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as_equivalence_functor (F : C ⥤ D) [is_equivalence F] :
F.as_equivalence.functor = F | rfl | lemma | category_theory.functor.as_equivalence_functor | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
as_equivalence_inverse (F : C ⥤ D) [is_equivalence F] :
F.as_equivalence.inverse = inv F | rfl | lemma | category_theory.functor.as_equivalence_inverse | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
as_equivalence_unit {F : C ⥤ D} [h : is_equivalence F] :
F.as_equivalence.unit_iso = @@is_equivalence.unit_iso _ _ h | rfl | lemma | category_theory.functor.as_equivalence_unit | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
as_equivalence_counit {F : C ⥤ D} [is_equivalence F] :
F.as_equivalence.counit_iso = is_equivalence.counit_iso | rfl | lemma | category_theory.functor.as_equivalence_counit | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_inv (F : C ⥤ D) [is_equivalence F] :
inv (inv F) = F | rfl | lemma | category_theory.functor.inv_inv | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [
"inv_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalence_trans (F : C ⥤ D) (G : D ⥤ E) [is_equivalence F] [is_equivalence G] :
is_equivalence (F ⋙ G) | is_equivalence.of_equivalence (equivalence.trans (as_equivalence F) (as_equivalence G)) | instance | category_theory.functor.is_equivalence_trans | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_inv (E : C ≌ D) : E.functor.inv = E.inverse | rfl | lemma | category_theory.equivalence.functor_inv | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inverse_inv (E : C ≌ D) : E.inverse.inv = E.functor | rfl | lemma | category_theory.equivalence.inverse_inv | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_as_equivalence (E : C ≌ D) : E.functor.as_equivalence = E | by { cases E, congr, } | lemma | category_theory.equivalence.functor_as_equivalence | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inverse_as_equivalence (E : C ≌ D) : E.inverse.as_equivalence = E.symm | by { cases E, congr, } | lemma | category_theory.equivalence.inverse_as_equivalence | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fun_inv_map (F : C ⥤ D) [is_equivalence F] (X Y : D) (f : X ⟶ Y) :
F.map (F.inv.map f) = F.as_equivalence.counit.app X ≫ f ≫ F.as_equivalence.counit_inv.app Y | begin
erw [nat_iso.naturality_2],
refl
end | lemma | category_theory.is_equivalence.fun_inv_map | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_fun_map (F : C ⥤ D) [is_equivalence F] (X Y : C) (f : X ⟶ Y) :
F.inv.map (F.map f) = F.as_equivalence.unit_inv.app X ≫ f ≫ F.as_equivalence.unit.app Y | begin
erw [nat_iso.naturality_1],
refl
end | lemma | category_theory.is_equivalence.inv_fun_map | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_iso {F G : C ⥤ D} (e : F ≅ G) (hF : is_equivalence F) : is_equivalence G | { inverse := hF.inverse,
unit_iso := hF.unit_iso ≪≫ nat_iso.hcomp e (iso.refl hF.inverse),
counit_iso := nat_iso.hcomp (iso.refl hF.inverse) e.symm ≪≫ hF.counit_iso,
functor_unit_iso_comp' := λ X, begin
dsimp [nat_iso.hcomp],
erw [id_comp, F.map_id, comp_id],
apply (cancel_epi (e.hom.app X)).mp,
s... | def | category_theory.is_equivalence.of_iso | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | When a functor `F` is an equivalence of categories, and `G` is isomorphic to `F`, then
`G` is also an equivalence of categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_iso_trans {F G H : C ⥤ D} (e : F ≅ G) (e' : G ≅ H) (hF : is_equivalence F) :
(of_iso e' (of_iso e hF)) = of_iso (e ≪≫ e') hF | begin
dsimp [of_iso],
congr' 1; ext X; dsimp [nat_iso.hcomp],
{ simp only [id_comp, assoc, functor.map_comp], },
{ simp only [functor.map_id, comp_id, id_comp, assoc], },
end | lemma | category_theory.is_equivalence.of_iso_trans | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [
"functor.map_id"
] | Compatibility of `of_iso` with the composition of isomorphisms of functors | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_iso_refl (F : C ⥤ D) (hF : is_equivalence F) : of_iso (iso.refl F) hF = hF | begin
unfreezingI { rcases hF with ⟨Finv, Funit, Fcounit, Fcomp⟩, },
dsimp [of_iso],
congr' 1; ext X; dsimp [nat_iso.hcomp],
{ simp only [comp_id, map_id], },
{ simp only [id_comp, map_id], },
end | lemma | category_theory.is_equivalence.of_iso_refl | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [
"map_id"
] | Compatibility of `of_iso` with identity isomorphisms of functors | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_of_iso {F G : C ⥤ D} (e : F ≅ G) : is_equivalence F ≃ is_equivalence G | { to_fun := of_iso e,
inv_fun := of_iso e.symm,
left_inv := λ hF, by rw [of_iso_trans, iso.self_symm_id, of_iso_refl],
right_inv := λ hF, by rw [of_iso_trans, iso.symm_self_id, of_iso_refl], } | def | category_theory.is_equivalence.equiv_of_iso | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [
"inv_fun"
] | When `F` and `G` are two isomorphic functors, then `F` is an equivalence iff `G` is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cancel_comp_right {E : Type*} [category E]
