statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
Q_f_0_eq (q : ℕ) : ((Q q).f 0 : X _[0] ⟶ X _[0]) = 0 | by simp only [homological_complex.sub_f_apply, homological_complex.id_f, Q, P_f_0_eq, sub_self] | lemma | algebraic_topology.dold_kan.Q_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.id_f",
"homological_complex.sub_f_apply"
] | All the `Q q` coincide with `0` in degree 0. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_P : Π (q n : ℕ), higher_faces_vanish q (((P q).f (n+1) : X _[n+1] ⟶ X _[n+1])) | | 0 := λ n j hj₁, by { exfalso, have hj₂ := fin.is_lt j, linarith, }
| (q+1) := λ n, by { unfold P, exact (of_P q n).induction, } | lemma | algebraic_topology.dold_kan.higher_faces_vanish.of_P | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/projections.lean | [
"algebraic_topology.dold_kan.faces",
"category_theory.idempotents.basic"
] | [
"fin.is_lt"
] | This lemma expresses the vanishing of
`(P q).f (n+1) ≫ X.δ k : X _[n+1] ⟶ X _[n]` when `k≠0` and `k≥n-q+2` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_P_eq_self {Y : C} {n q : ℕ} {φ : Y ⟶ X _[n+1]}
(v : higher_faces_vanish q φ) : φ ≫ (P q).f (n+1) = φ | begin
induction q with q hq,
{ unfold P,
apply comp_id, },
{ unfold P,
simp only [comp_add, homological_complex.comp_f, homological_complex.add_f_apply,
comp_id, ← assoc, hq v.of_succ, add_right_eq_self],
by_cases hqn : n<q,
{ exact v.of_succ.comp_Hσ_eq_zero hqn, },
{ cases nat.le.dest (... | lemma | algebraic_topology.dold_kan.higher_faces_vanish.comp_P_eq_self | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/projections.lean | [
"algebraic_topology.dold_kan.faces",
"category_theory.idempotents.basic"
] | [
"fin.coe_mk",
"fin.succ_mk",
"homological_complex.add_f_apply",
"homological_complex.comp_f"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_P_eq_self_iff {Y : C} {n q : ℕ} {φ : Y ⟶ X _[n+1]} :
φ ≫ (P q).f (n+1) = φ ↔ higher_faces_vanish q φ | begin
split,
{ intro hφ,
rw ← hφ,
apply higher_faces_vanish.of_comp,
apply higher_faces_vanish.of_P, },
{ exact higher_faces_vanish.comp_P_eq_self, },
end | lemma | algebraic_topology.dold_kan.comp_P_eq_self_iff | 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_f_idem (q n : ℕ) :
((P q).f n : X _[n] ⟶ _) ≫ ((P q).f n) = (P q).f n | begin
cases n,
{ rw [P_f_0_eq q, comp_id], },
{ exact (higher_faces_vanish.of_P q n).comp_P_eq_self, }
end | lemma | algebraic_topology.dold_kan.P_f_idem | 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_f_idem (q n : ℕ) :
((Q q).f n : X _[n] ⟶ _) ≫ ((Q q).f n) = (Q q).f n | idem_of_id_sub_idem _ (P_f_idem q n) | lemma | algebraic_topology.dold_kan.Q_f_idem | 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_idem (q : ℕ) : (P q : K[X] ⟶ K[X]) ≫ P q = P q | by { ext n, exact P_f_idem q n, } | lemma | algebraic_topology.dold_kan.P_idem | 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_idem (q : ℕ) : (Q q : K[X] ⟶ K[X]) ≫ Q q = Q q | by { ext n, exact Q_f_idem q n, } | lemma | algebraic_topology.dold_kan.Q_idem | 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 | |
nat_trans_P (q : ℕ) :
alternating_face_map_complex C ⟶ alternating_face_map_complex C | { app := λ X, P q,
naturality' := λ X Y f, begin
induction q with q hq,
{ unfold P,
dsimp only [alternating_face_map_complex],
rw [id_comp, comp_id], },
{ unfold P,
simp only [add_comp, comp_add, assoc, comp_id, hq],
congr' 1,
rw [← assoc, hq, assoc],
congr' 1,
ex... | def | algebraic_topology.dold_kan.nat_trans_P | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/projections.lean | [
"algebraic_topology.dold_kan.faces",
"category_theory.idempotents.basic"
] | [] | For each `q`, `P q` is a natural transformation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
P_f_naturality (q n : ℕ) {X Y : simplicial_object C} (f : X ⟶ Y) :
f.app (op [n]) ≫ (P q).f n = (P q).f n ≫ f.app (op [n]) | homological_complex.congr_hom ((nat_trans_P q).naturality f) n | lemma | algebraic_topology.dold_kan.P_f_naturality | 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_f_naturality (q n : ℕ) {X Y : simplicial_object C} (f : X ⟶ Y) :
f.app (op [n]) ≫ (Q q).f n = (Q q).f n ≫ f.app (op [n]) | begin
simp only [Q, homological_complex.sub_f_apply, homological_complex.id_f,
comp_sub, P_f_naturality, sub_comp, sub_left_inj],
dsimp,
simp only [comp_id, id_comp],
end | lemma | algebraic_topology.dold_kan.Q_f_naturality | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/projections.lean | [
"algebraic_topology.dold_kan.faces",
"category_theory.idempotents.basic"
] | [
"homological_complex.id_f",
"homological_complex.sub_f_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_trans_Q (q : ℕ) :
alternating_face_map_complex C ⟶ alternating_face_map_complex C | { app := λ X, Q q, } | def | algebraic_topology.dold_kan.nat_trans_Q | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/projections.lean | [
"algebraic_topology.dold_kan.faces",
"category_theory.idempotents.basic"
] | [] | For each `q`, `Q q` is a natural transformation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_P {D : Type*} [category D] [preadditive D]
(G : C ⥤ D) [G.additive] (X : simplicial_object C) (q n : ℕ) :
G.map ((P q : K[X] ⟶ _).f n) = (P q : K[((whiskering C D).obj G).obj X] ⟶ _).f n | begin
induction q with q hq,
{ unfold P,
apply G.map_id, },
{ unfold P,
simp only [comp_add, homological_complex.comp_f, homological_complex.add_f_apply,
comp_id, functor.map_add, functor.map_comp, hq, map_Hσ], }
end | lemma | algebraic_topology.dold_kan.map_P | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_Q {D : Type*} [category D] [preadditive D]
(G : C ⥤ D) [G.additive] (X : simplicial_object C) (q n : ℕ) :
G.map ((Q q : K[X] ⟶ _).f n) = (Q q : K[((whiskering C D).obj G).obj X] ⟶ _).f n | begin
rw [← add_right_inj (G.map ((P q : K[X] ⟶ _).f n)), ← G.map_add, map_P G X q n,
P_add_Q_f, P_add_Q_f],
apply G.map_id,
