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
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split_epi_category_imp_of_is_equivalence [is_equivalence F] [split_epi_category C] :
split_epi_category D | ⟨λ X Y f, begin
introI,
rw ← F.inv.is_split_epi_iff f,
apply is_split_epi_of_epi,
end⟩ | def | category_theory.functor.split_epi_category_imp_of_is_equivalence | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | If `F : C ⥤ D` is an equivalence of categories and `C` is a `split_epi_category`,
then `D` also is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strong_epi_map_of_strong_epi (adj : F ⊣ F') (f : A ⟶ B)
[h₁ : F'.preserves_monomorphisms] [h₂ : F.preserves_epimorphisms] [strong_epi f] :
strong_epi (F.map f) | ⟨infer_instance, λ X Y Z, by { introI, rw adj.has_lifting_property_iff, apply_instance, }⟩ | lemma | category_theory.adjunction.strong_epi_map_of_strong_epi | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [
"adj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strong_epi_map_of_is_equivalence [is_equivalence F] (f : A ⟶ B) [h : strong_epi f] :
strong_epi (F.map f) | F.as_equivalence.to_adjunction.strong_epi_map_of_strong_epi f | instance | category_theory.adjunction.strong_epi_map_of_is_equivalence | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strong_epi_map_iff_strong_epi_of_is_equivalence [is_equivalence F] :
strong_epi (F.map f) ↔ strong_epi f | begin
split,
{ introI,
have e : arrow.mk f ≅ arrow.mk (F.inv.map (F.map f)) :=
arrow.iso_of_nat_iso F.as_equivalence.unit_iso (arrow.mk f),
rw strong_epi.iff_of_arrow_iso e,
apply_instance, },
{ introI,
apply_instance, },
end | lemma | category_theory.functor.strong_epi_map_iff_strong_epi_of_is_equivalence | category_theory.functor | src/category_theory/functor/epi_mono.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.strong_epi",
"category_theory.lifting_properties.adjunction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_diagram : J ⥤ structured_arrow c.X K | { obj := λ j, structured_arrow.mk (c.π.app j),
map := λ j k g, structured_arrow.hom_mk g (by simpa) } | def | category_theory.structured_arrow_cone.to_diagram | category_theory.functor | src/category_theory/functor/flat.lean | [
"category_theory.limits.filtered_colimit_commutes_finite_limit",
"category_theory.limits.preserves.functor_category",
"category_theory.limits.bicones",
"category_theory.limits.comma",
"category_theory.limits.preserves.finite",
"category_theory.limits.shapes.finite_limits"
] | [] | Given a cone `c : cone K` and a map `f : X ⟶ c.X`, we can construct a cone of structured
arrows over `X` with `f` as the cone point. This is the underlying diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagram_to_cone {X : D} (G : J ⥤ structured_arrow X F) : cone (G ⋙ proj X F ⋙ F) | { X := X, π := { app := λ j, (G.obj j).hom } } | def | category_theory.structured_arrow_cone.diagram_to_cone | category_theory.functor | src/category_theory/functor/flat.lean | [
"category_theory.limits.filtered_colimit_commutes_finite_limit",
"category_theory.limits.preserves.functor_category",
"category_theory.limits.bicones",
"category_theory.limits.comma",
"category_theory.limits.preserves.finite",
"category_theory.limits.shapes.finite_limits"
] | [] | Given a diagram of `structured_arrow X F`s, we may obtain a cone with cone point `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_cone {X : D} (f : X ⟶ F.obj c.X) :
cone (to_diagram (F.map_cone c) ⋙ map f ⋙ pre _ K F) | { X := mk f, π := { app := λ j, hom_mk (c.π.app j) rfl,
naturality' := λ j k g, by { ext, dsimp, simp } } } | def | category_theory.structured_arrow_cone.to_cone | category_theory.functor | src/category_theory/functor/flat.lean | [
"category_theory.limits.filtered_colimit_commutes_finite_limit",
"category_theory.limits.preserves.functor_category",
"category_theory.limits.bicones",
"category_theory.limits.comma",
"category_theory.limits.preserves.finite",
"category_theory.limits.shapes.finite_limits"
] | [] | Given a cone `c : cone K` and a map `f : X ⟶ F.obj c.X`, we can construct a cone of structured
arrows over `X` with `f` as the cone point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
representably_flat (F : C ⥤ D) : Prop | (cofiltered : ∀ (X : D), is_cofiltered (structured_arrow X F)) | class | category_theory.representably_flat | category_theory.functor | src/category_theory/functor/flat.lean | [
"category_theory.limits.filtered_colimit_commutes_finite_limit",
"category_theory.limits.preserves.functor_category",
"category_theory.limits.bicones",
"category_theory.limits.comma",
"category_theory.limits.preserves.finite",
"category_theory.limits.shapes.finite_limits"
] | [] | A functor `F : C ⥤ D` is representably-flat functor if the comma category `(X/F)`
is cofiltered for each `X : C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
representably_flat.id : representably_flat (𝟭 C) | begin
constructor,
intro X,
haveI : nonempty (structured_arrow X (𝟭 C)) := ⟨structured_arrow.mk (𝟙 _)⟩,
rsufficesI : is_cofiltered_or_empty (structured_arrow X (𝟭 C)),
{ constructor },
constructor,
{ intros Y Z,
use structured_arrow.mk (𝟙 _),
use structured_arrow.hom_mk Y.hom (by erw [functor.... | instance | category_theory.representably_flat.id | category_theory.functor | src/category_theory/functor/flat.lean | [
"category_theory.limits.filtered_colimit_commutes_finite_limit",
"category_theory.limits.preserves.functor_category",
"category_theory.limits.bicones",
"category_theory.limits.comma",
"category_theory.limits.preserves.finite",
"category_theory.limits.shapes.finite_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
representably_flat.comp (F : C ⥤ D) (G : D ⥤ E)
[representably_flat F] [representably_flat G] : representably_flat (F ⋙ G) | begin
constructor,
intro X,
haveI : nonempty (structured_arrow X (F ⋙ G)),
{ have f₁ : structured_arrow X G := nonempty.some infer_instance,
have f₂ : structured_arrow f₁.right F := nonempty.some infer_instance,
exact ⟨structured_arrow.mk (f₁.hom ≫ G.map f₂.hom)⟩ },
rsufficesI : is_cofiltered_or_empty... | instance | category_theory.representably_flat.comp | category_theory.functor | src/category_theory/functor/flat.lean | [
"category_theory.limits.filtered_colimit_commutes_finite_limit",
"category_theory.limits.preserves.functor_category",
"category_theory.limits.bicones",
"category_theory.limits.comma",
"category_theory.limits.preserves.finite",
"category_theory.limits.shapes.finite_limits"
] | [
"nonempty.some"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cofiltered_of_has_finite_limits [has_finite_limits C] : is_cofiltered C | { cone_objs := λ A B, ⟨limits.prod A B, limits.prod.fst, limits.prod.snd, trivial⟩,
cone_maps := λ A B f g, ⟨equalizer f g, equalizer.ι f g, equalizer.condition f g⟩,
nonempty := ⟨⊤_ C⟩ } | lemma | category_theory.cofiltered_of_has_finite_limits | category_theory.functor | src/category_theory/functor/flat.lean | [
"category_theory.limits.filtered_colimit_commutes_finite_limit",
"category_theory.limits.preserves.functor_category",
"category_theory.limits.bicones",
"category_theory.limits.comma",
"category_theory.limits.preserves.finite",
"category_theory.limits.shapes.finite_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
flat_of_preserves_finite_limits [has_finite_limits C] (F : C ⥤ D)
[preserves_finite_limits F] : representably_flat F | ⟨λ X,
begin
haveI : has_finite_limits (structured_arrow X F) :=
begin
apply has_finite_limits_of_has_finite_limits_of_size.{v₁} (structured_arrow X F),
intros J sJ fJ, resetI, constructor
end,
exact cofiltered_of_has_finite_limits
end⟩ | lemma | category_theory.flat_of_preserves_finite_limits | category_theory.functor | src/category_theory/functor/flat.lean | [
"category_theory.limits.filtered_colimit_commutes_finite_limit",
"category_theory.limits.preserves.functor_category",
"category_theory.limits.bicones",
"category_theory.limits.comma",
"category_theory.limits.preserves.finite",
"category_theory.limits.shapes.finite_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift : s.X ⟶ F.obj c.X | let s' := is_cofiltered.cone (to_diagram s ⋙ structured_arrow.pre _ K F) in
s'.X.hom ≫ (F.map $ hc.lift $
(cones.postcompose ({ app := λ X, 𝟙 _, naturality' := by simp }
: (to_diagram s ⋙ pre s.X K F) ⋙ proj s.X F ⟶ K)).obj $
(structured_arrow.proj s.X F).map_cone s') | def | category_theory.preserves_finite_limits_of_flat.lift | category_theory.functor | src/category_theory/functor/flat.lean | [
"category_theory.limits.filtered_colimit_commutes_finite_limit",
"category_theory.limits.preserves.functor_category",
"category_theory.limits.bicones",
"category_theory.limits.comma",
"category_theory.limits.preserves.finite",
"category_theory.limits.shapes.finite_limits"
] | [
"lift"
] | (Implementation).