(F : C ⥤ D) (G : D ⥤ E) (hG : is_equivalence G) (hGF : is_equivalence (F ⋙ G)) :
is_equivalence F | of_iso ((functor.associator F G G.inv) ≪≫ nat_iso.hcomp (iso.refl F) hG.unit_iso.symm ≪≫
right_unitor F) (functor.is_equivalence_trans (F ⋙ G) (G.inv)) | def | category_theory.is_equivalence.cancel_comp_right | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | If `G` and `F ⋙ G` are equivalence of categories, then `F` is also an equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cancel_comp_left {E : Type*} [category E]
(F : C ⥤ D) (G : D ⥤ E) (hF : is_equivalence F) (hGF : is_equivalence (F ⋙ G)) :
is_equivalence G | of_iso ((functor.associator F.inv F G).symm ≪≫ nat_iso.hcomp hF.counit_iso (iso.refl G) ≪≫
left_unitor G) (functor.is_equivalence_trans F.inv (F ⋙ G)) | def | category_theory.is_equivalence.cancel_comp_left | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | If `F` and `F ⋙ G` are equivalence of categories, then `G` is also an equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ess_surj_of_equivalence (F : C ⥤ D) [is_equivalence F] : ess_surj F | ⟨λ Y, ⟨F.inv.obj Y, ⟨F.as_equivalence.counit_iso.app Y⟩⟩⟩ | lemma | category_theory.equivalence.ess_surj_of_equivalence | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | An equivalence is essentially surjective.
See <https://stacks.math.columbia.edu/tag/02C3>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
faithful_of_equivalence (F : C ⥤ D) [is_equivalence F] : faithful F | { map_injective' := λ X Y f g w,
begin
have p := congr_arg (@category_theory.functor.map _ _ _ _ F.inv _ _) w,
simpa only [cancel_epi, cancel_mono, is_equivalence.inv_fun_map] using p
end }. | instance | category_theory.equivalence.faithful_of_equivalence | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | An equivalence is faithful.
See <https://stacks.math.columbia.edu/tag/02C3>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
full_of_equivalence (F : C ⥤ D) [is_equivalence F] : full F | { preimage := λ X Y f, F.as_equivalence.unit.app X ≫ F.inv.map f ≫ F.as_equivalence.unit_inv.app Y,
witness' := λ X Y f, F.inv.map_injective $
by simpa only [is_equivalence.inv_fun_map, assoc, iso.inv_hom_id_app_assoc, iso.inv_hom_id_app]
using comp_id _ } | instance | category_theory.equivalence.full_of_equivalence | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | An equivalence is full.
See <https://stacks.math.columbia.edu/tag/02C3>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence_inverse (F : C ⥤ D) [full F] [faithful F]
[ess_surj F] : D ⥤ C | { obj := λ X, F.obj_preimage X,
map := λ X Y f, F.preimage ((F.obj_obj_preimage_iso X).hom ≫ f ≫ (F.obj_obj_preimage_iso Y).inv),
map_id' := λ X, begin apply F.map_injective, tidy end,
map_comp' := λ X Y Z f g, by apply F.map_injective; simp } | def | category_theory.equivalence.equivalence_inverse | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_fully_faithfully_ess_surj
(F : C ⥤ D) [full F] [faithful F] [ess_surj F] : is_equivalence F | is_equivalence.mk (equivalence_inverse F)
(nat_iso.of_components
(λ X, (F.preimage_iso $ F.obj_obj_preimage_iso $ F.obj X).symm)
(λ X Y f, by { apply F.map_injective, obviously }))
(nat_iso.of_components F.obj_obj_preimage_iso (by tidy)) | def | category_theory.equivalence.of_fully_faithfully_ess_surj | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | A functor which is full, faithful, and essentially surjective is an equivalence.
See <https://stacks.math.columbia.edu/tag/02C3>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor_map_inj_iff (e : C ≌ D) {X Y : C} (f g : X ⟶ Y) :
e.functor.map f = e.functor.map g ↔ f = g | ⟨λ h, e.functor.map_injective h, λ h, h ▸ rfl⟩ | lemma | category_theory.equivalence.functor_map_inj_iff | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inverse_map_inj_iff (e : C ≌ D) {X Y : D} (f g : X ⟶ Y) :
e.inverse.map f = e.inverse.map g ↔ f = g | functor_map_inj_iff e.symm f g | lemma | category_theory.equivalence.inverse_map_inj_iff | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ess_surj_induced_functor {C' : Type*} (e : C' ≃ D) : ess_surj (induced_functor e) | { mem_ess_image := λ Y, ⟨e.symm Y, by simp⟩, } | instance | category_theory.equivalence.ess_surj_induced_functor | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_functor_of_equiv {C' : Type*} (e : C' ≃ D) : is_equivalence (induced_functor e) | equivalence.of_fully_faithfully_ess_surj _ | instance | category_theory.equivalence.induced_functor_of_equiv | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fully_faithful_to_ess_image (F : C ⥤ D) [full F] [faithful F] :
is_equivalence F.to_ess_image | of_fully_faithfully_ess_surj F.to_ess_image | instance | category_theory.equivalence.fully_faithful_to_ess_image | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_to_hom {X Y : C} (p : X = Y) : X ⟶ Y | by rw p; exact 𝟙 _ | def | category_theory.eq_to_hom | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | An equality `X = Y` gives us a morphism `X ⟶ Y`.
