statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
map (F : C ⥤ D) : thin_skeleton C ⥤ thin_skeleton D | { obj := quotient.map F.obj $ λ X₁ X₂ ⟨hX⟩, ⟨F.map_iso hX⟩,
map := λ X Y, quotient.rec_on_subsingleton₂ X Y $
λ x y k, hom_of_le (k.le.elim (λ t, ⟨F.map t⟩)) } | def | category_theory.thin_skeleton.map | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [
"quotient.map"
] | A functor `C ⥤ D` computably lowers to a functor `thin_skeleton C ⥤ thin_skeleton D`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_to_thin_skeleton (F : C ⥤ D) : F ⋙ to_thin_skeleton D = to_thin_skeleton C ⋙ map F | rfl | lemma | category_theory.thin_skeleton.comp_to_thin_skeleton | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_nat_trans {F₁ F₂ : C ⥤ D} (k : F₁ ⟶ F₂) : map F₁ ⟶ map F₂ | { app := λ X, quotient.rec_on_subsingleton X (λ x, ⟨⟨⟨k.app x⟩⟩⟩) } | def | category_theory.thin_skeleton.map_nat_trans | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [] | Given a natural transformation `F₁ ⟶ F₂`, induce a natural transformation `map F₁ ⟶ map F₂`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map₂ (F : C ⥤ D ⥤ E) :
thin_skeleton C ⥤ thin_skeleton D ⥤ thin_skeleton E | { obj := λ x,
{ obj := λ y, quotient.map₂ (λ X Y, (F.obj X).obj Y)
(λ X₁ X₂ ⟨hX⟩ Y₁ Y₂ ⟨hY⟩, ⟨(F.obj X₁).map_iso hY ≪≫ (F.map_iso hX).app Y₂⟩) x y,
map := λ y₁ y₂, quotient.rec_on_subsingleton x $
λ X, quotient.rec_on_subsingleton₂ y₁ y₂ $
λ Y₁ Y₂ hY, hom_of_le (hY.le.eli... | def | category_theory.thin_skeleton.map₂ | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [
"quotient.map₂"
] | A functor `C ⥤ D ⥤ E` computably lowers to a functor
`thin_skeleton C ⥤ thin_skeleton D ⥤ thin_skeleton E` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_thin_skeleton_faithful : faithful (to_thin_skeleton C) | {} | instance | category_theory.thin_skeleton.to_thin_skeleton_faithful | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_thin_skeleton : thin_skeleton C ⥤ C | { obj := quotient.out,
map := λ x y, quotient.rec_on_subsingleton₂ x y $
λ X Y f,
(nonempty.some (quotient.mk_out X)).hom
≫ f.le.some
≫ (nonempty.some (quotient.mk_out Y)).inv } | def | category_theory.thin_skeleton.from_thin_skeleton | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [
"nonempty.some",
"quotient.mk_out",
"quotient.out"
] | Use `quotient.out` to create a functor out of the thin skeleton. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
from_thin_skeleton_equivalence : is_equivalence (from_thin_skeleton C) | { inverse := to_thin_skeleton C,
counit_iso := nat_iso.of_components (λ X, (nonempty.some (quotient.mk_out X))) (by tidy),
unit_iso :=
nat_iso.of_components
(λ x, quotient.rec_on_subsingleton x
(λ X, eq_to_iso (quotient.sound ⟨(nonempty.some (quotient.mk_out X)).symm⟩)))
(by tidy) } | instance | category_theory.thin_skeleton.from_thin_skeleton_equivalence | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [
"nonempty.some",
"quotient.mk_out"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equivalence : thin_skeleton C ≌ C | (from_thin_skeleton C).as_equivalence | def | category_theory.thin_skeleton.equivalence | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [] | The equivalence between the thin skeleton and the category itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_of_both_ways {X Y : C} (f : X ⟶ Y) (g : Y ⟶ X) : X ≈ Y | ⟨iso_of_both_ways f g⟩ | lemma | category_theory.thin_skeleton.equiv_of_both_ways | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thin_skeleton_partial_order : partial_order (thin_skeleton C) | { le_antisymm := quotient.ind₂
begin
rintros _ _ ⟨f⟩ ⟨g⟩,
apply quotient.sound (equiv_of_both_ways f g),
end,
..category_theory.thin_skeleton.preorder C } | instance | category_theory.thin_skeleton.thin_skeleton_partial_order | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [
"category_theory.thin_skeleton.preorder"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
skeletal : skeletal (thin_skeleton C) | λ X Y, quotient.induction_on₂ X Y $ λ x y h, h.elim $ λ i, i.1.le.antisymm i.2.le | lemma | category_theory.thin_skeleton.skeletal | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp_eq (F : E ⥤ D) (G : D ⥤ C) : map (F ⋙ G) = map F ⋙ map G | functor.eq_of_iso skeletal $
