statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
equiv_equiv_iso {A B : Fintype} : (A ≃ B) ≃ (A ≅ B) | { to_fun := λ e,
{ hom := e,
inv := e.symm, },
inv_fun := λ i,
{ to_fun := i.hom,
inv_fun := i.inv,
left_inv := iso.hom_inv_id_apply i,
right_inv := iso.inv_hom_id_apply i, },
left_inv := by tidy,
right_inv := by tidy, } | def | Fintype.equiv_equiv_iso | category_theory | src/category_theory/Fintype.lean | [
"category_theory.concrete_category.basic",
"category_theory.full_subcategory",
"category_theory.skeletal",
"category_theory.elementwise",
"data.fintype.card"
] | [
"Fintype",
"equiv_equiv_iso",
"inv_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
skeleton : Type u | ulift ℕ | def | Fintype.skeleton | category_theory | src/category_theory/Fintype.lean | [
"category_theory.concrete_category.basic",
"category_theory.full_subcategory",
"category_theory.skeletal",
"category_theory.elementwise",
"data.fintype.card"
] | [] | The "standard" skeleton for `Fintype`. This is the full subcategory of `Fintype` spanned by objects
of the form `ulift (fin n)` for `n : ℕ`. We parameterize the objects of `Fintype.skeleton`
directly as `ulift ℕ`, as the type `ulift (fin m) ≃ ulift (fin n)` is
nonempty if and only if `n = m`. Specifying universes, `ske... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk : ℕ → skeleton | ulift.up | def | Fintype.skeleton.mk | category_theory | src/category_theory/Fintype.lean | [
"category_theory.concrete_category.basic",
"category_theory.full_subcategory",
"category_theory.skeletal",
"category_theory.elementwise",
"data.fintype.card"
] | [] | Given any natural number `n`, this creates the associated object of `Fintype.skeleton`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
len : skeleton → ℕ | ulift.down | def | Fintype.skeleton.len | category_theory | src/category_theory/Fintype.lean | [
"category_theory.concrete_category.basic",
"category_theory.full_subcategory",
"category_theory.skeletal",
"category_theory.elementwise",
"data.fintype.card"
] | [] | Given any object of `Fintype.skeleton`, this returns the associated natural number. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext (X Y : skeleton) : X.len = Y.len → X = Y | ulift.ext _ _ | lemma | Fintype.skeleton.ext | category_theory | src/category_theory/Fintype.lean | [
"category_theory.concrete_category.basic",
"category_theory.full_subcategory",
"category_theory.skeletal",
"category_theory.elementwise",
"data.fintype.card"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_skeletal : skeletal skeleton.{u} | λ X Y ⟨h⟩, ext _ _ $ fin.equiv_iff_eq.mp $
nonempty.intro $
{ to_fun := λ x, (h.hom ⟨x⟩).down,
inv_fun := λ x, (h.inv ⟨x⟩).down,
left_inv := begin
intro a,
change ulift.down _ = _,
rw ulift.up_down,
change ((h.hom ≫ h.inv) _).down = _,
simpa,
end,
right_inv := begin
intro a,
change... | lemma | Fintype.skeleton.is_skeletal | category_theory | src/category_theory/Fintype.lean | [
"category_theory.concrete_category.basic",
"category_theory.full_subcategory",
"category_theory.skeletal",
"category_theory.elementwise",
"data.fintype.card"
] | [
"inv_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
incl : skeleton.{u} ⥤ Fintype.{u} | { obj := λ X, Fintype.of (ulift (fin X.len)),
map := λ _ _ f, f } | def | Fintype.skeleton.incl | category_theory | src/category_theory/Fintype.lean | [
"category_theory.concrete_category.basic",
"category_theory.full_subcategory",
"category_theory.skeletal",
"category_theory.elementwise",
"data.fintype.card"
] | [
"Fintype.of"
] | The canonical fully faithful embedding of `Fintype.skeleton` into `Fintype`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence : skeleton ≌ Fintype | incl.as_equivalence | def | Fintype.skeleton.equivalence | category_theory | src/category_theory/Fintype.lean | [
"category_theory.concrete_category.basic",
"category_theory.full_subcategory",
"category_theory.skeletal",
"category_theory.elementwise",
"data.fintype.card"
] | [
"Fintype"
] | The equivalence between `Fintype.skeleton` and `Fintype`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
incl_mk_nat_card (n : ℕ) : fintype.card (incl.obj (mk n)) = n | begin
convert finset.card_fin n,
apply fintype.of_equiv_card,
end | lemma | Fintype.skeleton.incl_mk_nat_card | category_theory | src/category_theory/Fintype.lean | [
"category_theory.concrete_category.basic",
"category_theory.full_subcategory",
"category_theory.skeletal",
"category_theory.elementwise",
"data.fintype.card"
] | [
"finset.card_fin",
"fintype.card",
"fintype.of_equiv_card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_skeleton : is_skeleton_of Fintype skeleton skeleton.incl | { skel := skeleton.is_skeletal,
eqv := by apply_instance } | def | Fintype.is_skeleton | category_theory | src/category_theory/Fintype.lean | [
"category_theory.concrete_category.basic",
"category_theory.full_subcategory",
"category_theory.skeletal",
"category_theory.elementwise",
"data.fintype.card"
] | [
"Fintype"
] | `Fintype.skeleton` is a skeleton of `Fintype`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