It is typically better to use this, rather than rewriting by the equality then using `𝟙 _`
which usually leads to dependent type theory hell. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_to_hom_refl (X : C) (p : X = X) : eq_to_hom p = 𝟙 X | rfl | lemma | category_theory.eq_to_hom_refl | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_to_hom_trans {X Y Z : C} (p : X = Y) (q : Y = Z) :
eq_to_hom p ≫ eq_to_hom q = eq_to_hom (p.trans q) | by { cases p, cases q, simp, } | lemma | category_theory.eq_to_hom_trans | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_eq_to_hom_iff {X Y Y' : C} (p : Y = Y') (f : X ⟶ Y) (g : X ⟶ Y') :
f ≫ eq_to_hom p = g ↔ f = g ≫ eq_to_hom p.symm | { mp := λ h, h ▸ by simp,
mpr := λ h, by simp [eq_whisker h (eq_to_hom p)] } | lemma | category_theory.comp_eq_to_hom_iff | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_to_hom_comp_iff {X X' Y : C} (p : X = X') (f : X ⟶ Y) (g : X' ⟶ Y) :
eq_to_hom p ≫ g = f ↔ g = eq_to_hom p.symm ≫ f | { mp := λ h, h ▸ by simp,
mpr := λ h, h ▸ by simp [whisker_eq _ h] } | lemma | category_theory.eq_to_hom_comp_iff | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_arg_mpr_hom_left {X Y Z : C} (p : X = Y) (q : Y ⟶ Z) :
(congr_arg (λ W : C, W ⟶ Z) p).mpr q = eq_to_hom p ≫ q | by { cases p, simp, } | lemma | category_theory.congr_arg_mpr_hom_left | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | If we (perhaps unintentionally) perform equational rewriting on
the source object of a morphism,
we can replace the resulting `_.mpr f` term by a composition with an `eq_to_hom`.
It may be advisable to introduce any necessary `eq_to_hom` morphisms manually,
rather than relying on this lemma firing. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
congr_arg_mpr_hom_right {X Y Z : C} (p : X ⟶ Y) (q : Z = Y) :
(congr_arg (λ W : C, X ⟶ W) q).mpr p = p ≫ eq_to_hom q.symm | by { cases q, simp, } | lemma | category_theory.congr_arg_mpr_hom_right | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | If we (perhaps unintentionally) perform equational rewriting on
the target object of a morphism,
we can replace the resulting `_.mpr f` term by a composition with an `eq_to_hom`.
It may be advisable to introduce any necessary `eq_to_hom` morphisms manually,
rather than relying on this lemma firing. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_to_iso {X Y : C} (p : X = Y) : X ≅ Y | ⟨eq_to_hom p, eq_to_hom p.symm, by simp, by simp⟩ | def | category_theory.eq_to_iso | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | An equality `X = Y` gives us an isomorphism `X ≅ Y`.
