statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
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map_augmented_cech_conerve {f g : arrow C}
[∀ n : ℕ, has_wide_pushout f.left (λ i : fin (n+1), f.right) (λ i, f.hom)]
[∀ n : ℕ, has_wide_pushout g.left (λ i : fin (n+1), g.right) (λ i, g.hom)]
(F : f ⟶ g) : f.augmented_cech_conerve ⟶ g.augmented_cech_conerve | { left := F.left,
right := map_cech_conerve F,
w' := by { ext, simp } } | def | category_theory.arrow.map_augmented_cech_conerve | algebraic_topology | src/algebraic_topology/cech_nerve.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products",
"category_theory.arrow"
] | [] | The morphism between augmented Čech conerves associated to a morphism of arrows. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cech_conerve : arrow C ⥤ cosimplicial_object C | { obj := λ f, f.cech_conerve,
map := λ f g F, arrow.map_cech_conerve F,
map_id' := λ i, by { ext, { dsimp, simp }, { dsimp, simp } },
map_comp' := λ f g h F G, by { ext, { simp }, { simp } } } | def | category_theory.cosimplicial_object.cech_conerve | algebraic_topology | src/algebraic_topology/cech_nerve.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products",
"category_theory.arrow"
] | [] | The Čech conerve construction, as a functor from `arrow C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
augmented_cech_conerve : arrow C ⥤ cosimplicial_object.augmented C | { obj := λ f, f.augmented_cech_conerve,
map := λ f g F, arrow.map_augmented_cech_conerve F,
map_id' := λ f, by { ext, { refl }, { dsimp, simp }, { dsimp, simp } },
map_comp' := λ f g h F G, by { ext, { refl }, { simp }, { simp } } } | def | category_theory.cosimplicial_object.augmented_cech_conerve | algebraic_topology | src/algebraic_topology/cech_nerve.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products",
"category_theory.arrow"
] | [] | The augmented Čech conerve construction, as a functor from `arrow C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence_left_to_right (F : arrow C) (X : cosimplicial_object.augmented C)
(G : F.augmented_cech_conerve ⟶ X) : F ⟶ augmented.to_arrow.obj X | { left := G.left,
right :=
(wide_pushout.ι (λ i, F.hom) 0 ≫ G.right.app (simplex_category.mk 0) : _),
w' := begin
have := G.w,
apply_fun (λ e, e.app (simplex_category.mk 0)) at this,
simpa only [category_theory.functor.id_map, augmented.to_arrow_obj_hom,
wide_pushout.arrow_ι_assoc (λ i, F.hom)... | def | category_theory.cosimplicial_object.equivalence_left_to_right | algebraic_topology | src/algebraic_topology/cech_nerve.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products",
"category_theory.arrow"
] | [
"category_theory.functor.id_map",
"simplex_category.mk"
] | A helper function used in defining the Čech conerve adjunction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence_right_to_left (F : arrow C) (X : cosimplicial_object.augmented C)
(G : F ⟶ augmented.to_arrow.obj X) : F.augmented_cech_conerve ⟶ X | { left := G.left,
right := { app := λ x, limits.wide_pushout.desc (G.left ≫ X.hom.app _)
(λ i, G.right ≫ X.right.map (simplex_category.const x i))
begin
rintros j,
rw ← arrow.w_assoc G,
have t := X.hom.naturality (x.const j),
dsimp at t ⊢,
simp only [category.id_com... | def | category_theory.cosimplicial_object.equivalence_right_to_left | algebraic_topology | src/algebraic_topology/cech_nerve.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products",
"category_theory.arrow"
] | [
"simplex_category.const"
] | A helper function used in defining the Čech conerve adjunction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cech_conerve_equiv (F : arrow C) (X : cosimplicial_object.augmented C) :
(F.augmented_cech_conerve ⟶ X) ≃ (F ⟶ augmented.to_arrow.obj X) | { to_fun := equivalence_left_to_right _ _,
inv_fun := equivalence_right_to_left _ _,
left_inv := begin
intro A,
dsimp,
ext _, { refl }, ext _ ⟨⟩, -- A bug in the `ext` tactic?
{ dsimp,
simp only [arrow.cech_conerve_map, wide_pushout.ι_desc, category.assoc,
← nat_trans.naturality, wide... | def | category_theory.cosimplicial_object.cech_conerve_equiv | algebraic_topology | src/algebraic_topology/cech_nerve.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products",
"category_theory.arrow"
] | [
"inv_fun"
] | A helper function used in defining the Čech conerve adjunction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cech_conerve_adjunction :
augmented_cech_conerve ⊣ (augmented.to_arrow : _ ⥤ arrow C) | adjunction.mk_of_hom_equiv
{ hom_equiv := cech_conerve_equiv,
hom_equiv_naturality_left_symm' := λ x y f g h, by { ext, { refl }, { simp }, { simp } },
hom_equiv_naturality_right' := λ x y f g h, by { ext, { simp }, { simp } } } | abbreviation | category_theory.cosimplicial_object.cech_conerve_adjunction | algebraic_topology | src/algebraic_topology/cech_nerve.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products",
"category_theory.arrow"
] | [] | The augmented Čech conerve construction is left adjoint to the `to_arrow` functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cech_nerve_terminal_from {C : Type u} [category.{v} C] [has_finite_products C] (X : C) :
simplicial_object C | { obj := λ n, ∏ (λ i : fin (n.unop.len + 1), X),
map := λ m n f, limits.pi.lift (λ i, limits.pi.π _ (f.unop.to_order_hom i)),
map_id' := λ f, limit.hom_ext $ λ j, by discrete_cases;
simpa only [limit.lift_π, category.id_comp],
map_comp' := λ m n o f g, limit.hom_ext $ λ j, by discrete_cases;
simpa only [c... | def | category_theory.cech_nerve_terminal_from | algebraic_topology | src/algebraic_topology/cech_nerve.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products",
"category_theory.arrow"
] | [] | Given an object `X : C`, the natural simplicial object sending `[n]` to `Xⁿ⁺¹`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_cospan (X : C) : wide_pullback_shape ι ⥤ C | wide_pullback_shape.wide_cospan (terminal C) (λ i : ι, X) (λ i, terminal.from X) | def | category_theory.cech_nerve_terminal_from.wide_cospan | algebraic_topology | src/algebraic_topology/cech_nerve.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products",
"category_theory.arrow"