Given a limit cone `c : cone K` and a cone `s : cone (K ⋙ F)` with `F` representably flat,
`s` can factor through `F.map_cone c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fac (x : J) : lift F hc s ≫ (F.map_cone c).π.app x = s.π.app x | by simpa [lift, ←functor.map_comp] | lemma | category_theory.preserves_finite_limits_of_flat.fac | category_theory.functor | src/category_theory/functor/flat.lean | [
"category_theory.limits.filtered_colimit_commutes_finite_limit",
"category_theory.limits.preserves.functor_category",
"category_theory.limits.bicones",
"category_theory.limits.comma",
"category_theory.limits.preserves.finite",
"category_theory.limits.shapes.finite_limits"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniq {K : J ⥤ C} {c : cone K} (hc : is_limit c)
(s : cone (K ⋙ F)) (f₁ f₂ : s.X ⟶ F.obj c.X)
(h₁ : ∀ (j : J), f₁ ≫ (F.map_cone c).π.app j = s.π.app j)
(h₂ : ∀ (j : J), f₂ ≫ (F.map_cone c).π.app j = s.π.app j) : f₁ = f₂ | begin
-- We can make two cones over the diagram of `s` via `f₁` and `f₂`.
let α₁ : to_diagram (F.map_cone c) ⋙ map f₁ ⟶ to_diagram s :=
{ app := λ X, eq_to_hom (by simp [←h₁]), naturality' := λ _ _ _, by { ext, simp } },
let α₂ : to_diagram (F.map_cone c) ⋙ map f₂ ⟶ to_diagram s :=
{ app := λ X, eq_to_hom (by... | lemma | category_theory.preserves_finite_limits_of_flat.uniq | category_theory.functor | src/category_theory/functor/flat.lean | [
"category_theory.limits.filtered_colimit_commutes_finite_limit",
"category_theory.limits.preserves.functor_category",
"category_theory.limits.bicones",
"category_theory.limits.comma",
"category_theory.limits.preserves.finite",
"category_theory.limits.shapes.finite_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_finite_limits_of_flat (F : C ⥤ D) [representably_flat F] :
preserves_finite_limits F | begin
apply preserves_finite_limits_of_preserves_finite_limits_of_size,
intros J _ _, constructor,
intros K, constructor,
intros c hc,
exactI { lift := preserves_finite_limits_of_flat.lift F hc,
fac' := preserves_finite_limits_of_flat.fac F hc,
uniq' := λ s m h, by
{ apply preserves_finite_limits_... | def | category_theory.preserves_finite_limits_of_flat | category_theory.functor | src/category_theory/functor/flat.lean | [
"category_theory.limits.filtered_colimit_commutes_finite_limit",
"category_theory.limits.preserves.functor_category",
"category_theory.limits.bicones",
"category_theory.limits.comma",
"category_theory.limits.preserves.finite",
"category_theory.limits.shapes.finite_limits"
] | [
"lift"
] | Representably flat functors preserve finite limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_finite_limits_iff_flat [has_finite_limits C] (F : C ⥤ D) :
representably_flat F ≃ preserves_finite_limits F | { to_fun := λ _, by exactI preserves_finite_limits_of_flat F,
inv_fun := λ _, by exactI flat_of_preserves_finite_limits F,
left_inv := λ _, proof_irrel _ _,
right_inv := λ x, by { cases x, unfold preserves_finite_limits_of_flat,
dunfold preserves_finite_limits_of_preserves_finite_limits_of_size, congr } } | def | category_theory.preserves_finite_limits_iff_flat | category_theory.functor | src/category_theory/functor/flat.lean | [
"category_theory.limits.filtered_colimit_commutes_finite_limit",
"category_theory.limits.preserves.functor_category",
"category_theory.limits.bicones",
"category_theory.limits.comma",
"category_theory.limits.preserves.finite",
"category_theory.limits.shapes.finite_limits"
] | [
"inv_fun"
] | If `C` is finitely cocomplete, then `F : C ⥤ D` is representably flat iff it preserves
finite limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Lan_evaluation_iso_colim (F : C ⥤ D) (X : D)
[∀ (X : D), has_colimits_of_shape (costructured_arrow F X) E] :
Lan F ⋙ (evaluation D E).obj X ≅
((whiskering_left _ _ E).obj (costructured_arrow.proj F X)) ⋙ colim | nat_iso.of_components (λ G, colim.map_iso (iso.refl _))
begin
intros G H i,
ext,
simp only [functor.comp_map, colimit.ι_desc_assoc, functor.map_iso_refl, evaluation_obj_map,
whiskering_left_obj_map, category.comp_id, Lan_map_app, category.assoc],
erw [colimit.ι_pre_assoc (Lan.diagram F H X) (costructured_ar... | def | category_theory.Lan_evaluation_iso_colim | category_theory.functor | src/category_theory/functor/flat.lean | [
"category_theory.limits.filtered_colimit_commutes_finite_limit",
"category_theory.limits.preserves.functor_category",
"category_theory.limits.bicones",
"category_theory.limits.comma",
"category_theory.limits.preserves.finite",
"category_theory.limits.shapes.finite_limits"
] | [] | (Implementation)
The evaluation of `Lan F` at `X` is the colimit over the costructured arrows over `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Lan_preserves_finite_limits_of_flat (F : C ⥤ D) [representably_flat F] :
preserves_finite_limits (Lan F.op : _ ⥤ (Dᵒᵖ ⥤ E)) | begin
apply preserves_finite_limits_of_preserves_finite_limits_of_size.{u₁},
intros J _ _, resetI,
apply preserves_limits_of_shape_of_evaluation (Lan F.op : (Cᵒᵖ ⥤ E) ⥤ (Dᵒᵖ ⥤ E)) J,
