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 |
|---|---|---|---|---|---|---|---|---|---|---|
equivalence_unit_iso : dold_kan.equivalence.unit_iso =
(ε : 𝟭 (simplicial_object C) ≅ N ⋙ Γ) | compatibility.equivalence_unit_iso_eq hε | lemma | category_theory.idempotents.dold_kan.equivalence_unit_iso | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/equivalence_pseudoabelian.lean | [
"algebraic_topology.dold_kan.equivalence_additive",
"algebraic_topology.dold_kan.compatibility",
"category_theory.idempotents.simplicial_object"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
higher_faces_vanish {Y : C} {n : ℕ} (q : ℕ) (φ : Y ⟶ X _[n+1]) : Prop | ∀ (j : fin (n+1)), (n+1 ≤ (j : ℕ) + q) → φ ≫ X.δ j.succ = 0 | def | algebraic_topology.dold_kan.higher_faces_vanish | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/faces.lean | [
"algebraic_topology.dold_kan.homotopies",
"tactic.ring_exp"
] | [] | A morphism `φ : Y ⟶ X _[n+1]` satisfies `higher_faces_vanish q φ`
when the compositions `φ ≫ X.δ j` are `0` for `j ≥ max 1 (n+2-q)`. When `q ≤ n+1`,
it basically means that the composition `φ ≫ X.δ j` are `0` for the `q` highest
possible values of a nonzero `j`. Otherwise, when `q ≥ n+2`, all the compositions
`φ ≫ X.δ ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_δ_eq_zero {Y : C} {n : ℕ} {q : ℕ} {φ : Y ⟶ X _[n+1]}
(v : higher_faces_vanish q φ) (j : fin (n+2)) (hj₁ : j ≠ 0) (hj₂ : n+2 ≤ (j : ℕ) + q) :
φ ≫ X.δ j = 0 | begin
obtain ⟨i, hi⟩ := fin.eq_succ_of_ne_zero hj₁,
subst hi,
apply v i,
rw [← @nat.add_le_add_iff_right 1, add_assoc],
simpa only [fin.coe_succ, add_assoc, add_comm 1] using hj₂,
end | lemma | algebraic_topology.dold_kan.higher_faces_vanish.comp_δ_eq_zero | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/faces.lean | [
"algebraic_topology.dold_kan.homotopies",
"tactic.ring_exp"
] | [
"fin.coe_succ",
"fin.eq_succ_of_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_succ {Y : C} {n q : ℕ} {φ : Y ⟶ X _[n+1]}
(v : higher_faces_vanish (q+1) φ) : higher_faces_vanish q φ | λ j hj, v j (by simpa only [← add_assoc] using le_add_right hj) | lemma | algebraic_topology.dold_kan.higher_faces_vanish.of_succ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/faces.lean | [
"algebraic_topology.dold_kan.homotopies",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_comp {Y Z : C} {q n : ℕ} {φ : Y ⟶ X _[n+1]}
(v : higher_faces_vanish q φ) (f : Z ⟶ Y) :
higher_faces_vanish q (f ≫ φ) | λ j hj,
by rw [assoc, v j hj, comp_zero] | lemma | algebraic_topology.dold_kan.higher_faces_vanish.of_comp | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/faces.lean | [
"algebraic_topology.dold_kan.homotopies",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_Hσ_eq {Y : C} {n a q : ℕ} {φ : Y ⟶ X _[n+1]}
(v : higher_faces_vanish q φ) (hnaq : n=a+q) : φ ≫ (Hσ q).f (n+1) =
- φ ≫ X.δ ⟨a+1, nat.succ_lt_succ (nat.lt_succ_iff.mpr (nat.le.intro hnaq.symm))⟩ ≫
X.σ ⟨a, nat.lt_succ_iff.mpr (nat.le.intro hnaq.symm)⟩ | begin
have hnaq_shift : Π d : ℕ, n+d=(a+d)+q,
{ intro d, rw [add_assoc, add_comm d, ← add_assoc, hnaq], },
rw [Hσ, homotopy.null_homotopic_map'_f (c_mk (n+2) (n+1) rfl) (c_mk (n+1) n rfl),
hσ'_eq hnaq (c_mk (n+1) n rfl), hσ'_eq (hnaq_shift 1) (c_mk (n+2) (n+1) rfl)],
simp only [alternating_face_map_complex.... | lemma | algebraic_topology.dold_kan.higher_faces_vanish.comp_Hσ_eq | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/faces.lean | [
"algebraic_topology.dold_kan.homotopies",
"tactic.ring_exp"
] | [
"fin.cast_le_mk",
"fin.cast_mk",
"fin.cast_succ",
"fin.cast_succ_mk",
"fin.coe_cast",
"fin.coe_cast_le",
"fin.coe_cast_succ",
"fin.coe_mk",
"fin.coe_pred",
"fin.ext_iff",
"fin.last",
"fin.le_iff_coe_le_coe",
"fin.nat_add_mk",
"fin.pred_eq_iff_eq_succ",
"fin.succ_mk",
"homotopy.null_hom... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_Hσ_eq_zero {Y : C} {n q : ℕ} {φ : Y ⟶ X _[n+1]}
(v : higher_faces_vanish q φ) (hqn : n<q) : φ ≫ (Hσ q).f (n+1) = 0 | begin
simp only [Hσ, homotopy.null_homotopic_map'_f (c_mk (n+2) (n+1) rfl) (c_mk (n+1) n rfl)],
rw [hσ'_eq_zero hqn (c_mk (n+1) n rfl), comp_zero, zero_add],
by_cases hqn' : n+1<q,
{ rw [hσ'_eq_zero hqn' (c_mk (n+2) (n+1) rfl), zero_comp, comp_zero], },
{ simp only [hσ'_eq (show n+1=0+q, by linarith) (c_mk (n... | lemma | algebraic_topology.dold_kan.higher_faces_vanish.comp_Hσ_eq_zero | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/faces.lean | [
"algebraic_topology.dold_kan.homotopies",
"tactic.ring_exp"
] | [
"fin.cast_le_mk",
"fin.cast_mk",
"fin.cast_nat_add",
"fin.cast_succ_mk",
"fin.coe_add_nat",
"fin.coe_one",
"fin.coe_pred",
"fin.coe_zero",
"fin.ext_iff",
"fin.last",
"fin.lt_iff_coe_lt_coe",
"fin.mk_one",
"fin.mk_zero",
"fin.pred_eq_iff_eq_succ",
"homotopy.null_homotopic_map'_f",
"neg_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induction {Y : C} {n q : ℕ} {φ : Y ⟶ X _[n+1]}
(v : higher_faces_vanish q φ) : higher_faces_vanish (q+1) (φ ≫ (𝟙 _ + Hσ q).f (n+1)) | begin
intros j hj₁,
dsimp,
simp only [comp_add, add_comp, comp_id],
-- when n < q, the result follows immediately from the assumption
by_cases hqn : n<q,
{ rw [v.comp_Hσ_eq_zero hqn, zero_comp, add_zero, v j (by linarith)], },
-- we now assume that n≥q, and write n=a+q
cases nat.le.dest (not_lt.mp hqn) ... | lemma | algebraic_topology.dold_kan.higher_faces_vanish.induction | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/faces.lean | [
"algebraic_topology.dold_kan.homotopies",
"tactic.ring_exp"
] | [
"by_contradiction",
"fin.coe_mk",
"fin.coe_succ",
"fin.eq_zero",
"fin.eta",
"fin.le_iff_coe_le_coe",
"fin.lt_iff_coe_lt_coe",
"fin.mk_one",
"fin.mk_zero",
"nat.succ_le_iff",
"ne.le_iff_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_δ₀ {Δ Δ' : simplex_category} (i : Δ' ⟶ Δ) [mono i] : Prop | (Δ.len = Δ'.len+1) ∧ (i.to_order_hom 0 ≠ 0) | def | algebraic_topology.dold_kan.is_δ₀ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [
"simplex_category"
] | `is_δ₀ i` is a simple condition used to check whether a monomorphism `i` in
`simplex_category` identifies to the coface map `δ 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iff {j : ℕ} {i : fin (j+2)} : is_δ₀ (simplex_category.δ i) ↔ i = 0 | begin
split,
{ rintro ⟨h₁, h₂⟩,
by_contradiction,
exact h₂ (fin.succ_above_ne_zero_zero h), },
{ rintro rfl,
exact ⟨rfl, fin.succ_ne_zero _⟩, },
end | lemma | algebraic_topology.dold_kan.is_δ₀.iff | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [
"by_contradiction",
"fin.succ_above_ne_zero_zero",
"fin.succ_ne_zero",
"simplex_category.δ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_δ₀ {n : ℕ} {i : [n] ⟶ [n+1]} [mono i] (hi : is_δ₀ i) :
i = simplex_category.δ 0 | begin
unfreezingI { obtain ⟨j, rfl⟩ := simplex_category.eq_δ_of_mono i, },
rw iff at hi,
rw hi,
end | lemma | algebraic_topology.dold_kan.is_δ₀.eq_δ₀ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [
"simplex_category.eq_δ_of_mono",
"simplex_category.δ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summand (Δ : simplex_categoryᵒᵖ) (A : splitting.index_set Δ) : C | K.X A.1.unop.len | def | algebraic_topology.dold_kan.Γ₀.obj.summand | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [] | In the definition of `(Γ₀.obj K).obj Δ` as a direct sum indexed by `A : splitting.index_set Δ`,