nat_iso.of_components (λ X, quotient.rec_on_subsingleton X (λ x, iso.refl _)) (by tidy) | lemma | category_theory.thin_skeleton.map_comp_eq | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id_eq : map (𝟭 C) = 𝟭 (thin_skeleton C) | functor.eq_of_iso skeletal $
nat_iso.of_components (λ X, quotient.rec_on_subsingleton X (λ x, iso.refl _)) (by tidy) | lemma | category_theory.thin_skeleton.map_id_eq | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_iso_eq {F₁ F₂ : D ⥤ C} (h : F₁ ≅ F₂) : map F₁ = map F₂ | functor.eq_of_iso skeletal { hom := map_nat_trans h.hom, inv := map_nat_trans h.inv } | lemma | category_theory.thin_skeleton.map_iso_eq | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thin_skeleton_is_skeleton : is_skeleton_of C (thin_skeleton C)
(from_thin_skeleton C) | { skel := skeletal,
eqv := thin_skeleton.from_thin_skeleton_equivalence C } | def | category_theory.thin_skeleton.thin_skeleton_is_skeleton | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [] | `from_thin_skeleton C` exhibits the thin skeleton as a skeleton. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_skeleton_of_inhabited :
inhabited (is_skeleton_of C (thin_skeleton C) (from_thin_skeleton C)) | ⟨thin_skeleton_is_skeleton⟩ | instance | category_theory.thin_skeleton.is_skeleton_of_inhabited | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_adjunction
(R : D ⥤ C) (L : C ⥤ D) (h : L ⊣ R) :
thin_skeleton.map L ⊣ thin_skeleton.map R | adjunction.mk_of_unit_counit
{ unit :=
{ app := λ X,
begin
letI := is_isomorphic_setoid C,
refine quotient.rec_on_subsingleton X (λ x, hom_of_le ⟨h.unit.app x⟩),
-- TODO: make quotient.rec_on_subsingleton' so the letI isn't needed
end },
counit :=
{ app := λ X,
begin
letI := is... | def | category_theory.thin_skeleton.lower_adjunction | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [] | An adjunction between thin categories gives an adjunction between their thin skeletons. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence.thin_skeleton_order_iso
[quiver.is_thin C] (e : C ≌ α) : thin_skeleton C ≃o α | ((thin_skeleton.equivalence C).trans e).to_order_iso | def | category_theory.equivalence.thin_skeleton_order_iso | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [
"quiver.is_thin"
] | When `e : C ≌ α` is a categorical equivalence from a thin category `C` to some partial order `α`,
the `thin_skeleton C` is order isomorphic to `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
structured_arrow (S : D) (T : C ⥤ D) | comma (functor.from_punit S) T | def | category_theory.structured_arrow | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | The category of `T`-structured arrows with domain `S : D` (here `T : C ⥤ D`),
has as its objects `D`-morphisms of the form `S ⟶ T Y`, for some `Y : C`,
and morphisms `C`-morphisms `Y ⟶ Y'` making the obvious triangle commute. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
proj (S : D) (T : C ⥤ D) : structured_arrow S T ⥤ C | comma.snd _ _ | def | category_theory.structured_arrow.proj | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | The obvious projection functor from structured arrows. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk (f : S ⟶ T.obj Y) : structured_arrow S T | ⟨⟨⟨⟩⟩, Y, f⟩ | def | category_theory.structured_arrow.mk | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | Construct a structured arrow from a morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_left (f : S ⟶ T.obj Y) : (mk f).left = ⟨⟨⟩⟩ | rfl | lemma | category_theory.structured_arrow.mk_left | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_right (f : S ⟶ T.obj Y) : (mk f).right = Y | rfl | lemma | category_theory.structured_arrow.mk_right | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_hom_eq_self (f : S ⟶ T.obj Y) : (mk f).hom = f | rfl | lemma | category_theory.structured_arrow.mk_hom_eq_self | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w {A B : structured_arrow S T} (f : A ⟶ B) : A.hom ≫ T.map f.right = B.hom | by { have := f.w; tidy } | lemma | category_theory.structured_arrow.w | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_mk {f f' : structured_arrow S T} (g : f.right ⟶ f'.right) (w : f.hom ≫ T.map g = f'.hom) :