end | lemma | algebraic_topology.dold_kan.map_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_is_eventually_constant {q n : ℕ} (hqn : n ≤ q) :
((P (q+1)).f n : X _[n] ⟶ _ ) = (P q).f n | begin
cases n,
{ simp only [P_f_0_eq], },
{ unfold P,
simp only [add_right_eq_self, comp_add, homological_complex.comp_f,
homological_complex.add_f_apply, comp_id],
exact (higher_faces_vanish.of_P q n).comp_Hσ_eq_zero
(nat.succ_le_iff.mp hqn), },
end | lemma | algebraic_topology.dold_kan.P_is_eventually_constant | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/p_infty.lean | [
"algebraic_topology.dold_kan.projections",
"category_theory.idempotents.functor_categories",
"category_theory.idempotents.functor_extension"
] | [
"homological_complex.add_f_apply",
"homological_complex.comp_f"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Q_is_eventually_constant {q n : ℕ} (hqn : n ≤ q) :
((Q (q+1)).f n : X _[n] ⟶ _ ) = (Q q).f n | by simp only [Q, homological_complex.sub_f_apply, P_is_eventually_constant hqn] | lemma | algebraic_topology.dold_kan.Q_is_eventually_constant | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/p_infty.lean | [
"algebraic_topology.dold_kan.projections",
"category_theory.idempotents.functor_categories",
"category_theory.idempotents.functor_extension"
] | [
"homological_complex.sub_f_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
P_infty : K[X] ⟶ K[X] | chain_complex.of_hom _ _ _ _ _ _
(λ n, ((P n).f n : X _[n] ⟶ _ ))
(λ n, by simpa only [← P_is_eventually_constant (show n ≤ n, by refl),
alternating_face_map_complex.obj_d_eq] using (P (n+1)).comm (n+1) n) | def | algebraic_topology.dold_kan.P_infty | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/p_infty.lean | [
"algebraic_topology.dold_kan.projections",
"category_theory.idempotents.functor_categories",
"category_theory.idempotents.functor_extension"
] | [
"chain_complex.of_hom",
"comm"
] | The endomorphism `P_infty : K[X] ⟶ K[X]` obtained from the `P q` by passing to the limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Q_infty : K[X] ⟶ K[X] | 𝟙 _ - P_infty | def | algebraic_topology.dold_kan.Q_infty | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/p_infty.lean | [
"algebraic_topology.dold_kan.projections",
"category_theory.idempotents.functor_categories",
"category_theory.idempotents.functor_extension"
] | [] | The endomorphism `Q_infty : K[X] ⟶ K[X]` obtained from the `Q q` by passing to the limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
P_infty_f_0 : (P_infty.f 0 : X _[0] ⟶ X _[0]) = 𝟙 _ | rfl | lemma | algebraic_topology.dold_kan.P_infty_f_0 | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/p_infty.lean | [
"algebraic_topology.dold_kan.projections",
"category_theory.idempotents.functor_categories",
"category_theory.idempotents.functor_extension"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
P_infty_f (n : ℕ) : (P_infty.f n : X _[n] ⟶ X _[n] ) = (P n).f n | rfl | lemma | algebraic_topology.dold_kan.P_infty_f | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/p_infty.lean | [
"algebraic_topology.dold_kan.projections",
"category_theory.idempotents.functor_categories",
"category_theory.idempotents.functor_extension"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Q_infty_f_0 : (Q_infty.f 0 : X _[0] ⟶ X _[0]) = 0 | by { dsimp [Q_infty], simp only [sub_self], } | lemma | algebraic_topology.dold_kan.Q_infty_f_0 | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/p_infty.lean | [
"algebraic_topology.dold_kan.projections",
"category_theory.idempotents.functor_categories",
"category_theory.idempotents.functor_extension"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Q_infty_f (n : ℕ) : (Q_infty.f n : X _[n] ⟶ X _[n] ) = (Q n).f n | rfl | lemma | algebraic_topology.dold_kan.Q_infty_f | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/p_infty.lean | [
"algebraic_topology.dold_kan.projections",
"category_theory.idempotents.functor_categories",
"category_theory.idempotents.functor_extension"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
P_infty_f_naturality (n : ℕ) {X Y : simplicial_object C} (f : X ⟶ Y) :
f.app (op [n]) ≫ P_infty.f n = P_infty.f n ≫ f.app (op [n]) | P_f_naturality n n f | lemma | algebraic_topology.dold_kan.P_infty_f_naturality | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/p_infty.lean | [
"algebraic_topology.dold_kan.projections",
"category_theory.idempotents.functor_categories",
"category_theory.idempotents.functor_extension"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Q_infty_f_naturality (n : ℕ) {X Y : simplicial_object C} (f : X ⟶ Y) :
f.app (op [n]) ≫ Q_infty.f n = Q_infty.f n ≫ f.app (op [n]) | Q_f_naturality n n f | lemma | algebraic_topology.dold_kan.Q_infty_f_naturality | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/p_infty.lean | [
"algebraic_topology.dold_kan.projections",
"category_theory.idempotents.functor_categories",
"category_theory.idempotents.functor_extension"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
P_infty_f_idem (n : ℕ) :
(P_infty.f n : X _[n] ⟶ _) ≫ (P_infty.f n) = P_infty.f n | by simp only [P_infty_f, P_f_idem] | lemma | algebraic_topology.dold_kan.P_infty_f_idem | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/p_infty.lean | [
"algebraic_topology.dold_kan.projections",
"category_theory.idempotents.functor_categories",
"category_theory.idempotents.functor_extension"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
P_infty_idem : (P_infty : K[X] ⟶ _) ≫ P_infty = P_infty | by { ext n, exact P_infty_f_idem n, } | lemma | algebraic_topology.dold_kan.P_infty_idem | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/p_infty.lean | [
"algebraic_topology.dold_kan.projections",
"category_theory.idempotents.functor_categories",
"category_theory.idempotents.functor_extension"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Q_infty_f_idem (n : ℕ) :