the summand `summand K Δ A` is `K.X A.1.len`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
obj₂ (K : chain_complex C ℕ) (Δ : simplex_categoryᵒᵖ) [has_finite_coproducts C] : C | ∐ (λ (A : splitting.index_set Δ), summand K Δ A) | def | algebraic_topology.dold_kan.Γ₀.obj.obj₂ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [
"chain_complex"
] | The functor `Γ₀` sends a chain complex `K` to the simplicial object which
sends `Δ` to the direct sum of the objects `summand K Δ A` for all `A : splitting.index_set Δ` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_mono (K : chain_complex C ℕ) {Δ' Δ : simplex_category} (i : Δ' ⟶ Δ) [mono i] :
K.X Δ.len ⟶ K.X Δ'.len | begin
by_cases Δ = Δ',
{ exact eq_to_hom (by congr'), },
{ by_cases is_δ₀ i,
{ exact K.d Δ.len Δ'.len, },
{ exact 0, }, },
end | def | algebraic_topology.dold_kan.Γ₀.obj.termwise.map_mono | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [
"chain_complex",
"simplex_category"
] | A monomorphism `i : Δ' ⟶ Δ` induces a morphism `K.X Δ.len ⟶ K.X Δ'.len` which
is the identity if `Δ = Δ'`, the differential on the complex `K` if `i = δ 0`, and
zero otherwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_mono_id : map_mono K (𝟙 Δ) = 𝟙 _ | by { unfold map_mono, simp only [eq_self_iff_true, eq_to_hom_refl, dite_eq_ite, if_true], } | lemma | algebraic_topology.dold_kan.Γ₀.obj.termwise.map_mono_id | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [
"dite_eq_ite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mono_δ₀' (hi : is_δ₀ i) : map_mono K i = K.d Δ.len Δ'.len | begin
unfold map_mono,
classical,
rw [dif_neg, dif_pos hi],
unfreezingI { rintro rfl, },
simpa only [self_eq_add_right, nat.one_ne_zero] using hi.1,
end | lemma | algebraic_topology.dold_kan.Γ₀.obj.termwise.map_mono_δ₀' | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mono_δ₀ {n : ℕ} : map_mono K (δ (0 : fin (n+2))) = K.d (n+1) n | map_mono_δ₀' K _ (by rw is_δ₀.iff) | lemma | algebraic_topology.dold_kan.Γ₀.obj.termwise.map_mono_δ₀ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mono_eq_zero (h₁ : Δ ≠ Δ') (h₂ : ¬is_δ₀ i) : map_mono K i = 0 | by { unfold map_mono, rw ne.def at h₁, split_ifs, refl, } | lemma | algebraic_topology.dold_kan.Γ₀.obj.termwise.map_mono_eq_zero | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mono_naturality : map_mono K i ≫ f.f Δ'.len = f.f Δ.len ≫ map_mono K' i | begin
unfold map_mono,
split_ifs,
{ unfreezingI { subst h, },
simp only [id_comp, eq_to_hom_refl, comp_id], },
{ rw homological_complex.hom.comm, },
{ rw [zero_comp, comp_zero], }
end | lemma | algebraic_topology.dold_kan.Γ₀.obj.termwise.map_mono_naturality | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [
"homological_complex.hom.comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mono_comp : map_mono K i ≫ map_mono K i' = map_mono K (i' ≫ i) | begin
/- case where i : Δ' ⟶ Δ is the identity -/
by_cases h₁ : Δ = Δ',
{ unfreezingI { subst h₁, },
simp only [simplex_category.eq_id_of_mono i,
comp_id, id_comp, map_mono_id K, eq_to_hom_refl], },
/- case where i' : Δ'' ⟶ Δ' is the identity -/
by_cases h₂ : Δ' = Δ'',
{ unfreezingI { subst h₂, },... | lemma | algebraic_topology.dold_kan.Γ₀.obj.termwise.map_mono_comp | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [
"by_contradiction",
"homological_complex.d_comp_d",
"nat.exists_eq_add_of_lt",
"simplex_category.eq_id_of_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (K : chain_complex C ℕ) {Δ' Δ : simplex_categoryᵒᵖ} (θ : Δ ⟶ Δ') :
obj₂ K Δ ⟶ obj₂ K Δ' | sigma.desc (λ A, termwise.map_mono K (image.ι (θ.unop ≫ A.e)) ≫
(sigma.ι (summand K Δ') (A.pull θ))) | def | algebraic_topology.dold_kan.Γ₀.obj.map | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [
"chain_complex"
] | The simplicial morphism on the simplicial object `Γ₀.obj K` induced by
a morphism `Δ' → Δ` in `simplex_category` is defined on each summand
associated to an `A : Γ_index_set Δ` in terms of the epi-mono factorisation
of `θ ≫ A.e`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_on_summand₀ {Δ Δ' : simplex_categoryᵒᵖ} (A : splitting.index_set Δ) {θ : Δ ⟶ Δ'}
{Δ'' : simplex_category} {e : Δ'.unop ⟶ Δ''} {i : Δ'' ⟶ A.1.unop} [epi e] [mono i]
(fac : e ≫ i = θ.unop ≫ A.e) :
(sigma.ι (summand K Δ) A) ≫ map K θ =
termwise.map_mono K i ≫ sigma.ι (summand K Δ') (splitting.index_set.mk e) | begin
simp only [map, colimit.ι_desc, cofan.mk_ι_app],
have h := simplex_category.image_eq fac,
unfreezingI { subst h, },
congr,
{ exact simplex_category.image_ι_eq fac, },
{ dsimp only [simplicial_object.splitting.index_set.pull],
congr,
exact simplex_category.factor_thru_image_eq fac, },
end | lemma | algebraic_topology.dold_kan.Γ₀.obj.map_on_summand₀ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [
"simplex_category",
"simplex_category.factor_thru_image_eq",
"simplex_category.image_eq",
"simplex_category.image_ι_eq",
"simplicial_object.splitting.index_set.pull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_on_summand₀' {Δ Δ' : simplex_categoryᵒᵖ} (A : splitting.index_set Δ) (θ : Δ ⟶ Δ') :
(sigma.ι (summand K Δ) A) ≫ map K θ =
termwise.map_mono K (image.ι (θ.unop ≫ A.e)) ≫ sigma.ι (summand K _) (A.pull θ) | map_on_summand₀ K A (A.fac_pull θ) | lemma | algebraic_topology.dold_kan.Γ₀.obj.map_on_summand₀' | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
obj (K : chain_complex C ℕ) : simplicial_object C | { obj := λ Δ, obj.obj₂ K Δ,
map := λ Δ Δ' θ, obj.map K θ,
map_id' := λ Δ, begin
ext A,
cases A,
have fac : A.e ≫ 𝟙 A.1.unop = (𝟙 Δ).unop ≫ A.e := by rw [unop_id, comp_id, id_comp],
erw [obj.map_on_summand₀ K A fac, obj.termwise.map_mono_id, id_comp, comp_id],
unfreezingI { rcases A with ⟨Δ', ⟨... | def | algebraic_topology.dold_kan.Γ₀.obj | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [
"chain_complex"
] | The functor `Γ₀ : chain_complex C ℕ ⥤ simplicial_object C`, on objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
splitting_map_eq_id (Δ : simplex_categoryᵒᵖ) :
(simplicial_object.splitting.map (Γ₀.obj K)
(λ (n : ℕ), sigma.ι (Γ₀.obj.summand K (op [n])) (splitting.index_set.id (op [n]))) Δ)
= 𝟙 _ | begin
ext A,
discrete_cases,
induction Δ using opposite.rec,
induction Δ with n,
dsimp,
simp only [colimit.ι_desc, cofan.mk_ι_app, comp_id, Γ₀.obj_map],
rw [Γ₀.obj.map_on_summand₀ K
(simplicial_object.splitting.index_set.id A.1) (show A.e ≫ 𝟙 _ = A.e.op.unop ≫ 𝟙 _, by refl),
Γ₀.obj.termwise.map_... | lemma | algebraic_topology.dold_kan.Γ₀.splitting_map_eq_id | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [
"opposite.rec",
"simplicial_object.splitting.index_set.id",
"simplicial_object.splitting.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
splitting (K : chain_complex C ℕ) : simplicial_object.splitting (Γ₀.obj K) | { N := λ n, K.X n,
ι := λ n, sigma.ι (Γ₀.obj.summand K (op [n])) (splitting.index_set.id (op [n])),
map_is_iso' := λ Δ, begin
rw Γ₀.splitting_map_eq_id,
apply is_iso.id,
end, } | def | algebraic_topology.dold_kan.Γ₀.splitting | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [
"chain_complex",
"simplicial_object.splitting"
] | By construction, the simplicial `Γ₀.obj K` is equipped with a splitting. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