] | [] | The diagram `option ι ⥤ C` sending `none` to the terminal object and `some j` to `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_to_wide_cospan_none (X Y : C) : unique (Y ⟶ (wide_cospan ι X).obj none) | by unfold wide_cospan; dsimp; apply_instance | instance | category_theory.cech_nerve_terminal_from.unique_to_wide_cospan_none | algebraic_topology | src/algebraic_topology/cech_nerve.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products",
"category_theory.arrow"
] | [
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wide_cospan.limit_cone [fintype ι] (X : C) : limit_cone (wide_cospan ι X) | { cone :=
{ X := ∏ (λ i : ι, X),
π :=
{ app := λ X, option.cases_on X (terminal.from _) (λ i, limit.π _ ⟨i⟩),
naturality' := λ i j f,
begin
cases f,
{ cases i,
all_goals { dsimp, simp }},
{ dsimp,
simp only [terminal.comp_from],
exact subsingleton.elim _ _... | def | category_theory.cech_nerve_terminal_from.wide_cospan.limit_cone | algebraic_topology | src/algebraic_topology/cech_nerve.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products",
"category_theory.arrow"
] | [
"fintype",
"lift"
] | The product `Xᶥ` is the vertex of a limit cone on `wide_cospan ι X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_wide_pullback [finite ι] (X : C) :
has_wide_pullback (arrow.mk (terminal.from X)).right
(λ i : ι, (arrow.mk (terminal.from X)).left) (λ i, (arrow.mk (terminal.from X)).hom) | begin
casesI nonempty_fintype ι,
exact ⟨⟨wide_cospan.limit_cone ι X⟩⟩,
end | instance | category_theory.cech_nerve_terminal_from.has_wide_pullback | algebraic_topology | src/algebraic_topology/cech_nerve.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products",
"category_theory.arrow"
] | [
"finite",
"nonempty_fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso (X : C) :
(arrow.mk (terminal.from X)).cech_nerve ≅ cech_nerve_terminal_from X | iso.symm (nat_iso.of_components (λ m, ((limit.is_limit _).cone_point_unique_up_to_iso
(wide_cospan.limit_cone (fin (m.unop.len + 1)) X).2).symm) $ λ m n f, wide_pullback.hom_ext _ _ _
(begin
intro j,
simp only [category.assoc],
dunfold cech_nerve_terminal_from wide_pullback.π pi.lift,
erw [wide_pullback.lift_... | def | category_theory.cech_nerve_terminal_from.iso | algebraic_topology | src/algebraic_topology/cech_nerve.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products",
"category_theory.arrow"
] | [
"iso"
] | Given an object `X : C`, the Čech nerve of the hom to the terminal object `X ⟶ ⊤_ C` is
naturally isomorphic to a simplicial object sending `[n]` to `Xⁿ⁺¹` (when `C` is `G-Set`, this is
`EG`, the universal cover of the classifying space of `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extra_degeneracy (X : simplicial_object.augmented C) | (s' : point.obj X ⟶ (drop.obj X) _[0])
(s : Π (n : ℕ), (drop.obj X) _[n] ⟶ (drop.obj X) _[n+1])
(s'_comp_ε' : s' ≫ X.hom.app (op [0]) = 𝟙 _)
(s₀_comp_δ₁' : s 0 ≫ (drop.obj X).δ 1 = X.hom.app (op [0]) ≫ s')
(s_comp_δ₀' : Π (n : ℕ), s n ≫ (drop.obj X).δ 0 = 𝟙 _)
(s_comp_δ' : Π (n : ℕ) (i : fin (n+2)), s (n+1) ≫ (drop.o... | structure | simplicial_object.augmented.extra_degeneracy | algebraic_topology | src/algebraic_topology/extra_degeneracy.lean | [
"algebraic_topology.alternating_face_map_complex",
"algebraic_topology.simplicial_set",
"algebraic_topology.cech_nerve",
"algebra.homology.homotopy",
"tactic.fin_cases"
] | [] | The datum of an extra degeneracy is a technical condition on
augmented simplicial objects. The morphisms `s'` and `s n` of the
structure formally behave like extra degeneracies `σ (-1)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map {D : Type*} [category D]
{X : simplicial_object.augmented C} (ed : extra_degeneracy X) (F : C ⥤ D) :
extra_degeneracy (((whiskering _ _).obj F).obj X) | { s' := F.map ed.s',
s := λ n, F.map (ed.s n),
s'_comp_ε' := by { dsimp, erw [comp_id, ← F.map_comp, ed.s'_comp_ε, F.map_id], },
s₀_comp_δ₁' := by { dsimp, erw [comp_id, ← F.map_comp, ← F.map_comp, ed.s₀_comp_δ₁], },
s_comp_δ₀' := λ n, by { dsimp, erw [← F.map_comp, ed.s_comp_δ₀, F.map_id], },
s_comp_δ' := λ ... | def | simplicial_object.augmented.extra_degeneracy.map | algebraic_topology | src/algebraic_topology/extra_degeneracy.lean | [
"algebraic_topology.alternating_face_map_complex",
"algebraic_topology.simplicial_set",
"algebraic_topology.cech_nerve",
"algebra.homology.homotopy",
"tactic.fin_cases"
] | [] | If `ed` is an extra degeneracy for `X : simplicial_object.augmented C` and
`F : C ⥤ D` is a functor, then `ed.map F` is an extra degeneracy for the
augmented simplical object in `D` obtained by applying `F` to `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_iso {X Y : simplicial_object.augmented C} (e : X ≅ Y) (ed : extra_degeneracy X) :
extra_degeneracy Y | { s' := (point.map_iso e).inv ≫ ed.s' ≫ (drop.map_iso e).hom.app (op [0]),
s := λ n, (drop.map_iso e).inv.app (op [n]) ≫ ed.s n ≫ (drop.map_iso e).hom.app (op [n+1]),
s'_comp_ε' := by simpa only [functor.map_iso, assoc, w₀, ed.s'_comp_ε_assoc]
using (point.map_iso e).inv_hom_id,
s₀_comp_δ₁' := begin
have ... | def | simplicial_object.augmented.extra_degeneracy.of_iso | algebraic_topology | src/algebraic_topology/extra_degeneracy.lean | [
"algebraic_topology.alternating_face_map_complex",
"algebraic_topology.simplicial_set",
"algebraic_topology.cech_nerve",
"algebra.homology.homotopy",
"tactic.fin_cases"
] | [] | If `X` and `Y` are isomorphic augmented simplicial objects, then an extra
degeneracy for `X` gives also an extra degeneracy for `Y` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
shift_fun {n : ℕ} {X : Type*} [has_zero X] (f : fin n → X) (i : fin (n+1)) : X | dite (i = 0) (λ h, 0) (λ h, f (i.pred h)) | def | sSet.augmented.standard_simplex.shift_fun | algebraic_topology | src/algebraic_topology/extra_degeneracy.lean | [
"algebraic_topology.alternating_face_map_complex",
"algebraic_topology.simplicial_set",
"algebraic_topology.cech_nerve",
"algebra.homology.homotopy",
"tactic.fin_cases"
] | [] | When `[has_zero X]`, the shift of a map `f : fin n → X`
is a map `fin (n+1) → X` which sends `0` to `0` and `i.succ` to `f i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
shift_fun_0 {n : ℕ} {X : Type*} [has_zero X] (f : fin n → X) : shift_fun f 0 = 0 | rfl | lemma | sSet.augmented.standard_simplex.shift_fun_0 | algebraic_topology | src/algebraic_topology/extra_degeneracy.lean | [
"algebraic_topology.alternating_face_map_complex",
"algebraic_topology.simplicial_set",
"algebraic_topology.cech_nerve",