discrete_fintype {α : Type*} [fintype α] : fintype (discrete α) | fintype.of_equiv α (discrete_equiv.symm) | instance | category_theory.discrete_fintype | category_theory | src/category_theory/fin_category.lean | [
"data.fintype.card",
"category_theory.discrete_category",
"category_theory.opposites",
"category_theory.category.ulift"
] | [
"fintype",
"fintype.of_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete_hom_fintype {α : Type*} (X Y : discrete α) : fintype (X ⟶ Y) | by { apply ulift.fintype } | instance | category_theory.discrete_hom_fintype | category_theory | src/category_theory/fin_category.lean | [
"data.fintype.card",
"category_theory.discrete_category",
"category_theory.opposites",
"category_theory.category.ulift"
] | [
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fin_category (J : Type v) [small_category J] | (fintype_obj : fintype J . tactic.apply_instance)
(fintype_hom : Π (j j' : J), fintype (j ⟶ j') . tactic.apply_instance) | class | category_theory.fin_category | category_theory | src/category_theory/fin_category.lean | [
"data.fintype.card",
"category_theory.discrete_category",
"category_theory.opposites",
"category_theory.category.ulift"
] | [
"fintype"
] | A category with a `fintype` of objects, and a `fintype` for each morphism space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fin_category_discrete_of_fintype (J : Type v) [fintype J] :
fin_category (discrete J) | { } | instance | category_theory.fin_category_discrete_of_fintype | category_theory | src/category_theory/fin_category.lean | [
"data.fintype.card",
"category_theory.discrete_category",
"category_theory.opposites",
"category_theory.category.ulift"
] | [
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
obj_as_type : Type | induced_category α (fintype.equiv_fin α).symm | abbreviation | category_theory.fin_category.obj_as_type | category_theory | src/category_theory/fin_category.lean | [
"data.fintype.card",
"category_theory.discrete_category",
"category_theory.opposites",
"category_theory.category.ulift"
] | [
"fintype.equiv_fin"
] | A fin_category `α` is equivalent to a category with objects in `Type`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
obj_as_type_equiv : obj_as_type α ≌ α | (induced_functor (fintype.equiv_fin α).symm).as_equivalence | def | category_theory.fin_category.obj_as_type_equiv | category_theory | src/category_theory/fin_category.lean | [
"data.fintype.card",
"category_theory.discrete_category",
"category_theory.opposites",
"category_theory.category.ulift"
] | [
"fintype.equiv_fin"
] | The constructed category is indeed equivalent to `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
as_type : Type | fin (fintype.card α) | abbreviation | category_theory.fin_category.as_type | category_theory | src/category_theory/fin_category.lean | [
"data.fintype.card",
"category_theory.discrete_category",
"category_theory.opposites",
"category_theory.category.ulift"
] | [
"fintype.card"
] | A fin_category `α` is equivalent to a fin_category with in `Type`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
category_as_type : small_category (as_type α) | { hom := λ i j, fin (fintype.card (@quiver.hom (obj_as_type α) _ i j)),
id := λ i, fintype.equiv_fin _ (𝟙 i),
comp := λ i j k f g, fintype.equiv_fin _
((fintype.equiv_fin _).symm f ≫ (fintype.equiv_fin _).symm g) } | instance | category_theory.fin_category.category_as_type | category_theory | src/category_theory/fin_category.lean | [
"data.fintype.card",
"category_theory.discrete_category",
"category_theory.opposites",
"category_theory.category.ulift"
] | [
"fintype.card",
"fintype.equiv_fin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
as_type_to_obj_as_type : as_type α ⥤ obj_as_type α | { obj := id, map := λ i j, (fintype.equiv_fin _).symm } | def | category_theory.fin_category.as_type_to_obj_as_type | category_theory | src/category_theory/fin_category.lean | [
"data.fintype.card",
"category_theory.discrete_category",
"category_theory.opposites",
"category_theory.category.ulift"
] | [
"fintype.equiv_fin"
] | The "identity" functor from `as_type α` to `obj_as_type α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
obj_as_type_to_as_type : obj_as_type α ⥤ as_type α | { obj := id, map := λ i j, fintype.equiv_fin _ } | def | category_theory.fin_category.obj_as_type_to_as_type | category_theory | src/category_theory/fin_category.lean | [
"data.fintype.card",
"category_theory.discrete_category",
"category_theory.opposites",
"category_theory.category.ulift"
] | [
"fintype.equiv_fin"
] | The "identity" functor from `obj_as_type α` to `as_type α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
as_type_equiv_obj_as_type : as_type α ≌ obj_as_type α | equivalence.mk (as_type_to_obj_as_type α) (obj_as_type_to_as_type α)
(nat_iso.of_components iso.refl $ λ _ _ _, by { dsimp, simp })
(nat_iso.of_components iso.refl $ λ _ _ _, by { dsimp, simp }) | def | category_theory.fin_category.as_type_equiv_obj_as_type | category_theory | src/category_theory/fin_category.lean | [