It is typically better to use this, rather than rewriting by the equality then using `iso.refl _`
which usually leads to dependent type theory hell. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_to_iso.hom {X Y : C} (p : X = Y) : (eq_to_iso p).hom = eq_to_hom p | rfl | lemma | category_theory.eq_to_iso.hom | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_to_iso.inv {X Y : C} (p : X = Y) : (eq_to_iso p).inv = eq_to_hom p.symm | rfl | lemma | category_theory.eq_to_iso.inv | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_to_iso_refl {X : C} (p : X = X) : eq_to_iso p = iso.refl X | rfl | lemma | category_theory.eq_to_iso_refl | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_to_iso_trans {X Y Z : C} (p : X = Y) (q : Y = Z) :
eq_to_iso p ≪≫ eq_to_iso q = eq_to_iso (p.trans q) | by ext; simp | lemma | category_theory.eq_to_iso_trans | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_to_hom_op {X Y : C} (h : X = Y) :
(eq_to_hom h).op = eq_to_hom (congr_arg op h.symm) | by { cases h, refl, } | lemma | category_theory.eq_to_hom_op | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_to_hom_unop {X Y : Cᵒᵖ} (h : X = Y) :
(eq_to_hom h).unop = eq_to_hom (congr_arg unop h.symm) | by { cases h, refl, } | lemma | category_theory.eq_to_hom_unop | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_eq_to_hom {X Y : C} (h : X = Y) : inv (eq_to_hom h) = eq_to_hom h.symm | by { ext, simp, } | lemma | category_theory.inv_eq_to_hom | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {F G : C ⥤ D} (h_obj : ∀ X, F.obj X = G.obj X)
(h_map : ∀ X Y f, F.map f = eq_to_hom (h_obj X) ≫ G.map f ≫ eq_to_hom (h_obj Y).symm) :
F = G | begin
cases F with F_obj _ _ _, cases G with G_obj _ _ _,
obtain rfl : F_obj = G_obj, by { ext X, apply h_obj },
congr,
funext X Y f,
simpa using h_map X Y f
end | lemma | category_theory.functor.ext | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | Proving equality between functors. This isn't an extensionality lemma,
because usually you don't really want to do this. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
conj_eq_to_hom_iff_heq {W X Y Z : C} (f : W ⟶ X) (g : Y ⟶ Z) (h : W = Y) (h' : X = Z) :
f = eq_to_hom h ≫ g ≫ eq_to_hom h'.symm ↔ f == g | by { cases h, cases h', simp } | lemma | category_theory.functor.conj_eq_to_hom_iff_heq | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | Two morphisms are conjugate via eq_to_hom if and only if they are heterogeneously equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hext {F G : C ⥤ D} (h_obj : ∀ X, F.obj X = G.obj X)
(h_map : ∀ X Y (f : X ⟶ Y), F.map f == G.map f) : F = G | functor.ext h_obj (λ _ _ f,
(conj_eq_to_hom_iff_heq _ _ (h_obj _) (h_obj _)).2 $ h_map _ _ f) | lemma | category_theory.functor.hext | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [
"functor.ext"
] | Proving equality between functors using heterogeneous equality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
congr_obj {F G : C ⥤ D} (h : F = G) (X) : F.obj X = G.obj X | by subst h | lemma | category_theory.functor.congr_obj | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_hom {F G : C ⥤ D} (h : F = G) {X Y} (f : X ⟶ Y) :
F.map f = eq_to_hom (congr_obj h X) ≫ G.map f ≫ eq_to_hom (congr_obj h Y).symm | by subst h; simp | lemma | category_theory.functor.congr_hom | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_inv_of_congr_hom (F G : C ⥤ D) {X Y : C} (e : X ≅ Y)
(hX : F.obj X = G.obj X) (hY : F.obj Y = G.obj Y)
(h₂ : F.map e.hom = eq_to_hom (by rw hX) ≫ G.map e.hom ≫ eq_to_hom (by rw hY)) :
F.map e.inv = eq_to_hom (by rw hY) ≫ G.map e.inv ≫ eq_to_hom (by rw hX) | by simp only [← is_iso.iso.inv_hom e, functor.map_inv, h₂, is_iso.inv_comp,
inv_eq_to_hom, category.assoc] | lemma | category_theory.functor.congr_inv_of_congr_hom | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_map (F : C ⥤ D) {X Y : C} {f g : X ⟶ Y} (h : f = g) :
F.map f = F.map g | by rw h | lemma | category_theory.functor.congr_map | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp_heq (hx : F.obj X = G.obj X) (hy : F.obj Y = G.obj Y) (hz : F.obj Z = G.obj Z)
(hf : F.map f == G.map f) (hg : F.map g == G.map g) : F.map (f ≫ g) == G.map (f ≫ g) | by { rw [F.map_comp, G.map_comp], congr' } | lemma | category_theory.functor.map_comp_heq | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp_heq' (hobj : ∀ X : C, F.obj X = G.obj X)
(hmap : ∀ {X Y} (f : X ⟶ Y), F.map f == G.map f) :
F.map (f ≫ g) == G.map (f ≫ g) | by rw functor.hext hobj (λ _ _, hmap) | lemma | category_theory.functor.map_comp_heq' | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
precomp_map_heq (H : E ⥤ C)
(hmap : ∀ {X Y} (f : X ⟶ Y), F.map f == G.map f) {X Y : E} (f : X ⟶ Y) :
(H ⋙ F).map f == (H ⋙ G).map f | hmap _ | lemma | category_theory.functor.precomp_map_heq | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
postcomp_map_heq (H : D ⥤ E) (hx : F.obj X = G.obj X) (hy : F.obj Y = G.obj Y)
(hmap : F.map f == G.map f) : (F ⋙ H).map f == (G ⋙ H).map f | by { dsimp, congr' } | lemma | category_theory.functor.postcomp_map_heq | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
postcomp_map_heq' (H : D ⥤ E) (hobj : ∀ X : C, F.obj X = G.obj X)
(hmap : ∀ {X Y} (f : X ⟶ Y), F.map f == G.map f) :
(F ⋙ H).map f == (G ⋙ H).map f | by rw functor.hext hobj (λ _ _, hmap) | lemma | category_theory.functor.postcomp_map_heq' | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hcongr_hom {F G : C ⥤ D} (h : F = G) {X Y} (f : X ⟶ Y) : F.map f == G.map f | by subst h | lemma | category_theory.functor.hcongr_hom | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_to_hom_map (F : C ⥤ D) {X Y : C} (p : X = Y) :
F.map (eq_to_hom p) = eq_to_hom (congr_arg F.obj p) | by cases p; simp | lemma | category_theory.eq_to_hom_map | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | This is not always a good idea as a `@[simp]` lemma,
as we lose the ability to use results that interact with `F`,
e.g. the naturality of a natural transformation.