intro K,
haveI : is_filtered (costructured_arrow F.op K) :=
is_filtered.of_equivalence (structured_arrow_op_equivalence F (... | instance | category_theory.Lan_preserves_finite_limits_of_flat | category_theory.functor | src/category_theory/functor/flat.lean | [
"category_theory.limits.filtered_colimit_commutes_finite_limit",
"category_theory.limits.preserves.functor_category",
"category_theory.limits.bicones",
"category_theory.limits.comma",
"category_theory.limits.preserves.finite",
"category_theory.limits.shapes.finite_limits"
] | [] | If `F : C ⥤ D` is a representably flat functor between small categories, then the functor
`Lan F.op` that takes presheaves over `C` to presheaves over `D` preserves finite limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Lan_flat_of_flat (F : C ⥤ D) [representably_flat F] :
representably_flat (Lan F.op : _ ⥤ (Dᵒᵖ ⥤ E)) | flat_of_preserves_finite_limits _ | instance | category_theory.Lan_flat_of_flat | category_theory.functor | src/category_theory/functor/flat.lean | [
"category_theory.limits.filtered_colimit_commutes_finite_limit",
"category_theory.limits.preserves.functor_category",
"category_theory.limits.bicones",
"category_theory.limits.comma",
"category_theory.limits.preserves.finite",
"category_theory.limits.shapes.finite_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Lan_preserves_finite_limits_of_preserves_finite_limits (F : C ⥤ D)
[preserves_finite_limits F] : preserves_finite_limits (Lan F.op : _ ⥤ (Dᵒᵖ ⥤ E)) | begin
haveI := flat_of_preserves_finite_limits F,
apply_instance
end | instance | category_theory.Lan_preserves_finite_limits_of_preserves_finite_limits | category_theory.functor | src/category_theory/functor/flat.lean | [
"category_theory.limits.filtered_colimit_commutes_finite_limit",
"category_theory.limits.preserves.functor_category",
"category_theory.limits.bicones",
"category_theory.limits.comma",
"category_theory.limits.preserves.finite",
"category_theory.limits.shapes.finite_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
flat_iff_Lan_flat (F : C ⥤ D) :
representably_flat F ↔ representably_flat (Lan F.op : _ ⥤ (Dᵒᵖ ⥤ Type u₁)) | ⟨λ H, by exactI infer_instance, λ H,
begin
resetI,
haveI := preserves_finite_limits_of_flat (Lan F.op : _ ⥤ (Dᵒᵖ ⥤ Type u₁)),
haveI : preserves_finite_limits F :=
begin
apply preserves_finite_limits_of_preserves_finite_limits_of_size.{u₁},
intros, resetI, apply preserves_limit_of_Lan_preserves_lim... | lemma | category_theory.flat_iff_Lan_flat | category_theory.functor | src/category_theory/functor/flat.lean | [
"category_theory.limits.filtered_colimit_commutes_finite_limit",
"category_theory.limits.preserves.functor_category",
"category_theory.limits.bicones",
"category_theory.limits.comma",
"category_theory.limits.preserves.finite",
"category_theory.limits.shapes.finite_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_finite_limits_iff_Lan_preserves_finite_limits (F : C ⥤ D) :
preserves_finite_limits F ≃ preserves_finite_limits (Lan F.op : _ ⥤ (Dᵒᵖ ⥤ Type u₁)) | { to_fun := λ _, by exactI infer_instance,
inv_fun := λ _,
begin
apply preserves_finite_limits_of_preserves_finite_limits_of_size.{u₁},
intros, resetI, apply preserves_limit_of_Lan_preserves_limit
end,
left_inv := λ x,
begin
cases x, unfold preserves_finite_limits_of_flat,
dunfold preserves_fi... | def | category_theory.preserves_finite_limits_iff_Lan_preserves_finite_limits | category_theory.functor | src/category_theory/functor/flat.lean | [
"category_theory.limits.filtered_colimit_commutes_finite_limit",
"category_theory.limits.preserves.functor_category",
"category_theory.limits.bicones",
"category_theory.limits.comma",
"category_theory.limits.preserves.finite",
"category_theory.limits.shapes.finite_limits"
] | [
"category_theory.Lan_preserves_finite_limits_of_flat",
"category_theory.Lan_preserves_finite_limits_of_preserves_finite_limits",
"inv_fun"
] | If `C` is finitely complete, then `F : C ⥤ D` preserves finite limits iff
`Lan F.op : (Cᵒᵖ ⥤ Type*) ⥤ (Dᵒᵖ ⥤ Type*)` preserves finite limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
full (F : C ⥤ D) | (preimage : ∀ {X Y : C} (f : (F.obj X) ⟶ (F.obj Y)), X ⟶ Y)
(witness' : ∀ {X Y : C} (f : (F.obj X) ⟶ (F.obj Y)), F.map (preimage f) = f . obviously) | class | category_theory.full | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | A functor `F : C ⥤ D` is full if for each `X Y : C`, `F.map` is surjective.
In fact, we use a constructive definition, so the `full F` typeclass contains data,
specifying a particular preimage of each `f : F.obj X ⟶ F.obj Y`.
See <https://stacks.math.columbia.edu/tag/001C>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
faithful (F : C ⥤ D) : Prop | (map_injective' [] : ∀ {X Y : C}, function.injective (@functor.map _ _ _ _ F X Y) . obviously) | class | category_theory.faithful | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | A functor `F : C ⥤ D` is faithful if for each `X Y : C`, `F.map` is injective.