"algebra.homology.homotopy",
"tactic.fin_cases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
shift_fun_succ {n : ℕ} {X : Type*} [has_zero X] (f : fin n → X)
(i : fin n) : shift_fun f i.succ = f i | begin
dsimp [shift_fun],
split_ifs,
{ exfalso,
simpa only [fin.ext_iff, fin.coe_succ] using h, },
{ simp only [fin.pred_succ], },
end | lemma | sSet.augmented.standard_simplex.shift_fun_succ | algebraic_topology | src/algebraic_topology/extra_degeneracy.lean | [
"algebraic_topology.alternating_face_map_complex",
"algebraic_topology.simplicial_set",
"algebraic_topology.cech_nerve",
"algebra.homology.homotopy",
"tactic.fin_cases"
] | [
"fin.coe_succ",
"fin.ext_iff",
"fin.pred_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
shift {n : ℕ} {Δ : simplex_category} (f : [n] ⟶ Δ) : [n+1] ⟶ Δ | simplex_category.hom.mk
{ to_fun := shift_fun f.to_order_hom,
monotone' := λ i₁ i₂ hi, begin
by_cases h₁ : i₁ = 0,
{ subst h₁,
simp only [shift_fun_0, fin.zero_le], },
{ have h₂ : i₂ ≠ 0 := by { intro h₂, subst h₂, exact h₁ (le_antisymm hi (fin.zero_le _)), },
cases fin.eq_succ_of_ne_zero h₁ w... | def | sSet.augmented.standard_simplex.shift | algebraic_topology | src/algebraic_topology/extra_degeneracy.lean | [
"algebraic_topology.alternating_face_map_complex",
"algebraic_topology.simplicial_set",
"algebraic_topology.cech_nerve",
"algebra.homology.homotopy",
"tactic.fin_cases"
] | [
"fin.eq_succ_of_ne_zero",
"fin.zero_le",
"simplex_category",
"simplex_category.hom.mk"
] | The shift of a morphism `f : [n] → Δ` in `simplex_category` corresponds to
the monotone map which sends `0` to `0` and `i.succ` to `f.to_order_hom i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extra_degeneracy (Δ : simplex_category) :
simplicial_object.augmented.extra_degeneracy (standard_simplex.obj Δ) | { s' := λ x, simplex_category.hom.mk (order_hom.const _ 0),
s := λ n f, shift f,
s'_comp_ε' := by { ext1 j, fin_cases j, },
s₀_comp_δ₁' := by { ext x j, fin_cases j, refl, },
s_comp_δ₀' := λ n, begin
ext φ i : 4,
dsimp [simplicial_object.δ, simplex_category.δ, sSet.standard_simplex],
simp only [shif... | def | sSet.augmented.standard_simplex.extra_degeneracy | algebraic_topology | src/algebraic_topology/extra_degeneracy.lean | [
"algebraic_topology.alternating_face_map_complex",
"algebraic_topology.simplicial_set",
"algebraic_topology.cech_nerve",
"algebra.homology.homotopy",
"tactic.fin_cases"
] | [
"fin.eq_succ_of_ne_zero",
"fin.succ_pred_above_succ",
"fin.succ_succ_above_succ",
"fin.succ_succ_above_zero",
"order_hom.const",
"sSet.standard_simplex",
"simplex_category",
"simplex_category.hom.mk",
"simplex_category.δ",
"simplex_category.σ",
"simplicial_object.augmented.extra_degeneracy"
] | The obvious extra degeneracy on the standard simplex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonempty_extra_degeneracy_standard_simplex (Δ : simplex_category) :
nonempty (simplicial_object.augmented.extra_degeneracy (standard_simplex.obj Δ)) | ⟨standard_simplex.extra_degeneracy Δ⟩ | instance | sSet.augmented.standard_simplex.nonempty_extra_degeneracy_standard_simplex | algebraic_topology | src/algebraic_topology/extra_degeneracy.lean | [
"algebraic_topology.alternating_face_map_complex",
"algebraic_topology.simplicial_set",
"algebraic_topology.cech_nerve",
"algebra.homology.homotopy",
"tactic.fin_cases"
] | [
"simplex_category",
"simplicial_object.augmented.extra_degeneracy"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extra_degeneracy.s (n : ℕ) :
f.cech_nerve.obj (op [n]) ⟶ f.cech_nerve.obj (op [n + 1]) | wide_pullback.lift (wide_pullback.base _)
(λ i, dite (i = 0) (λ h, wide_pullback.base _ ≫ S.section_)
(λ h, wide_pullback.π _ (i.pred h)))
(λ i, begin
split_ifs,
{ subst h,
simp only [assoc, split_epi.id, comp_id], },
{ simp only [wide_pullback.π_arrow], },
end) | def | category_theory.arrow.augmented_cech_nerve.extra_degeneracy.s | algebraic_topology | src/algebraic_topology/extra_degeneracy.lean | [
"algebraic_topology.alternating_face_map_complex",
"algebraic_topology.simplicial_set",
"algebraic_topology.cech_nerve",
"algebra.homology.homotopy",
"tactic.fin_cases"
] | [] | The extra degeneracy map on the Čech nerve of a split epi. It is
given on the `0`-projection by the given section of the split epi,
and by shifting the indices on the other projections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extra_degeneracy.s_comp_π_0 (n : ℕ) :
extra_degeneracy.s f S n ≫ wide_pullback.π _ 0 = wide_pullback.base _ ≫ S.section_ | by { dsimp [extra_degeneracy.s], simpa only [wide_pullback.lift_π], } | lemma | category_theory.arrow.augmented_cech_nerve.extra_degeneracy.s_comp_π_0 | algebraic_topology | src/algebraic_topology/extra_degeneracy.lean | [
"algebraic_topology.alternating_face_map_complex",
"algebraic_topology.simplicial_set",
"algebraic_topology.cech_nerve",
"algebra.homology.homotopy",
"tactic.fin_cases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extra_degeneracy.s_comp_π_succ (n : ℕ) (i : fin (n+1)) :
extra_degeneracy.s f S n ≫ wide_pullback.π _ i.succ = wide_pullback.π _ i | begin
dsimp [extra_degeneracy.s],
simp only [wide_pullback.lift_π],
split_ifs,
{ exfalso,
simpa only [fin.ext_iff, fin.coe_succ, fin.coe_zero, nat.succ_ne_zero] using h, },
{ congr,
apply fin.pred_succ, },
end | lemma | category_theory.arrow.augmented_cech_nerve.extra_degeneracy.s_comp_π_succ | algebraic_topology | src/algebraic_topology/extra_degeneracy.lean | [
"algebraic_topology.alternating_face_map_complex",
"algebraic_topology.simplicial_set",
"algebraic_topology.cech_nerve",
"algebra.homology.homotopy",
"tactic.fin_cases"
] | [
"fin.coe_succ",
"fin.coe_zero",
"fin.ext_iff",
"fin.pred_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extra_degeneracy.s_comp_base (n : ℕ) :
extra_degeneracy.s f S n ≫ wide_pullback.base _ = wide_pullback.base _ | by apply wide_pullback.lift_base | lemma | category_theory.arrow.augmented_cech_nerve.extra_degeneracy.s_comp_base | algebraic_topology | src/algebraic_topology/extra_degeneracy.lean | [
"algebraic_topology.alternating_face_map_complex",
"algebraic_topology.simplicial_set",
"algebraic_topology.cech_nerve",
"algebra.homology.homotopy",
"tactic.fin_cases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extra_degeneracy :
simplicial_object.augmented.extra_degeneracy f.augmented_cech_nerve | { s' := S.section_ ≫ wide_pullback.lift f.hom (λ i, 𝟙 _) (λ i, by rw id_comp),