In some files it may be appropriate to use `local attribute [simp] eq_to_hom_map`, however. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_to_iso_map (F : C ⥤ D) {X Y : C} (p : X = Y) :
F.map_iso (eq_to_iso p) = eq_to_iso (congr_arg F.obj p) | by ext; cases p; simp | lemma | category_theory.eq_to_iso_map | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | See the note on `eq_to_hom_map` regarding using this as a `simp` lemma. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_to_hom_app {F G : C ⥤ D} (h : F = G) (X : C) :
(eq_to_hom h : F ⟶ G).app X = eq_to_hom (functor.congr_obj h X) | by subst h; refl | lemma | category_theory.eq_to_hom_app | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_trans.congr {F G : C ⥤ D} (α : F ⟶ G) {X Y : C} (h : X = Y) :
α.app X = F.map (eq_to_hom h) ≫ α.app Y ≫ G.map (eq_to_hom h.symm) | by { rw [α.naturality_assoc], simp [eq_to_hom_map], } | lemma | category_theory.nat_trans.congr | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_conj_eq_to_hom {X Y : C} (f : X ⟶ Y) :
f = eq_to_hom rfl ≫ f ≫ eq_to_hom rfl | by simp only [category.id_comp, eq_to_hom_refl, category.comp_id] | lemma | category_theory.eq_conj_eq_to_hom | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dcongr_arg {ι : Type*} {F G : ι → C} (α : ∀ i, F i ⟶ G i) {i j : ι} (h : i = j) :
α i = eq_to_hom (congr_arg F h) ≫ α j ≫ eq_to_hom (congr_arg G h.symm) | by { subst h, simp } | lemma | category_theory.dcongr_arg | category_theory | src/category_theory/eq_to_hom.lean | [
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
essentially_small (C : Type u) [category.{v} C] : Prop | (equiv_small_category : ∃ (S : Type w) (_ : small_category S), by exactI nonempty (C ≌ S)) | class | category_theory.essentially_small | category_theory | src/category_theory/essentially_small.lean | [
"logic.small.basic",
"category_theory.category.ulift",
"category_theory.skeletal"
] | [] | A category is `essentially_small.{w}` if there exists
an equivalence to some `S : Type w` with `[small_category S]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
essentially_small.mk' {C : Type u} [category.{v} C] {S : Type w} [small_category S]
(e : C ≌ S) : essentially_small.{w} C | ⟨⟨S, _, ⟨e⟩⟩⟩ | lemma | category_theory.essentially_small.mk' | category_theory | src/category_theory/essentially_small.lean | [
"logic.small.basic",
"category_theory.category.ulift",
"category_theory.skeletal"
] | [] | Constructor for `essentially_small C` from an explicit small category witness. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
small_model (C : Type u) [category.{v} C] [essentially_small.{w} C] : Type w | classical.some (@essentially_small.equiv_small_category C _ _) | def | category_theory.small_model | category_theory | src/category_theory/essentially_small.lean | [
"logic.small.basic",
"category_theory.category.ulift",
"category_theory.skeletal"
] | [] | An arbitrarily chosen small model for an essentially small category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
small_category_small_model
(C : Type u) [category.{v} C] [essentially_small.{w} C] : small_category (small_model C) | classical.some (classical.some_spec (@essentially_small.equiv_small_category C _ _)) | instance | category_theory.small_category_small_model | category_theory | src/category_theory/essentially_small.lean | [
"logic.small.basic",
"category_theory.category.ulift",
"category_theory.skeletal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_small_model (C : Type u) [category.{v} C] [essentially_small.{w} C] : C ≌ small_model C | nonempty.some (classical.some_spec (classical.some_spec
(@essentially_small.equiv_small_category C _ _))) | def | category_theory.equiv_small_model | category_theory | src/category_theory/essentially_small.lean | [
"logic.small.basic",
"category_theory.category.ulift",
"category_theory.skeletal"
] | [
"nonempty.some"
] | The (noncomputable) categorical equivalence between
an essentially small category and its small model. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
essentially_small_congr {C : Type u} [category.{v} C] {D : Type u'} [category.{v'} D]
(e : C ≌ D) : essentially_small.{w} C ↔ essentially_small.{w} D | begin
fsplit,
{ rintro ⟨S, 𝒮, ⟨f⟩⟩,
resetI,
exact essentially_small.mk' (e.symm.trans f), },
{ rintro ⟨S, 𝒮, ⟨f⟩⟩,
resetI,
exact essentially_small.mk' (e.trans f), },
end | lemma | category_theory.essentially_small_congr | category_theory | src/category_theory/essentially_small.lean | [
"logic.small.basic",
"category_theory.category.ulift",
"category_theory.skeletal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete.essentially_small_of_small {α : Type u} [small.{w} α] :
essentially_small.{w} (discrete α) | ⟨⟨discrete (shrink α), ⟨infer_instance, ⟨discrete.equivalence (equiv_shrink _)⟩⟩⟩⟩ | lemma | category_theory.discrete.essentially_small_of_small | category_theory | src/category_theory/essentially_small.lean | [
"logic.small.basic",
"category_theory.category.ulift",
"category_theory.skeletal"
] | [
"equiv_shrink",
"shrink"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
essentially_small_self : essentially_small.{max w v u} C | essentially_small.mk' (as_small.equiv : C ≌ as_small.{w} C) | lemma | category_theory.essentially_small_self | category_theory | src/category_theory/essentially_small.lean | [
"logic.small.basic",
"category_theory.category.ulift",
"category_theory.skeletal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_small (C : Type u) [category.{v} C] : Prop | (hom_small : ∀ X Y : C, small.{w} (X ⟶ Y) . tactic.apply_instance) | class | category_theory.locally_small | category_theory | src/category_theory/essentially_small.lean | [
"logic.small.basic",
"category_theory.category.ulift",
"category_theory.skeletal"
] | [] | A category is `w`-locally small if every hom set is `w`-small.