splitting_iso_hom_eq_id (Δ : simplex_categoryᵒᵖ) : ((splitting K).iso Δ).hom = 𝟙 _ | splitting_map_eq_id K Δ | lemma | algebraic_topology.dold_kan.Γ₀.splitting_iso_hom_eq_id | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
obj.map_on_summand {Δ Δ' : simplex_categoryᵒᵖ} (A : splitting.index_set Δ) (θ : Δ ⟶ Δ')
{Δ'' : simplex_category}
{e : Δ'.unop ⟶ Δ''} {i : Δ'' ⟶ A.1.unop} [epi e] [mono i]
(fac : e ≫ i = θ.unop ≫ A.e) : (Γ₀.splitting K).ι_summand A ≫ (Γ₀.obj K).map θ =
Γ₀.obj.termwise.map_mono K i ≫ (Γ₀.splitting K).ι_summand (s... | begin
dsimp only [simplicial_object.splitting.ι_summand,
simplicial_object.splitting.ι_coprod],
simp only [assoc, Γ₀.splitting_iso_hom_eq_id, id_comp, comp_id],
exact Γ₀.obj.map_on_summand₀ K A fac,
end | lemma | algebraic_topology.dold_kan.Γ₀.obj.map_on_summand | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [
"simplex_category",
"simplicial_object.splitting.ι_coprod",
"simplicial_object.splitting.ι_summand"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
obj.map_on_summand' {Δ Δ' : simplex_categoryᵒᵖ} (A : splitting.index_set Δ) (θ : Δ ⟶ Δ') :
(splitting K).ι_summand A ≫ (obj K).map θ =
obj.termwise.map_mono K (image.ι (θ.unop ≫ A.e)) ≫ (splitting K).ι_summand (A.pull θ) | by { apply obj.map_on_summand, apply image.fac, } | lemma | algebraic_topology.dold_kan.Γ₀.obj.map_on_summand' | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
obj.map_mono_on_summand_id {Δ Δ' : simplex_category} (i : Δ' ⟶ Δ) [mono i] :
(splitting K).ι_summand (splitting.index_set.id (op Δ)) ≫ (obj K).map i.op =
obj.termwise.map_mono K i ≫ (splitting K).ι_summand (splitting.index_set.id (op Δ')) | obj.map_on_summand K (splitting.index_set.id (op Δ)) i.op (rfl : 𝟙 _ ≫ i = i ≫ 𝟙 _) | lemma | algebraic_topology.dold_kan.Γ₀.obj.map_mono_on_summand_id | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [
"simplex_category"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
obj.map_epi_on_summand_id {Δ Δ' : simplex_category } (e : Δ' ⟶ Δ) [epi e] :
(Γ₀.splitting K).ι_summand (splitting.index_set.id (op Δ)) ≫ (Γ₀.obj K).map e.op =
(Γ₀.splitting K).ι_summand (splitting.index_set.mk e) | by simpa only [Γ₀.obj.map_on_summand K (splitting.index_set.id (op Δ)) e.op
(rfl : e ≫ 𝟙 Δ = e ≫ 𝟙 Δ), Γ₀.obj.termwise.map_mono_id] using id_comp _ | lemma | algebraic_topology.dold_kan.Γ₀.obj.map_epi_on_summand_id | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [
"simplex_category"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map {K K' : chain_complex C ℕ} (f : K ⟶ K') : obj K ⟶ obj K' | { app := λ Δ, (Γ₀.splitting K).desc Δ (λ A, f.f A.1.unop.len ≫ (Γ₀.splitting K').ι_summand A),
naturality' := λ Δ' Δ θ, begin
apply (Γ₀.splitting K).hom_ext',
intro A,
simp only [(splitting K).ι_desc_assoc, obj.map_on_summand'_assoc K _ θ,
(splitting K).ι_desc, assoc, obj.map_on_summand' K' _ θ],
... | def | algebraic_topology.dold_kan.Γ₀.map | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [
"chain_complex"
] | The functor `Γ₀ : chain_complex C ℕ ⥤ simplicial_object C`, on morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Γ₀' : chain_complex C ℕ ⥤ simplicial_object.split C | { obj := λ K, simplicial_object.split.mk' (Γ₀.splitting K),
map := λ K K' f,
{ F := Γ₀.map f,
f := f.f,
comm' := λ n, by { dsimp, simpa only [← splitting.ι_summand_id,
(Γ₀.splitting K).ι_desc], }, }, } | def | algebraic_topology.dold_kan.Γ₀' | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [
"chain_complex",
"simplicial_object.split",
"simplicial_object.split.mk'"
] | The functor `Γ₀' : chain_complex C ℕ ⥤ simplicial_object.split C`
that induces `Γ₀ : chain_complex C ℕ ⥤ simplicial_object C`, which
shall be the inverse functor of the Dold-Kan equivalence for
abelian or pseudo-abelian categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Γ₀ : chain_complex C ℕ ⥤ simplicial_object C | Γ₀' ⋙ split.forget _ | def | algebraic_topology.dold_kan.Γ₀ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [
"chain_complex"
] | The functor `Γ₀ : chain_complex C ℕ ⥤ simplicial_object C`, which is
the inverse functor of the Dold-Kan equivalence when `C` is an abelian
category, or more generally a pseudoabelian category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Γ₂ : karoubi (chain_complex C ℕ) ⥤ karoubi (simplicial_object C) | (category_theory.idempotents.functor_extension₂ _ _).obj Γ₀ | def | algebraic_topology.dold_kan.Γ₂ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [
"category_theory.idempotents.functor_extension₂",
"chain_complex"
] | The extension of `Γ₀ : chain_complex C ℕ ⥤ simplicial_object C`
on the idempotent completions. It shall be an equivalence of categories
for any additive category `C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
higher_faces_vanish.on_Γ₀_summand_id (K : chain_complex C ℕ) (n : ℕ) :
higher_faces_vanish (n+1) ((Γ₀.splitting K).ι_summand (splitting.index_set.id (op [n+1]))) | begin
intros j hj,
have eq := Γ₀.obj.map_mono_on_summand_id K (simplex_category.δ j.succ),
rw [Γ₀.obj.termwise.map_mono_eq_zero K, zero_comp] at eq, rotate,
{ intro h,
exact (nat.succ_ne_self n) (congr_arg simplex_category.len h), },
{ exact λ h, fin.succ_ne_zero j (by simpa only [is_δ₀.iff] using h), },
... | lemma | algebraic_topology.dold_kan.higher_faces_vanish.on_Γ₀_summand_id | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [
"chain_complex",
"fin.succ_ne_zero",
"simplex_category.len",
"simplex_category.δ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
P_infty_on_Γ₀_splitting_summand_eq_self
(K : chain_complex C ℕ) {n : ℕ} :
(Γ₀.splitting K).ι_summand (splitting.index_set.id (op [n])) ≫ (P_infty : K[Γ₀.obj K] ⟶ _).f n =
(Γ₀.splitting K).ι_summand (splitting.index_set.id (op [n])) | begin
rw P_infty_f,
cases n,
{ simpa only [P_f_0_eq] using comp_id _, },
{ exact (higher_faces_vanish.on_Γ₀_summand_id K n).comp_P_eq_self, },
end | lemma | algebraic_topology.dold_kan.P_infty_on_Γ₀_splitting_summand_eq_self | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_gamma.lean | [
"algebraic_topology.dold_kan.split_simplicial_object"
] | [
"chain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
N₁ : simplicial_object C ⥤ karoubi (chain_complex C ℕ) | { obj := λ X,
{ X := alternating_face_map_complex.obj X,
p := P_infty,
idem := P_infty_idem, },
map := λ X Y f,
{ f := P_infty ≫ alternating_face_map_complex.map f,
comm := by { ext, simp }, },
map_id' := λ X, by { ext, dsimp, simp },
map_comp' := λ X Y Z f g, by { ext, simp } } | def | algebraic_topology.dold_kan.N₁ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_n.lean | [
"algebraic_topology.dold_kan.p_infty"
] | [
"chain_complex",
"comm"
] | The functor `simplicial_object C ⥤ karoubi (chain_complex C ℕ)` which maps
`X` to the formal direct factor of `K[X]` defined by `P_infty`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
N₂ : karoubi (simplicial_object C) ⥤ karoubi (chain_complex C ℕ) | (functor_extension₁ _ _).obj N₁ | def | algebraic_topology.dold_kan.N₂ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/functor_n.lean | [
"algebraic_topology.dold_kan.p_infty"
] | [
"chain_complex"