s := λ n, extra_degeneracy.s f S n,
s'_comp_ε' :=
by simp only [augmented_cech_nerve_hom_app, assoc, wide_pullback.lift_base, split_epi.id],
s₀_comp_δ₁' := begin
dsimp [cech_nerve, simplicial_object.δ, simplex_category.δ],
... | def | category_theory.arrow.augmented_cech_nerve.extra_degeneracy | algebraic_topology | src/algebraic_topology/extra_degeneracy.lean | [
"algebraic_topology.alternating_face_map_complex",
"algebraic_topology.simplicial_set",
"algebraic_topology.cech_nerve",
"algebra.homology.homotopy",
"tactic.fin_cases"
] | [
"fin.eq_succ_of_ne_zero",
"fin.succ_pred_above_succ",
"fin.succ_succ_above_succ",
"fin.succ_succ_above_zero",
"simplex_category.δ",
"simplex_category.σ",
"simplicial_object.augmented.extra_degeneracy"
] | The augmented Čech nerve associated to a split epimorphism has an extra degeneracy. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopy_equiv {C : Type*} [category C]
[preadditive C] [has_zero_object C] {X : simplicial_object.augmented C}
(ed : extra_degeneracy X) :
homotopy_equiv (algebraic_topology.alternating_face_map_complex.obj (drop.obj X))
((chain_complex.single₀ C).obj (point.obj X)) | { hom := alternating_face_map_complex.ε.app X,
inv := (chain_complex.from_single₀_equiv _ _).inv_fun ed.s',
homotopy_inv_hom_id := homotopy.of_eq (by { ext, exact ed.s'_comp_ε, }),
homotopy_hom_inv_id :=
{ hom := λ i j, begin
by_cases i+1 = j,
{ exact (-ed.s i) ≫ eq_to_hom (by congr'), },
{ ex... | def | simplicial_object.augmented.extra_degeneracy.homotopy_equiv | algebraic_topology | src/algebraic_topology/extra_degeneracy.lean | [
"algebraic_topology.alternating_face_map_complex",
"algebraic_topology.simplicial_set",
"algebraic_topology.cech_nerve",
"algebra.homology.homotopy",
"tactic.fin_cases"
] | [
"algebraic_topology.alternating_face_map_complex.obj",
"chain_complex.from_single₀_equiv",
"chain_complex.single₀",
"chain_complex.to_single₀_equiv",
"comm",
"fin.coe_one",
"fin.coe_succ",
"fin.coe_zero",
"homotopy.d_next_succ_chain_complex",
"homotopy.d_next_zero_chain_complex",
"homotopy.of_eq... | If `C` is a preadditive category and `X` is an augmented simplicial object
in `C` that has an extra degeneracy, then the augmentation on the alternating
face map complex of `X` is an homotopy equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
obj_X : Π n : ℕ, subobject (X.obj (op (simplex_category.mk n))) | | 0 := ⊤
| (n+1) := finset.univ.inf (λ k : fin (n+1), kernel_subobject (X.δ k.succ)) | def | algebraic_topology.normalized_Moore_complex.obj_X | algebraic_topology | src/algebraic_topology/Moore_complex.lean | [
"algebra.homology.homological_complex",
"algebraic_topology.simplicial_object",
"category_theory.abelian.basic"
] | [
"simplex_category.mk"
] | The normalized Moore complex in degree `n`, as a subobject of `X n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
obj_d : Π n : ℕ, (obj_X X (n+1) : C) ⟶ (obj_X X n : C) | | 0 := subobject.arrow _ ≫ X.δ (0 : fin 2) ≫ inv ((⊤ : subobject _).arrow)
| (n+1) :=
begin
-- The differential is `subobject.arrow _ ≫ X.δ (0 : fin (n+3))`,
-- factored through the intersection of the kernels.
refine factor_thru _ (arrow _ ≫ X.δ (0 : fin (n+3))) _,
-- We now need to show that it factors!
-- ... | def | algebraic_topology.normalized_Moore_complex.obj_d | algebraic_topology | src/algebraic_topology/Moore_complex.lean | [
"algebra.homology.homological_complex",
"algebraic_topology.simplicial_object",
"category_theory.abelian.basic"
] | [
"fin.zero_le",
"finset.univ"
] | The differentials in the normalized Moore complex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
d_squared (n : ℕ) : obj_d X (n+1) ≫ obj_d X n = 0 | begin
-- It's a pity we need to do a case split here;
-- after the first simp the proofs are almost identical
cases n; dsimp,
{ simp only [subobject.factor_thru_arrow_assoc],
slice_lhs 2 3 { erw ←X.δ_comp_δ (fin.zero_le (0 : fin (0 + 2))), },
rw ←factor_thru_arrow _ _ (finset_inf_arrow_factors finset.un... | lemma | algebraic_topology.normalized_Moore_complex.d_squared | algebraic_topology | src/algebraic_topology/Moore_complex.lean | [
"algebra.homology.homological_complex",
"algebraic_topology.simplicial_object",
"category_theory.abelian.basic"
] | [
"fin.zero_le",
"finset.univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
obj (X : simplicial_object C) : chain_complex C ℕ | chain_complex.of (λ n, (obj_X X n : C)) -- the coercion here picks a representative of the subobject
(obj_d X) (d_squared X) | def | algebraic_topology.normalized_Moore_complex.obj | algebraic_topology | src/algebraic_topology/Moore_complex.lean | [
"algebra.homology.homological_complex",
"algebraic_topology.simplicial_object",
"category_theory.abelian.basic"
] | [
"chain_complex",
"chain_complex.of"
] | The normalized Moore complex functor, on objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map (f : X ⟶ Y) : obj X ⟶ obj Y | chain_complex.of_hom _ _ _ _ _ _
(λ n, begin
refine factor_thru _ (arrow _ ≫ f.app (op (simplex_category.mk n))) _,
cases n; dsimp,
{ apply top_factors, },
{ refine (finset_inf_factors _).mpr (λ i m, _),
apply kernel_subobject_factors,
slice_lhs 2 3 { erw ←f.naturality, },
rw ←factor... | def | algebraic_topology.normalized_Moore_complex.map | algebraic_topology | src/algebraic_topology/Moore_complex.lean | [
"algebra.homology.homological_complex",
"algebraic_topology.simplicial_object",
"category_theory.abelian.basic"
] | [
"chain_complex.of_hom",
"finset.univ",
"simplex_category.mk"
] | The normalized Moore complex functor, on morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normalized_Moore_complex : simplicial_object C ⥤ chain_complex C ℕ | { obj := obj,
map := λ X Y f, map f,
map_id' := λ X, by { ext n, cases n; { dsimp, simp, }, },
map_comp' := λ X Y Z f g, by { ext n, cases n; simp, }, } | def | algebraic_topology.normalized_Moore_complex | algebraic_topology | src/algebraic_topology/Moore_complex.lean | [
"algebra.homology.homological_complex",
"algebraic_topology.simplicial_object",
"category_theory.abelian.basic"
] | [
"chain_complex"
] | The (normalized) Moore complex of a simplicial object `X` in an abelian category `C`.
The `n`-th object is intersection of
the kernels of `X.δ i : X.obj n ⟶ X.obj (n-1)`, for `i = 1, ..., n`.