(Q_infty.f n : X _[n] ⟶ _) ≫ (Q_infty.f n) = Q_infty.f n | Q_f_idem _ _ | lemma | algebraic_topology.dold_kan.Q_infty_f_idem | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/p_infty.lean | [
"algebraic_topology.dold_kan.projections",
"category_theory.idempotents.functor_categories",
"category_theory.idempotents.functor_extension"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Q_infty_idem : (Q_infty : K[X] ⟶ _) ≫ Q_infty = Q_infty | by { ext n, exact Q_infty_f_idem n, } | lemma | algebraic_topology.dold_kan.Q_infty_idem | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/p_infty.lean | [
"algebraic_topology.dold_kan.projections",
"category_theory.idempotents.functor_categories",
"category_theory.idempotents.functor_extension"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
P_infty_f_comp_Q_infty_f (n : ℕ) :
(P_infty.f n : X _[n] ⟶ _) ≫ Q_infty.f n = 0 | begin
dsimp only [Q_infty],
simp only [homological_complex.sub_f_apply, homological_complex.id_f, comp_sub, comp_id,
P_infty_f_idem, sub_self],
end | lemma | algebraic_topology.dold_kan.P_infty_f_comp_Q_infty_f | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/p_infty.lean | [
"algebraic_topology.dold_kan.projections",
"category_theory.idempotents.functor_categories",
"category_theory.idempotents.functor_extension"
] | [
"homological_complex.id_f",
"homological_complex.sub_f_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
P_infty_comp_Q_infty :
(P_infty : K[X] ⟶ _) ≫ Q_infty = 0 | by { ext n, apply P_infty_f_comp_Q_infty_f, } | lemma | algebraic_topology.dold_kan.P_infty_comp_Q_infty | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/p_infty.lean | [
"algebraic_topology.dold_kan.projections",
"category_theory.idempotents.functor_categories",
"category_theory.idempotents.functor_extension"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Q_infty_f_comp_P_infty_f (n : ℕ) :
(Q_infty.f n : X _[n] ⟶ _) ≫ P_infty.f n = 0 | begin
dsimp only [Q_infty],
simp only [homological_complex.sub_f_apply, homological_complex.id_f, sub_comp, id_comp,
P_infty_f_idem, sub_self],
end | lemma | algebraic_topology.dold_kan.Q_infty_f_comp_P_infty_f | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/p_infty.lean | [
"algebraic_topology.dold_kan.projections",
"category_theory.idempotents.functor_categories",
"category_theory.idempotents.functor_extension"
] | [
"homological_complex.id_f",
"homological_complex.sub_f_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Q_infty_comp_P_infty :
(Q_infty : K[X] ⟶ _) ≫ P_infty = 0 | by { ext n, apply Q_infty_f_comp_P_infty_f, } | lemma | algebraic_topology.dold_kan.Q_infty_comp_P_infty | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/p_infty.lean | [
"algebraic_topology.dold_kan.projections",
"category_theory.idempotents.functor_categories",
"category_theory.idempotents.functor_extension"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
P_infty_add_Q_infty :
(P_infty : K[X] ⟶ _) + Q_infty = 𝟙 _ | by { dsimp only [Q_infty], simp only [add_sub_cancel'_right], } | lemma | algebraic_topology.dold_kan.P_infty_add_Q_infty | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/p_infty.lean | [
"algebraic_topology.dold_kan.projections",
"category_theory.idempotents.functor_categories",
"category_theory.idempotents.functor_extension"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
P_infty_f_add_Q_infty_f (n : ℕ) :
(P_infty.f n : X _[n] ⟶ _ ) + Q_infty.f n = 𝟙 _ | homological_complex.congr_hom (P_infty_add_Q_infty) n | lemma | algebraic_topology.dold_kan.P_infty_f_add_Q_infty_f | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/p_infty.lean | [
"algebraic_topology.dold_kan.projections",
"category_theory.idempotents.functor_categories",
"category_theory.idempotents.functor_extension"
] | [
"homological_complex.congr_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_trans_P_infty :
alternating_face_map_complex C ⟶ alternating_face_map_complex C | { app := λ _, P_infty,
naturality' := λ X Y f, by { ext n, exact P_infty_f_naturality n f, }, } | def | algebraic_topology.dold_kan.nat_trans_P_infty | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/p_infty.lean | [
"algebraic_topology.dold_kan.projections",
"category_theory.idempotents.functor_categories",
"category_theory.idempotents.functor_extension"
] | [] | `P_infty` induces a natural transformation, i.e. an endomorphism of
the functor `alternating_face_map_complex C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_trans_P_infty_f (n : ℕ) | nat_trans_P_infty C ◫ 𝟙 (homological_complex.eval _ _ n) | def | algebraic_topology.dold_kan.nat_trans_P_infty_f | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/p_infty.lean | [
"algebraic_topology.dold_kan.projections",
"category_theory.idempotents.functor_categories",
"category_theory.idempotents.functor_extension"
] | [
"homological_complex.eval"
] | The natural transformation in each degree that is induced by `nat_trans_P_infty`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_P_infty_f {D : Type*} [category D] [preadditive D]
(G : C ⥤ D) [G.additive] (X : simplicial_object C) (n : ℕ) :
(P_infty : K[((whiskering C D).obj G).obj X] ⟶ _).f n =
G.map ((P_infty : alternating_face_map_complex.obj X ⟶ _).f n) | by simp only [P_infty_f, map_P] | lemma | algebraic_topology.dold_kan.map_P_infty_f | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/p_infty.lean | [
"algebraic_topology.dold_kan.projections",
"category_theory.idempotents.functor_categories",
"category_theory.idempotents.functor_extension"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
karoubi_P_infty_f {Y : karoubi (simplicial_object C)} (n : ℕ) :
((P_infty : K[(karoubi_functor_category_embedding _ _).obj Y] ⟶ _).f n).f =
Y.p.app (op [n]) ≫ (P_infty : K[Y.X] ⟶ _).f n | begin
-- We introduce P_infty endomorphisms P₁, P₂, P₃, P₄ on various objects Y₁, Y₂, Y₃, Y₄.