"data.fintype.card",
"category_theory.discrete_category",
"category_theory.opposites",
"category_theory.category.ulift"
] | [] | The constructed category (`as_type α`) is equivalent to `obj_as_type α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
as_type_fin_category : fin_category (as_type α) | {} | instance | category_theory.fin_category.as_type_fin_category | category_theory | src/category_theory/fin_category.lean | [
"data.fintype.card",
"category_theory.discrete_category",
"category_theory.opposites",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_as_type : as_type α ≌ α | (as_type_equiv_obj_as_type α).trans (obj_as_type_equiv α) | def | category_theory.fin_category.equiv_as_type | category_theory | src/category_theory/fin_category.lean | [
"data.fintype.card",
"category_theory.discrete_category",
"category_theory.opposites",
"category_theory.category.ulift"
] | [] | The constructed category (`as_type α`) is indeed equivalent to `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fin_category_opposite {J : Type v} [small_category J] [fin_category J] :
fin_category Jᵒᵖ | { fintype_obj := fintype.of_equiv _ equiv_to_opposite,
fintype_hom := λ j j', fintype.of_equiv _ (op_equiv j j').symm, } | instance | category_theory.fin_category_opposite | category_theory | src/category_theory/fin_category.lean | [
"data.fintype.card",
"category_theory.discrete_category",
"category_theory.opposites",
"category_theory.category.ulift"
] | [
"fintype.of_equiv"
] | The opposite of a finite category is finite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fin_category_ulift {J : Type v} [small_category J] [fin_category J] :
fin_category.{(max w v)} (ulift_hom.{w (max w v)} (ulift.{w v} J)) | { fintype_obj := ulift.fintype J } | instance | category_theory.fin_category_ulift | category_theory | src/category_theory/fin_category.lean | [
"data.fintype.card",
"category_theory.discrete_category",
"category_theory.opposites",
"category_theory.category.ulift"
] | [] | Applying `ulift` to morphisms and objects of a category preserves finiteness. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
induced_category : Type u₁ | C | def | category_theory.induced_category | category_theory | src/category_theory/full_subcategory.lean | [
"category_theory.functor.fully_faithful"
] | [] | `induced_category D F`, where `F : C → D`, is a typeclass synonym for `C`,
which provides a category structure so that the morphisms `X ⟶ Y` are the morphisms
in `D` from `F X` to `F Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
induced_category.has_coe_to_sort {α : Sort*} [has_coe_to_sort D α] :
has_coe_to_sort (induced_category D F) α | ⟨λ c, ↥(F c)⟩ | instance | category_theory.induced_category.has_coe_to_sort | category_theory | src/category_theory/full_subcategory.lean | [
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_category.category : category.{v} (induced_category D F) | { hom := λ X Y, F X ⟶ F Y,
id := λ X, 𝟙 (F X),
comp := λ _ _ _ f g, f ≫ g } | instance | category_theory.induced_category.category | category_theory | src/category_theory/full_subcategory.lean | [
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_functor : induced_category D F ⥤ D | { obj := F, map := λ x y f, f } | def | category_theory.induced_functor | category_theory | src/category_theory/full_subcategory.lean | [
"category_theory.functor.fully_faithful"
] | [] | The forgetful functor from an induced category to the original category,
forgetting the extra data. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
induced_category.full : full (induced_functor F) | { preimage := λ x y f, f } | instance | category_theory.induced_category.full | category_theory | src/category_theory/full_subcategory.lean | [
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_category.faithful : faithful (induced_functor F) | {} | instance | category_theory.induced_category.faithful | category_theory | src/category_theory/full_subcategory.lean | [
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
full_subcategory | (obj : C)
(property : Z obj) | structure | category_theory.full_subcategory | category_theory | src/category_theory/full_subcategory.lean | [
"category_theory.functor.fully_faithful"
] | [] | A subtype-like structure for full subcategories. Morphisms just ignore the property. We don't use
actual subtypes since the simp-normal form `↑X` of `X.val` does not work well for full
subcategories.
See <https://stacks.math.columbia.edu/tag/001D>. We do not define 'strictly full' subcategories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
full_subcategory.category : category.{v} (full_subcategory Z) | induced_category.category full_subcategory.obj | instance | category_theory.full_subcategory.category | category_theory | src/category_theory/full_subcategory.lean | [
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
full_subcategory_inclusion : full_subcategory Z ⥤ C | induced_functor full_subcategory.obj | def | category_theory.full_subcategory_inclusion | category_theory | src/category_theory/full_subcategory.lean | [