The differentials are induced from `X.δ 0`,
which maps each of these intersections of kernels to the next. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normalized_Moore_complex_obj_d (X : simplicial_object C) (n : ℕ) :
((normalized_Moore_complex C).obj X).d (n+1) n = normalized_Moore_complex.obj_d X n | by apply chain_complex.of_d | lemma | algebraic_topology.normalized_Moore_complex_obj_d | algebraic_topology | src/algebraic_topology/Moore_complex.lean | [
"algebra.homology.homological_complex",
"algebraic_topology.simplicial_object",
"category_theory.abelian.basic"
] | [
"chain_complex.of_d"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nerve (C : Type u) [category.{v} C] : sSet.{max u v} | { obj := λ Δ, (simplex_category.to_Cat.obj Δ.unop) ⥤ C,
map := λ Δ₁ Δ₂ f x, simplex_category.to_Cat.map f.unop ⋙ x,
map_id' := λ Δ, begin
rw [unop_id, functor.map_id],
ext x,
apply functor.id_comp,
end, } | def | category_theory.nerve | algebraic_topology | src/algebraic_topology/nerve.lean | [
"algebraic_topology.simplicial_set"
] | [
"functor.map_id"
] | The nerve of a category | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nerve_functor : Cat ⥤ sSet | { obj := λ C, nerve C,
map := λ C C' F,
{ app := λ Δ x, x ⋙ F, },
map_id' := λ C, begin
ext Δ x,
apply functor.comp_id,
end, } | def | category_theory.nerve_functor | algebraic_topology | src/algebraic_topology/nerve.lean | [
"algebraic_topology.simplicial_set"
] | [
"sSet"
] | The nerve of a category, as a functor `Cat ⥤ sSet` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simplex_category | ℕ | def | simplex_category | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [] | The simplex category:
* objects are natural numbers `n : ℕ`
* morphisms from `n` to `m` are monotone functions `fin (n+1) → fin (m+1)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk (n : ℕ) : simplex_category | n | def | simplex_category.mk | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"simplex_category"
] | Interpet a natural number as an object of the simplex category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
len (n : simplex_category) : ℕ | n | def | simplex_category.len | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"simplex_category"
] | The length of an object of `simplex_category`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext (a b : simplex_category) : a.len = b.len → a = b | id | lemma | simplex_category.ext | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"simplex_category"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
len_mk (n : ℕ) : [n].len = n | rfl | lemma | simplex_category.len_mk | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_len (n : simplex_category) : [n.len] = n | rfl | lemma | simplex_category.mk_len | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"simplex_category"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rec {F : Π (Δ : simplex_category), Sort*} (h : ∀ (n : ℕ), F [n]) :
Π X, F X | λ n, h n.len | def | simplex_category.rec | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"simplex_category"
] | A recursor for `simplex_category`. Use it as `induction Δ using simplex_category.rec`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom (a b : simplex_category) | fin (a.len + 1) →o fin (b.len + 1) | def | simplex_category.hom | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"simplex_category"
] | Morphisms in the simplex_category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk {a b : simplex_category} (f : fin (a.len + 1) →o fin (b.len + 1)) :
simplex_category.hom a b | f | def | simplex_category.hom.mk | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"simplex_category",
"simplex_category.hom"
] | Make a moprhism in `simplex_category` from a monotone map of fin's. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_order_hom {a b : simplex_category} (f : simplex_category.hom a b) :
fin (a.len + 1) →o fin (b.len + 1) | f | def | simplex_category.hom.to_order_hom | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"simplex_category",
"simplex_category.hom"
] | Recover the monotone map from a morphism in the simplex category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {a b : simplex_category} (f g : simplex_category.hom a b) :
f.to_order_hom = g.to_order_hom → f = g | id | lemma | simplex_category.hom.ext | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"simplex_category",
"simplex_category.hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_to_order_hom {a b : simplex_category}
(f : simplex_category.hom a b) : mk (f.to_order_hom) = f | rfl | lemma | simplex_category.hom.mk_to_order_hom | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"simplex_category",
"simplex_category.hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_order_hom_mk {a b : simplex_category}
(f : fin (a.len + 1) →o fin (b.len + 1)) : (mk f).to_order_hom = f | rfl | lemma | simplex_category.hom.to_order_hom_mk | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"simplex_category"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_to_order_hom_apply {a b : simplex_category}
(f : fin (a.len + 1) →o fin (b.len + 1)) (i : fin (a.len + 1)) :
(mk f).to_order_hom i = f i | rfl | lemma | simplex_category.hom.mk_to_order_hom_apply | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"simplex_category"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id (a : simplex_category) :
simplex_category.hom a a | mk order_hom.id | def | simplex_category.hom.id | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"order_hom.id",
"simplex_category",
"simplex_category.hom"
] | Identity morphisms of `simplex_category`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp {a b c : simplex_category} (f : simplex_category.hom b c)
(g : simplex_category.hom a b) : simplex_category.hom a c | mk $ f.to_order_hom.comp g.to_order_hom | def | simplex_category.hom.comp | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"simplex_category",
"simplex_category.hom"
] | Composition of morphisms of `simplex_category`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
small_category : small_category.{0} simplex_category | { hom := λ n m, simplex_category.hom n m,
id := λ m, simplex_category.hom.id _,
comp := λ _ _ _ f g, simplex_category.hom.comp g f, } | instance | simplex_category.small_category | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"simplex_category",
"simplex_category.hom",
"simplex_category.hom.comp",
"simplex_category.hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const (x : simplex_category) (i : fin (x.len+1)) : [0] ⟶ x | hom.mk $ ⟨λ _, i, by tauto⟩ | def | simplex_category.const | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"simplex_category"
] | The constant morphism from [0]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
const_comp (x y : simplex_category) (i : fin (x.len + 1)) (f : x ⟶ y) :
const x i ≫ f = const y (f.to_order_hom i) | rfl | lemma | simplex_category.const_comp | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"simplex_category"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_hom {n m : ℕ} (f : (fin (n+1)) →o (fin (m+1))) : [n] ⟶ [m] | simplex_category.hom.mk f | def | simplex_category.mk_hom | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"simplex_category.hom.mk"