let Y₁ := (karoubi_functor_category_embedding _ _).obj Y,
let Y₂ := Y.X,
let Y₃ := (((whiskering _ _).obj (to_karoubi C)).obj Y.X),
let Y₄ := (karoubi_functor_category_embedding _ _).obj ((to_karoubi _).obj Y.X),
l... | lemma | algebraic_topology.dold_kan.karoubi_P_infty_f | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/p_infty.lean | [
"algebraic_topology.dold_kan.projections",
"category_theory.idempotents.functor_categories",
"category_theory.idempotents.functor_extension"
] | [] | Given an object `Y : karoubi (simplicial_object C)`, this lemma
computes `P_infty` for the associated object in `simplicial_object (karoubi C)`
in terms of `P_infty` for `Y.X : simplicial_object C` and `Y.p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
π_summand [has_zero_morphisms C] {Δ : simplex_categoryᵒᵖ} (A : index_set Δ) :
X.obj Δ ⟶ s.N A.1.unop.len | begin
refine (s.iso Δ).inv ≫ sigma.desc (λ B, _),
by_cases B = A,
{ exact eq_to_hom (by { subst h, refl, }), },
{ exact 0, },
end | def | simplicial_object.splitting.π_summand | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/split_simplicial_object.lean | [
"algebraic_topology.split_simplicial_object",
"algebraic_topology.dold_kan.degeneracies",
"algebraic_topology.dold_kan.functor_n"
] | [] | The projection on a summand of the coproduct decomposition given
by a splitting of a simplicial object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_π_summand_eq_id [has_zero_morphisms C] {Δ : simplex_categoryᵒᵖ} (A : index_set Δ) :
s.ι_summand A ≫ s.π_summand A = 𝟙 _ | begin
dsimp [ι_summand, π_summand],
simp only [summand, assoc, is_iso.hom_inv_id_assoc],
erw [colimit.ι_desc, cofan.mk_ι_app],
dsimp,
simp only [eq_self_iff_true, if_true],
end | lemma | simplicial_object.splitting.ι_π_summand_eq_id | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/split_simplicial_object.lean | [
"algebraic_topology.split_simplicial_object",
"algebraic_topology.dold_kan.degeneracies",
"algebraic_topology.dold_kan.functor_n"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι_π_summand_eq_zero [has_zero_morphisms C] {Δ : simplex_categoryᵒᵖ} (A B : index_set Δ)
(h : B ≠ A) : s.ι_summand A ≫ s.π_summand B = 0 | begin
dsimp [ι_summand, π_summand],
simp only [summand, assoc, is_iso.hom_inv_id_assoc],
erw [colimit.ι_desc, cofan.mk_ι_app],
apply dif_neg,
exact h.symm,
end | lemma | simplicial_object.splitting.ι_π_summand_eq_zero | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/split_simplicial_object.lean | [
"algebraic_topology.split_simplicial_object",
"algebraic_topology.dold_kan.degeneracies",
"algebraic_topology.dold_kan.functor_n"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decomposition_id (Δ : simplex_categoryᵒᵖ) :
𝟙 (X.obj Δ) = ∑ (A : index_set Δ), s.π_summand A ≫ s.ι_summand A | begin
apply s.hom_ext',
intro A,
rw [comp_id, comp_sum, finset.sum_eq_single A, ι_π_summand_eq_id_assoc],
{ intros B h₁ h₂,
rw [s.ι_π_summand_eq_zero_assoc _ _ h₂, zero_comp], },
{ simp only [finset.mem_univ, not_true, is_empty.forall_iff], },
end | lemma | simplicial_object.splitting.decomposition_id | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/split_simplicial_object.lean | [
"algebraic_topology.split_simplicial_object",
"algebraic_topology.dold_kan.degeneracies",
"algebraic_topology.dold_kan.functor_n"
] | [
"finset.mem_univ",
"is_empty.forall_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
σ_comp_π_summand_id_eq_zero {n : ℕ} (i : fin (n+1)) :
X.σ i ≫ s.π_summand (index_set.id (op [n+1])) = 0 | begin
apply s.hom_ext',
intro A,
dsimp only [simplicial_object.σ],
rw [comp_zero, s.ι_summand_epi_naturality_assoc A (simplex_category.σ i).op,
ι_π_summand_eq_zero],
symmetry,
change ¬ (A.epi_comp (simplex_category.σ i).op).eq_id,
rw index_set.eq_id_iff_len_eq,
have h := simplex_category.len_le_of_e... | lemma | simplicial_object.splitting.σ_comp_π_summand_id_eq_zero | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/split_simplicial_object.lean | [
"algebraic_topology.split_simplicial_object",
"algebraic_topology.dold_kan.degeneracies",
"algebraic_topology.dold_kan.functor_n"
] | [
"simplex_category.len_le_of_epi",
"simplex_category.σ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι_summand_comp_P_infty_eq_zero {X : simplicial_object C}
(s : simplicial_object.splitting X)
{n : ℕ} (A : simplicial_object.splitting.index_set (op [n]))
(hA : ¬ A.eq_id) :
s.ι_summand A ≫ P_infty.f n = 0 | begin
rw simplicial_object.splitting.index_set.eq_id_iff_mono at hA,
rw [simplicial_object.splitting.ι_summand_eq, assoc,
degeneracy_comp_P_infty X n A.e hA, comp_zero],
end | lemma | simplicial_object.splitting.ι_summand_comp_P_infty_eq_zero | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/split_simplicial_object.lean | [
"algebraic_topology.split_simplicial_object",
"algebraic_topology.dold_kan.degeneracies",
"algebraic_topology.dold_kan.functor_n"
] | [
"simplicial_object.splitting",
"simplicial_object.splitting.index_set",
"simplicial_object.splitting.index_set.eq_id_iff_mono",
"simplicial_object.splitting.ι_summand_eq"
] | If a simplicial object `X` in an additive category is split,
then `P_infty` vanishes on all the summands of `X _[n]` which do
not correspond to the identity of `[n]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_P_infty_eq_zero_iff {Z : C} {n : ℕ} (f : Z ⟶ X _[n]) :
f ≫ P_infty.f n = 0 ↔ f ≫ s.π_summand (index_set.id (op [n])) = 0 | begin
split,
{ intro h,
cases n,
{ dsimp at h,
rw [comp_id] at h,
rw [h, zero_comp], },
{ have h' := f ≫= P_infty_f_add_Q_infty_f (n+1),
dsimp at h',
rw [comp_id, comp_add, h, zero_add] at h',
rw [← h', assoc, Q_infty_f, decomposition_Q, preadditive.sum_comp,
preadd... | lemma | simplicial_object.splitting.comp_P_infty_eq_zero_iff | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/split_simplicial_object.lean | [
"algebraic_topology.split_simplicial_object",
"algebraic_topology.dold_kan.degeneracies",
"algebraic_topology.dold_kan.functor_n"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
P_infty_comp_π_summand_id (n : ℕ) :
P_infty.f n ≫ s.π_summand (index_set.id (op [n])) = s.π_summand (index_set.id (op [n])) | begin
conv_rhs { rw ← id_comp (s.π_summand _), },
symmetry,
rw [← sub_eq_zero, ← sub_comp, ← comp_P_infty_eq_zero_iff, sub_comp, id_comp,
P_infty_f_idem, sub_self],
end | lemma | simplicial_object.splitting.P_infty_comp_π_summand_id | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/split_simplicial_object.lean | [
"algebraic_topology.split_simplicial_object",
"algebraic_topology.dold_kan.degeneracies",