"category_theory.functor.fully_faithful"
] | [] | The forgetful functor from a full subcategory into the original category
("forgetting" the condition). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
full_subcategory_inclusion.obj {X} :
(full_subcategory_inclusion Z).obj X = X.obj | rfl | lemma | category_theory.full_subcategory_inclusion.obj | category_theory | src/category_theory/full_subcategory.lean | [
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
full_subcategory_inclusion.map {X Y} {f : X ⟶ Y} :
(full_subcategory_inclusion Z).map f = f | rfl | lemma | category_theory.full_subcategory_inclusion.map | category_theory | src/category_theory/full_subcategory.lean | [
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
full_subcategory.full : full (full_subcategory_inclusion Z) | induced_category.full _ | instance | category_theory.full_subcategory.full | category_theory | src/category_theory/full_subcategory.lean | [
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
full_subcategory.faithful : faithful (full_subcategory_inclusion Z) | induced_category.faithful _ | instance | category_theory.full_subcategory.faithful | category_theory | src/category_theory/full_subcategory.lean | [
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
full_subcategory.map (h : ∀ ⦃X⦄, Z X → Z' X) : full_subcategory Z ⥤ full_subcategory Z' | { obj := λ X, ⟨X.1, h X.2⟩,
map := λ X Y f, f } | def | category_theory.full_subcategory.map | category_theory | src/category_theory/full_subcategory.lean | [
"category_theory.functor.fully_faithful"
] | [] | An implication of predicates `Z → Z'` induces a functor between full subcategories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
full_subcategory.map_inclusion (h : ∀ ⦃X⦄, Z X → Z' X) :
full_subcategory.map h ⋙ full_subcategory_inclusion Z' = full_subcategory_inclusion Z | rfl | lemma | category_theory.full_subcategory.map_inclusion | category_theory | src/category_theory/full_subcategory.lean | [
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
full_subcategory.lift (F : C ⥤ D) (hF : ∀ X, P (F.obj X)) : C ⥤ full_subcategory P | { obj := λ X, ⟨F.obj X, hF X⟩,
map := λ X Y f, F.map f } | def | category_theory.full_subcategory.lift | category_theory | src/category_theory/full_subcategory.lean | [
"category_theory.functor.fully_faithful"
] | [] | A functor which maps objects to objects satisfying a certain property induces a lift through
the full subcategory of objects satisfying that property. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
full_subcategory.lift_comp_inclusion (F : C ⥤ D) (hF : ∀ X, P (F.obj X)) :
full_subcategory.lift P F hF ⋙ full_subcategory_inclusion P ≅ F | nat_iso.of_components (λ X, iso.refl _) (by simp) | def | category_theory.full_subcategory.lift_comp_inclusion | category_theory | src/category_theory/full_subcategory.lean | [
"category_theory.functor.fully_faithful"
] | [] | Composing the lift of a functor through a full subcategory with the inclusion yields the
original functor. Unfortunately, this is not true by definition, so we only get a natural
isomorphism, but it is pointwise definitionally true, see
`full_subcategory.inclusion_obj_lift_obj` and `full_subcategory.inclusi... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
full_subcategory.inclusion_obj_lift_obj (F : C ⥤ D) (hF : ∀ X, P (F.obj X)) {X : C} :
(full_subcategory_inclusion P).obj ((full_subcategory.lift P F hF).obj X) = F.obj X | rfl | lemma | category_theory.full_subcategory.inclusion_obj_lift_obj | category_theory | src/category_theory/full_subcategory.lean | [
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
full_subcategory.inclusion_map_lift_map (F : C ⥤ D) (hF : ∀ X, P (F.obj X)) {X Y : C}
(f : X ⟶ Y) :
(full_subcategory_inclusion P).map ((full_subcategory.lift P F hF).map f) = F.map f | rfl | lemma | category_theory.full_subcategory.inclusion_map_lift_map | category_theory | src/category_theory/full_subcategory.lean | [
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
full_subcategory.lift_comp_map (F : C ⥤ D) (hF : ∀ X, P (F.obj X)) (h : ∀ ⦃X⦄, P X → Q X) :
full_subcategory.lift P F hF ⋙ full_subcategory.map h =
full_subcategory.lift Q F (λ X, h (hF X)) | rfl | lemma | category_theory.full_subcategory.lift_comp_map | category_theory | src/category_theory/full_subcategory.lean | [
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_separating (𝒢 : set C) : Prop | ∀ ⦃X Y : C⦄ (f g : X ⟶ Y), (∀ (G ∈ 𝒢) (h : G ⟶ X), h ≫ f = h ≫ g) → f = g | def | category_theory.is_separating | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | We say that `𝒢` is a separating set if the functors `C(G, -)` for `G ∈ 𝒢` are collectively
faithful, i.e., if `h ≫ f = h ≫ g` for all `h` with domain in `𝒢` implies `f = g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_coseparating (𝒢 : set C) : Prop | ∀ ⦃X Y : C⦄ (f g : X ⟶ Y), (∀ (G ∈ 𝒢) (h : Y ⟶ G), f ≫ h = g ≫ h) → f = g | def | category_theory.is_coseparating | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | We say that `𝒢` is a coseparating set if the functors `C(-, G)` for `G ∈ 𝒢` are collectively