See <https://stacks.math.columbia.edu/tag/001C>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_injective (F : C ⥤ D) [faithful F] : function.injective $ @functor.map _ _ _ _ F X Y | faithful.map_injective F | lemma | category_theory.functor.map_injective | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_iso_injective (F : C ⥤ D) [faithful F] :
function.injective $ @functor.map_iso _ _ _ _ F X Y | λ i j h, iso.ext (map_injective F (congr_arg iso.hom h : _)) | lemma | category_theory.functor.map_iso_injective | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage (F : C ⥤ D) [full F] (f : F.obj X ⟶ F.obj Y) : X ⟶ Y | full.preimage.{v₁ v₂} f | def | category_theory.functor.preimage | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | The specified preimage of a morphism under a full functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_preimage (F : C ⥤ D) [full F] {X Y : C} (f : F.obj X ⟶ F.obj Y) :
F.map (preimage F f) = f | by unfold preimage; obviously | lemma | category_theory.functor.image_preimage | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_surjective (F : C ⥤ D) [full F] : function.surjective (@functor.map _ _ _ _ F X Y) | λ f, ⟨F.preimage f, F.image_preimage f⟩ | lemma | category_theory.functor.map_surjective | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
full_of_exists (F : C ⥤ D)
(h : ∀ (X Y : C) (f : F.obj X ⟶ F.obj Y), ∃ p, F.map p = f) : full F | by { choose p hp using h, exact ⟨p, hp⟩ } | def | category_theory.functor.full_of_exists | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | Deduce that `F` is full from the existence of preimages, using choice. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
full_of_surjective (F : C ⥤ D)
(h : ∀ (X Y : C), function.surjective (@functor.map _ _ _ _ F X Y)) : full F | full_of_exists _ h | def | category_theory.functor.full_of_surjective | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | Deduce that `F` is full from surjectivity of `F.map`, using choice. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preimage_id : F.preimage (𝟙 (F.obj X)) = 𝟙 X | F.map_injective (by simp) | lemma | category_theory.preimage_id | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_comp (f : F.obj X ⟶ F.obj Y) (g : F.obj Y ⟶ F.obj Z) :
F.preimage (f ≫ g) = F.preimage f ≫ F.preimage g | F.map_injective (by simp) | lemma | category_theory.preimage_comp | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_map (f : X ⟶ Y) :
F.preimage (F.map f) = f | F.map_injective (by simp) | lemma | category_theory.preimage_map | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_iso (f : (F.obj X) ≅ (F.obj Y)) : X ≅ Y | { hom := F.preimage f.hom,
inv := F.preimage f.inv,
hom_inv_id' := F.map_injective (by simp),
inv_hom_id' := F.map_injective (by simp), } | def | category_theory.functor.preimage_iso | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | If `F : C ⥤ D` is fully faithful, every isomorphism `F.obj X ≅ F.obj Y` has a preimage. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preimage_iso_map_iso (f : X ≅ Y) :
F.preimage_iso (F.map_iso f) = f | by { ext, simp, } | lemma | category_theory.functor.preimage_iso_map_iso | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_of_fully_faithful (f : X ⟶ Y) [is_iso (F.map f)] : is_iso f | ⟨⟨F.preimage (inv (F.map f)),
⟨F.map_injective (by simp), F.map_injective (by simp)⟩⟩⟩ | lemma | category_theory.is_iso_of_fully_faithful | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | If the image of a morphism under a fully faithful functor in an isomorphism,
then the original morphisms is also an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_of_fully_faithful {X Y} : (X ⟶ Y) ≃ (F.obj X ⟶ F.obj Y) | { to_fun := λ f, F.map f,
inv_fun := λ f, F.preimage f,
left_inv := λ f, by simp,
right_inv := λ f, by simp } | def | category_theory.equiv_of_fully_faithful | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [
"inv_fun"
] | If `F` is fully faithful, we have an equivalence of hom-sets `X ⟶ Y` and `F X ⟶ F Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_equiv_of_fully_faithful {X Y} : (X ≅ Y) ≃ (F.obj X ≅ F.obj Y) | { to_fun := λ f, F.map_iso f,
inv_fun := λ f, F.preimage_iso f,
left_inv := λ f, by simp,
right_inv := λ f, by { ext, simp, } } | def | category_theory.iso_equiv_of_fully_faithful | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [
"inv_fun"
] | If `F` is fully faithful, we have an equivalence of iso-sets `X ≅ Y` and `F X ≅ F Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_trans_of_comp_fully_faithful (α : F ⋙ H ⟶ G ⋙ H) : F ⟶ G | { app := λ X, (equiv_of_fully_faithful H).symm (α.app X),
naturality' := λ X Y f, by { dsimp, apply H.map_injective, simpa using α.naturality f, } } | def | category_theory.nat_trans_of_comp_fully_faithful | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | We can construct a natural transformation between functors by constructing a
natural transformation between those functors composed with a fully faithful functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_iso_of_comp_fully_faithful (i : F ⋙ H ≅ G ⋙ H) : F ≅ G | nat_iso.of_components
(λ X, (iso_equiv_of_fully_faithful H).symm (i.app X))
(λ X Y f, by { dsimp, apply H.map_injective, simpa using i.hom.naturality f, }) | def | category_theory.nat_iso_of_comp_fully_faithful | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | We can construct a natural isomorphism between functors by constructing a natural isomorphism
between those functors composed with a fully faithful functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_iso_of_comp_fully_faithful_hom (i : F ⋙ H ≅ G ⋙ H) :
(nat_iso_of_comp_fully_faithful H i).hom = nat_trans_of_comp_fully_faithful H i.hom | by { ext, simp [nat_iso_of_comp_fully_faithful], } | lemma | category_theory.nat_iso_of_comp_fully_faithful_hom | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_iso_of_comp_fully_faithful_inv (i : F ⋙ H ≅ G ⋙ H) :
(nat_iso_of_comp_fully_faithful H i).inv = nat_trans_of_comp_fully_faithful H i.inv | by { ext, simp [←preimage_comp], dsimp, simp, } | lemma | category_theory.nat_iso_of_comp_fully_faithful_inv | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_trans.equiv_of_comp_fully_faithful : (F ⟶ G) ≃ (F ⋙ H ⟶ G ⋙ H) | { to_fun := λ α, α ◫ 𝟙 H,
inv_fun := nat_trans_of_comp_fully_faithful H,
left_inv := by tidy,
right_inv := by tidy, } | def | category_theory.nat_trans.equiv_of_comp_fully_faithful | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [
"inv_fun"
] | Horizontal composition with a fully faithful functor induces a bijection on
natural transformations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_iso.equiv_of_comp_fully_faithful : (F ≅ G) ≃ (F ⋙ H ≅ G ⋙ H) | { to_fun := λ e, nat_iso.hcomp e (iso.refl H),