"algebraic_topology.dold_kan.functor_n"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
π_summand_comp_ι_summand_comp_P_infty_eq_P_infty (n : ℕ) :
s.π_summand (index_set.id (op [n])) ≫ s.ι_summand (index_set.id (op [n])) ≫ P_infty.f n =
P_infty.f n | begin
conv_rhs { rw ← id_comp (P_infty.f n), },
erw [s.decomposition_id, preadditive.sum_comp],
rw [fintype.sum_eq_single (index_set.id (op [n])), assoc],
rintros A (hA : ¬A.eq_id),
rw [assoc, s.ι_summand_comp_P_infty_eq_zero A hA, comp_zero],
end | lemma | simplicial_object.splitting.π_summand_comp_ι_summand_comp_P_infty_eq_P_infty | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/split_simplicial_object.lean | [
"algebraic_topology.split_simplicial_object",
"algebraic_topology.dold_kan.degeneracies",
"algebraic_topology.dold_kan.functor_n"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
d (i j : ℕ) : s.N i ⟶ s.N j | s.ι_summand (index_set.id (op [i])) ≫ K[X].d i j ≫ s.π_summand (index_set.id (op [j])) | def | simplicial_object.splitting.d | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/split_simplicial_object.lean | [
"algebraic_topology.split_simplicial_object",
"algebraic_topology.dold_kan.degeneracies",
"algebraic_topology.dold_kan.functor_n"
] | [] | The differentials `s.d i j : s.N i ⟶ s.N j` on nondegenerate simplices of a split
simplicial object are induced by the differentials on the alternating face map complex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_summand_comp_d_comp_π_summand_eq_zero (j k : ℕ) (A : index_set (op [j])) (hA : ¬A.eq_id) :
s.ι_summand A ≫ K[X].d j k ≫ s.π_summand (index_set.id (op [k])) = 0 | begin
rw A.eq_id_iff_mono at hA,
rw [← assoc, ← s.comp_P_infty_eq_zero_iff, assoc, ← P_infty.comm j k, s.ι_summand_eq, assoc,
degeneracy_comp_P_infty_assoc X j A.e hA, zero_comp, comp_zero],
end | lemma | simplicial_object.splitting.ι_summand_comp_d_comp_π_summand_eq_zero | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/split_simplicial_object.lean | [
"algebraic_topology.split_simplicial_object",
"algebraic_topology.dold_kan.degeneracies",
"algebraic_topology.dold_kan.functor_n"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nondeg_complex : chain_complex C ℕ | { X := s.N,
d := s.d,
shape' := λ i j hij, by simp only [d, K[X].shape i j hij, zero_comp, comp_zero],
d_comp_d' := λ i j k hij hjk, begin
simp only [d, assoc],
have eq : K[X].d i j ≫ 𝟙 (X.obj (op [j])) ≫ K[X].d j k ≫
s.π_summand (index_set.id (op [k])) = 0 :=
by erw [id_comp, homological_com... | def | simplicial_object.splitting.nondeg_complex | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/split_simplicial_object.lean | [
"algebraic_topology.split_simplicial_object",
"algebraic_topology.dold_kan.degeneracies",
"algebraic_topology.dold_kan.functor_n"
] | [
"chain_complex",
"finset.mem_compl",
"finset.mem_singleton"
] | If `s` is a splitting of a simplicial object `X` in a preadditive category,
`s.nondeg_complex` is a chain complex which is given in degree `n` by
the nondegenerate `n`-simplices of `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_karoubi_nondeg_complex_iso_N₁ : (to_karoubi _).obj s.nondeg_complex ≅ N₁.obj X | { hom :=
{ f :=
{ f := λ n, s.ι_summand (index_set.id (op [n])) ≫ P_infty.f n,
comm' := λ i j hij, begin
dsimp,
rw [assoc, assoc, assoc, π_summand_comp_ι_summand_comp_P_infty_eq_P_infty,
homological_complex.hom.comm],
end, },
comm := by { ext n, dsimp, rw [id_comp, assoc,... | def | simplicial_object.splitting.to_karoubi_nondeg_complex_iso_N₁ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/split_simplicial_object.lean | [
"algebraic_topology.split_simplicial_object",
"algebraic_topology.dold_kan.degeneracies",
"algebraic_topology.dold_kan.functor_n"
] | [
"comm",
"finset.mem_univ",
"homological_complex.comp_f",
"homological_complex.hom.comm",
"is_empty.forall_iff"
] | The chain complex `s.nondeg_complex` attached to a splitting of a simplicial object `X`
becomes isomorphic to the normalized Moore complex `N₁.obj X` defined as a formal direct
factor in the category `karoubi (chain_complex C ℕ)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nondeg_complex_functor : split C ⥤ chain_complex C ℕ | { obj := λ S, S.s.nondeg_complex,
map := λ S₁ S₂ Φ,
{ f := Φ.f,
comm' := λ i j hij, begin
dsimp,
erw [← ι_summand_naturality_symm_assoc Φ (splitting.index_set.id (op [i])),
((alternating_face_map_complex C).map Φ.F).comm_assoc i j],
simp only [assoc],
congr' 2,
apply S₁.s.h... | def | simplicial_object.split.nondeg_complex_functor | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/split_simplicial_object.lean | [
"algebraic_topology.split_simplicial_object",
"algebraic_topology.dold_kan.degeneracies",
"algebraic_topology.dold_kan.functor_n"
] | [
"chain_complex"
] | The functor which sends a split simplicial object in a preadditive category to
the chain complex which consists of nondegenerate simplices. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_karoubi_nondeg_complex_functor_iso_N₁ :
nondeg_complex_functor ⋙ to_karoubi (chain_complex C ℕ) ≅ forget C ⋙ dold_kan.N₁ | nat_iso.of_components (λ S, S.s.to_karoubi_nondeg_complex_iso_N₁)
(λ S₁ S₂ Φ, begin
ext n,
dsimp,
simp only [karoubi.comp_f, to_karoubi_map_f, homological_complex.comp_f,
nondeg_complex_functor_map_f, splitting.to_karoubi_nondeg_complex_iso_N₁_hom_f_f,
N₁_map_f, alternating_face_map_complex.ma... | def | simplicial_object.split.to_karoubi_nondeg_complex_functor_iso_N₁ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/split_simplicial_object.lean | [
"algebraic_topology.split_simplicial_object",
"algebraic_topology.dold_kan.degeneracies",
"algebraic_topology.dold_kan.functor_n"
] | [
"chain_complex",
"homological_complex.comp_f"
] | The natural isomorphism (in `karoubi (chain_complex C ℕ)`) between the chain complex
of nondegenerate simplices of a split simplicial object and the normalized Moore complex
defined as a formal direct factor of the alternating face map complex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl_trans_symm_aux (x : I × I) : ℝ | if (x.2 : ℝ) ≤ 1/2 then
x.1 * 2 * x.2
else
x.1 * (2 - 2 * x.2) | def | path.homotopy.refl_trans_symm_aux | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [] | Auxilliary function for `refl_trans_symm` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_refl_trans_symm_aux : continuous refl_trans_symm_aux | begin
refine continuous_if_le _ _ (continuous.continuous_on _) (continuous.continuous_on _) _,
{ continuity },
{ continuity },
{ continuity },
{ continuity },