] | Make a morphism `[n] ⟶ [m]` from a monotone map between fin's.
This is useful for constructing morphisms beetween `[n]` directly
without identifying `n` with `[n].len`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_zero_zero (f : [0] ⟶ [0]) : f = 𝟙 _ | by { ext : 2, dsimp, apply subsingleton.elim } | lemma | simplex_category.hom_zero_zero | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
δ {n} (i : fin (n+2)) : [n] ⟶ [n+1] | mk_hom (fin.succ_above i).to_order_hom | def | simplex_category.δ | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"fin.succ_above"
] | The `i`-th face map from `[n]` to `[n+1]` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
σ {n} (i : fin (n+1)) : [n+1] ⟶ [n] | mk_hom
{ to_fun := fin.pred_above i,
monotone' := fin.pred_above_right_monotone i } | def | simplex_category.σ | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"fin.pred_above",
"fin.pred_above_right_monotone"
] | The `i`-th degeneracy map from `[n+1]` to `[n]` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
δ_comp_δ {n} {i j : fin (n+2)} (H : i ≤ j) :
δ i ≫ δ j.succ = δ j ≫ δ i.cast_succ | begin
ext k,
dsimp [δ, fin.succ_above],
simp only [order_embedding.to_order_hom_coe,
order_embedding.coe_of_strict_mono,
function.comp_app,
simplex_category.hom.to_order_hom_mk,
order_hom.comp_coe],
rcases i with ⟨i, _⟩,
rcases j with ⟨j, _⟩,
rcases k with ⟨k, _⟩,
split_ifs; { simp at *; l... | lemma | simplex_category.δ_comp_δ | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"fin.succ_above",
"order_embedding.coe_of_strict_mono",
"simplex_category.hom.to_order_hom_mk"
] | The generic case of the first simplicial identity | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
δ_comp_δ' {n} {i : fin (n+2)} {j : fin (n+3)} (H : i.cast_succ < j) :
δ i ≫ δ j = δ (j.pred (λ hj, by simpa only [hj, fin.not_lt_zero] using H)) ≫ δ i.cast_succ | begin
rw ← δ_comp_δ,
{ rw fin.succ_pred, },
{ simpa only [fin.le_iff_coe_le_coe, ← nat.lt_succ_iff, nat.succ_eq_add_one, ← fin.coe_succ,
j.succ_pred, fin.lt_iff_coe_lt_coe] using H, },
end | lemma | simplex_category.δ_comp_δ' | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"fin.coe_succ",
"fin.le_iff_coe_le_coe",
"fin.lt_iff_coe_lt_coe",
"fin.not_lt_zero",
"fin.succ_pred",
"nat.lt_succ_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
δ_comp_δ'' {n} {i : fin (n+3)} {j : fin (n+2)} (H : i ≤ j.cast_succ) :
δ (i.cast_lt (nat.lt_of_le_of_lt (fin.le_iff_coe_le_coe.mp H) j.is_lt)) ≫ δ j.succ =
δ j ≫ δ i | begin
rw δ_comp_δ,
{ refl, },
{ exact H, },
end | lemma | simplex_category.δ_comp_δ'' | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
δ_comp_δ_self {n} {i : fin (n+2)} : δ i ≫ δ i.cast_succ = δ i ≫ δ i.succ | (δ_comp_δ (le_refl i)).symm | lemma | simplex_category.δ_comp_δ_self | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [] | The special case of the first simplicial identity | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
δ_comp_δ_self' {n} {i : fin (n+2)} {j : fin (n+3)} (H : j = i.cast_succ) :
δ i ≫ δ j = δ i ≫ δ i.succ | by { subst H, rw δ_comp_δ_self, } | lemma | simplex_category.δ_comp_δ_self' | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
δ_comp_σ_of_le {n} {i : fin (n+2)} {j : fin (n+1)} (H : i ≤ j.cast_succ) :
δ i.cast_succ ≫ σ j.succ = σ j ≫ δ i | begin
ext k,
suffices : ite (j.succ.cast_succ < ite (k < i) k.cast_succ k.succ)
(ite (k < i) (k:ℕ) (k + 1) - 1) (ite (k < i) k (k + 1)) =
ite ((if h : (j:ℕ) < k
then k.pred (by { rintro rfl, exact nat.not_lt_zero _ h })
else k.cast_lt (by { cases j, cases k, simp only [len_mk], linarith })... | lemma | simplex_category.δ_comp_σ_of_le | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"fin.cast_succ_mk",
"fin.mk_le_mk",
"fin.pred_above",
"fin.succ_above",
"nat.le_of_pred_lt"
] | The second simplicial identity | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
δ_comp_σ_self {n} {i : fin (n+1)} :
δ i.cast_succ ≫ σ i = 𝟙 [n] | begin
ext j,
suffices : ite (fin.cast_succ i < ite (j < i) (fin.cast_succ j) j.succ)
(ite (j < i) (j:ℕ) (j + 1) - 1) (ite (j < i) j (j + 1)) = j,
{ dsimp [δ, σ, fin.succ_above, fin.pred_above], simpa [fin.pred_above] with push_cast },
rcases i with ⟨i, _⟩,
rcases j with ⟨j, _⟩,
dsimp,
split_ifs; { sim... | lemma | simplex_category.δ_comp_σ_self | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"fin.cast_succ",
"fin.pred_above",
"fin.succ_above"
] | The first part of the third simplicial identity | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
δ_comp_σ_self' {n} {j : fin (n+2)} {i : fin (n+1)} (H : j = i.cast_succ) :
δ j ≫ σ i = 𝟙 [n] | by { subst H, rw δ_comp_σ_self, } | lemma | simplex_category.δ_comp_σ_self' | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
δ_comp_σ_succ {n} {i : fin (n+1)} :
δ i.succ ≫ σ i = 𝟙 [n] | begin
ext j,
rcases i with ⟨i, _⟩,
rcases j with ⟨j, _⟩,
dsimp [δ, σ, fin.succ_above, fin.pred_above],
simp [fin.pred_above] with push_cast,
split_ifs; { simp at *; linarith, },
end | lemma | simplex_category.δ_comp_σ_succ | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"fin.pred_above",
"fin.succ_above"
] | The second part of the third simplicial identity | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
δ_comp_σ_succ' {n} (j : fin (n+2)) (i : fin (n+1)) (H : j = i.succ) :
δ j ≫ σ i = 𝟙 [n] | by { subst H, rw δ_comp_σ_succ, } | lemma | simplex_category.δ_comp_σ_succ' | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
δ_comp_σ_of_gt {n} {i : fin (n+2)} {j : fin (n+1)} (H : j.cast_succ < i) :
δ i.succ ≫ σ j.cast_succ = σ j ≫ δ i | begin
ext k,
dsimp [δ, σ, fin.succ_above, fin.pred_above],
rcases i with ⟨i, _⟩,
rcases j with ⟨j, _⟩,
rcases k with ⟨k, _⟩,
simp only [fin.mk_lt_mk, fin.cast_succ_mk] at H,
suffices : ite (_ < ite (k < i + 1) _ _) _ _ =
ite _ (ite (j < k) (k - 1) k) (ite (j < k) (k - 1) k + 1),
{ simpa [apply_dite ... | lemma | simplex_category.δ_comp_σ_of_gt | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"apply_dite",
"fin.cast_succ",
"fin.cast_succ_mk",
"fin.mk_lt_mk",
"fin.pred_above",
"fin.succ_above",
"nat.add_succ_sub_one",
"nat.le_pred_of_lt",
"one_ne_zero"
] | The fourth simplicial identity | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
δ_comp_σ_of_gt' {n} {i : fin (n+3)} {j : fin (n+2)} (H : j.succ < i) :
δ i ≫ σ j = σ (j.cast_lt ((add_lt_add_iff_right 1).mp (lt_of_lt_of_le
(by simpa only [fin.val_eq_coe, ← fin.coe_succ]
using fin.lt_iff_coe_lt_coe.mp H) i.is_le))) ≫
δ (i.pred (λ hi, by simpa only [fin.not_lt_zero, hi] using H)) | begin
rw ← δ_comp_σ_of_gt,
{ simpa only [fin.succ_pred], },
{ rw [fin.cast_succ_cast_lt, ← fin.succ_lt_succ_iff, fin.succ_pred],
exact H, },
end | lemma | simplex_category.δ_comp_σ_of_gt' | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"fin.cast_succ_cast_lt",
"fin.coe_succ",
"fin.not_lt_zero",
"fin.succ_lt_succ_iff",
"fin.succ_pred",
"fin.val_eq_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
σ_comp_σ {n} {i j : fin (n+1)} (H : i ≤ j) :
σ i.cast_succ ≫ σ j = σ j.succ ≫ σ i | begin
ext k,
dsimp [σ, fin.pred_above],
rcases i with ⟨i, _⟩,
rcases j with ⟨j, _⟩,
rcases k with ⟨k, _⟩,
simp only [fin.mk_le_mk] at H,
-- At this point `simp with push_cast` makes good progress, but neither `simp?` nor `squeeze_simp`
-- return usable sets of lemmas.