inv_fun := nat_iso_of_comp_fully_faithful H,
left_inv := by tidy,
right_inv := by tidy, } | def | category_theory.nat_iso.equiv_of_comp_fully_faithful | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [
"inv_fun"
] | Horizontal composition with a fully faithful functor induces a bijection on
natural isomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
full.id : full (𝟭 C) | { preimage := λ _ _ f, f } | instance | category_theory.full.id | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
faithful.id : faithful (𝟭 C) | by obviously | instance | category_theory.faithful.id | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
faithful.comp [faithful F] [faithful G] : faithful (F ⋙ G) | { map_injective' := λ _ _ _ _ p, F.map_injective (G.map_injective p) } | instance | category_theory.faithful.comp | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
faithful.of_comp [faithful $ F ⋙ G] : faithful F | { map_injective' := λ X Y, (F ⋙ G).map_injective.of_comp } | lemma | category_theory.faithful.of_comp | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
full.of_iso [full F] (α : F ≅ F') : full F' | { preimage := λ X Y f, F.preimage ((α.app X).hom ≫ f ≫ (α.app Y).inv),
witness' := λ X Y f, by simp [←nat_iso.naturality_1 α], } | def | category_theory.full.of_iso | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | If `F` is full, and naturally isomorphic to some `F'`, then `F'` is also full. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
faithful.of_iso [faithful F] (α : F ≅ F') : faithful F' | { map_injective' := λ X Y f f' h, F.map_injective
(by rw [←nat_iso.naturality_1 α.symm, h, nat_iso.naturality_1 α.symm]) } | lemma | category_theory.faithful.of_iso | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
faithful.of_comp_iso {H : C ⥤ E} [ℋ : faithful H] (h : F ⋙ G ≅ H) : faithful F | @faithful.of_comp _ _ _ _ _ _ F G (faithful.of_iso h.symm) | lemma | category_theory.faithful.of_comp_iso | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
faithful.of_comp_eq {H : C ⥤ E} [ℋ : faithful H] (h : F ⋙ G = H) : faithful F | @faithful.of_comp _ _ _ _ _ _ F G (h.symm ▸ ℋ) | lemma | category_theory.faithful.of_comp_eq | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
faithful.div (F : C ⥤ E) (G : D ⥤ E) [faithful G]
(obj : C → D) (h_obj : ∀ X, G.obj (obj X) = F.obj X)
(map : Π {X Y}, (X ⟶ Y) → (obj X ⟶ obj Y))
(h_map : ∀ {X Y} {f : X ⟶ Y}, G.map (map f) == F.map f) :
C ⥤ D | { obj := obj,
map := @map,
map_id' :=
begin
assume X,
apply G.map_injective,
apply eq_of_heq,
transitivity F.map (𝟙 X), from h_map,
rw [F.map_id, G.map_id, h_obj X]
end,
map_comp' :=
begin
assume X Y Z f g,
apply G.map_injective,
apply eq_of_heq,
transitivity F.map (f ≫ ... | def | category_theory.faithful.div | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | “Divide” a functor by a faithful functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
faithful.div_comp (F : C ⥤ E) [faithful F] (G : D ⥤ E) [faithful G]
(obj : C → D) (h_obj : ∀ X, G.obj (obj X) = F.obj X)
(map : Π {X Y}, (X ⟶ Y) → (obj X ⟶ obj Y))
(h_map : ∀ {X Y} {f : X ⟶ Y}, G.map (map f) == F.map f) :
(faithful.div F G obj @h_obj @map @h_map) ⋙ G = F | begin
casesI F with F_obj _ _ _, casesI G with G_obj _ _ _,
unfold faithful.div functor.comp,
unfold_projs at h_obj,
have: F_obj = G_obj ∘ obj := (funext h_obj).symm,
substI this,
congr,
funext,
exact eq_of_heq h_map
end | lemma | category_theory.faithful.div_comp | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [
"functor.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
faithful.div_faithful (F : C ⥤ E) [faithful F] (G : D ⥤ E) [faithful G]
(obj : C → D) (h_obj : ∀ X, G.obj (obj X) = F.obj X)
(map : Π {X Y}, (X ⟶ Y) → (obj X ⟶ obj Y))
(h_map : ∀ {X Y} {f : X ⟶ Y}, G.map (map f) == F.map f) :
faithful (faithful.div F G obj @h_obj @map @h_map) | (faithful.div_comp F G _ h_obj _ @h_map).faithful_of_comp | lemma | category_theory.faithful.div_faithful | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
full.comp [full F] [full G] : full (F ⋙ G) | { preimage := λ _ _ f, F.preimage (G.preimage f) } | instance | category_theory.full.comp | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
full.of_comp_faithful [full $ F ⋙ G] [faithful G] : full F | { preimage := λ X Y f, (F ⋙ G).preimage (G.map f),
witness' := λ X Y f, G.map_injective ((F ⋙ G).image_preimage _) } | def | category_theory.full.of_comp_faithful | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | If `F ⋙ G` is full and `G` is faithful, then `F` is full. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
full.of_comp_faithful_iso {F : C ⥤ D} {G : D ⥤ E} {H : C ⥤ E} [full H] [faithful G]
(h : F ⋙ G ≅ H) : full F | @full.of_comp_faithful _ _ _ _ _ _ F G (full.of_iso h.symm) _ | def | category_theory.full.of_comp_faithful_iso | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | If `F ⋙ G` is full and `G` is faithful, then `F` is full. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fully_faithful_cancel_right {F G : C ⥤ D} (H : D ⥤ E)
[full H] [faithful H] (comp_iso: F ⋙ H ≅ G ⋙ H) : F ≅ G | nat_iso.of_components
(λ X, H.preimage_iso (comp_iso.app X))
(λ X Y f, H.map_injective (by simpa using comp_iso.hom.naturality f)) | def | category_theory.fully_faithful_cancel_right | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | Given a natural isomorphism between `F ⋙ H` and `G ⋙ H` for a fully faithful functor `H`, we
can 'cancel' it to give a natural iso between `F` and `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fully_faithful_cancel_right_hom_app {F G : C ⥤ D} {H : D ⥤ E}
[full H] [faithful H] (comp_iso: F ⋙ H ≅ G ⋙ H) (X : C) :
(fully_faithful_cancel_right H comp_iso).hom.app X = H.preimage (comp_iso.hom.app X) | rfl | lemma | category_theory.fully_faithful_cancel_right_hom_app | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fully_faithful_cancel_right_inv_app {F G : C ⥤ D} {H : D ⥤ E}
[full H] [faithful H] (comp_iso: F ⋙ H ≅ G ⋙ H) (X : C) :
(fully_faithful_cancel_right H comp_iso).inv.app X = H.preimage (comp_iso.inv.app X) | rfl | lemma | category_theory.fully_faithful_cancel_right_inv_app | category_theory.functor | src/category_theory/functor/fully_faithful.lean | [
"category_theory.natural_isomorphism",
"logic.equiv.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functorial (F : C → D) : Type (max v₁ v₂ u₁ u₂) | (map : Π {X Y : C}, (X ⟶ Y) → ((F X) ⟶ (F Y)))
(map_id' : ∀ (X : C), map (𝟙 X) = 𝟙 (F X) . obviously)
(map_comp' : ∀ {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z), map (f ≫ g) = (map f) ≫ (map g) . obviously) | class | category_theory.functorial | category_theory.functor | src/category_theory/functor/functorial.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (F : C → D) [functorial.{v₁ v₂} F] {X Y : C} (f : X ⟶ Y) : F X ⟶ F Y | functorial.map.{v₁ v₂} f | def | category_theory.map | category_theory.functor | src/category_theory/functor/functorial.lean | [