See `shrink_homs C` for a category instance where every hom set has been replaced by a small model. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
locally_small_congr {C : Type u} [category.{v} C] {D : Type u'} [category.{v'} D]
(e : C ≌ D) : locally_small.{w} C ↔ locally_small.{w} D | begin
fsplit,
{ rintro ⟨L⟩,
fsplit,
intros X Y,
specialize L (e.inverse.obj X) (e.inverse.obj Y),
refine (small_congr _).mpr L,
exact equiv_of_fully_faithful e.inverse, },
{ rintro ⟨L⟩,
fsplit,
intros X Y,
specialize L (e.functor.obj X) (e.functor.obj Y),
refine (small_congr _)... | lemma | category_theory.locally_small_congr | category_theory | src/category_theory/essentially_small.lean | [
"logic.small.basic",
"category_theory.category.ulift",
"category_theory.skeletal"
] | [
"small_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_small_self (C : Type u) [category.{v} C] : locally_small.{v} C | {} | instance | category_theory.locally_small_self | category_theory | src/category_theory/essentially_small.lean | [
"logic.small.basic",
"category_theory.category.ulift",
"category_theory.skeletal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_small_of_essentially_small
(C : Type u) [category.{v} C] [essentially_small.{w} C] : locally_small.{w} C | (locally_small_congr (equiv_small_model C)).mpr (category_theory.locally_small_self _) | instance | category_theory.locally_small_of_essentially_small | category_theory | src/category_theory/essentially_small.lean | [
"logic.small.basic",
"category_theory.category.ulift",
"category_theory.skeletal"
] | [
"category_theory.locally_small_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
shrink_homs (C : Type u) | C | def | category_theory.shrink_homs | category_theory | src/category_theory/essentially_small.lean | [
"logic.small.basic",
"category_theory.category.ulift",
"category_theory.skeletal"
] | [] | We define a type alias `shrink_homs C` for `C`. When we have `locally_small.{w} C`,
we'll put a `category.{w}` instance on `shrink_homs C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_shrink_homs {C' : Type*} (X : C') : shrink_homs C' | X | def | category_theory.shrink_homs.to_shrink_homs | category_theory | src/category_theory/essentially_small.lean | [
"logic.small.basic",
"category_theory.category.ulift",
"category_theory.skeletal"
] | [] | Help the typechecker by explicitly translating from `C` to `shrink_homs C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
from_shrink_homs {C' : Type*} (X : shrink_homs C') : C' | X | def | category_theory.shrink_homs.from_shrink_homs | category_theory | src/category_theory/essentially_small.lean | [
"logic.small.basic",
"category_theory.category.ulift",
"category_theory.skeletal"
] | [] | Help the typechecker by explicitly translating from `shrink_homs C` to `C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_from (X : C') : from_shrink_homs (to_shrink_homs X) = X | rfl | lemma | category_theory.shrink_homs.to_from | category_theory | src/category_theory/essentially_small.lean | [
"logic.small.basic",
"category_theory.category.ulift",
"category_theory.skeletal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_to (X : shrink_homs C') : to_shrink_homs (from_shrink_homs X) = X | rfl | lemma | category_theory.shrink_homs.from_to | category_theory | src/category_theory/essentially_small.lean | [
"logic.small.basic",
"category_theory.category.ulift",
"category_theory.skeletal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor : C ⥤ shrink_homs C | { obj := λ X, to_shrink_homs X,
map := λ X Y f, equiv_shrink (X ⟶ Y) f, } | def | category_theory.shrink_homs.functor | category_theory | src/category_theory/essentially_small.lean | [
"logic.small.basic",
"category_theory.category.ulift",
"category_theory.skeletal"
] | [
"equiv_shrink"
] | Implementation of `shrink_homs.equivalence`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inverse : shrink_homs C ⥤ C | { obj := λ X, from_shrink_homs X,
map := λ X Y f, (equiv_shrink (from_shrink_homs X ⟶ from_shrink_homs Y)).symm f, } | def | category_theory.shrink_homs.inverse | category_theory | src/category_theory/essentially_small.lean | [
"logic.small.basic",
"category_theory.category.ulift",
"category_theory.skeletal"
] | [
"equiv_shrink"
] | Implementation of `shrink_homs.equivalence`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence : C ≌ shrink_homs C | equivalence.mk (functor C) (inverse C)
(nat_iso.of_components (λ X, iso.refl X) (by tidy))
(nat_iso.of_components (λ X, iso.refl X) (by tidy)) | def | category_theory.shrink_homs.equivalence | category_theory | src/category_theory/essentially_small.lean | [
"logic.small.basic",
"category_theory.category.ulift",
"category_theory.skeletal"
] | [] | The categorical equivalence between `C` and `shrink_homs C`, when `C` is locally small. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
essentially_small_iff (C : Type u) [category.{v} C] :
essentially_small.{w} C ↔ small.{w} (skeleton C) ∧ locally_small.{w} C | begin
-- This theorem is the only bit of real work in this file.