-- To avoid using a non-terminal ... | lemma | simplex_category.σ_comp_σ | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"fin.mk_le_mk",
"fin.pred_above",
"subtype.mk_lt_mk"
] | The fifth simplicial identity | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
skeletal_functor : simplex_category ⥤ NonemptyFinLinOrd.{v} | { obj := λ a, NonemptyFinLinOrd.of $ ulift (fin (a.len + 1)),
map := λ a b f,
⟨λ i, ulift.up (f.to_order_hom i.down), λ i j h, f.to_order_hom.monotone h⟩,
map_id' := λ a, by { ext, simp, },
map_comp' := λ a b c f g, by { ext, simp, }, } | def | simplex_category.skeletal_functor | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"NonemptyFinLinOrd.of",
"simplex_category"
] | The functor that exhibits `simplex_category` as skeleton
of `NonemptyFinLinOrd` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
skeletal_functor.coe_map
{Δ₁ Δ₂ : simplex_category} (f : Δ₁ ⟶ Δ₂) :
coe_fn (skeletal_functor.{v}.map f) = ulift.up ∘ f.to_order_hom ∘ ulift.down | rfl | lemma | simplex_category.skeletal_functor.coe_map | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"simplex_category"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
skeletal : skeletal simplex_category | λ X Y ⟨I⟩,
begin
suffices : fintype.card (fin (X.len+1)) = fintype.card (fin (Y.len+1)),
{ ext, simpa },
{ apply fintype.card_congr,
refine equiv.ulift.symm.trans (((skeletal_functor ⋙ forget _).map_iso I).to_equiv.trans _),
apply equiv.ulift }
end | lemma | simplex_category.skeletal | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"equiv.ulift",
"fintype.card",
"fintype.card_congr",
"simplex_category"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalence : is_equivalence (skeletal_functor.{v}) | equivalence.of_fully_faithfully_ess_surj skeletal_functor | instance | simplex_category.skeletal_functor.is_equivalence | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
skeletal_equivalence : simplex_category ≌ NonemptyFinLinOrd.{v} | functor.as_equivalence skeletal_functor | def | simplex_category.skeletal_equivalence | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"simplex_category"
] | The equivalence that exhibits `simplex_category` as skeleton
of `NonemptyFinLinOrd` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_skeleton_of : is_skeleton_of NonemptyFinLinOrd simplex_category skeletal_functor.{v} | { skel := skeletal,
eqv := skeletal_functor.is_equivalence } | def | simplex_category.is_skeleton_of | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"NonemptyFinLinOrd",
"simplex_category"
] | `simplex_category` is a skeleton of `NonemptyFinLinOrd`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
truncated (n : ℕ) | full_subcategory (λ a : simplex_category, a.len ≤ n) | def | simplex_category.truncated | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"simplex_category"
] | The truncated simplex category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inclusion {n : ℕ} : simplex_category.truncated n ⥤ simplex_category | full_subcategory_inclusion _ | def | simplex_category.truncated.inclusion | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"simplex_category",
"simplex_category.truncated"
] | The fully faithful inclusion of the truncated simplex category into the usual
simplex category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono_iff_injective {n m : simplex_category} {f : n ⟶ m} :
mono f ↔ function.injective f.to_order_hom | begin
rw ← functor.mono_map_iff_mono skeletal_equivalence.functor.{0},
dsimp only [skeletal_equivalence, functor.as_equivalence_functor],
rw [NonemptyFinLinOrd.mono_iff_injective, skeletal_functor.coe_map,
function.injective.of_comp_iff ulift.up_injective,
function.injective.of_comp_iff' _ ulift.down_bije... | theorem | simplex_category.mono_iff_injective | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"NonemptyFinLinOrd.mono_iff_injective",
"function.injective.of_comp_iff",
"function.injective.of_comp_iff'",
"simplex_category",
"ulift.down_bijective",
"ulift.up_injective"
] | A morphism in `simplex_category` is a monomorphism precisely when it is an injective function | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
epi_iff_surjective {n m : simplex_category} {f: n ⟶ m} :
epi f ↔ function.surjective f.to_order_hom | begin
rw ← functor.epi_map_iff_epi skeletal_equivalence.functor.{0},
dsimp only [skeletal_equivalence, functor.as_equivalence_functor],
rw [NonemptyFinLinOrd.epi_iff_surjective, skeletal_functor.coe_map,
function.surjective.of_comp_iff' ulift.up_bijective,
function.surjective.of_comp_iff _ ulift.down_surj... | lemma | simplex_category.epi_iff_surjective | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"NonemptyFinLinOrd.epi_iff_surjective",
"function.surjective.of_comp_iff",
"function.surjective.of_comp_iff'",
"simplex_category",
"ulift.down_surjective",
"ulift.up_bijective"
] | A morphism in `simplex_category` is an epimorphism if and only if it is a surjective function | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
len_le_of_mono {x y : simplex_category} {f : x ⟶ y} :
mono f → (x.len ≤ y.len) | begin
intro hyp_f_mono,
have f_inj : function.injective f.to_order_hom.to_fun,
{ exact mono_iff_injective.elim_left (hyp_f_mono) },
simpa using fintype.card_le_of_injective f.to_order_hom.to_fun f_inj,
end | lemma | simplex_category.len_le_of_mono | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"fintype.card_le_of_injective",
"simplex_category"
] | A monomorphism in `simplex_category` must increase lengths | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_of_mono {n m : ℕ} {f : [n] ⟶ [m]} : (category_theory.mono f) → (n ≤ m) | len_le_of_mono | lemma | simplex_category.le_of_mono | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"category_theory.mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
len_le_of_epi {x y : simplex_category} {f : x ⟶ y} :
epi f → y.len ≤ x.len | begin
intro hyp_f_epi,
have f_surj : function.surjective f.to_order_hom.to_fun,
{ exact epi_iff_surjective.elim_left (hyp_f_epi) },
simpa using fintype.card_le_of_surjective f.to_order_hom.to_fun f_surj,
end | lemma | simplex_category.len_le_of_epi | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"fintype.card_le_of_surjective",
"simplex_category"
] | An epimorphism in `simplex_category` must decrease lengths | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_of_epi {n m : ℕ} {f : [n] ⟶ [m]} : epi f → (m ≤ n) | len_le_of_epi | lemma | simplex_category.le_of_epi | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_of_bijective {x y : simplex_category} {f : x ⟶ y}
(hf : function.bijective (f.to_order_hom.to_fun)) : is_iso f | begin
haveI : is_iso ((forget simplex_category).map f) := (is_iso_iff_bijective _).mpr hf,
exact is_iso_of_reflects_iso f (forget simplex_category),
end | lemma | simplex_category.is_iso_of_bijective | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"simplex_category"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso_of_iso {x y : simplex_category} (e : x ≅ y) :
fin (x.len+1) ≃o fin (y.len+1) | equiv.to_order_iso
{ to_fun := e.hom.to_order_hom,
inv_fun := e.inv.to_order_hom,