"category_theory.functor.basic"
] | [] | If `F : C → D` (just a function) has `[functorial F]`,
we can write `map F f : F X ⟶ F Y` for the action of `F` on a morphism `f : X ⟶ Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_as_map {F : C → D} [functorial.{v₁ v₂} F] {X Y : C} {f : X ⟶ Y} :
functorial.map.{v₁ v₂} f = map F f | rfl | lemma | category_theory.map_as_map | category_theory.functor | src/category_theory/functor/functorial.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functorial.map_id {F : C → D} [functorial.{v₁ v₂} F] {X : C} :
map F (𝟙 X) = 𝟙 (F X) | functorial.map_id' X | lemma | category_theory.functorial.map_id | category_theory.functor | src/category_theory/functor/functorial.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functorial.map_comp
{F : C → D} [functorial.{v₁ v₂} F] {X Y Z : C} {f : X ⟶ Y} {g : Y ⟶ Z} :
map F (f ≫ g) = map F f ≫ map F g | functorial.map_comp' f g | lemma | category_theory.functorial.map_comp | category_theory.functor | src/category_theory/functor/functorial.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of (F : C → D) [I : functorial.{v₁ v₂} F] : C ⥤ D | { obj := F,
..I } | def | category_theory.functor.of | category_theory.functor | src/category_theory/functor/functorial.lean | [
"category_theory.functor.basic"
] | [] | Bundle a functorial function as a functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_functorial_obj (F : C ⥤ D) {X Y : C} (f : X ⟶ Y) : map F.obj f = F.map f | rfl | lemma | category_theory.map_functorial_obj | category_theory.functor | src/category_theory/functor/functorial.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functorial_id : functorial.{v₁ v₁} (id : C → C) | { map := λ X Y f, f } | instance | category_theory.functorial_id | category_theory.functor | src/category_theory/functor/functorial.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functorial_comp (F : C → D) [functorial.{v₁ v₂} F] (G : D → E) [functorial.{v₂ v₃} G] :
functorial.{v₁ v₃} (G ∘ F) | { ..(functor.of F ⋙ functor.of G) } | def | category_theory.functorial_comp | category_theory.functor | src/category_theory/functor/functorial.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom : Cᵒᵖ × C ⥤ Type v | { obj := λ p, unop p.1 ⟶ p.2,
map := λ X Y f, λ h, f.1.unop ≫ h ≫ f.2 } | def | category_theory.functor.hom | category_theory.functor | src/category_theory/functor/hom.lean | [
"category_theory.products.basic",
"category_theory.types"
] | [] | `functor.hom` is the hom-pairing, sending `(X, Y)` to `X ⟶ Y`, contravariant in `X` and
covariant in `Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_inv_iso [h : is_equivalence H] (i : F ≅ G ⋙ H) : F ⋙ H.inv ≅ G | iso_whisker_right i H.inv ≪≫ (associator G H H.inv) ≪≫
iso_whisker_left G h.unit_iso.symm ≪≫ eq_to_iso (functor.comp_id G) | def | category_theory.comp_inv_iso | category_theory.functor | src/category_theory/functor/inv_isos.lean | [
"category_theory.eq_to_hom"
] | [] | Construct an isomorphism `F ⋙ H.inv ≅ G` from an isomorphism `F ≅ G ⋙ H`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_comp_inv [h : is_equivalence H] (i : G ⋙ H ≅ F) : G ≅ F ⋙ H.inv | (comp_inv_iso i.symm).symm | def | category_theory.iso_comp_inv | category_theory.functor | src/category_theory/functor/inv_isos.lean | [
"category_theory.eq_to_hom"
] | [] | Construct an isomorphism `G ≅ F ⋙ H.inv` from an isomorphism `G ⋙ H ≅ F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_comp_iso [h : is_equivalence G] (i : F ≅ G ⋙ H) : G.inv ⋙ F ≅ H | iso_whisker_left G.inv i ≪≫ (associator G.inv G H).symm ≪≫
iso_whisker_right h.counit_iso H ≪≫ eq_to_iso (functor.id_comp H) | def | category_theory.inv_comp_iso | category_theory.functor | src/category_theory/functor/inv_isos.lean | [
"category_theory.eq_to_hom"
] | [] | Construct an isomorphism `G.inv ⋙ F ≅ H` from an isomorphism `F ≅ G ⋙ H`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_inv_comp [h : is_equivalence G] (i : G ⋙ H ≅ F) : H ≅ G.inv ⋙ F | (inv_comp_iso i.symm).symm | def | category_theory.iso_inv_comp | category_theory.functor | src/category_theory/functor/inv_isos.lean | [
"category_theory.eq_to_hom"
] | [] | Construct an isomorphism `H ≅ G.inv ⋙ F` from an isomorphism `G ⋙ H ≅ F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor.left_derived (F : C ⥤ D) [F.additive] (n : ℕ) : C ⥤ D | projective_resolutions C ⋙ F.map_homotopy_category _ ⋙ homotopy_category.homology_functor D _ n | def | category_theory.functor.left_derived | category_theory.functor | src/category_theory/functor/left_derived.lean | [
"category_theory.preadditive.projective_resolution"
] | [
"homotopy_category.homology_functor"
] | The left derived functors of an additive functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor.left_derived_obj_iso (F : C ⥤ D) [F.additive] (n : ℕ)
{X : C} (P : ProjectiveResolution X) :
(F.left_derived n).obj X ≅
(homology_functor D _ n).obj ((F.map_homological_complex _).obj P.complex) | (homotopy_category.homology_functor D _ n).map_iso
(homotopy_category.iso_of_homotopy_equiv
(F.map_homotopy_equiv (ProjectiveResolution.homotopy_equiv _ P)))
≪≫ (homotopy_category.homology_factors D _ n).app _ | def | category_theory.functor.left_derived_obj_iso | category_theory.functor | src/category_theory/functor/left_derived.lean | [
"category_theory.preadditive.projective_resolution"
] | [
"homology_functor",
"homotopy_category.homology_factors",
"homotopy_category.homology_functor",
"homotopy_category.iso_of_homotopy_equiv"
] | We can compute a left derived functor using a chosen projective resolution. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor.left_derived_obj_projective_zero (F : C ⥤ D) [F.additive]
(X : C) [projective X] :
(F.left_derived 0).obj X ≅ F.obj X | F.left_derived_obj_iso 0 (ProjectiveResolution.self X) ≪≫
(homology_functor _ _ _).map_iso ((chain_complex.single₀_map_homological_complex F).app X) ≪≫
(chain_complex.homology_functor_0_single₀ D).app (F.obj X) | def | category_theory.functor.left_derived_obj_projective_zero | category_theory.functor | src/category_theory/functor/left_derived.lean | [
"category_theory.preadditive.projective_resolution"
] | [
"chain_complex.homology_functor_0_single₀",
"chain_complex.single₀_map_homological_complex",
"homology_functor"
] | The 0-th derived functor of `F` on a projective object `X` is just `F.obj X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor.left_derived_obj_projective_succ (F : C ⥤ D) [F.additive] (n : ℕ)
(X : C) [projective X] :
(F.left_derived (n+1)).obj X ≅ 0 | F.left_derived_obj_iso (n+1) (ProjectiveResolution.self X) ≪≫
(homology_functor _ _ _).map_iso ((chain_complex.single₀_map_homological_complex F).app X) ≪≫