faithful, i.e., if `f ≫ h = g ≫ h` for all `h` with codomain in `𝒢` implies `f = g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_detecting (𝒢 : set C) : Prop | ∀ ⦃X Y : C⦄ (f : X ⟶ Y), (∀ (G ∈ 𝒢) (h : G ⟶ Y), ∃! (h' : G ⟶ X), h' ≫ f = h) → is_iso f | def | category_theory.is_detecting | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | We say that `𝒢` is a detecting set if the functors `C(G, -)` collectively reflect isomorphisms,
i.e., if any `h` with domain in `𝒢` uniquely factors through `f`, then `f` is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_codetecting (𝒢 : set C) : Prop | ∀ ⦃X Y : C⦄ (f : X ⟶ Y), (∀ (G ∈ 𝒢) (h : X ⟶ G), ∃! (h' : Y ⟶ G), f ≫ h' = h) → is_iso f | def | category_theory.is_codetecting | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | We say that `𝒢` is a codetecting set if the functors `C(-, G)` collectively reflect
isomorphisms, i.e., if any `h` with codomain in `G` uniquely factors through `f`, then `f` is
an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_separating_op_iff (𝒢 : set C) : is_separating 𝒢.op ↔ is_coseparating 𝒢 | begin
refine ⟨λ h𝒢 X Y f g hfg, _, λ h𝒢 X Y f g hfg, _⟩,
{ refine quiver.hom.op_inj (h𝒢 _ _ (λ G hG h, quiver.hom.unop_inj _)),
simpa only [unop_comp, quiver.hom.unop_op] using hfg _ (set.mem_op.1 hG) _ },
{ refine quiver.hom.unop_inj (h𝒢 _ _ (λ G hG h, quiver.hom.op_inj _)),
simpa only [op_comp, quiv... | lemma | category_theory.is_separating_op_iff | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"quiver.hom.op_inj",
"quiver.hom.op_unop",
"quiver.hom.unop_inj",
"quiver.hom.unop_op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coseparating_op_iff (𝒢 : set C) : is_coseparating 𝒢.op ↔ is_separating 𝒢 | begin
refine ⟨λ h𝒢 X Y f g hfg, _, λ h𝒢 X Y f g hfg, _⟩,
{ refine quiver.hom.op_inj (h𝒢 _ _ (λ G hG h, quiver.hom.unop_inj _)),
simpa only [unop_comp, quiver.hom.unop_op] using hfg _ (set.mem_op.1 hG) _ },
{ refine quiver.hom.unop_inj (h𝒢 _ _ (λ G hG h, quiver.hom.op_inj _)),
simpa only [op_comp, quiv... | lemma | category_theory.is_coseparating_op_iff | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"quiver.hom.op_inj",
"quiver.hom.op_unop",
"quiver.hom.unop_inj",
"quiver.hom.unop_op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coseparating_unop_iff (𝒢 : set Cᵒᵖ) : is_coseparating 𝒢.unop ↔ is_separating 𝒢 | by rw [← is_separating_op_iff, set.unop_op] | lemma | category_theory.is_coseparating_unop_iff | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"set.unop_op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_separating_unop_iff (𝒢 : set Cᵒᵖ) : is_separating 𝒢.unop ↔ is_coseparating 𝒢 | by rw [← is_coseparating_op_iff, set.unop_op] | lemma | category_theory.is_separating_unop_iff | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"set.unop_op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_detecting_op_iff (𝒢 : set C) : is_detecting 𝒢.op ↔ is_codetecting 𝒢 | begin
refine ⟨λ h𝒢 X Y f hf, _, λ h𝒢 X Y f hf, _⟩,
{ refine (is_iso_op_iff _).1 (h𝒢 _ (λ G hG h, _)),
obtain ⟨t, ht, ht'⟩ := hf (unop G) (set.mem_op.1 hG) h.unop,
exact ⟨t.op, quiver.hom.unop_inj ht, λ y hy,
quiver.hom.unop_inj (ht' _ (quiver.hom.op_inj hy))⟩ },
{ refine (is_iso_unop_iff _).1 (h�... | lemma | category_theory.is_detecting_op_iff | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"quiver.hom.op_inj",
"quiver.hom.unop_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_codetecting_op_iff (𝒢 : set C) : is_codetecting 𝒢.op ↔ is_detecting 𝒢 | begin
refine ⟨λ h𝒢 X Y f hf, _, λ h𝒢 X Y f hf, _⟩,
{ refine (is_iso_op_iff _).1 (h𝒢 _ (λ G hG h, _)),
obtain ⟨t, ht, ht'⟩ := hf (unop G) (set.mem_op.1 hG) h.unop,
exact ⟨t.op, quiver.hom.unop_inj ht, λ y hy,
quiver.hom.unop_inj (ht' _ (quiver.hom.op_inj hy))⟩ },
{ refine (is_iso_unop_iff _).1 (h�... | lemma | category_theory.is_codetecting_op_iff | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"quiver.hom.op_inj",
"quiver.hom.unop_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_detecting_unop_iff (𝒢 : set Cᵒᵖ) : is_detecting 𝒢.unop ↔ is_codetecting 𝒢 | by rw [← is_codetecting_op_iff, set.unop_op] | lemma | category_theory.is_detecting_unop_iff | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"set.unop_op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_codetecting_unop_iff {𝒢 : set Cᵒᵖ} : is_codetecting 𝒢.unop ↔ is_detecting 𝒢 | by rw [← is_detecting_op_iff, set.unop_op] | lemma | category_theory.is_codetecting_unop_iff | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"set.unop_op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_detecting.is_separating [has_equalizers C] {𝒢 : set C} (h𝒢 : is_detecting 𝒢) :
is_separating 𝒢 | λ X Y f g hfg,
have is_iso (equalizer.ι f g), from h𝒢 _ (λ G hG h, equalizer.exists_unique _ (hfg _ hG _)),
by exactI eq_of_epi_equalizer | lemma | category_theory.is_detecting.is_separating | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_codetecting.is_coseparating [has_coequalizers C] {𝒢 : set C} :