intros x hx,
norm_num [hx, mul_assoc],
end | lemma | path.homotopy.continuous_refl_trans_symm_aux | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [
"continuity",
"continuous",
"continuous.continuous_on",
"continuous_if_le",
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl_trans_symm_aux_mem_I (x : I × I) : refl_trans_symm_aux x ∈ I | begin
dsimp only [refl_trans_symm_aux],
split_ifs,
{ split,
{ apply mul_nonneg,
{ apply mul_nonneg,
{ unit_interval },
{ norm_num } },
{ unit_interval } },
{ rw [mul_assoc],
apply mul_le_one,
{ unit_interval },
{ apply mul_nonneg,
{ norm_num },
... | lemma | path.homotopy.refl_trans_symm_aux_mem_I | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [
"mul_assoc",
"mul_le_one",
"unit_interval",
"unit_interval.le_one",
"unit_interval.nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl_trans_symm (p : path x₀ x₁) : homotopy (path.refl x₀) (p.trans p.symm) | { to_fun := λ x, p ⟨refl_trans_symm_aux x, refl_trans_symm_aux_mem_I x⟩,
continuous_to_fun := by continuity,
map_zero_left' := by norm_num [refl_trans_symm_aux],
map_one_left' := λ x, begin
dsimp only [refl_trans_symm_aux, path.coe_to_continuous_map, path.trans],
change _ = ite _ _ _,
split_ifs,
{... | def | path.homotopy.refl_trans_symm | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [
"continuity",
"homotopy",
"path",
"path.coe_to_continuous_map",
"path.extend",
"path.refl",
"path.symm",
"path.trans",
"set.Icc_extend_of_mem",
"set.mem_singleton_iff",
"unit_interval.le_one",
"unit_interval.mul_pos_mem_iff",
"unit_interval.two_mul_sub_one_mem_iff",
"zero_lt_two"
] | For any path `p` from `x₀` to `x₁`, we have a homotopy from the constant path based at `x₀` to
`p.trans p.symm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl_symm_trans (p : path x₀ x₁) : homotopy (path.refl x₁) (p.symm.trans p) | (refl_trans_symm p.symm).cast rfl $ congr_arg _ path.symm_symm | def | path.homotopy.refl_symm_trans | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [
"homotopy",
"path",
"path.refl",
"path.symm_symm"
] | For any path `p` from `x₀` to `x₁`, we have a homotopy from the constant path based at `x₁` to
`p.symm.trans p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans_refl_reparam_aux (t : I) : ℝ | if (t : ℝ) ≤ 1/2 then
2 * t
else
1 | def | path.homotopy.trans_refl_reparam_aux | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [] | Auxilliary function for `trans_refl_reparam` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_trans_refl_reparam_aux : continuous trans_refl_reparam_aux | begin
refine continuous_if_le _ _ (continuous.continuous_on _) (continuous.continuous_on _) _;
[continuity, continuity, continuity, continuity, skip],
intros x hx,
norm_num [hx]
end | lemma | path.homotopy.continuous_trans_refl_reparam_aux | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [
"continuity",
"continuous",
"continuous.continuous_on",
"continuous_if_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_refl_reparam_aux_mem_I (t : I) : trans_refl_reparam_aux t ∈ I | begin
unfold trans_refl_reparam_aux,
split_ifs; split; linarith [unit_interval.le_one t, unit_interval.nonneg t]
end | lemma | path.homotopy.trans_refl_reparam_aux_mem_I | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [
"unit_interval.le_one",
"unit_interval.nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_refl_reparam_aux_zero : trans_refl_reparam_aux 0 = 0 | by norm_num [trans_refl_reparam_aux] | lemma | path.homotopy.trans_refl_reparam_aux_zero | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_refl_reparam_aux_one : trans_refl_reparam_aux 1 = 1 | by norm_num [trans_refl_reparam_aux] | lemma | path.homotopy.trans_refl_reparam_aux_one | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_refl_reparam (p : path x₀ x₁) : p.trans (path.refl x₁) =
p.reparam (λ t, ⟨trans_refl_reparam_aux t, trans_refl_reparam_aux_mem_I t⟩) (by continuity)
(subtype.ext trans_refl_reparam_aux_zero) (subtype.ext trans_refl_reparam_aux_one) | begin
ext,
unfold trans_refl_reparam_aux,
simp only [path.trans_apply, not_le, coe_to_fun, function.comp_app],
split_ifs,
{ refl },
{ simp }
end | lemma | path.homotopy.trans_refl_reparam | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [
"continuity",
"path",
"path.refl",
"path.trans_apply",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_refl (p : path x₀ x₁) : homotopy (p.trans (path.refl x₁)) p | ((homotopy.reparam p (λ t, ⟨trans_refl_reparam_aux t, trans_refl_reparam_aux_mem_I t⟩)
(by continuity) (subtype.ext trans_refl_reparam_aux_zero)
(subtype.ext trans_refl_reparam_aux_one)).cast rfl (trans_refl_reparam p).symm).symm | def | path.homotopy.trans_refl | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [
"continuity",
"homotopy",
"path",
"path.refl",
"subtype.ext"
] | For any path `p` from `x₀` to `x₁`, we have a homotopy from `p.trans (path.refl x₁)` to `p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl_trans (p : path x₀ x₁) : homotopy ((path.refl x₀).trans p) p | (trans_refl p.symm).symm₂.cast (by simp) (by simp) | def | path.homotopy.refl_trans | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [
"homotopy",
"path",
"path.refl"
] | For any path `p` from `x₀` to `x₁`, we have a homotopy from `(path.refl x₀).trans p` to `p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans_assoc_reparam_aux (t : I) : ℝ | if (t : ℝ) ≤ 1/4 then
2 * t
else if (t : ℝ) ≤ 1/2 then
t + 1/4
else
1/2 * (t + 1) | def | path.homotopy.trans_assoc_reparam_aux | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [] | Auxilliary function for `trans_assoc_reparam`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_trans_assoc_reparam_aux : continuous trans_assoc_reparam_aux | begin
refine continuous_if_le _ _ (continuous.continuous_on _)
(continuous_if_le _ _ (continuous.continuous_on _)
(continuous.continuous_on _) _).continuous_on _;
[continuity, continuity, continuity, continuity, continuity, continuity, continuity,
skip, skip];
{ intros x hx,
norm_num [hx], ... | lemma | path.homotopy.continuous_trans_assoc_reparam_aux | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [
"continuity",
"continuous",
"continuous.continuous_on",
"continuous_if_le",
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_assoc_reparam_aux_mem_I (t : I) : trans_assoc_reparam_aux t ∈ I | begin
unfold trans_assoc_reparam_aux,
split_ifs; split; linarith [unit_interval.le_one t, unit_interval.nonneg t]
end | lemma | path.homotopy.trans_assoc_reparam_aux_mem_I | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [
"unit_interval.le_one",
"unit_interval.nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_assoc_reparam_aux_zero : trans_assoc_reparam_aux 0 = 0 | by norm_num [trans_assoc_reparam_aux] | lemma | path.homotopy.trans_assoc_reparam_aux_zero | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_assoc_reparam_aux_one : trans_assoc_reparam_aux 1 = 1 | by norm_num [trans_assoc_reparam_aux] | lemma | path.homotopy.trans_assoc_reparam_aux_one | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : path x₀ x₁) (q : path x₁ x₂) (r : path x₂ x₃) :
(p.trans q).trans r = (p.trans (q.trans r)).reparam
(λ t, ⟨trans_assoc_reparam_aux t, trans_assoc_reparam_aux_mem_I t⟩)
(by continuity) (subtype.ext trans_assoc_reparam_aux_zero)
(subtype.ext trans_assoc_reparam_aux... | begin
ext,
simp only [trans_assoc_reparam_aux, path.trans_apply, mul_inv_cancel_left₀, not_le,
function.comp_app, ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite,
subtype.coe_mk, path.coe_to_fun],
-- TODO: why does split_ifs not reduce the ifs??????
split_ifs with h₁ h₂ h₃ h₄... | lemma | path.homotopy.trans_assoc_reparam | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [
"bit0_eq_zero",
"continuity",
"mul_inv_cancel_left₀",
"mul_ite",
"one_div",
"one_ne_zero",
"path",
"path.coe_to_fun",
"path.trans_apply",
"ring",
"subtype.coe_mk",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_assoc {x₀ x₁ x₂ x₃ : X} (p : path x₀ x₁) (q : path x₁ x₂) (r : path x₂ x₃) :
homotopy ((p.trans q).trans r) (p.trans (q.trans r)) | ((homotopy.reparam (p.trans (q.trans r))
(λ t, ⟨trans_assoc_reparam_aux t, trans_assoc_reparam_aux_mem_I t⟩)
(by continuity) (subtype.ext trans_assoc_reparam_aux_zero)
(subtype.ext trans_assoc_reparam_aux_one)).cast rfl (trans_assoc_reparam p q r).symm).symm | def | path.homotopy.trans_assoc | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [
"continuity",
"homotopy",
"path",
"subtype.ext"
] | For paths `p q r`, we have a homotopy from `(p.trans q).trans r` to `p.trans (q.trans r)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fundamental_groupoid (X : Type u) | X | def | fundamental_groupoid | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [] | The fundamental groupoid of a space `X` is defined to be a type synonym for `X`, and we subsequently
put a `category_theory.groupoid` structure on it. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_eq (x y z : fundamental_groupoid X) (p : x ⟶ y) (q : y ⟶ z) : p ≫ q = p.comp q | rfl | lemma | comp_eq | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [
"fundamental_groupoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_eq_path_refl (x : fundamental_groupoid X) : 𝟙 x = ⟦path.refl x⟧ | rfl | lemma | id_eq_path_refl | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [
"fundamental_groupoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fundamental_groupoid_functor : Top ⥤ category_theory.Groupoid | { obj := λ X, { α := fundamental_groupoid X },
map := λ X Y f,
{ obj := f,
map := λ x y p, p.map_fn f,
map_id' := λ X, rfl,
map_comp' := λ x y z p q, quotient.induction_on₂ p q $ λ a b,
by simp [comp_eq, ← path.homotopic.map_lift, ← path.homotopic.comp_lift] },
map_id' := begin
intro X,
... | def | fundamental_groupoid_functor | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [
"Top",
"category_theory.Groupoid",
"comp_eq",
"fundamental_groupoid",
"path.homotopic.comp_lift",
"path.homotopic.map_lift",
"path.map_map",
"quotient.eq",
"quotient.map_mk"
] | The functor sending a topological space `X` to its fundamental groupoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_eq {X Y : Top} {x₀ x₁ : X} (f : C(X, Y)) (p : path.homotopic.quotient x₀ x₁) :
(πₘ f).map p = p.map_fn f | rfl | lemma | map_eq | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [
"Top",
"path.homotopic.quotient"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_top {X : Top} (x : πₓ X) : X | x | def | to_top | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [
"Top"
] | Help the typechecker by converting a point in a groupoid back to a point in
the underlying topological space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
from_top {X : Top} (x : X) : πₓ X | x | def | from_top | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [
"Top"
] | Help the typechecker by converting a point in a topological space to a
point in the fundamental groupoid of that space | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_path {X : Top} {x₀ x₁ : πₓ X} (p : x₀ ⟶ x₁) :
path.homotopic.quotient x₀ x₁ | p | def | to_path | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [
"Top",
"path.homotopic.quotient"
] | Help the typechecker by converting an arrow in the fundamental groupoid of
a topological space back to a path in that space (i.e., `path.homotopic.quotient`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
from_path {X : Top} {x₀ x₁ : X} (p : path.homotopic.quotient x₀ x₁) : (x₀ ⟶ x₁) | p | def | from_path | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/basic.lean | [
"category_theory.category.Groupoid",
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.homotopy.path"
] | [
"Top",
"path.homotopic.quotient"
] | Help the typechecker by convering a path in a topological space to an arrow in the
fundamental groupoid of that space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fundamental_group (X : Type u) [topological_space X] (x : X) | @Aut (fundamental_groupoid X) _ x | def | fundamental_group | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/fundamental_group.lean | [
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.path_connected",
"topology.homotopy.path",
"algebraic_topology.fundamental_groupoid.basic"
] | [
"fundamental_groupoid",
"topological_space"
] | The fundamental group is the automorphism group (vertex group) of the basepoint
in the fundamental groupoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fundamental_group_mul_equiv_of_path (p : path x₀ x₁) :
fundamental_group X x₀ ≃* fundamental_group X x₁ | Aut.Aut_mul_equiv_of_iso (as_iso ⟦p⟧) | def | fundamental_group.fundamental_group_mul_equiv_of_path | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/fundamental_group.lean | [
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.path_connected",
"topology.homotopy.path",
"algebraic_topology.fundamental_groupoid.basic"
] | [
"fundamental_group",
"path"
] | Get an isomorphism between the fundamental groups at two points given a path | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fundamental_group_mul_equiv_of_path_connected [path_connected_space X] :