f ⟶ f' | { left := eq_to_hom (by ext),
right := g,
w' := by { dsimp, simpa using w.symm, }, } | def | category_theory.structured_arrow.hom_mk | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | To construct a morphism of structured arrows,
we need a morphism of the objects underlying the target,
and to check that the triangle commutes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_mk' {F : C ⥤ D} {X : D} {Y : C}
(U : structured_arrow X F) (f : U.right ⟶ Y) :
U ⟶ mk (U.hom ≫ F.map f) | { left := eq_to_hom (by ext), right := f } | def | category_theory.structured_arrow.hom_mk' | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | Given a structured arrow `X ⟶ F(U)`, and an arrow `U ⟶ Y`, we can construct a morphism of
structured arrow given by `(X ⟶ F(U)) ⟶ (X ⟶ F(U) ⟶ F(Y))`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_mk {f f' : structured_arrow S T} (g : f.right ≅ f'.right)
(w : f.hom ≫ T.map g.hom = f'.hom) : f ≅ f' | comma.iso_mk (eq_to_iso (by ext)) g (by simpa [eq_to_hom_map] using w.symm) | def | category_theory.structured_arrow.iso_mk | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | To construct an isomorphism of structured arrows,
we need an isomorphism of the objects underlying the target,
and to check that the triangle commutes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {A B : structured_arrow S T} (f g : A ⟶ B) : f.right = g.right → f = g | comma_morphism.ext _ _ (subsingleton.elim _ _) | lemma | category_theory.structured_arrow.ext | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {A B : structured_arrow S T} (f g : A ⟶ B) : f = g ↔ f.right = g.right | ⟨λ h, h ▸ rfl, ext f g⟩ | lemma | category_theory.structured_arrow.ext_iff | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj_faithful : faithful (proj S T) | { map_injective' := λ X Y, ext } | instance | category_theory.structured_arrow.proj_faithful | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_of_mono_right {A B : structured_arrow S T} (f : A ⟶ B) [h : mono f.right] : mono f | (proj S T).mono_of_mono_map h | lemma | category_theory.structured_arrow.mono_of_mono_right | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | The converse of this is true with additional assumptions, see `mono_iff_mono_right`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
epi_of_epi_right {A B : structured_arrow S T} (f : A ⟶ B) [h : epi f.right] : epi f | (proj S T).epi_of_epi_map h | lemma | category_theory.structured_arrow.epi_of_epi_right | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_hom_mk {A B : structured_arrow S T} (f : A.right ⟶ B.right) (w) [h : mono f] :
mono (hom_mk f w) | (proj S T).mono_of_mono_map h | instance | category_theory.structured_arrow.mono_hom_mk | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_hom_mk {A B : structured_arrow S T} (f : A.right ⟶ B.right) (w) [h : epi f] :
epi (hom_mk f w) | (proj S T).epi_of_epi_map h | instance | category_theory.structured_arrow.epi_hom_mk | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_mk (f : structured_arrow S T) : f = mk f.hom | by { cases f, congr, ext, } | lemma | category_theory.structured_arrow.eq_mk | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | Eta rule for structured arrows. Prefer `structured_arrow.eta`, since equality of objects tends
to cause problems. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eta (f : structured_arrow S T) : f ≅ mk f.hom | iso_mk (iso.refl _) (by tidy) | def | category_theory.structured_arrow.eta | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | Eta rule for structured arrows. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map (f : S ⟶ S') : structured_arrow S' T ⥤ structured_arrow S T | comma.map_left _ ((functor.const _).map f) | def | category_theory.structured_arrow.map | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [
"functor.const"
] | A morphism between source objects `S ⟶ S'`
contravariantly induces a functor between structured arrows,
`structured_arrow S' T ⥤ structured_arrow S T`.