is_codetecting 𝒢 → is_coseparating 𝒢 | by simpa only [← is_separating_op_iff, ← is_detecting_op_iff] using is_detecting.is_separating | lemma | category_theory.is_codetecting.is_coseparating | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_separating.is_detecting [balanced C] {𝒢 : set C} (h𝒢 : is_separating 𝒢) :
is_detecting 𝒢 | begin
intros X Y f hf,
refine (is_iso_iff_mono_and_epi _).2 ⟨⟨λ Z g h hgh, h𝒢 _ _ (λ G hG i, _)⟩, ⟨λ Z g h hgh, _⟩⟩,
{ obtain ⟨t, -, ht⟩ := hf G hG (i ≫ g ≫ f),
rw [ht (i ≫ g) (category.assoc _ _ _), ht (i ≫ h) (hgh.symm ▸ category.assoc _ _ _)] },
{ refine h𝒢 _ _ (λ G hG i, _),
obtain ⟨t, rfl, -⟩ := ... | lemma | category_theory.is_separating.is_detecting | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"balanced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coseparating.is_codetecting [balanced C] {𝒢 : set C} :
is_coseparating 𝒢 → is_codetecting 𝒢 | by simpa only [← is_detecting_op_iff, ← is_separating_op_iff] using is_separating.is_detecting | lemma | category_theory.is_coseparating.is_codetecting | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"balanced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_detecting_iff_is_separating [has_equalizers C] [balanced C] (𝒢 : set C) :
is_detecting 𝒢 ↔ is_separating 𝒢 | ⟨is_detecting.is_separating, is_separating.is_detecting⟩ | lemma | category_theory.is_detecting_iff_is_separating | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"balanced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_codetecting_iff_is_coseparating [has_coequalizers C] [balanced C] {𝒢 : set C} :
is_codetecting 𝒢 ↔ is_coseparating 𝒢 | ⟨is_codetecting.is_coseparating, is_coseparating.is_codetecting⟩ | lemma | category_theory.is_codetecting_iff_is_coseparating | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"balanced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_separating.mono {𝒢 : set C} (h𝒢 : is_separating 𝒢) {ℋ : set C} (h𝒢ℋ : 𝒢 ⊆ ℋ) :
is_separating ℋ | λ X Y f g hfg, h𝒢 _ _ $ λ G hG h, hfg _ (h𝒢ℋ hG) _ | lemma | category_theory.is_separating.mono | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coseparating.mono {𝒢 : set C} (h𝒢 : is_coseparating 𝒢) {ℋ : set C} (h𝒢ℋ : 𝒢 ⊆ ℋ) :
is_coseparating ℋ | λ X Y f g hfg, h𝒢 _ _ $ λ G hG h, hfg _ (h𝒢ℋ hG) _ | lemma | category_theory.is_coseparating.mono | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_detecting.mono {𝒢 : set C} (h𝒢 : is_detecting 𝒢) {ℋ : set C} (h𝒢ℋ : 𝒢 ⊆ ℋ) :
is_detecting ℋ | λ X Y f hf, h𝒢 _ $ λ G hG h, hf _ (h𝒢ℋ hG) _ | lemma | category_theory.is_detecting.mono | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_codetecting.mono {𝒢 : set C} (h𝒢 : is_codetecting 𝒢) {ℋ : set C} (h𝒢ℋ : 𝒢 ⊆ ℋ) :
is_codetecting ℋ | λ X Y f hf, h𝒢 _ $ λ G hG h, hf _ (h𝒢ℋ hG) _ | lemma | category_theory.is_codetecting.mono | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thin_of_is_separating_empty (h : is_separating (∅ : set C)) : quiver.is_thin C | λ _ _, ⟨λ f g, h _ _ $ λ G, false.elim⟩ | lemma | category_theory.thin_of_is_separating_empty | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"quiver.is_thin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_separating_empty_of_thin [quiver.is_thin C] : is_separating (∅ : set C) | λ X Y f g hfg, subsingleton.elim _ _ | lemma | category_theory.is_separating_empty_of_thin | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"quiver.is_thin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thin_of_is_coseparating_empty (h : is_coseparating (∅ : set C)) : quiver.is_thin C | λ _ _, ⟨λ f g, h _ _ $ λ G, false.elim⟩ | lemma | category_theory.thin_of_is_coseparating_empty | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"quiver.is_thin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coseparating_empty_of_thin [quiver.is_thin C] :
is_coseparating (∅ : set C) | λ X Y f g hfg, subsingleton.elim _ _ | lemma | category_theory.is_coseparating_empty_of_thin | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"quiver.is_thin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
groupoid_of_is_detecting_empty (h : is_detecting (∅ : set C)) {X Y : C} (f : X ⟶ Y) :
is_iso f | h _ $ λ G, false.elim | lemma | category_theory.groupoid_of_is_detecting_empty | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_detecting_empty_of_groupoid [∀ {X Y : C} (f : X ⟶ Y), is_iso f] :
is_detecting (∅ : set C) | λ X Y f hf, infer_instance | lemma | category_theory.is_detecting_empty_of_groupoid | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
groupoid_of_is_codetecting_empty (h : is_codetecting (∅ : set C)) {X Y : C} (f : X ⟶ Y) :
is_iso f | h _ $ λ G, false.elim | lemma | category_theory.groupoid_of_is_codetecting_empty | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_codetecting_empty_of_groupoid [∀ {X Y : C} (f : X ⟶ Y), is_iso f] :
is_codetecting (∅ : set C) | λ X Y f hf, infer_instance | lemma | category_theory.is_codetecting_empty_of_groupoid | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_separating_iff_epi (𝒢 : set C)
[Π (A : C), has_coproduct (λ f : Σ G : 𝒢, (G : C) ⟶ A, (f.1 : C))] :