] | The extension of `N₁` to the Karoubi envelope of `simplicial_object C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Γ₀_nondeg_complex_iso (K : chain_complex C ℕ) : (Γ₀.splitting K).nondeg_complex ≅ K | homological_complex.hom.iso_of_components (λ n, iso.refl _)
begin
rintros _ n (rfl : n+1=_),
dsimp,
simp only [id_comp, comp_id, alternating_face_map_complex.obj_d_eq,
preadditive.sum_comp, preadditive.comp_sum],
rw fintype.sum_eq_single (0 : fin (n+2)),
{ simp only [fin.coe_zero, pow_zero, one_zsmul],
... | def | algebraic_topology.dold_kan.Γ₀_nondeg_complex_iso | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/gamma_comp_n.lean | [
"algebraic_topology.dold_kan.functor_gamma",
"category_theory.idempotents.homological_complex"
] | [
"chain_complex",
"fin.coe_zero",
"homological_complex.hom.iso_of_components",
"pow_zero",
"simplex_category.len"
] | The isomorphism `(Γ₀.splitting K).nondeg_complex ≅ K` for all `K : chain_complex C ℕ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Γ₀'_comp_nondeg_complex_functor :
Γ₀' ⋙ split.nondeg_complex_functor ≅ 𝟭 (chain_complex C ℕ) | nat_iso.of_components Γ₀_nondeg_complex_iso
(λ X Y f, by { ext n, dsimp, simp only [comp_id, id_comp], }) | def | algebraic_topology.dold_kan.Γ₀'_comp_nondeg_complex_functor | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/gamma_comp_n.lean | [
"algebraic_topology.dold_kan.functor_gamma",
"category_theory.idempotents.homological_complex"
] | [
"chain_complex"
] | The natural isomorphism `(Γ₀.splitting K).nondeg_complex ≅ K` for `K : chain_complex C ℕ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
N₁Γ₀ : Γ₀ ⋙ N₁ ≅ to_karoubi (chain_complex C ℕ) | calc Γ₀ ⋙ N₁ ≅ Γ₀' ⋙ split.forget C ⋙ N₁ : functor.associator _ _ _
... ≅ Γ₀' ⋙ split.nondeg_complex_functor ⋙ to_karoubi _ :
iso_whisker_left Γ₀' split.to_karoubi_nondeg_complex_functor_iso_N₁.symm
... ≅ (Γ₀' ⋙ split.nondeg_complex_functor) ⋙ to_karoubi _ : (functor.associator _ _ _).symm
... ≅ 𝟭 _ ⋙ to_karoubi (ch... | def | algebraic_topology.dold_kan.N₁Γ₀ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/gamma_comp_n.lean | [
"algebraic_topology.dold_kan.functor_gamma",
"category_theory.idempotents.homological_complex"
] | [
"chain_complex"
] | The natural isomorphism `Γ₀ ⋙ N₁ ≅ to_karoubi (chain_complex C ℕ)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
N₁Γ₀_app (K : chain_complex C ℕ) :
N₁Γ₀.app K = (Γ₀.splitting K).to_karoubi_nondeg_complex_iso_N₁.symm
≪≫ (to_karoubi _).map_iso (Γ₀_nondeg_complex_iso K) | begin
ext1,
dsimp [N₁Γ₀],
erw [id_comp, comp_id, comp_id],
refl,
end | lemma | algebraic_topology.dold_kan.N₁Γ₀_app | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/gamma_comp_n.lean | [
"algebraic_topology.dold_kan.functor_gamma",
"category_theory.idempotents.homological_complex"
] | [
"chain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
N₁Γ₀_hom_app (K : chain_complex C ℕ) :
N₁Γ₀.hom.app K = (Γ₀.splitting K).to_karoubi_nondeg_complex_iso_N₁.inv
≫ (to_karoubi _).map (Γ₀_nondeg_complex_iso K).hom | by { change (N₁Γ₀.app K).hom = _, simpa only [N₁Γ₀_app], } | lemma | algebraic_topology.dold_kan.N₁Γ₀_hom_app | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/gamma_comp_n.lean | [
"algebraic_topology.dold_kan.functor_gamma",
"category_theory.idempotents.homological_complex"
] | [
"chain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
N₁Γ₀_inv_app (K : chain_complex C ℕ) :
N₁Γ₀.inv.app K = (to_karoubi _).map (Γ₀_nondeg_complex_iso K).inv ≫
(Γ₀.splitting K).to_karoubi_nondeg_complex_iso_N₁.hom | by { change (N₁Γ₀.app K).inv = _, simpa only [N₁Γ₀_app], } | lemma | algebraic_topology.dold_kan.N₁Γ₀_inv_app | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/gamma_comp_n.lean | [
"algebraic_topology.dold_kan.functor_gamma",
"category_theory.idempotents.homological_complex"
] | [
"chain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
N₁Γ₀_hom_app_f_f (K : chain_complex C ℕ) (n : ℕ) :
(N₁Γ₀.hom.app K).f.f n = (Γ₀.splitting K).to_karoubi_nondeg_complex_iso_N₁.inv.f.f n | by { rw N₁Γ₀_hom_app, apply comp_id, } | lemma | algebraic_topology.dold_kan.N₁Γ₀_hom_app_f_f | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/gamma_comp_n.lean | [
"algebraic_topology.dold_kan.functor_gamma",
"category_theory.idempotents.homological_complex"
] | [
"chain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
N₁Γ₀_inv_app_f_f (K : chain_complex C ℕ) (n : ℕ) :
(N₁Γ₀.inv.app K).f.f n = (Γ₀.splitting K).to_karoubi_nondeg_complex_iso_N₁.hom.f.f n | by { rw N₁Γ₀_inv_app, apply id_comp, } | lemma | algebraic_topology.dold_kan.N₁Γ₀_inv_app_f_f | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/gamma_comp_n.lean | [
"algebraic_topology.dold_kan.functor_gamma",
"category_theory.idempotents.homological_complex"
] | [
"chain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
N₂Γ₂_to_karoubi : to_karoubi (chain_complex C ℕ) ⋙ Γ₂ ⋙ N₂ = Γ₀ ⋙ N₁ | begin
have h := functor.congr_obj (functor_extension₂_comp_whiskering_left_to_karoubi
(chain_complex C ℕ) (simplicial_object C)) Γ₀,
have h' := functor.congr_obj (functor_extension₁_comp_whiskering_left_to_karoubi
(simplicial_object C) (chain_complex C ℕ)) N₁,
dsimp [N₂, Γ₂, functor_extension₁] at h h' ⊢,... | lemma | algebraic_topology.dold_kan.N₂Γ₂_to_karoubi | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/gamma_comp_n.lean | [
"algebraic_topology.dold_kan.functor_gamma",
"category_theory.idempotents.homological_complex"
] | [
"chain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
N₂Γ₂_to_karoubi_iso : to_karoubi (chain_complex C ℕ) ⋙ Γ₂ ⋙ N₂ ≅ Γ₀ ⋙ N₁ | eq_to_iso (N₂Γ₂_to_karoubi) | def | algebraic_topology.dold_kan.N₂Γ₂_to_karoubi_iso | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/gamma_comp_n.lean | [
"algebraic_topology.dold_kan.functor_gamma",
"category_theory.idempotents.homological_complex"
] | [
"chain_complex"
] | Compatibility isomorphism between `to_karoubi _ ⋙ Γ₂ ⋙ N₂` and `Γ₀ ⋙ N₁` which
are functors `chain_complex C ℕ ⥤ karoubi (chain_complex C ℕ)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
N₂Γ₂ : Γ₂ ⋙ N₂ ≅ 𝟭 (karoubi (chain_complex C ℕ)) | ((whiskering_left _ _ _).obj (to_karoubi (chain_complex C ℕ))).preimage_iso
(N₂Γ₂_to_karoubi_iso ≪≫ N₁Γ₀) | def | algebraic_topology.dold_kan.N₂Γ₂ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/gamma_comp_n.lean | [
"algebraic_topology.dold_kan.functor_gamma",
"category_theory.idempotents.homological_complex"
] | [
"chain_complex"
] | The counit isomorphism of the Dold-Kan equivalence for additive categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
N₂Γ₂_compatible_with_N₁Γ₀ (K : chain_complex C ℕ) :
N₂Γ₂.hom.app ((to_karoubi _).obj K) = N₂Γ₂_to_karoubi_iso.hom.app K ≫ N₁Γ₀.hom.app K | congr_app (((whiskering_left _ _ (karoubi (chain_complex C ℕ ))).obj
(to_karoubi (chain_complex C ℕ))).image_preimage
(N₂Γ₂_to_karoubi_iso.hom ≫ N₁Γ₀.hom : _ ⟶ to_karoubi _ ⋙ 𝟭 _)) K | lemma | algebraic_topology.dold_kan.N₂Γ₂_compatible_with_N₁Γ₀ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/gamma_comp_n.lean | [
"algebraic_topology.dold_kan.functor_gamma",
"category_theory.idempotents.homological_complex"
] | [
"chain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
N₂Γ₂_inv_app_f_f (X : karoubi (chain_complex C ℕ)) (n : ℕ) :
(N₂Γ₂.inv.app X).f.f n =