(chain_complex.homology_functor_succ_single₀ D n).app (F.obj X) ≪≫
(functor.zero_obj _).iso_zero | def | category_theory.functor.left_derived_obj_projective_succ | category_theory.functor | src/category_theory/functor/left_derived.lean | [
"category_theory.preadditive.projective_resolution"
] | [
"chain_complex.homology_functor_succ_single₀",
"chain_complex.single₀_map_homological_complex",
"homology_functor"
] | The higher derived functors vanish on projective objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor.left_derived_map_eq (F : C ⥤ D) [F.additive] (n : ℕ) {X Y : C} (f : X ⟶ Y)
{P : ProjectiveResolution X} {Q : ProjectiveResolution Y} (g : P.complex ⟶ Q.complex)
(w : g ≫ Q.π = P.π ≫ (chain_complex.single₀ C).map f) :
(F.left_derived n).map f =
(F.left_derived_obj_iso n P).hom ≫
(homology_functor D _... | begin
dsimp only [functor.left_derived, functor.left_derived_obj_iso],
dsimp, simp only [category.comp_id, category.id_comp],
rw [←homology_functor_map, homotopy_category.homology_functor_map_factors],
simp only [←functor.map_comp],
congr' 1,
apply homotopy_category.eq_of_homotopy,
apply functor.map_homot... | lemma | category_theory.functor.left_derived_map_eq | category_theory.functor | src/category_theory/functor/left_derived.lean | [
"category_theory.preadditive.projective_resolution"
] | [
"chain_complex.single₀",
"homology_functor",
"homotopy.trans",
"homotopy_category.eq_of_homotopy",
"homotopy_category.homology_functor_map_factors",
"homotopy_category.homotopy_out_map"
] | We can compute a left derived functor on a morphism using a lift of that morphism
to a chain map between chosen projective resolutions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_trans.left_derived {F G : C ⥤ D} [F.additive] [G.additive] (α : F ⟶ G) (n : ℕ) :
F.left_derived n ⟶ G.left_derived n | whisker_left (projective_resolutions C)
(whisker_right (nat_trans.map_homotopy_category α _)
(homotopy_category.homology_functor D _ n)) | def | category_theory.nat_trans.left_derived | category_theory.functor | src/category_theory/functor/left_derived.lean | [
"category_theory.preadditive.projective_resolution"
] | [
"homotopy_category.homology_functor"
] | The natural transformation between left-derived functors induced by a natural transformation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_trans.left_derived_id (F : C ⥤ D) [F.additive] (n : ℕ) :
nat_trans.left_derived (𝟙 F) n = 𝟙 (F.left_derived n) | by { simp [nat_trans.left_derived], refl, } | lemma | category_theory.nat_trans.left_derived_id | category_theory.functor | src/category_theory/functor/left_derived.lean | [
"category_theory.preadditive.projective_resolution"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_trans.left_derived_comp
{F G H : C ⥤ D} [F.additive] [G.additive] [H.additive]
(α : F ⟶ G) (β : G ⟶ H) (n : ℕ) :
nat_trans.left_derived (α ≫ β) n = nat_trans.left_derived α n ≫ nat_trans.left_derived β n | by simp [nat_trans.left_derived] | lemma | category_theory.nat_trans.left_derived_comp | category_theory.functor | src/category_theory/functor/left_derived.lean | [
"category_theory.preadditive.projective_resolution"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_trans.left_derived_eq {F G : C ⥤ D} [F.additive] [G.additive] (α : F ⟶ G) (n : ℕ)
{X : C} (P : ProjectiveResolution X) :
(nat_trans.left_derived α n).app X =
(F.left_derived_obj_iso n P).hom ≫
(homology_functor D _ n).map ((nat_trans.map_homological_complex α _).app P.complex) ≫
(G.left_derive... | begin
symmetry,
dsimp [nat_trans.left_derived, functor.left_derived_obj_iso],
simp only [category.comp_id, category.id_comp],
rw [←homology_functor_map, homotopy_category.homology_functor_map_factors],
simp only [←functor.map_comp],
congr' 1,
apply homotopy_category.eq_of_homotopy,
simp only [nat_trans.... | lemma | category_theory.nat_trans.left_derived_eq | category_theory.functor | src/category_theory/functor/left_derived.lean | [
"category_theory.preadditive.projective_resolution"
] | [
"homology_functor",
"homotopy.comp_left_id",
"homotopy_category.eq_of_homotopy",
"homotopy_category.homology_functor_map_factors"
] | A component of the natural transformation between left-derived functors can be computed
using a chosen projective resolution. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_isomorphisms (F : C ⥤ D) : Prop | (reflects : Π {A B : C} (f : A ⟶ B) [is_iso (F.map f)], is_iso f) | class | category_theory.reflects_isomorphisms | category_theory.functor | src/category_theory/functor/reflects_isomorphisms.lean | [
"category_theory.balanced",
"category_theory.functor.epi_mono",
"category_theory.functor.fully_faithful"
] | [] | Define what it means for a functor `F : C ⥤ D` to reflect isomorphisms: for any
morphism `f : A ⟶ B`, if `F.map f` is an isomorphism then `f` is as well.
Note that we do not assume or require that `F` is faithful. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_of_reflects_iso {A B : C} (f : A ⟶ B) (F : C ⥤ D)
[is_iso (F.map f)] [reflects_isomorphisms F] :
is_iso f | reflects_isomorphisms.reflects F f | lemma | category_theory.is_iso_of_reflects_iso | category_theory.functor | src/category_theory/functor/reflects_isomorphisms.lean | [
"category_theory.balanced",
"category_theory.functor.epi_mono",
"category_theory.functor.fully_faithful"
] | [] | If `F` reflects isos and `F.map f` is an iso, then `f` is an iso. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_full_and_faithful (F : C ⥤ D) [full F] [faithful F] : reflects_isomorphisms F | { reflects := λ X Y f i, by exactI
⟨⟨F.preimage (inv (F.map f)), ⟨F.map_injective (by simp), F.map_injective (by simp)⟩⟩⟩ } | instance | category_theory.of_full_and_faithful | category_theory.functor | src/category_theory/functor/reflects_isomorphisms.lean | [
"category_theory.balanced",
"category_theory.functor.epi_mono",
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_isomorphisms_of_reflects_monomorphisms_of_reflects_epimorphisms [balanced C]
(F : C ⥤ D) [reflects_monomorphisms F] [reflects_epimorphisms F] : reflects_isomorphisms F | { reflects := λ A B f hf,
begin
resetI,
haveI : epi f := epi_of_epi_map F infer_instance,
haveI : mono f := mono_of_mono_map F infer_instance,
exact is_iso_of_mono_of_epi f
end } | instance | category_theory.reflects_isomorphisms_of_reflects_monomorphisms_of_reflects_epimorphisms | category_theory.functor | src/category_theory/functor/reflects_isomorphisms.lean | [
"category_theory.balanced",
"category_theory.functor.epi_mono",
"category_theory.functor.fully_faithful"
] | [
"balanced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_thin_iff : quiver.is_thin C ↔ ∀ c : C, subsingleton (c ⟶ c) | begin
refine ⟨λ h c, h c c, λ h c d, subsingleton.intro $ λ f g, _⟩,
haveI := h d,