Ideally this would be described as a 2-functor from `D`
(promoted to a 2-category with equations as 2-morphisms)
to `Cat`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_mk {f : S' ⟶ T.obj Y} (g : S ⟶ S') :
(map g).obj (mk f) = mk (g ≫ f) | rfl | lemma | category_theory.structured_arrow.map_mk | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id {f : structured_arrow S T} : (map (𝟙 S)).obj f = f | by { rw eq_mk f, simp, } | lemma | category_theory.structured_arrow.map_id | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [
"map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp {f : S ⟶ S'} {f' : S' ⟶ S''} {h : structured_arrow S'' T} :
(map (f ≫ f')).obj h = (map f).obj ((map f').obj h) | by { rw eq_mk h, simp, } | lemma | category_theory.structured_arrow.map_comp | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [
"map_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj_reflects_iso : reflects_isomorphisms (proj S T) | { reflects := λ Y Z f t, by exactI
⟨⟨structured_arrow.hom_mk (inv ((proj S T).map f)) (by simp), by tidy⟩⟩ } | instance | category_theory.structured_arrow.proj_reflects_iso | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_id_initial [full T] [faithful T] : is_initial (mk (𝟙 (T.obj Y))) | { desc := λ c, hom_mk (T.preimage c.X.hom) (by { dsimp, simp, }),
uniq' := λ c m _, begin
ext,
apply T.map_injective,
simpa only [hom_mk_right, T.image_preimage, ←w m] using (category.id_comp _).symm,
end } | def | category_theory.structured_arrow.mk_id_initial | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | The identity structured arrow is initial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pre (S : D) (F : B ⥤ C) (G : C ⥤ D) : structured_arrow S (F ⋙ G) ⥤ structured_arrow S G | comma.pre_right _ F G | def | category_theory.structured_arrow.pre | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | The functor `(S, F ⋙ G) ⥤ (S, G)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
post (S : C) (F : B ⥤ C) (G : C ⥤ D) :
structured_arrow S F ⥤ structured_arrow (G.obj S) (F ⋙ G) | { obj := λ X, structured_arrow.mk (G.map X.hom),
map := λ X Y f, structured_arrow.hom_mk f.right
(by simp [functor.comp_map, ←G.map_comp, ← f.w]) } | def | category_theory.structured_arrow.post | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | The functor `(S, F) ⥤ (G(S), F ⋙ G)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
small_proj_preimage_of_locally_small {𝒢 : set C} [small.{v₁} 𝒢] [locally_small.{v₁} D] :
small.{v₁} ((proj S T).obj ⁻¹' 𝒢) | begin
suffices : (proj S T).obj ⁻¹' 𝒢 = set.range (λ f : Σ G : 𝒢, S ⟶ T.obj G, mk f.2),
{ rw this, apply_instance },
exact set.ext (λ X, ⟨λ h, ⟨⟨⟨_, h⟩, X.hom⟩, (eq_mk _).symm⟩, by tidy⟩)
end | instance | category_theory.structured_arrow.small_proj_preimage_of_locally_small | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [
"set.ext",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
costructured_arrow (S : C ⥤ D) (T : D) | comma S (functor.from_punit T) | def | category_theory.costructured_arrow | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | The category of `S`-costructured arrows with target `T : D` (here `S : C ⥤ D`),
has as its objects `D`-morphisms of the form `S Y ⟶ T`, for some `Y : C`,
and morphisms `C`-morphisms `Y ⟶ Y'` making the obvious triangle commute. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
proj (S : C ⥤ D) (T : D) : costructured_arrow S T ⥤ C | comma.fst _ _ | def | category_theory.costructured_arrow.proj | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | The obvious projection functor from costructured arrows. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk (f : S.obj Y ⟶ T) : costructured_arrow S T | ⟨Y, ⟨⟨⟩⟩, f⟩ | def | category_theory.costructured_arrow.mk | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | Construct a costructured arrow from a morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_left (f : S.obj Y ⟶ T) : (mk f).left = Y | rfl | lemma | category_theory.costructured_arrow.mk_left | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_right (f : S.obj Y ⟶ T) : (mk f).right = ⟨⟨⟩⟩ | rfl | lemma | category_theory.costructured_arrow.mk_right | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_hom_eq_self (f : S.obj Y ⟶ T) : (mk f).hom = f | rfl | lemma | category_theory.costructured_arrow.mk_hom_eq_self | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w {A B : costructured_arrow S T} (f : A ⟶ B) :
S.map f.left ≫ B.hom = A.hom | by tidy | lemma | category_theory.costructured_arrow.w | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_mk {f f' : costructured_arrow S T} (g : f.left ⟶ f'.left) (w : S.map g ≫ f'.hom = f.hom) :
f ⟶ f' | { left := g,
right := eq_to_hom (by ext),
w' := by simpa [eq_to_hom_map] using w, } | def | category_theory.costructured_arrow.hom_mk | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | To construct a morphism of costructured arrows,
we need a morphism of the objects underlying the source,
and to check that the triangle commutes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_mk {f f' : costructured_arrow S T} (g : f.left ≅ f'.left)