fsplit,
{ intro h,
fsplit,
{ rcases h with ⟨S, 𝒮, ⟨e⟩⟩,
resetI,
refine ⟨⟨skeleton S, ⟨_⟩⟩⟩,
exact e.skeleton_equiv, },
{ resetI, apply_instance, }, },
{ rintro ⟨⟨S, ⟨e⟩⟩, L⟩,
resetI,
let e' := (shrink_homs.eq... | theorem | category_theory.essentially_small_iff | category_theory | src/category_theory/essentially_small.lean | [
"logic.small.basic",
"category_theory.category.ulift",
"category_theory.skeletal"
] | [] | A category is essentially small if and only if
the underlying type of its skeleton (i.e. the "set" of isomorphism classes) is small,
and it is locally small. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
locally_small_of_thin {C : Type u} [category.{v} C] [quiver.is_thin C] :
locally_small.{w} C | {} | instance | category_theory.locally_small_of_thin | category_theory | src/category_theory/essentially_small.lean | [
"logic.small.basic",
"category_theory.category.ulift",
"category_theory.skeletal"
] | [
"quiver.is_thin"
] | Any thin category is locally small. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
essentially_small_iff_of_thin
{C : Type u} [category.{v} C] [quiver.is_thin C] :
essentially_small.{w} C ↔ small.{w} (skeleton C) | by simp [essentially_small_iff, category_theory.locally_small_of_thin] | theorem | category_theory.essentially_small_iff_of_thin | category_theory | src/category_theory/essentially_small.lean | [
"logic.small.basic",
"category_theory.category.ulift",
"category_theory.skeletal"
] | [
"category_theory.locally_small_of_thin",
"quiver.is_thin"
] | A thin category is essentially small if and only if the underlying type of its skeleton is small. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ess_image (F : C ⥤ D) : set D | λ Y, ∃ (X : C), nonempty (F.obj X ≅ Y) | def | category_theory.functor.ess_image | category_theory | src/category_theory/essential_image.lean | [
"category_theory.natural_isomorphism",
"category_theory.full_subcategory"
] | [] | The essential image of a functor `F` consists of those objects in the target category which are
isomorphic to an object in the image of the function `F.obj`. In other words, this is the closure
under isomorphism of the function `F.obj`.