is_separating 𝒢 ↔ ∀ A : C, epi (sigma.desc (@sigma.snd 𝒢 (λ G, (G : C) ⟶ A))) | begin
refine ⟨λ h A, ⟨λ Z u v huv, h _ _ (λ G hG f, _)⟩, λ h X Y f g hh, _⟩,
{ simpa using (sigma.ι (λ f : Σ G : 𝒢, (G : C) ⟶ A, (f.1 : C)) ⟨⟨G, hG⟩, f⟩) ≫= huv },
{ haveI := h X,
refine (cancel_epi (sigma.desc (@sigma.snd 𝒢 (λ G, (G : C) ⟶ X)))).1 (colimit.hom_ext (λ j, _)),
simpa using hh j.as.1.1 j.a... | lemma | category_theory.is_separating_iff_epi | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coseparating_iff_mono (𝒢 : set C)
[Π (A : C), has_product (λ f : Σ G : 𝒢, A ⟶ (G : C), (f.1 : C))] :
is_coseparating 𝒢 ↔ ∀ A : C, mono (pi.lift (@sigma.snd 𝒢 (λ G, A ⟶ (G : C)))) | begin
refine ⟨λ h A, ⟨λ Z u v huv, h _ _ (λ G hG f, _)⟩, λ h X Y f g hh, _⟩,
{ simpa using huv =≫ (pi.π (λ f : Σ G : 𝒢, A ⟶ (G : C), (f.1 : C)) ⟨⟨G, hG⟩, f⟩) },
{ haveI := h Y,
refine (cancel_mono (pi.lift (@sigma.snd 𝒢 (λ G, Y ⟶ (G : C))))).1 (limit.hom_ext (λ j, _)),
simpa using hh j.as.1.1 j.as.1.2 j... | lemma | category_theory.is_coseparating_iff_mono | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_initial_of_is_coseparating [well_powered C] [has_limits C] {𝒢 : set C} [small.{v₁} 𝒢]
(h𝒢 : is_coseparating 𝒢) : has_initial C | begin
haveI := has_products_of_shape_of_small C 𝒢,
haveI := λ A, has_products_of_shape_of_small.{v₁} C (Σ G : 𝒢, A ⟶ (G : C)),
letI := complete_lattice_of_complete_semilattice_Inf (subobject (pi_obj (coe : 𝒢 → C))),
suffices : ∀ A : C, unique (((⊥ : subobject (pi_obj (coe : 𝒢 → C))) : C) ⟶ A),
{ exactI ha... | lemma | category_theory.has_initial_of_is_coseparating | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"bot_le",
"complete_lattice_of_complete_semilattice_Inf",
"unique"
] | An ingredient of the proof of the Special Adjoint Functor Theorem: a complete well-powered
category with a small coseparating set has an initial object.
In fact, it follows from the Special Adjoint Functor Theorem that `C` is already cocomplete,
see `has_colimits_of_has_limits_of_is_coseparating`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_terminal_of_is_separating [well_powered Cᵒᵖ] [has_colimits C] {𝒢 : set C} [small.{v₁} 𝒢]
(h𝒢 : is_separating 𝒢) : has_terminal C | begin
haveI : small.{v₁} 𝒢.op := small_of_injective (set.op_equiv_self 𝒢).injective,
haveI : has_initial Cᵒᵖ := has_initial_of_is_coseparating ((is_coseparating_op_iff _).2 h𝒢),
exact has_terminal_of_has_initial_op
end | lemma | category_theory.has_terminal_of_is_separating | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"set.op_equiv_self",
"small_of_injective"
] | An ingredient of the proof of the Special Adjoint Functor Theorem: a cocomplete well-copowered
category with a small separating set has a terminal object.
In fact, it follows from the Special Adjoint Functor Theorem that `C` is already complete, see
`has_limits_of_has_colimits_of_is_separating`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_le_of_is_detecting {𝒢 : set C} (h𝒢 : is_detecting 𝒢) {X : C} (P Q : subobject X)
(h₁ : P ≤ Q) (h₂ : ∀ (G ∈ 𝒢) {f : G ⟶ X}, Q.factors f → P.factors f) : P = Q | begin
suffices : is_iso (of_le _ _ h₁),
{ exactI le_antisymm h₁ (le_of_comm (inv (of_le _ _ h₁)) (by simp)) },
refine h𝒢 _ (λ G hG f, _),
have : P.factors (f ≫ Q.arrow) := h₂ _ hG ((factors_iff _ _).2 ⟨_, rfl⟩),
refine ⟨factor_thru _ _ this, _, λ g (hg : g ≫ _ = f), _⟩,
{ simp only [← cancel_mono Q.arrow, ... | lemma | category_theory.subobject.eq_of_le_of_is_detecting | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_eq_of_is_detecting [has_pullbacks C] {𝒢 : set C} (h𝒢 : is_detecting 𝒢) {X : C}
(P Q : subobject X) (h : ∀ (G ∈ 𝒢) {f : G ⟶ X}, P.factors f → Q.factors f) : P ⊓ Q = P | eq_of_le_of_is_detecting h𝒢 _ _ _root_.inf_le_left (λ G hG f hf, (inf_factors _).2 ⟨hf, h _ hG hf⟩) | lemma | category_theory.subobject.inf_eq_of_is_detecting | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_is_detecting [has_pullbacks C] {𝒢 : set C} (h𝒢 : is_detecting 𝒢) {X : C}
(P Q : subobject X) (h : ∀ (G ∈ 𝒢) {f : G ⟶ X}, P.factors f ↔ Q.factors f) : P = Q | calc P = P ⊓ Q : eq.symm $ inf_eq_of_is_detecting h𝒢 _ _ $ λ G hG f hf, (h G hG).1 hf
... = Q ⊓ P : inf_comm
... = Q : inf_eq_of_is_detecting h𝒢 _ _ $ λ G hG f hf, (h G hG).2 hf | lemma | category_theory.subobject.eq_of_is_detecting | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"inf_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
well_powered_of_is_detecting [has_pullbacks C] {𝒢 : set C} [small.{v₁} 𝒢]
(h𝒢 : is_detecting 𝒢) : well_powered C | ⟨λ X, @small_of_injective _ _ _ (λ P : subobject X, { f : Σ G : 𝒢, G.1 ⟶ X | P.factors f.2 }) $
λ P Q h, subobject.eq_of_is_detecting h𝒢 _ _ (by simpa [set.ext_iff] using h)⟩ | lemma | category_theory.well_powered_of_is_detecting | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"set.ext_iff",
"small_of_injective"
] | A category with pullbacks and a small detecting set is well-powered. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_coseparating_proj_preimage {𝒢 : set C} (h𝒢 : is_coseparating 𝒢) :
is_coseparating ((proj S T).obj ⁻¹' 𝒢) | begin