(w : S.map g.hom ≫ f'.hom = f.hom) : f ≅ f' | comma.iso_mk g (eq_to_iso (by ext)) (by simpa [eq_to_hom_map] using w) | def | category_theory.costructured_arrow.iso_mk | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | To construct an isomorphism of costructured arrows,
we need an isomorphism of the objects underlying the source,
and to check that the triangle commutes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {A B : costructured_arrow S T} (f g : A ⟶ B) (h : f.left = g.left) : f = g | comma_morphism.ext _ _ h (subsingleton.elim _ _) | lemma | category_theory.costructured_arrow.ext | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {A B : costructured_arrow S T} (f g : A ⟶ B) : f = g ↔ f.left = g.left | ⟨λ h, h ▸ rfl, ext f g⟩ | lemma | category_theory.costructured_arrow.ext_iff | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_of_mono_left {A B : costructured_arrow S T} (f : A ⟶ B) [h : mono f.left] : mono f | (proj S T).mono_of_mono_map h | lemma | category_theory.costructured_arrow.mono_of_mono_left | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_of_epi_left {A B : costructured_arrow S T} (f : A ⟶ B) [h : epi f.left] : epi f | (proj S T).epi_of_epi_map h | lemma | category_theory.costructured_arrow.epi_of_epi_left | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | The converse of this is true with additional assumptions, see `epi_iff_epi_left`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono_hom_mk {A B : costructured_arrow S T} (f : A.left ⟶ B.left) (w) [h : mono f] :
mono (hom_mk f w) | (proj S T).mono_of_mono_map h | instance | category_theory.costructured_arrow.mono_hom_mk | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_hom_mk {A B : costructured_arrow S T} (f : A.left ⟶ B.left) (w) [h : epi f] :
epi (hom_mk f w) | (proj S T).epi_of_epi_map h | instance | category_theory.costructured_arrow.epi_hom_mk | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_mk (f : costructured_arrow S T) : f = mk f.hom | by { cases f, congr, ext, } | lemma | category_theory.costructured_arrow.eq_mk | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | Eta rule for costructured arrows. Prefer `costructured_arrow.eta`, as equality of objects tends
to cause problems. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eta (f : costructured_arrow S T) : f ≅ mk f.hom | iso_mk (iso.refl _) (by tidy) | def | category_theory.costructured_arrow.eta | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | Eta rule for costructured arrows. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map (f : T ⟶ T') : costructured_arrow S T ⥤ costructured_arrow S T' | comma.map_right _ ((functor.const _).map f) | def | category_theory.costructured_arrow.map | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [
"functor.const"
] | A morphism between target objects `T ⟶ T'`
covariantly induces a functor between costructured arrows,
`costructured_arrow S T ⥤ costructured_arrow S T'`.
Ideally this would be described as a 2-functor from `D`
(promoted to a 2-category with equations as 2-morphisms)
to `Cat`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_mk {f : S.obj Y ⟶ T} (g : T ⟶ T') :
(map g).obj (mk f) = mk (f ≫ g) | rfl | lemma | category_theory.costructured_arrow.map_mk | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id {f : costructured_arrow S T} : (map (𝟙 T)).obj f = f | by { rw eq_mk f, simp, } | lemma | category_theory.costructured_arrow.map_id | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [
"map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp {f : T ⟶ T'} {f' : T' ⟶ T''} {h : costructured_arrow S T} :
(map (f ≫ f')).obj h = (map f').obj ((map f).obj h) | by { rw eq_mk h, simp, } | lemma | category_theory.costructured_arrow.map_comp | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [
"map_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj_reflects_iso : reflects_isomorphisms (proj S T) | { reflects := λ Y Z f t, by exactI
⟨⟨costructured_arrow.hom_mk (inv ((proj S T).map f)) (by simp), by tidy⟩⟩ } | instance | category_theory.costructured_arrow.proj_reflects_iso | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_id_terminal [full S] [faithful S] : is_terminal (mk (𝟙 (S.obj Y))) | { lift := λ c, hom_mk (S.preimage c.X.hom) (by { dsimp, simp, }),
uniq' := begin
rintros c m -,
ext,
apply S.map_injective,
simpa only [hom_mk_left, S.image_preimage, ←w m] using (category.comp_id _).symm,
end } | def | category_theory.costructured_arrow.mk_id_terminal | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [
"lift"
] | The identity costructured arrow is terminal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pre (F : B ⥤ C) (G : C ⥤ D) (S : D) : costructured_arrow (F ⋙ G) S ⥤ costructured_arrow G S | comma.pre_left F G _ | def | category_theory.costructured_arrow.pre | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | The functor `(F ⋙ G, S) ⥤ (G, S)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
post (F : B ⥤ C) (G : C ⥤ D) (S : C) :
costructured_arrow F S ⥤ costructured_arrow (F ⋙ G) (G.obj S) | { obj := λ X, costructured_arrow.mk (G.map X.hom),