This is the "non-evil" way of describing the image of a functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ess_image.witness {Y : D} (h : Y ∈ F.ess_image) : C | h.some | def | category_theory.functor.ess_image.witness | category_theory | src/category_theory/essential_image.lean | [
"category_theory.natural_isomorphism",
"category_theory.full_subcategory"
] | [] | Get the witnessing object that `Y` is in the subcategory given by `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ess_image.get_iso {Y : D} (h : Y ∈ F.ess_image) : F.obj h.witness ≅ Y | classical.choice h.some_spec | def | category_theory.functor.ess_image.get_iso | category_theory | src/category_theory/essential_image.lean | [
"category_theory.natural_isomorphism",
"category_theory.full_subcategory"
] | [] | Extract the isomorphism between `F.obj h.witness` and `Y` itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ess_image.of_iso {Y Y' : D} (h : Y ≅ Y') (hY : Y ∈ ess_image F) :
Y' ∈ ess_image F | hY.imp (λ B, nonempty.map (≪≫ h)) | lemma | category_theory.functor.ess_image.of_iso | category_theory | src/category_theory/essential_image.lean | [
"category_theory.natural_isomorphism",
"category_theory.full_subcategory"
] | [
"nonempty.map"
] | Being in the essential image is a "hygenic" property: it is preserved under isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ess_image.of_nat_iso {F' : C ⥤ D} (h : F ≅ F') {Y : D} (hY : Y ∈ ess_image F) :
Y ∈ ess_image F' | hY.imp (λ X, nonempty.map (λ t, h.symm.app X ≪≫ t)) | lemma | category_theory.functor.ess_image.of_nat_iso | category_theory | src/category_theory/essential_image.lean | [
"category_theory.natural_isomorphism",
"category_theory.full_subcategory"
] | [
"nonempty.map"
] | If `Y` is in the essential image of `F` then it is in the essential image of `F'` as long as
`F ≅ F'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ess_image_eq_of_nat_iso {F' : C ⥤ D} (h : F ≅ F') :
ess_image F = ess_image F' | funext (λ _, propext ⟨ess_image.of_nat_iso h, ess_image.of_nat_iso h.symm⟩) | lemma | category_theory.functor.ess_image_eq_of_nat_iso | category_theory | src/category_theory/essential_image.lean | [
"category_theory.natural_isomorphism",
"category_theory.full_subcategory"
] | [] | Isomorphic functors have equal essential images. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
obj_mem_ess_image (F : D ⥤ C) (Y : D) : F.obj Y ∈ ess_image F | ⟨Y, ⟨iso.refl _⟩⟩ | lemma | category_theory.functor.obj_mem_ess_image | category_theory | src/category_theory/essential_image.lean | [
"category_theory.natural_isomorphism",
"category_theory.full_subcategory"
] | [] | An object in the image is in the essential image. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ess_image_subcategory (F : C ⥤ D) | full_subcategory F.ess_image | def | category_theory.functor.ess_image_subcategory | category_theory | src/category_theory/essential_image.lean | [
"category_theory.natural_isomorphism",
"category_theory.full_subcategory"
] | [] | The essential image of a functor, interpreted of a full subcategory of the target category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ess_image_inclusion (F : C ⥤ D) : F.ess_image_subcategory ⥤ D | full_subcategory_inclusion _ | def | category_theory.functor.ess_image_inclusion | category_theory | src/category_theory/essential_image.lean | [
"category_theory.natural_isomorphism",
"category_theory.full_subcategory"
] | [] | The essential image as a subcategory has a fully faithful inclusion into the target category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_ess_image (F : C ⥤ D) : C ⥤ F.ess_image_subcategory | full_subcategory.lift _ F (obj_mem_ess_image _) | def | category_theory.functor.to_ess_image | category_theory | src/category_theory/essential_image.lean | [
"category_theory.natural_isomorphism",
"category_theory.full_subcategory"
] | [] | Given a functor `F : C ⥤ D`, we have an (essentially surjective) functor from `C` to the essential
image of `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_ess_image_comp_essential_image_inclusion (F : C ⥤ D) :
F.to_ess_image ⋙ F.ess_image_inclusion ≅ F | full_subcategory.lift_comp_inclusion _ _ _ | def | category_theory.functor.to_ess_image_comp_essential_image_inclusion | category_theory | src/category_theory/essential_image.lean | [
"category_theory.natural_isomorphism",
"category_theory.full_subcategory"
] | [] | The functor `F` factorises through its essential image, where the first functor is essentially
surjective and the second is fully faithful. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ess_surj (F : C ⥤ D) : Prop | (mem_ess_image [] (Y : D) : Y ∈ F.ess_image) | class | category_theory.ess_surj | category_theory | src/category_theory/essential_image.lean | [
"category_theory.natural_isomorphism",
"category_theory.full_subcategory"
] | [] | A functor `F : C ⥤ D` is essentially surjective if every object of `D` is in the essential image
of `F`. In other words, for every `Y : D`, there is some `X : C` with `F.obj X ≅ Y`.
See <https://stacks.math.columbia.edu/tag/001C>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor.obj_preimage (Y : D) : C | (ess_surj.mem_ess_image F Y).witness | def | category_theory.functor.obj_preimage | category_theory | src/category_theory/essential_image.lean | [
"category_theory.natural_isomorphism",
"category_theory.full_subcategory"
] | [] | Given an essentially surjective functor, we can find a preimage for every object `Y` in the
codomain. Applying the functor to this preimage will yield an object isomorphic to `Y`, see
`obj_obj_preimage_iso`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor.obj_obj_preimage_iso (Y : D) : F.obj (F.obj_preimage Y) ≅ Y | (ess_surj.mem_ess_image F Y).get_iso | def | category_theory.functor.obj_obj_preimage_iso | category_theory | src/category_theory/essential_image.lean | [
"category_theory.natural_isomorphism",
"category_theory.full_subcategory"
] | [] | Applying an essentially surjective functor to a preimage of `Y` yields an object that is
isomorphic to `Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
faithful.to_ess_image (F : C ⥤ D) [faithful F] : faithful F.to_ess_image | faithful.of_comp_iso F.to_ess_image_comp_essential_image_inclusion | instance | category_theory.faithful.to_ess_image | category_theory | src/category_theory/essential_image.lean | [
"category_theory.natural_isomorphism",
"category_theory.full_subcategory"
] | [] | The induced functor of a faithful functor is faithful | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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