X.p.f n ≫ (Γ₀.splitting X.X).ι_summand (splitting.index_set.id (op [n])) | begin
dsimp only [N₂Γ₂, functor.preimage_iso, iso.trans],
simp only [whiskering_left_obj_preimage_app, N₂Γ₂_to_karoubi_iso_inv, functor.id_map,
nat_trans.comp_app, eq_to_hom_app, functor.comp_map, assoc, karoubi.comp_f,
karoubi.eq_to_hom_f, eq_to_hom_refl, comp_id, karoubi.comp_p_assoc, N₂_map_f_f,
homo... | lemma | algebraic_topology.dold_kan.N₂Γ₂_inv_app_f_f | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/gamma_comp_n.lean | [
"algebraic_topology.dold_kan.functor_gamma",
"category_theory.idempotents.homological_complex"
] | [
"chain_complex",
"homological_complex.comp_f"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
c | complex_shape.down ℕ | abbreviation | algebraic_topology.dold_kan.c | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/homotopies.lean | [
"algebra.homology.homotopy",
"algebraic_topology.dold_kan.notations"
] | [
"complex_shape.down"
] | As we are using chain complexes indexed by `ℕ`, we shall need the relation
`c` such `c m n` if and only if `n+1=m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
c_mk (i j : ℕ) (h : j+1 = i) : c.rel i j | complex_shape.down_mk i j h | lemma | algebraic_topology.dold_kan.c_mk | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/homotopies.lean | [
"algebra.homology.homotopy",
"algebraic_topology.dold_kan.notations"
] | [
"complex_shape.down_mk"
] | Helper when we need some `c.rel i j` (i.e. `complex_shape.down ℕ`),
e.g. `c_mk n (n+1) rfl` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cs_down_0_not_rel_left (j : ℕ) : ¬c.rel 0 j | begin
intro hj,
dsimp at hj,
apply nat.not_succ_le_zero j,
rw [nat.succ_eq_add_one, hj],
end | lemma | algebraic_topology.dold_kan.cs_down_0_not_rel_left | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/homotopies.lean | [
"algebra.homology.homotopy",
"algebraic_topology.dold_kan.notations"
] | [] | This lemma is meant to be used with `null_homotopic_map'_f_of_not_rel_left` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hσ (q : ℕ) (n : ℕ) : X _[n] ⟶ X _[n+1] | if n<q
then 0
else (-1 : ℤ)^(n-q) • X.σ ⟨n-q, nat.sub_lt_succ n q⟩ | def | algebraic_topology.dold_kan.hσ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/homotopies.lean | [
"algebra.homology.homotopy",
"algebraic_topology.dold_kan.notations"
] | [] | The sequence of maps which gives the null homotopic maps `Hσ` that shall be in
the inductive construction of the projections `P q : K[X] ⟶ K[X]` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hσ' (q : ℕ) : Π n m, c.rel m n → (K[X].X n ⟶ K[X].X m) | λ n m hnm, (hσ q n) ≫ eq_to_hom (by congr') | def | algebraic_topology.dold_kan.hσ' | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/homotopies.lean | [
"algebra.homology.homotopy",
"algebraic_topology.dold_kan.notations"
] | [] | We can turn `hσ` into a datum that can be passed to `null_homotopic_map'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hσ'_eq_zero {q n m : ℕ} (hnq : n<q) (hnm : c.rel m n) :
(hσ' q n m hnm : X _[n] ⟶ X _[m])= 0 | by { simp only [hσ', hσ], split_ifs, exact zero_comp, } | lemma | algebraic_topology.dold_kan.hσ'_eq_zero | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/homotopies.lean | [
"algebra.homology.homotopy",
"algebraic_topology.dold_kan.notations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hσ'_eq {q n a m : ℕ} (ha : n=a+q) (hnm : c.rel m n) :
(hσ' q n m hnm : X _[n] ⟶ X _[m]) =
((-1 : ℤ)^a • X.σ ⟨a, nat.lt_succ_iff.mpr (nat.le.intro (eq.symm ha))⟩) ≫
eq_to_hom (by congr') | begin
simp only [hσ', hσ],
split_ifs,
{ exfalso, linarith, },
{ have h' := tsub_eq_of_eq_add ha,
congr', }
end | lemma | algebraic_topology.dold_kan.hσ'_eq | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/homotopies.lean | [
"algebra.homology.homotopy",
"algebraic_topology.dold_kan.notations"
] | [
"tsub_eq_of_eq_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hσ'_eq' {q n a : ℕ} (ha : n=a+q) :
(hσ' q n (n+1) rfl : X _[n] ⟶ X _[n+1]) =
(-1 : ℤ)^a • X.σ ⟨a, nat.lt_succ_iff.mpr (nat.le.intro (eq.symm ha))⟩ | by rw [hσ'_eq ha rfl, eq_to_hom_refl, comp_id] | lemma | algebraic_topology.dold_kan.hσ'_eq' | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/homotopies.lean | [
"algebra.homology.homotopy",
"algebraic_topology.dold_kan.notations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Hσ (q : ℕ) : K[X] ⟶ K[X] | null_homotopic_map' (hσ' q) | def | algebraic_topology.dold_kan.Hσ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/homotopies.lean | [
"algebra.homology.homotopy",
"algebraic_topology.dold_kan.notations"
] | [] | The null homotopic map $(hσ q) ∘ d + d ∘ (hσ q)$ | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopy_Hσ_to_zero (q : ℕ) : homotopy (Hσ q : K[X] ⟶ K[X]) 0 | null_homotopy' (hσ' q) | def | algebraic_topology.dold_kan.homotopy_Hσ_to_zero | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/homotopies.lean | [
"algebra.homology.homotopy",
"algebraic_topology.dold_kan.notations"
] | [
"homotopy"
] | `Hσ` is null homotopic | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hσ_eq_zero (q : ℕ) : (Hσ q : K[X] ⟶ K[X]).f 0 = 0 | begin
unfold Hσ,
rw null_homotopic_map'_f_of_not_rel_left (c_mk 1 0 rfl) cs_down_0_not_rel_left,
cases q,
{ rw hσ'_eq (show 0=0+0, by refl) (c_mk 1 0 rfl),
simp only [pow_zero, fin.mk_zero, one_zsmul, eq_to_hom_refl, category.comp_id],
erw chain_complex.of_d,
simp only [alternating_face_map_complex.... | lemma | algebraic_topology.dold_kan.Hσ_eq_zero | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/homotopies.lean | [
"algebra.homology.homotopy",
"algebraic_topology.dold_kan.notations"
] | [
"chain_complex.of_d",
"fin.coe_one",
"fin.coe_zero",
"fin.mk_zero",
"neg_smul",
"pow_one",
"pow_zero"
] | In degree `0`, the null homotopic map `Hσ` is zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hσ'_naturality (q : ℕ) (n m : ℕ) (hnm : c.rel m n)
{X Y : simplicial_object C} (f : X ⟶ Y) :
f.app (op [n]) ≫ hσ' q n m hnm = hσ' q n m hnm ≫ f.app (op [m]) | begin
have h : n+1 = m := hnm,
subst h,
simp only [hσ', eq_to_hom_refl, comp_id],
unfold hσ,
split_ifs,
{ rw [zero_comp, comp_zero], },
{ simp only [zsmul_comp, comp_zsmul],
erw f.naturality,
refl, },
end | lemma | algebraic_topology.dold_kan.hσ'_naturality | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/homotopies.lean | [
"algebra.homology.homotopy",
"algebraic_topology.dold_kan.notations"
] | [] | The maps `hσ' q n m hnm` are natural on the simplicial object | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_trans_Hσ (q : ℕ) :
alternating_face_map_complex C ⟶ alternating_face_map_complex C | { app := λ X, Hσ q,
naturality' := λ X Y f, begin
unfold Hσ,
rw [null_homotopic_map'_comp, comp_null_homotopic_map'],
congr,
ext n m hnm,
simp only [alternating_face_map_complex_map_f, hσ'_naturality],
end, } | def | algebraic_topology.dold_kan.nat_trans_Hσ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/homotopies.lean | [
"algebra.homology.homotopy",
"algebraic_topology.dold_kan.notations"
] | [] | For each q, `Hσ q` is a natural transformation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_hσ' {D : Type*} [category D] [preadditive D]
(G : C ⥤ D) [G.additive] (X : simplicial_object C)
(q n m : ℕ) (hnm : c.rel m n) :
