quotient",
"combinatorics.quiver.path"
] | [
"quiver.path",
"quiver.path.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of : V ⥤q (paths V) | { obj := λ X, X,
map := λ X Y f, f.to_path, } | def | category_theory.paths.of | category_theory | src/category_theory/path_category.lean | [
"category_theory.eq_to_hom",
"category_theory.quotient",
"combinatorics.quiver.path"
] | [] | The inclusion of a quiver `V` into its path category, as a prefunctor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift {C} [category C] (φ : V ⥤q C) : paths V ⥤ C | { obj := φ.obj,
map := λ X Y f, @quiver.path.rec V _ X (λ Y f, φ.obj X ⟶ φ.obj Y) (𝟙 $ φ.obj X)
(λ Y Z p f ihp, ihp ≫ (φ.map f)) Y f,
map_id' := λ X, by { refl, },
map_comp' := λ X Y Z f g, by
{ induction g with _ _ g' p ih _ _ _,
{ rw category.comp_id, refl, },
{ have : f ≫ g'.cons p... | def | category_theory.paths.lift | category_theory | src/category_theory/path_category.lean | [
"category_theory.eq_to_hom",
"category_theory.quotient",
"combinatorics.quiver.path"
] | [
"ih",
"lift",
"quiver.path.comp_cons"
] | Any prefunctor from `V` lifts to a functor from `paths V` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_nil {C} [category C] (φ : V ⥤q C) (X : V) :
(lift φ).map (quiver.path.nil) = 𝟙 (φ.obj X) | rfl | lemma | category_theory.paths.lift_nil | category_theory | src/category_theory/path_category.lean | [
"category_theory.eq_to_hom",
"category_theory.quotient",
"combinatorics.quiver.path"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_cons {C} [category C] (φ : V ⥤q C) {X Y Z : V}
(p : quiver.path X Y) (f : Y ⟶ Z) :
(lift φ).map (p.cons f) = (lift φ).map p ≫ (φ.map f) | rfl | lemma | category_theory.paths.lift_cons | category_theory | src/category_theory/path_category.lean | [
"category_theory.eq_to_hom",
"category_theory.quotient",
"combinatorics.quiver.path"
] | [
"lift",
"quiver.path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_to_path {C} [category C] (φ : V ⥤q C) {X Y : V} (f : X ⟶ Y) :
(lift φ).map f.to_path = φ.map f | by {dsimp [quiver.hom.to_path,lift], simp, } | lemma | category_theory.paths.lift_to_path | category_theory | src/category_theory/path_category.lean | [
"category_theory.eq_to_hom",
"category_theory.quotient",
"combinatorics.quiver.path"
] | [
"lift",
"quiver.hom.to_path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_spec {C} [category C] (φ : V ⥤q C) :
of ⋙q (lift φ).to_prefunctor = φ | begin
apply prefunctor.ext, rotate,
{ rintro X, refl, },
{ rintro X Y f, rcases φ with ⟨φo,φm⟩,
dsimp [lift, quiver.hom.to_path],
simp only [category.id_comp], },
end | lemma | category_theory.paths.lift_spec | category_theory | src/category_theory/path_category.lean | [
"category_theory.eq_to_hom",
"category_theory.quotient",
"combinatorics.quiver.path"
] | [
"lift",
"prefunctor.ext",
"quiver.hom.to_path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_unique {C} [category C] (φ : V ⥤q C) (Φ : paths V ⥤ C)
(hΦ : of ⋙q Φ.to_prefunctor = φ) : Φ = lift φ | begin
subst_vars,
apply functor.ext, rotate,
{ rintro X, refl, },
{ rintro X Y f,
dsimp [lift],
induction f with _ _ p f' ih,
{ simp only [category.comp_id], apply functor.map_id, },
{ simp only [category.comp_id, category.id_comp] at ih ⊢,
have : Φ.map (p.cons f') = Φ.map p ≫ (Φ.map (f'.t... | lemma | category_theory.paths.lift_unique | category_theory | src/category_theory/path_category.lean | [
"category_theory.eq_to_hom",
"category_theory.quotient",
"combinatorics.quiver.path"
] | [
"functor.ext",
"functor.map_id",
"ih",
"lift",
"lift_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_functor {C} [category C]
{F G : paths V ⥤ C}
(h_obj : F.obj = G.obj)
(h : ∀ (a b : V) (e : a ⟶ b), F.map e.to_path =
eq_to_hom (congr_fun h_obj a) ≫ G.map e.to_path ≫ eq_to_hom (congr_fun h_obj.symm b)) :
F = G | begin
ext X Y f,
{ induction f with Y' Z' g e ih,
{ erw [F.map_id, G.map_id, category.id_comp, eq_to_hom_trans, eq_to_hom_refl], },
{ erw [F.map_comp g e.to_path, G.map_comp g e.to_path, ih, h],
simp only [category.id_comp, eq_to_hom_refl, eq_to_hom_trans_assoc, category.assoc], }, },
{ intro X, rw ... | lemma | category_theory.paths.ext_functor | category_theory | src/category_theory/path_category.lean | [
"category_theory.eq_to_hom",
"category_theory.quotient",
"combinatorics.quiver.path"
] | [
"ih"
] | Two functors out of a path category are equal when they agree on singleton paths. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prefunctor.map_path_comp' (F : V ⥤q W) {X Y Z : paths V} (f : X ⟶ Y) (g : Y ⟶ Z) :
F.map_path (f ≫ g) = (F.map_path f).comp (F.map_path g) | prefunctor.map_path_comp _ _ _ | lemma | category_theory.prefunctor.map_path_comp' | category_theory | src/category_theory/path_category.lean | [
"category_theory.eq_to_hom",
"category_theory.quotient",
"combinatorics.quiver.path"
] | [
"prefunctor.map_path_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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