map := λ X Y f, costructured_arrow.hom_mk f.left
(by simp [functor.comp_map, ←G.map_comp, ← f.w]), } | def | category_theory.costructured_arrow.post | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | The functor `(F, S) ⥤ (F ⋙ G, G(S))`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
small_proj_preimage_of_locally_small {𝒢 : set C} [small.{v₁} 𝒢] [locally_small.{v₁} D] :
small.{v₁} ((proj S T).obj ⁻¹' 𝒢) | begin
suffices : (proj S T).obj ⁻¹' 𝒢 = set.range (λ f : Σ G : 𝒢, S.obj G ⟶ T, mk f.2),
{ rw this, apply_instance },
exact set.ext (λ X, ⟨λ h, ⟨⟨⟨_, h⟩, X.hom⟩, (eq_mk _).symm⟩, by tidy⟩)
end | instance | category_theory.costructured_arrow.small_proj_preimage_of_locally_small | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [
"set.ext",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_costructured_arrow (F : C ⥤ D) (d : D) :
(structured_arrow d F)ᵒᵖ ⥤ costructured_arrow F.op (op d) | { obj := λ X, @costructured_arrow.mk _ _ _ _ _ (op X.unop.right) F.op X.unop.hom.op,
map := λ X Y f, costructured_arrow.hom_mk (f.unop.right.op)
begin
dsimp,
rw [← op_comp, ← f.unop.w, functor.const_obj_map],
erw category.id_comp,
end } | def | category_theory.structured_arrow.to_costructured_arrow | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | For a functor `F : C ⥤ D` and an object `d : D`, we obtain a contravariant functor from the
category of structured arrows `d ⟶ F.obj c` to the category of costructured arrows
`F.op.obj c ⟶ (op d)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_costructured_arrow' (F : C ⥤ D) (d : D) :
(structured_arrow (op d) F.op)ᵒᵖ ⥤ costructured_arrow F d | { obj := λ X, @costructured_arrow.mk _ _ _ _ _ (unop X.unop.right) F X.unop.hom.unop,
map := λ X Y f, costructured_arrow.hom_mk f.unop.right.unop
begin
dsimp,
rw [← quiver.hom.unop_op (F.map (quiver.hom.unop f.unop.right)), ← unop_comp, ← F.op_map,
← f.unop.w, functor.const_obj_map],
erw category.... | def | category_theory.structured_arrow.to_costructured_arrow' | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [
"quiver.hom.unop",
"quiver.hom.unop_op"
] | For a functor `F : C ⥤ D` and an object `d : D`, we obtain a contravariant functor from the
category of structured arrows `op d ⟶ F.op.obj c` to the category of costructured arrows
`F.obj c ⟶ d`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_structured_arrow (F : C ⥤ D) (d : D) :
(costructured_arrow F d)ᵒᵖ ⥤ structured_arrow (op d) F.op | { obj := λ X, @structured_arrow.mk _ _ _ _ _ (op X.unop.left) F.op X.unop.hom.op,
map := λ X Y f, structured_arrow.hom_mk f.unop.left.op
begin
dsimp,
rw [← op_comp, f.unop.w, functor.const_obj_map],
erw category.comp_id,
end } | def | category_theory.costructured_arrow.to_structured_arrow | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [] | For a functor `F : C ⥤ D` and an object `d : D`, we obtain a contravariant functor from the
category of costructured arrows `F.obj c ⟶ d` to the category of structured arrows
`op d ⟶ F.op.obj c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_structured_arrow' (F : C ⥤ D) (d : D) :
(costructured_arrow F.op (op d))ᵒᵖ ⥤ structured_arrow d F | { obj := λ X, @structured_arrow.mk _ _ _ _ _ (unop X.unop.left) F X.unop.hom.unop,
map := λ X Y f, structured_arrow.hom_mk (f.unop.left.unop)
begin
dsimp,
rw [← quiver.hom.unop_op (F.map f.unop.left.unop), ← unop_comp, ← F.op_map,
f.unop.w, functor.const_obj_map],
erw category.comp_id,
end } | def | category_theory.costructured_arrow.to_structured_arrow' | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [
"quiver.hom.unop_op"
] | For a functor `F : C ⥤ D` and an object `d : D`, we obtain a contravariant functor from the
category of costructured arrows `F.op.obj c ⟶ op d` to the category of structured arrows
`d ⟶ F.obj c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
structured_arrow_op_equivalence (F : C ⥤ D) (d : D) :
(structured_arrow d F)ᵒᵖ ≌ costructured_arrow F.op (op d) | equivalence.mk (structured_arrow.to_costructured_arrow F d)
(costructured_arrow.to_structured_arrow' F d).right_op
(nat_iso.of_components (λ X, (@structured_arrow.iso_mk _ _ _ _ _ _
(structured_arrow.mk (unop X).hom) (unop X) (iso.refl _) (by tidy)).op)
(λ X Y f, quiver.hom.unop_inj $ begin ext, dsimp, simp... | def | category_theory.structured_arrow_op_equivalence | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [
"quiver.hom.unop_inj"
] | For a functor `F : C ⥤ D` and an object `d : D`, the category of structured arrows `d ⟶ F.obj c`
is contravariantly equivalent to the category of costructured arrows `F.op.obj c ⟶ op d`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
costructured_arrow_op_equivalence (F : C ⥤ D) (d : D) :
(costructured_arrow F d)ᵒᵖ ≌ structured_arrow (op d) F.op | equivalence.mk (costructured_arrow.to_structured_arrow F d)
(structured_arrow.to_costructured_arrow' F d).right_op
(nat_iso.of_components (λ X, (@costructured_arrow.iso_mk _ _ _ _ _ _
(costructured_arrow.mk (unop X).hom) (unop X) (iso.refl _) (by tidy)).op)