(hσ' q n m hnm : K[((whiskering _ _).obj G).obj X].X n ⟶ _) =
G.map (hσ' q n m hnm : K[X].X n ⟶ _) | begin
unfold hσ' hσ,
split_ifs,
{ simp only [functor.map_zero, zero_comp], },
{ simpa only [eq_to_hom_map, functor.map_comp, functor.map_zsmul], },
end | lemma | algebraic_topology.dold_kan.map_hσ' | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/homotopies.lean | [
"algebra.homology.homotopy",
"algebraic_topology.dold_kan.notations"
] | [] | The maps `hσ' q n m hnm` are compatible with the application of additive functors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_Hσ {D : Type*} [category D] [preadditive D]
(G : C ⥤ D) [G.additive] (X : simplicial_object C) (q n : ℕ) :
(Hσ q : K[((whiskering C D).obj G).obj X] ⟶ _).f n =
G.map ((Hσ q : K[X] ⟶ _).f n) | begin
unfold Hσ,
have eq := homological_complex.congr_hom (map_null_homotopic_map' G (hσ' q)) n,
simp only [functor.map_homological_complex_map_f, ← map_hσ'] at eq,
rw eq,
let h := (functor.congr_obj (map_alternating_face_map_complex G) X).symm,
congr',
end | lemma | algebraic_topology.dold_kan.map_Hσ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/homotopies.lean | [
"algebra.homology.homotopy",
"algebraic_topology.dold_kan.notations"
] | [
"homological_complex.congr_hom"
] | The null homotopic maps `Hσ` are compatible with the application of additive functors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopy_P_to_id : Π (q : ℕ),
homotopy (P q : K[X] ⟶ _) (𝟙 _) | | 0 := homotopy.refl _
| (q+1) := begin
refine homotopy.trans (homotopy.of_eq _)
(homotopy.trans
(homotopy.add (homotopy_P_to_id q) (homotopy.comp_left (homotopy_Hσ_to_zero q) (P q)))
(homotopy.of_eq _)),
{ unfold P, simp only [comp_add, comp_id], },
{ simp only [add_zero, comp_zer... | def | algebraic_topology.dold_kan.homotopy_P_to_id | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/homotopy_equivalence.lean | [
"algebraic_topology.dold_kan.normalized"
] | [
"homotopy",
"homotopy.add",
"homotopy.comp_left",
"homotopy.of_eq",
"homotopy.refl",
"homotopy.trans"
] | Inductive construction of homotopies from `P q` to `𝟙 _` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopy_Q_to_zero (q : ℕ) : homotopy (Q q : K[X] ⟶ _) 0 | homotopy.equiv_sub_zero.to_fun (homotopy_P_to_id X q).symm | def | algebraic_topology.dold_kan.homotopy_Q_to_zero | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/homotopy_equivalence.lean | [
"algebraic_topology.dold_kan.normalized"
] | [
"homotopy"
] | The complement projection `Q q` to `P q` is homotopic to zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopy_P_to_id_eventually_constant {q n : ℕ} (hqn : n<q):
((homotopy_P_to_id X (q+1)).hom n (n+1) : X _[n] ⟶ X _[n+1]) =
(homotopy_P_to_id X q).hom n (n+1) | begin
unfold homotopy_P_to_id,
simp only [homotopy_Hσ_to_zero, hσ'_eq_zero hqn (c_mk (n+1) n rfl), homotopy.trans_hom,
pi.add_apply, homotopy.of_eq_hom, pi.zero_apply, homotopy.add_hom, homotopy.comp_left_hom,
homotopy.null_homotopy'_hom, complex_shape.down_rel, eq_self_iff_true, dite_eq_ite,
if_true, c... | lemma | algebraic_topology.dold_kan.homotopy_P_to_id_eventually_constant | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/homotopy_equivalence.lean | [
"algebraic_topology.dold_kan.normalized"
] | [
"dite_eq_ite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homotopy_P_infty_to_id :
homotopy (P_infty : K[X] ⟶ _) (𝟙 _) | { hom := λ i j, (homotopy_P_to_id X (j+1)).hom i j,
zero' := λ i j hij, homotopy.zero _ i j hij,
comm := λ n, begin
cases n,
{ simpa only [homotopy.d_next_zero_chain_complex, homotopy.prev_d_chain_complex, P_f_0_eq,
zero_add, homological_complex.id_f, P_infty_f] using (homotopy_P_to_id X 2).comm 0,... | def | algebraic_topology.dold_kan.homotopy_P_infty_to_id | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/homotopy_equivalence.lean | [
"algebraic_topology.dold_kan.normalized"
] | [
"comm",
"homological_complex.id_f",
"homotopy",
"homotopy.d_next_succ_chain_complex",
"homotopy.d_next_zero_chain_complex",
"homotopy.prev_d_chain_complex",
"lt_add_one"
] | Construction of the homotopy from `P_infty` to the identity using eventually
(termwise) constant homotopies from `P q` to the identity for all `q` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopy_equiv_normalized_Moore_complex_alternating_face_map_complex {A : Type*}
[category A] [abelian A] {Y : simplicial_object A} :
homotopy_equiv ((normalized_Moore_complex A).obj Y) ((alternating_face_map_complex A).obj Y) | { hom := inclusion_of_Moore_complex_map Y,
inv := P_infty_to_normalized_Moore_complex Y,
homotopy_hom_inv_id := homotopy.of_eq (split_mono_inclusion_of_Moore_complex_map Y).id,
homotopy_inv_hom_id := homotopy.trans
(homotopy.of_eq (P_infty_to_normalized_Moore_complex_comp_inclusion_of_Moore_complex_map Y))
... | def | algebraic_topology.dold_kan.homotopy_equiv_normalized_Moore_complex_alternating_face_map_complex | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/homotopy_equivalence.lean | [
"algebraic_topology.dold_kan.normalized"
] | [
"homotopy.of_eq",
"homotopy.trans",
"homotopy_equiv"
] | The inclusion of the Moore complex in the alternating face map complex
is an homotopy equivalence | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
higher_faces_vanish.inclusion_of_Moore_complex_map (n : ℕ) :
higher_faces_vanish (n+1) ((inclusion_of_Moore_complex_map X).f (n+1)) | λ j hj,
begin
dsimp [inclusion_of_Moore_complex_map],
rw [← factor_thru_arrow _ _ (finset_inf_arrow_factors finset.univ
_ j (by simp only [finset.mem_univ])), assoc, kernel_subobject_arrow_comp, comp_zero],
end | lemma | algebraic_topology.dold_kan.higher_faces_vanish.inclusion_of_Moore_complex_map | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/normalized.lean | [
"algebraic_topology.dold_kan.functor_n"
] | [
"finset.mem_univ",
"finset.univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
factors_normalized_Moore_complex_P_infty (n : ℕ) :
subobject.factors (normalized_Moore_complex.obj_X X n) (P_infty.f n) | begin
cases n,
{ apply top_factors, },
{ rw [P_infty_f, normalized_Moore_complex.obj_X, finset_inf_factors],
intros i hi,
apply kernel_subobject_factors,
exact (higher_faces_vanish.of_P (n+1) n) i (le_add_self), }
end | lemma | algebraic_topology.dold_kan.factors_normalized_Moore_complex_P_infty | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/normalized.lean | [
"algebraic_topology.dold_kan.functor_n"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
P_infty_to_normalized_Moore_complex (X : simplicial_object A) : K[X] ⟶ N[X] | chain_complex.of_hom _ _ _ _ _ _
(λ n, factor_thru _ _ (factors_normalized_Moore_complex_P_infty n))
(λ n, begin
rw [← cancel_mono (normalized_Moore_complex.obj_X X n).arrow, assoc, assoc,
factor_thru_arrow, ← inclusion_of_Moore_complex_map_f,
← normalized_Moore_complex_obj_d, ← (inclusion_of_Moore_... | def | algebraic_topology.dold_kan.P_infty_to_normalized_Moore_complex | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/normalized.lean | [
"algebraic_topology.dold_kan.functor_n"
] | [
"chain_complex.of_hom"
] | P_infty factors through the normalized Moore complex | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
P_infty_to_normalized_Moore_complex_comp_inclusion_of_Moore_complex_map