refine λ X Y f g hfg, ext _ _ (h𝒢 _ _ (λ G hG h, _)),
exact congr_arg comma_morphism.right (hfg (mk (Y.hom ≫ T.map h)) hG (hom_mk h rfl))
end | lemma | category_theory.structured_arrow.is_coseparating_proj_preimage | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_separating_proj_preimage {𝒢 : set C} (h𝒢 : is_separating 𝒢) :
is_separating ((proj S T).obj ⁻¹' 𝒢) | begin
refine λ X Y f g hfg, ext _ _ (h𝒢 _ _ (λ G hG h, _)),
convert congr_arg comma_morphism.left (hfg (mk (S.map h ≫ X.hom)) hG (hom_mk h rfl))
end | lemma | category_theory.costructured_arrow.is_separating_proj_preimage | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_separator (G : C) : Prop | is_separating ({G} : set C) | def | category_theory.is_separator | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | We say that `G` is a separator if the functor `C(G, -)` is faithful. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_coseparator (G : C) : Prop | is_coseparating ({G} : set C) | def | category_theory.is_coseparator | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | We say that `G` is a coseparator if the functor `C(-, G)` is faithful. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_detector (G : C) : Prop | is_detecting ({G} : set C) | def | category_theory.is_detector | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | We say that `G` is a detector if the functor `C(G, -)` reflects isomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_codetector (G : C) : Prop | is_codetecting ({G} : set C) | def | category_theory.is_codetector | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | We say that `G` is a codetector if the functor `C(-, G)` reflects isomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_separator_op_iff (G : C) : is_separator (op G) ↔ is_coseparator G | by rw [is_separator, is_coseparator, ← is_separating_op_iff, set.singleton_op] | lemma | category_theory.is_separator_op_iff | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"set.singleton_op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coseparator_op_iff (G : C) : is_coseparator (op G) ↔ is_separator G | by rw [is_separator, is_coseparator, ← is_coseparating_op_iff, set.singleton_op] | lemma | category_theory.is_coseparator_op_iff | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"set.singleton_op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coseparator_unop_iff (G : Cᵒᵖ) : is_coseparator (unop G) ↔ is_separator G | by rw [is_separator, is_coseparator, ← is_coseparating_unop_iff, set.singleton_unop] | lemma | category_theory.is_coseparator_unop_iff | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"set.singleton_unop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_separator_unop_iff (G : Cᵒᵖ) : is_separator (unop G) ↔ is_coseparator G | by rw [is_separator, is_coseparator, ← is_separating_unop_iff, set.singleton_unop] | lemma | category_theory.is_separator_unop_iff | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"set.singleton_unop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_detector_op_iff (G : C) : is_detector (op G) ↔ is_codetector G | by rw [is_detector, is_codetector, ← is_detecting_op_iff, set.singleton_op] | lemma | category_theory.is_detector_op_iff | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"set.singleton_op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_codetector_op_iff (G : C) : is_codetector (op G) ↔ is_detector G | by rw [is_detector, is_codetector, ← is_codetecting_op_iff, set.singleton_op] | lemma | category_theory.is_codetector_op_iff | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"set.singleton_op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_codetector_unop_iff (G : Cᵒᵖ) : is_codetector (unop G) ↔ is_detector G | by rw [is_detector, is_codetector, ← is_codetecting_unop_iff, set.singleton_unop] | lemma | category_theory.is_codetector_unop_iff | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"set.singleton_unop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_detector_unop_iff (G : Cᵒᵖ) : is_detector (unop G) ↔ is_codetector G | by rw [is_detector, is_codetector, ← is_detecting_unop_iff, set.singleton_unop] | lemma | category_theory.is_detector_unop_iff | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"set.singleton_unop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_detector.is_separator [has_equalizers C] {G : C} : is_detector G → is_separator G | is_detecting.is_separating | lemma | category_theory.is_detector.is_separator | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_codetector.is_coseparator [has_coequalizers C] {G : C} :
is_codetector G → is_coseparator G | is_codetecting.is_coseparating | lemma | category_theory.is_codetector.is_coseparator | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_separator.is_detector [balanced C] {G : C} : is_separator G → is_detector G | is_separating.is_detecting | lemma | category_theory.is_separator.is_detector | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"balanced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.