(λ X Y f, quiver.hom.unop_inj $ begin ext, dsimp, ... | def | category_theory.costructured_arrow_op_equivalence | category_theory | src/category_theory/structured_arrow.lean | [
"category_theory.punit",
"category_theory.comma",
"category_theory.limits.shapes.terminal",
"category_theory.essentially_small"
] | [
"quiver.hom.unop_inj"
] | For a functor `F : C ⥤ D` and an object `d : D`, the category of costructured arrows
`F.obj c ⟶ d` is contravariantly equivalent to the category of structured arrows
`op d ⟶ F.op.obj c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_subterminal (A : C) : Prop | ∀ ⦃Z : C⦄ (f g : Z ⟶ A), f = g | def | category_theory.is_subterminal | category_theory | src/category_theory/subterminal.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.subobject.mono_over"
] | [] | An object `A` is subterminal iff for any `Z`, there is at most one morphism `Z ⟶ A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_subterminal.def : is_subterminal A ↔ ∀ ⦃Z : C⦄ (f g : Z ⟶ A), f = g | iff.rfl | lemma | category_theory.is_subterminal.def | category_theory | src/category_theory/subterminal.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.subobject.mono_over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_subterminal.mono_is_terminal_from (hA : is_subterminal A) {T : C} (hT : is_terminal T) :
mono (hT.from A) | { right_cancellation := λ Z g h _, hA _ _ } | lemma | category_theory.is_subterminal.mono_is_terminal_from | category_theory | src/category_theory/subterminal.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.subobject.mono_over"
] | [] | If `A` is subterminal, the unique morphism from it to a terminal object is a monomorphism.
The converse of `is_subterminal_of_mono_is_terminal_from`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_subterminal.mono_terminal_from [has_terminal C] (hA : is_subterminal A) :
mono (terminal.from A) | hA.mono_is_terminal_from terminal_is_terminal | lemma | category_theory.is_subterminal.mono_terminal_from | category_theory | src/category_theory/subterminal.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.subobject.mono_over"
] | [] | If `A` is subterminal, the unique morphism from it to the terminal object is a monomorphism.
The converse of `is_subterminal_of_mono_terminal_from`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_subterminal_of_mono_is_terminal_from {T : C} (hT : is_terminal T) [mono (hT.from A)] :
is_subterminal A | λ Z f g, by { rw ← cancel_mono (hT.from A), apply hT.hom_ext } | lemma | category_theory.is_subterminal_of_mono_is_terminal_from | category_theory | src/category_theory/subterminal.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.subobject.mono_over"
] | [] | If the unique morphism from `A` to a terminal object is a monomorphism, `A` is subterminal.
The converse of `is_subterminal.mono_is_terminal_from`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_subterminal_of_mono_terminal_from [has_terminal C] [mono (terminal.from A)] :
is_subterminal A | λ Z f g, by { rw ← cancel_mono (terminal.from A), apply subsingleton.elim } | lemma | category_theory.is_subterminal_of_mono_terminal_from | category_theory | src/category_theory/subterminal.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.subobject.mono_over"
] | [] | If the unique morphism from `A` to the terminal object is a monomorphism, `A` is subterminal.
The converse of `is_subterminal.mono_terminal_from`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_subterminal_of_is_terminal {T : C} (hT : is_terminal T) : is_subterminal T | λ Z f g, hT.hom_ext _ _ | lemma | category_theory.is_subterminal_of_is_terminal | category_theory | src/category_theory/subterminal.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.subobject.mono_over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_subterminal_of_terminal [has_terminal C] : is_subterminal (⊤_ C) | λ Z f g, subsingleton.elim _ _ | lemma | category_theory.is_subterminal_of_terminal | category_theory | src/category_theory/subterminal.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.subobject.mono_over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_subterminal.is_iso_diag (hA : is_subterminal A) [has_binary_product A A] :
is_iso (diag A) | ⟨⟨limits.prod.fst, ⟨by simp, by { rw is_subterminal.def at hA, tidy }⟩⟩⟩ | lemma | category_theory.is_subterminal.is_iso_diag | category_theory | src/category_theory/subterminal.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.subobject.mono_over"
] | [] | If `A` is subterminal, its diagonal morphism is an isomorphism.
The converse of `is_subterminal_of_is_iso_diag`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_subterminal_of_is_iso_diag [has_binary_product A A] [is_iso (diag A)] :
is_subterminal A | λ Z f g,
begin
have : (limits.prod.fst : A ⨯ A ⟶ _) = limits.prod.snd,
{ simp [←cancel_epi (diag A)] },
rw [←prod.lift_fst f g, this, prod.lift_snd],
end | lemma | category_theory.is_subterminal_of_is_iso_diag | category_theory | src/category_theory/subterminal.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.subobject.mono_over"
] | [] | If the diagonal morphism of `A` is an isomorphism, then it is subterminal.