calc f = f ≫ (inv g ≫ g) : by simp only [inv_eq_inv, is_iso.inv_hom_id, category.comp_id]
... = f ≫ (inv f ≫ g) : by congr
... = g : by simp only [inv_eq_inv, is_iso.hom_inv_id_assoc],
end | lemma | category_theory.groupoid.is_thin_iff | category_theory.groupoid | src/category_theory/groupoid/basic.lean | [
"category_theory.groupoid",
"combinatorics.quiver.basic"
] | [
"quiver.is_thin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_totally_disconnected | ∀ (c d : C), (c ⟶ d) → c = d | def | category_theory.groupoid.is_totally_disconnected | category_theory.groupoid | src/category_theory/groupoid/basic.lean | [
"category_theory.groupoid",
"combinatorics.quiver.basic"
] | [
"is_totally_disconnected"
] | A subgroupoid is totally disconnected if it only has loops. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quiver.hom.to_pos_path {X Y : V} (f : X ⟶ Y) :
((category_theory.paths.category_paths $ quiver.symmetrify V).hom X Y) | f.to_pos.to_path | abbreviation | category_theory.groupoid.free.quiver.hom.to_pos_path | category_theory.groupoid | src/category_theory/groupoid/free_groupoid.lean | [
"category_theory.category.basic",
"category_theory.functor.basic",
"category_theory.groupoid",
"tactic.nth_rewrite",
"category_theory.path_category",
"category_theory.quotient",
"combinatorics.quiver.symmetric"
] | [
"category_theory.paths.category_paths",
"quiver.symmetrify"
] | Shorthand for the "forward" arrow corresponding to `f` in `paths $ symmetrify V` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quiver.hom.to_neg_path {X Y : V} (f : X ⟶ Y) :
((category_theory.paths.category_paths $ quiver.symmetrify V).hom Y X) | f.to_neg.to_path | abbreviation | category_theory.groupoid.free.quiver.hom.to_neg_path | category_theory.groupoid | src/category_theory/groupoid/free_groupoid.lean | [
"category_theory.category.basic",
"category_theory.functor.basic",
"category_theory.groupoid",
"tactic.nth_rewrite",
"category_theory.path_category",
"category_theory.quotient",
"combinatorics.quiver.symmetric"
] | [
"category_theory.paths.category_paths",
"quiver.symmetrify"
] | Shorthand for the "forward" arrow corresponding to `f` in `paths $ symmetrify V` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
red_step : hom_rel (paths (quiver.symmetrify V))
| step (X Z : quiver.symmetrify V) (f : X ⟶ Z) :
red_step (𝟙 X) (f.to_path ≫ (quiver.reverse f).to_path) | inductive | category_theory.groupoid.free.red_step | category_theory.groupoid | src/category_theory/groupoid/free_groupoid.lean | [
"category_theory.category.basic",
"category_theory.functor.basic",
"category_theory.groupoid",
"tactic.nth_rewrite",
"category_theory.path_category",
"category_theory.quotient",
"combinatorics.quiver.symmetric"
] | [
"hom_rel",
"quiver.reverse",
"quiver.symmetrify",
"to_path"
] | The "reduction" relation | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.category_theory.free_groupoid (V) [Q : quiver V] | quotient (@red_step V Q) | def | category_theory.free_groupoid | category_theory.groupoid | src/category_theory/groupoid/free_groupoid.lean | [
"category_theory.category.basic",
"category_theory.functor.basic",
"category_theory.groupoid",
"tactic.nth_rewrite",
"category_theory.path_category",
"category_theory.quotient",
"combinatorics.quiver.symmetric"
] | [
"quiver"
] | The underlying vertices of the free groupoid | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
congr_reverse {X Y : paths $ quiver.symmetrify V} (p q : X ⟶ Y) :
quotient.comp_closure red_step p q →
quotient.comp_closure red_step (p.reverse) (q.reverse) | begin
rintro ⟨XW, pp, qq, WY, _, Z, f⟩,
have : quotient.comp_closure red_step (WY.reverse ≫ 𝟙 _ ≫ XW.reverse)
(WY.reverse ≫ (f.to_path ≫ (quiver.reverse f).to_path) ≫ XW.reverse),
{ apply quotient.comp_closure.intro,
apply red_step.step, },
simpa only [category_struct.comp, category_struct.id, quiver.p... | lemma | category_theory.groupoid.free.congr_reverse | category_theory.groupoid | src/category_theory/groupoid/free_groupoid.lean | [
"category_theory.category.basic",
"category_theory.functor.basic",
"category_theory.groupoid",
"tactic.nth_rewrite",
"category_theory.path_category",
"category_theory.quotient",
"combinatorics.quiver.symmetric"
] | [
"quiver.path.comp_assoc",
"quiver.path.nil_comp",
"quiver.path.reverse",
"quiver.path.reverse_comp",
"quiver.path.reverse_to_path",
"quiver.reverse",
"quiver.reverse_reverse",
"quiver.symmetrify",
"to_path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_comp_reverse {X Y : paths $ quiver.symmetrify V} (p : X ⟶ Y) :
quot.mk (@quotient.comp_closure _ _ red_step _ _) (p ≫ p.reverse) =
quot.mk (@quotient.comp_closure _ _ red_step _ _) (𝟙 X) | begin
apply quot.eqv_gen_sound,
induction p with _ _ q f ih,
{ apply eqv_gen.refl, },
{ simp only [quiver.path.reverse],
fapply eqv_gen.trans,
{ exact q ≫ q.reverse, },
{ apply eqv_gen.symm, apply eqv_gen.rel,
have : quotient.comp_closure
red_step (q ≫ (𝟙 _) ≫ q.reverse)
... | lemma | category_theory.groupoid.free.congr_comp_reverse | category_theory.groupoid | src/category_theory/groupoid/free_groupoid.lean | [
"category_theory.category.basic",
"category_theory.functor.basic",
"category_theory.groupoid",
"tactic.nth_rewrite",
"category_theory.path_category",
"category_theory.quotient",
"combinatorics.quiver.symmetric"
] | [
"ih",
"quiver.path.comp_assoc",
"quiver.path.reverse",
"quiver.reverse",
"quiver.symmetrify",
"to_path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_reverse_comp {X Y : paths $ quiver.symmetrify V} (p : X ⟶ Y) :
quot.mk (@quotient.comp_closure _ _ red_step _ _) (p.reverse ≫ p) =
quot.mk (@quotient.comp_closure _ _ red_step _ _) (𝟙 Y) | begin
nth_rewrite 1 ←quiver.path.reverse_reverse p,
apply congr_comp_reverse,
end | lemma | category_theory.groupoid.free.congr_reverse_comp | category_theory.groupoid | src/category_theory/groupoid/free_groupoid.lean | [
"category_theory.category.basic",
"category_theory.functor.basic",
"category_theory.groupoid",
"tactic.nth_rewrite",
"category_theory.path_category",
"category_theory.quotient",
"combinatorics.quiver.symmetric"
] | [
"quiver.symmetrify"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_inv {X Y : free_groupoid V} (f : X ⟶ Y) : Y ⟶ X | quot.lift_on f
(λ pp, quot.mk _ $ pp.reverse)
(λ pp qq con, quot.sound $ congr_reverse pp qq con) | def | category_theory.groupoid.free.quot_inv | category_theory.groupoid | src/category_theory/groupoid/free_groupoid.lean | [
"category_theory.category.basic",
"category_theory.functor.basic",
"category_theory.groupoid",
"tactic.nth_rewrite",
"category_theory.path_category",
"category_theory.quotient",
"combinatorics.quiver.symmetric"
] | [
"con"
] | The inverse of an arrow in the free groupoid | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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