(X : simplicial_object A) :
P_infty_to_normalized_Moore_complex X ≫ inclusion_of_Moore_complex_map X = P_infty | by tidy | lemma | algebraic_topology.dold_kan.P_infty_to_normalized_Moore_complex_comp_inclusion_of_Moore_complex_map | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/normalized.lean | [
"algebraic_topology.dold_kan.functor_n"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
P_infty_to_normalized_Moore_complex_naturality {X Y : simplicial_object A} (f : X ⟶ Y) :
alternating_face_map_complex.map f ≫ P_infty_to_normalized_Moore_complex Y =
P_infty_to_normalized_Moore_complex X ≫ normalized_Moore_complex.map f | by tidy | lemma | algebraic_topology.dold_kan.P_infty_to_normalized_Moore_complex_naturality | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/normalized.lean | [
"algebraic_topology.dold_kan.functor_n"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
P_infty_comp_P_infty_to_normalized_Moore_complex (X : simplicial_object A) :
P_infty ≫ P_infty_to_normalized_Moore_complex X = P_infty_to_normalized_Moore_complex X | by tidy | lemma | algebraic_topology.dold_kan.P_infty_comp_P_infty_to_normalized_Moore_complex | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/normalized.lean | [
"algebraic_topology.dold_kan.functor_n"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inclusion_of_Moore_complex_map_comp_P_infty (X : simplicial_object A) :
inclusion_of_Moore_complex_map X ≫ P_infty = inclusion_of_Moore_complex_map X | begin
ext n,
cases n,
{ dsimp, simp only [comp_id], },
{ exact (higher_faces_vanish.inclusion_of_Moore_complex_map n).comp_P_eq_self, },
end | lemma | algebraic_topology.dold_kan.inclusion_of_Moore_complex_map_comp_P_infty | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/normalized.lean | [
"algebraic_topology.dold_kan.functor_n"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_mono_inclusion_of_Moore_complex_map (X : simplicial_object A) :
split_mono (inclusion_of_Moore_complex_map X) | { retraction := P_infty_to_normalized_Moore_complex X,
id' := by simp only [← cancel_mono (inclusion_of_Moore_complex_map X), assoc, id_comp,
P_infty_to_normalized_Moore_complex_comp_inclusion_of_Moore_complex_map,
inclusion_of_Moore_complex_map_comp_P_infty], } | def | algebraic_topology.dold_kan.split_mono_inclusion_of_Moore_complex_map | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/normalized.lean | [
"algebraic_topology.dold_kan.functor_n"
] | [] | `inclusion_of_Moore_complex_map X` is a split mono. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
N₁_iso_normalized_Moore_complex_comp_to_karoubi :
N₁ ≅ (normalized_Moore_complex A ⋙ to_karoubi _) | { hom :=
{ app := λ X,
{ f := P_infty_to_normalized_Moore_complex X,
comm := by erw [comp_id, P_infty_comp_P_infty_to_normalized_Moore_complex] },
naturality' := λ X Y f, by simp only [functor.comp_map, normalized_Moore_complex_map,
P_infty_to_normalized_Moore_complex_naturality, karoubi.hom_ext, ... | def | algebraic_topology.dold_kan.N₁_iso_normalized_Moore_complex_comp_to_karoubi | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/normalized.lean | [
"algebraic_topology.dold_kan.functor_n"
] | [
"comm",
"homological_complex.comp_f"
] | When the category `A` is abelian,
the functor `N₁ : simplicial_object A ⥤ karoubi (chain_complex A ℕ)` defined
using `P_infty` identifies to the composition of the normalized Moore complex functor
and the inclusion in the Karoubi envelope. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
P_infty_comp_map_mono_eq_zero (X : simplicial_object C) {n : ℕ}
{Δ' : simplex_category} (i : Δ' ⟶ [n]) [hi : mono i] (h₁ : Δ'.len ≠ n) (h₂ : ¬is_δ₀ i) :
P_infty.f n ≫ X.map i.op = 0 | begin
unfreezingI { induction Δ' using simplex_category.rec with m, },
obtain ⟨k, hk⟩ := nat.exists_eq_add_of_lt (len_lt_of_mono i
(λ h, by { rw ← h at h₁, exact h₁ rfl, })),
simp only [len_mk] at hk,
cases k,
{ change n = m + 1 at hk,
unfreezingI { subst hk, obtain ⟨j, rfl⟩ := eq_δ_of_mono i, },
... | lemma | algebraic_topology.dold_kan.P_infty_comp_map_mono_eq_zero | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/n_comp_gamma.lean | [
"algebraic_topology.dold_kan.gamma_comp_n",
"algebraic_topology.dold_kan.n_reflects_iso"
] | [
"by_contra",
"fin.coe_succ",
"fin.coe_zero",
"fin.ext_iff",
"fin.zero_le",
"nat.exists_eq_add_of_lt",
"nat.lt_one_iff",
"simplex_category",
"simplex_category.rec",
"simplex_category.δ_comp_δ''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Γ₀_obj_termwise_map_mono_comp_P_infty (X : simplicial_object C) {Δ Δ' : simplex_category}
(i : Δ ⟶ Δ') [mono i] :
Γ₀.obj.termwise.map_mono (alternating_face_map_complex.obj X) i ≫ P_infty.f (Δ.len) =
P_infty.f (Δ'.len) ≫ X.map i.op | begin
unfreezingI
{ induction Δ using simplex_category.rec with n,
induction Δ' using simplex_category.rec with n', },
dsimp,
/- We start with the case `i` is an identity -/
by_cases n = n',
{ unfreezingI { subst h, },
simp only [simplex_category.eq_id_of_mono i, Γ₀.obj.termwise.map_mono_id, op_id, ... | lemma | algebraic_topology.dold_kan.Γ₀_obj_termwise_map_mono_comp_P_infty | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/n_comp_gamma.lean | [
"algebraic_topology.dold_kan.gamma_comp_n",
"algebraic_topology.dold_kan.n_reflects_iso"
] | [
"by_contradiction",
"fin.coe_zero",
"finset.mem_univ",
"is_empty.forall_iff",
"pow_zero",
"simplex_category",
"simplex_category.eq_id_of_mono",
"simplex_category.len",
"simplex_category.rec",
"simplex_category.δ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_trans : (N₁ : simplicial_object C ⥤ _) ⋙ Γ₂ ⟶ to_karoubi _ | { app := λ X,
{ f :=
{ app := λ Δ, (Γ₀.splitting K[X]).desc Δ (λ A, P_infty.f A.1.unop.len ≫ X.map (A.e.op)),
naturality' := λ Δ Δ' θ, begin
apply (Γ₀.splitting K[X]).hom_ext',
intro A,
change _ ≫ (Γ₀.obj K[X]).map θ ≫ _ = _,
simp only [splitting.ι_desc_assoc, assoc,
... | def | algebraic_topology.dold_kan.Γ₂N₁.nat_trans | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/n_comp_gamma.lean | [
"algebraic_topology.dold_kan.gamma_comp_n",
"algebraic_topology.dold_kan.n_reflects_iso"
] | [
"comm",
"hom_ext",
"homological_complex.comp_f",
"quiver.hom.unop_inj"
] | The natural transformation `N₁ ⋙ Γ₂ ⟶ to_karoubi (simplicial_object C)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compatibility_Γ₂N₁_Γ₂N₂ : to_karoubi (simplicial_object C) ⋙ N₂ ⋙ Γ₂ ≅ N₁ ⋙ Γ₂ | eq_to_iso (functor.congr_obj (functor_extension₁_comp_whiskering_left_to_karoubi _ _) (N₁ ⋙ Γ₂)) | def | algebraic_topology.dold_kan.compatibility_Γ₂N₁_Γ₂N₂ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/n_comp_gamma.lean | [
"algebraic_topology.dold_kan.gamma_comp_n",
"algebraic_topology.dold_kan.n_reflects_iso"
] | [] | The compatibility isomorphism relating `N₂ ⋙ Γ₂` and `N₁ ⋙ Γ₂`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_trans : (N₂ : karoubi (simplicial_object C) ⥤ _) ⋙ Γ₂ ⟶ 𝟭 _ | ((whiskering_left _ _ _).obj _).preimage (compatibility_Γ₂N₁_Γ₂N₂.hom ≫ Γ₂N₁.nat_trans) | def | algebraic_topology.dold_kan.Γ₂N₂.nat_trans | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/n_comp_gamma.lean | [
"algebraic_topology.dold_kan.gamma_comp_n",
"algebraic_topology.dold_kan.n_reflects_iso"