left_inv := λ i, by simpa only using congr_arg (λ φ, (hom.to_order_hom φ) i) e.hom_inv_id',
right_inv := λ i, by simpa only using congr_arg (λ φ, (hom.to_order_hom φ) i) e.inv_hom_id', }
e.hom.to_order_hom.monotone e... | def | simplex_category.order_iso_of_iso | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"equiv.to_order_iso",
"inv_fun",
"simplex_category"
] | An isomorphism in `simplex_category` induces an `order_iso`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_eq_iso_refl {x : simplex_category} (e : x ≅ x) :
e = iso.refl x | begin
have h : (finset.univ : finset (fin (x.len+1))).card = x.len+1 := finset.card_fin (x.len+1),
have eq₁ := finset.order_emb_of_fin_unique' h
(λ i, finset.mem_univ ((order_iso_of_iso e) i)),
have eq₂ := finset.order_emb_of_fin_unique' h
(λ i, finset.mem_univ ((order_iso_of_iso (iso.refl x)) i)),
ext1... | lemma | simplex_category.iso_eq_iso_refl | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"finset",
"finset.card_fin",
"finset.mem_univ",
"finset.order_emb_of_fin_unique'",
"finset.univ",
"order_embedding.to_order_hom",
"simplex_category"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_id_of_is_iso {x : simplex_category} (f : x ⟶ x) [hf : is_iso f] : f = 𝟙 _ | congr_arg (λ (φ : _ ≅ _), φ.hom) (iso_eq_iso_refl (as_iso f)) | lemma | simplex_category.eq_id_of_is_iso | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"simplex_category"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_σ_comp_of_not_injective' {n : ℕ} {Δ' : simplex_category} (θ : mk (n+1) ⟶ Δ')
(i : fin (n+1)) (hi : θ.to_order_hom i.cast_succ = θ.to_order_hom i.succ):
∃ (θ' : mk n ⟶ Δ'), θ = σ i ≫ θ' | begin
use δ i.succ ≫ θ,
ext1, ext1, ext1 x,
simp only [hom.to_order_hom_mk, function.comp_app, order_hom.comp_coe,
hom.comp, small_category_comp, σ, mk_hom, order_hom.coe_fun_mk],
by_cases h' : x ≤ i.cast_succ,
{ rw fin.pred_above_below i x h',
have eq := fin.cast_succ_cast_pred (gt_of_gt_of_ge (fin.c... | lemma | simplex_category.eq_σ_comp_of_not_injective' | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"fin.cast_succ_cast_pred",
"fin.cast_succ_lt_last",
"fin.coe_cast_succ",
"fin.coe_succ",
"fin.coe_zero",
"fin.ext_iff",
"fin.le_cast_succ_iff",
"fin.le_iff_coe_le_coe",
"fin.lt_iff_coe_lt_coe",
"fin.lt_succ",
"fin.pred_above_above",
"fin.pred_above_below",
"fin.pred_succ",
"fin.succ_above_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_σ_comp_of_not_injective {n : ℕ} {Δ' : simplex_category} (θ : mk (n+1) ⟶ Δ')
(hθ : ¬function.injective θ.to_order_hom) :
∃ (i : fin (n+1)) (θ' : mk n ⟶ Δ'), θ = σ i ≫ θ' | begin
simp only [function.injective, exists_prop, not_forall] at hθ,
-- as θ is not injective, there exists `x<y` such that `θ x = θ y`
-- and then, `θ x = θ (x+1)`
have hθ₂ : ∃ (x y : fin (n+2)), (hom.to_order_hom θ) x = (hom.to_order_hom θ) y ∧ x<y,
{ rcases hθ with ⟨x, y, ⟨h₁, h₂⟩⟩,
by_cases x<y,
{... | lemma | simplex_category.eq_σ_comp_of_not_injective | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"exists_prop",
"fin.cast_succ_cast_pred",
"fin.cast_succ_lt_iff_succ_le",
"fin.cast_succ_lt_succ",
"fin.le_last",
"not_forall",
"simplex_category"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_comp_δ_of_not_surjective' {n : ℕ} {Δ : simplex_category} (θ : Δ ⟶ mk (n+1))
(i : fin (n+2)) (hi : ∀ x, θ.to_order_hom x ≠ i) :
∃ (θ' : Δ ⟶ (mk n)), θ = θ' ≫ δ i | begin
by_cases i < fin.last (n+1),
{ use θ ≫ σ (fin.cast_pred i),
ext1, ext1, ext1 x,
simp only [hom.to_order_hom_mk, function.comp_app,
order_hom.comp_coe, hom.comp, small_category_comp],
by_cases h' : θ.to_order_hom x ≤ i,
{ simp only [σ, mk_hom, hom.to_order_hom_mk, order_hom.coe_fun_mk],
... | lemma | simplex_category.eq_comp_δ_of_not_surjective' | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"fin.cast_pred",
"fin.cast_succ_cast_pred",
"fin.coe_cast_pred_le_self",
"fin.coe_cast_succ",
"fin.coe_pred",
"fin.last",
"fin.le_iff_coe_le_coe",
"fin.le_last",
"fin.lt_iff_coe_lt_coe",
"fin.pred_above_above",
"fin.pred_above_below",
"fin.pred_above_last",
"fin.succ_above_above",
"fin.suc... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_comp_δ_of_not_surjective {n : ℕ} {Δ : simplex_category} (θ : Δ ⟶ mk (n+1))
(hθ : ¬function.surjective θ.to_order_hom) :
∃ (i : fin (n+2)) (θ' : Δ ⟶ (mk n)), θ = θ' ≫ δ i | begin
cases not_forall.mp hθ with i hi,
use i,
exact eq_comp_δ_of_not_surjective' θ i (not_exists.mp hi),
end | lemma | simplex_category.eq_comp_δ_of_not_surjective | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"simplex_category"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_id_of_mono {x : simplex_category} (i : x ⟶ x) [mono i] : i = 𝟙 _ | begin
suffices : is_iso i,
{ haveI := this, apply eq_id_of_is_iso, },
apply is_iso_of_bijective,
dsimp,
rw [fintype.bijective_iff_injective_and_card i.to_order_hom, ← mono_iff_injective,
eq_self_iff_true, and_true],
apply_instance,
end | lemma | simplex_category.eq_id_of_mono | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"fintype.bijective_iff_injective_and_card",
"simplex_category"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_id_of_epi {x : simplex_category} (i : x ⟶ x) [epi i] : i = 𝟙 _ | begin
suffices : is_iso i,
{ haveI := this, apply eq_id_of_is_iso, },
apply is_iso_of_bijective,
dsimp,
rw [fintype.bijective_iff_surjective_and_card i.to_order_hom, ← epi_iff_surjective,
eq_self_iff_true, and_true],
apply_instance,
end | lemma | simplex_category.eq_id_of_epi | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"fintype.bijective_iff_surjective_and_card",
"simplex_category"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_σ_of_epi {n : ℕ} (θ : mk (n+1) ⟶ mk n) [epi θ] : ∃ (i : fin (n+1)), θ = σ i | begin
rcases eq_σ_comp_of_not_injective θ _ with ⟨i, θ', h⟩, swap,
{ by_contradiction,
simpa only [nat.one_ne_zero, add_le_iff_nonpos_right, nonpos_iff_eq_zero]
using le_of_mono (mono_iff_injective.mpr h), },
use i,
haveI : epi (σ i ≫ θ') := by { rw ← h, apply_instance, },
haveI := category_theory.e... | lemma | simplex_category.eq_σ_of_epi | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"by_contradiction",
"category_theory.epi_of_epi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_δ_of_mono {n : ℕ} (θ : mk n ⟶ mk (n+1)) [mono θ] : ∃ (i : fin (n+2)), θ = δ i | begin
rcases eq_comp_δ_of_not_surjective θ _ with ⟨i, θ', h⟩, swap,
{ by_contradiction,
simpa only [add_le_iff_nonpos_right, nonpos_iff_eq_zero]
using le_of_epi (epi_iff_surjective.mpr h), },
use i,
haveI : mono (θ' ≫ δ i) := by { rw ← h, apply_instance, },
haveI := category_theory.mono_of_mono θ' (... | lemma | simplex_category.eq_δ_of_mono | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"by_contradiction",
"category_theory.mono_of_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
len_lt_of_mono {Δ' Δ : simplex_category} (i : Δ' ⟶ Δ) [hi : mono i]
(hi' : Δ ≠ Δ') : Δ'.len < Δ.len | begin
cases lt_or_eq_of_le (len_le_of_mono hi),
{ exact h, },
{ exfalso,
exact hi' (by { ext, exact h.symm,}), },
end | lemma | simplex_category.len_lt_of_mono | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"simplex_category"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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