The converse of `is_subterminal.is_iso_diag`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_subterminal.iso_diag (hA : is_subterminal A) [has_binary_product A A] :
A ⨯ A ≅ A | begin
letI := is_subterminal.is_iso_diag hA,
apply (as_iso (diag A)).symm,
end | def | category_theory.is_subterminal.iso_diag | category_theory | src/category_theory/subterminal.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.subobject.mono_over"
] | [] | If `A` is subterminal, it is isomorphic to `A ⨯ A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subterminals (C : Type u₁) [category.{v₁} C] | full_subcategory (λ (A : C), is_subterminal A) | def | category_theory.subterminals | category_theory | src/category_theory/subterminal.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.subobject.mono_over"
] | [] | The (full sub)category of subterminal objects.
TODO: If `C` is the category of sheaves on a topological space `X`, this category is equivalent
to the lattice of open subsets of `X`. More generally, if `C` is a topos, this is the lattice of
"external truth values". | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subterminal_inclusion : subterminals C ⥤ C | full_subcategory_inclusion _ | def | category_theory.subterminal_inclusion | category_theory | src/category_theory/subterminal.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.subobject.mono_over"
] | [] | The inclusion of the subterminal objects into the original category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subterminals_thin (X Y : subterminals C) : subsingleton (X ⟶ Y) | ⟨λ f g, Y.2 f g⟩ | instance | category_theory.subterminals_thin | category_theory | src/category_theory/subterminal.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.subobject.mono_over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subterminals_equiv_mono_over_terminal [has_terminal C] :
subterminals C ≌ mono_over (⊤_ C) | { functor :=
{ obj := λ X, ⟨over.mk (terminal.from X.1), X.2.mono_terminal_from⟩,
map := λ X Y f, mono_over.hom_mk f (by ext1 ⟨⟨⟩⟩) },
inverse :=
{ obj := λ X, ⟨X.obj.left, λ Z f g, by { rw ← cancel_mono X.arrow, apply subsingleton.elim }⟩,
map := λ X Y f, f.1 },
unit_iso :=
{ hom := { app := λ X, 𝟙 ... | def | category_theory.subterminals_equiv_mono_over_terminal | category_theory | src/category_theory/subterminal.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.subobject.mono_over"
] | [] | The category of subterminal objects is equivalent to the category of monomorphisms to the terminal
object (which is in turn equivalent to the subobjects of the terminal object). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subterminals_to_mono_over_terminal_comp_forget [has_terminal C] :
(subterminals_equiv_mono_over_terminal C).functor ⋙ mono_over.forget _ ⋙ over.forget _ =
subterminal_inclusion C | rfl | lemma | category_theory.subterminals_to_mono_over_terminal_comp_forget | category_theory | src/category_theory/subterminal.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.subobject.mono_over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_over_terminal_to_subterminals_comp [has_terminal C] :
(subterminals_equiv_mono_over_terminal C).inverse ⋙ subterminal_inclusion C =
mono_over.forget _ ⋙ over.forget _ | rfl | lemma | category_theory.mono_over_terminal_to_subterminals_comp | category_theory | src/category_theory/subterminal.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.subobject.mono_over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thin_category : category C | {}. | def | category_theory.thin_category | category_theory | src/category_theory/thin.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | Construct a category instance from a category_struct, using the fact that
hom spaces are subsingletons to prove the axioms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor_thin : quiver.is_thin (D ⥤ C) | λ _ _, ⟨λ α β, nat_trans.ext α β (funext (λ _, subsingleton.elim _ _))⟩ | instance | category_theory.functor_thin | category_theory | src/category_theory/thin.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [
"quiver.is_thin"
] | If `C` is a thin category, then `D ⥤ C` is a thin category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_of_both_ways {X Y : C} (f : X ⟶ Y) (g : Y ⟶ X) : X ≅ Y | { hom := f, inv := g } | def | category_theory.iso_of_both_ways | category_theory | src/category_theory/thin.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | To show `X ≅ Y` in a thin category, it suffices to just give any morphism in each direction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subsingleton_iso {X Y : C} : subsingleton (X ≅ Y) | ⟨by { intros i₁ i₂, ext1, apply subsingleton.elim }⟩ | instance | category_theory.subsingleton_iso | category_theory | src/category_theory/thin.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
types : large_category (Type u) | { hom := λ a b, (a → b),
id := λ a, id,
comp := λ _ _ _ f g, g ∘ f } | instance | category_theory.types | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.