] | [] | The natural transformation `N₂ ⋙ Γ₂ ⟶ 𝟭 (simplicial_object C)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_trans_app_f_app (P : karoubi (simplicial_object C)) :
Γ₂N₂.nat_trans.app P = (N₂ ⋙ Γ₂).map P.decomp_id_i ≫
(compatibility_Γ₂N₁_Γ₂N₂.hom ≫ Γ₂N₁.nat_trans).app P.X ≫ P.decomp_id_p | whiskering_left_obj_preimage_app ((compatibility_Γ₂N₁_Γ₂N₂.hom ≫ Γ₂N₁.nat_trans)) P | lemma | algebraic_topology.dold_kan.Γ₂N₂.nat_trans_app_f_app | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/n_comp_gamma.lean | [
"algebraic_topology.dold_kan.gamma_comp_n",
"algebraic_topology.dold_kan.n_reflects_iso"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compatibility_Γ₂N₁_Γ₂N₂_nat_trans (X : simplicial_object C) :
Γ₂N₁.nat_trans.app X = (compatibility_Γ₂N₁_Γ₂N₂.app X).inv ≫
Γ₂N₂.nat_trans.app ((to_karoubi _).obj X) | begin
rw [← cancel_epi (compatibility_Γ₂N₁_Γ₂N₂.app X).hom, iso.hom_inv_id_assoc],
exact congr_app (((whiskering_left _ _ _).obj _).image_preimage
(compatibility_Γ₂N₁_Γ₂N₂.hom ≫ Γ₂N₁.nat_trans : _ ⟶ to_karoubi _ ⋙ 𝟭 _ )).symm X,
end | lemma | algebraic_topology.dold_kan.compatibility_Γ₂N₁_Γ₂N₂_nat_trans | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/n_comp_gamma.lean | [
"algebraic_topology.dold_kan.gamma_comp_n",
"algebraic_topology.dold_kan.n_reflects_iso"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
identity_N₂_objectwise (P : karoubi (simplicial_object C)) :
N₂Γ₂.inv.app (N₂.obj P) ≫ N₂.map (Γ₂N₂.nat_trans.app P) = 𝟙 (N₂.obj P) | begin
ext n,
have eq₁ : (N₂Γ₂.inv.app (N₂.obj P)).f.f n = P_infty.f n ≫ P.p.app (op [n]) ≫
(Γ₀.splitting (N₂.obj P).X).ι_summand (splitting.index_set.id (op [n])),
{ simp only [N₂Γ₂_inv_app_f_f, N₂_obj_p_f, assoc], },
have eq₂ : (Γ₀.splitting (N₂.obj P).X).ι_summand (splitting.index_set.id (op [n])) ≫
(... | lemma | algebraic_topology.dold_kan.identity_N₂_objectwise | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/n_comp_gamma.lean | [
"algebraic_topology.dold_kan.gamma_comp_n",
"algebraic_topology.dold_kan.n_reflects_iso"
] | [
"homological_complex.comp_f"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
identity_N₂ :
((𝟙 (N₂ : karoubi (simplicial_object C) ⥤ _ ) ◫ N₂Γ₂.inv) ≫
(Γ₂N₂.nat_trans ◫ 𝟙 N₂) : N₂ ⟶ N₂) = 𝟙 N₂ | by { ext P : 2, dsimp, rw [Γ₂.map_id, N₂.map_id, comp_id, id_comp, identity_N₂_objectwise P], } | lemma | algebraic_topology.dold_kan.identity_N₂ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/n_comp_gamma.lean | [
"algebraic_topology.dold_kan.gamma_comp_n",
"algebraic_topology.dold_kan.n_reflects_iso"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Γ₂N₂ : 𝟭 _ ≅ (N₂ : karoubi (simplicial_object C) ⥤ _) ⋙ Γ₂ | (as_iso Γ₂N₂.nat_trans).symm | def | algebraic_topology.dold_kan.Γ₂N₂ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/n_comp_gamma.lean | [
"algebraic_topology.dold_kan.gamma_comp_n",
"algebraic_topology.dold_kan.n_reflects_iso"
] | [] | The unit isomorphism of the Dold-Kan equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Γ₂N₁ : to_karoubi _ ≅ (N₁ : simplicial_object C ⥤ _) ⋙ Γ₂ | (as_iso Γ₂N₁.nat_trans).symm | def | algebraic_topology.dold_kan.Γ₂N₁ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/n_comp_gamma.lean | [
"algebraic_topology.dold_kan.gamma_comp_n",
"algebraic_topology.dold_kan.n_reflects_iso"
] | [] | The natural isomorphism `to_karoubi (simplicial_object C) ≅ N₁ ⋙ Γ₂`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compatibility_N₂_N₁_karoubi :
N₂ ⋙ (karoubi_chain_complex_equivalence C ℕ).functor =
karoubi_functor_category_embedding simplex_categoryᵒᵖ C ⋙ N₁ ⋙
(karoubi_chain_complex_equivalence (karoubi C) ℕ).functor ⋙
functor.map_homological_complex (karoubi_karoubi.equivalence C).inverse _ | begin
refine category_theory.functor.ext (λ P, _) (λ P Q f, _),
{ refine homological_complex.ext _ _,
{ ext n,
{ dsimp,
simp only [karoubi_P_infty_f, comp_id, P_infty_f_naturality, id_comp], },
{ refl, }, },
{ rintros _ n (rfl : n+1 = _),
ext,
have h := (alternating_face_map_... | lemma | algebraic_topology.dold_kan.compatibility_N₂_N₁_karoubi | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/n_reflects_iso.lean | [
"algebraic_topology.dold_kan.functor_n",
"algebraic_topology.dold_kan.decomposition",
"category_theory.idempotents.homological_complex",
"category_theory.idempotents.karoubi_karoubi"
] | [
"category_theory.functor.ext",
"comm",
"homological_complex.eq_to_hom_f",
"homological_complex.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
P : ℕ → (K[X] ⟶ K[X]) | | 0 := 𝟙 _
| (q+1) := P q ≫ (𝟙 _ + Hσ q) | def | algebraic_topology.dold_kan.P | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/projections.lean | [
"algebraic_topology.dold_kan.faces",
"category_theory.idempotents.basic"
] | [] | This is the inductive definition of the projections `P q : K[X] ⟶ K[X]`,
with `P 0 := 𝟙 _` and `P (q+1) := P q ≫ (𝟙 _ + Hσ q)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
P_f_0_eq (q : ℕ) : ((P q).f 0 : X _[0] ⟶ X _[0]) = 𝟙 _ | begin
induction q with q hq,
{ refl, },
{ unfold P,
simp only [homological_complex.add_f_apply, homological_complex.comp_f,
homological_complex.id_f, id_comp, hq, Hσ_eq_zero, add_zero], },
end | lemma | algebraic_topology.dold_kan.P_f_0_eq | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/projections.lean | [
"algebraic_topology.dold_kan.faces",
"category_theory.idempotents.basic"
] | [
"homological_complex.add_f_apply",
"homological_complex.comp_f",
"homological_complex.id_f"
] | All the `P q` coincide with `𝟙 _` in degree 0. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Q (q : ℕ) : K[X] ⟶ K[X] | 𝟙 _ - P q | def | algebraic_topology.dold_kan.Q | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/projections.lean | [
"algebraic_topology.dold_kan.faces",
"category_theory.idempotents.basic"
] | [] | `Q q` is the complement projection associated to `P q` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
P_add_Q (q : ℕ) : P q + Q q = 𝟙 K[X] | by { rw Q, abel, } | lemma | algebraic_topology.dold_kan.P_add_Q | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/projections.lean | [
"algebraic_topology.dold_kan.faces",
"category_theory.idempotents.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
P_add_Q_f (q n : ℕ) : (P q).f n + (Q q).f n = 𝟙 (X _[n]) | homological_complex.congr_hom (P_add_Q q) n | lemma | algebraic_topology.dold_kan.P_add_Q_f | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/projections.lean | [
"algebraic_topology.dold_kan.faces",
"category_theory.idempotents.basic"
] | [
"homological_complex.congr_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Q_eq_zero : (Q 0 : K[X] ⟶ _) = 0 | sub_self _ | lemma | algebraic_topology.dold_kan.Q_eq_zero | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/projections.lean | [
"algebraic_topology.dold_kan.faces",
"category_theory.idempotents.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Q_eq (q : ℕ) : (Q (q+1) : K[X] ⟶ _) = Q q - P q ≫ Hσ q | by { unfold Q P, simp only [comp_add, comp_id], abel, } | lemma | algebraic_topology.dold_kan.Q_eq | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/projections.lean | [
"algebraic_topology.dold_kan.faces",
"category_theory.idempotents.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.