(fundamental_group X x₀) ≃* (fundamental_group X x₁) | fundamental_group_mul_equiv_of_path (path_connected_space.some_path x₀ x₁) | def | fundamental_group.fundamental_group_mul_equiv_of_path_connected | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/fundamental_group.lean | [
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.path_connected",
"topology.homotopy.path",
"algebraic_topology.fundamental_groupoid.basic"
] | [
"fundamental_group",
"path_connected_space",
"path_connected_space.some_path"
] | The fundamental group of a path connected space is independent of the choice of basepoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_arrow {X : Top} {x : X} (p : fundamental_group X x) : x ⟶ x | p.hom | abbreviation | fundamental_group.to_arrow | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/fundamental_group.lean | [
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.path_connected",
"topology.homotopy.path",
"algebraic_topology.fundamental_groupoid.basic"
] | [
"Top",
"fundamental_group"
] | An element of the fundamental group as an arrow in the fundamental groupoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_path {X : Top} {x : X} (p : fundamental_group X x) :
path.homotopic.quotient x x | to_arrow p | abbreviation | fundamental_group.to_path | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/fundamental_group.lean | [
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.path_connected",
"topology.homotopy.path",
"algebraic_topology.fundamental_groupoid.basic"
] | [
"Top",
"fundamental_group",
"path.homotopic.quotient",
"to_path"
] | An element of the fundamental group as a quotient of homotopic paths. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
from_arrow {X : Top} {x : X} (p : x ⟶ x) : fundamental_group X x | ⟨p, category_theory.groupoid.inv p⟩ | abbreviation | fundamental_group.from_arrow | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/fundamental_group.lean | [
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.path_connected",
"topology.homotopy.path",
"algebraic_topology.fundamental_groupoid.basic"
] | [
"Top",
"fundamental_group"
] | An element of the fundamental group, constructed from an arrow in the fundamental groupoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
from_path {X : Top} {x : X} (p : path.homotopic.quotient x x) :
fundamental_group X x | from_arrow p | abbreviation | fundamental_group.from_path | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/fundamental_group.lean | [
"category_theory.groupoid",
"topology.category.Top.basic",
"topology.path_connected",
"topology.homotopy.path",
"algebraic_topology.fundamental_groupoid.basic"
] | [
"Top",
"from_path",
"fundamental_group",
"path.homotopic.quotient"
] | An element of the fundamental gorup, constructed from a quotient of homotopic paths. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
path01 : path (0 : I) 1 | { to_fun := id, source' := rfl, target' := rfl } | def | unit_interval.path01 | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/induced_maps.lean | [
"topology.homotopy.equiv",
"category_theory.equivalence",
"algebraic_topology.fundamental_groupoid.product"
] | [
"path"
] | The path 0 ⟶ 1 in I | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
upath01 : path (ulift.up 0 : ulift.{u} I) (ulift.up 1) | { to_fun := ulift.up, source' := rfl, target' := rfl } | def | unit_interval.upath01 | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/induced_maps.lean | [
"topology.homotopy.equiv",
"category_theory.equivalence",
"algebraic_topology.fundamental_groupoid.product"
] | [
"path"
] | The path 0 ⟶ 1 in ulift I | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uhpath01 : @from_top (Top.of $ ulift.{u} I) (ulift.up (0 : I)) ⟶ from_top (ulift.up 1) | ⟦upath01⟧ | def | unit_interval.uhpath01 | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/induced_maps.lean | [
"topology.homotopy.equiv",
"category_theory.equivalence",
"algebraic_topology.fundamental_groupoid.product"
] | [
"Top.of",
"from_top"
] | The homotopy path class of 0 → 1 in `ulift I` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hcast {X : Top} {x₀ x₁ : X} (hx : x₀ = x₁) : from_top x₀ ⟶ from_top x₁ | eq_to_hom hx | abbreviation | continuous_map.homotopy.hcast | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/induced_maps.lean | [
"topology.homotopy.equiv",
"category_theory.equivalence",
"algebraic_topology.fundamental_groupoid.product"
] | [
"Top",
"from_top"
] | Abbreviation for `eq_to_hom` that accepts points in a topological space | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hcast_def {X : Top} {x₀ x₁ : X} (hx₀ : x₀ = x₁) : hcast hx₀ = eq_to_hom hx₀ | rfl | lemma | continuous_map.homotopy.hcast_def | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/induced_maps.lean | [
"topology.homotopy.equiv",
"category_theory.equivalence",
"algebraic_topology.fundamental_groupoid.product"
] | [
"Top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
heq_path_of_eq_image : (πₘ f).map ⟦p⟧ == (πₘ g).map ⟦q⟧ | by { simp only [map_eq, ← path.homotopic.map_lift], apply path.homotopic.hpath_hext, exact hfg, } | lemma | continuous_map.homotopy.heq_path_of_eq_image | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/induced_maps.lean | [
"topology.homotopy.equiv",
"category_theory.equivalence",
"algebraic_topology.fundamental_groupoid.product"
] | [
"map_eq",
"path.homotopic.hpath_hext",
"path.homotopic.map_lift"
] | If `f(p(t) = g(q(t))` for two paths `p` and `q`, then the induced path homotopy classes
`f(p)` and `g(p)` are the same as well, despite having a priori different types | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
start_path : f x₀ = g x₂ | by { convert hfg 0; simp only [path.source], } | lemma | continuous_map.homotopy.start_path | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/induced_maps.lean | [
"topology.homotopy.equiv",
"category_theory.equivalence",
"algebraic_topology.fundamental_groupoid.product"
] | [
"path.source"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
end_path : f x₁ = g x₃ | by { convert hfg 1; simp only [path.target], } | lemma | continuous_map.homotopy.end_path | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/induced_maps.lean | [
"topology.homotopy.equiv",
"category_theory.equivalence",
"algebraic_topology.fundamental_groupoid.product"
] | [
"path.target"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_path_of_eq_image :
(πₘ f).map ⟦p⟧ = hcast (start_path hfg) ≫ (πₘ g).map ⟦q⟧ ≫ hcast (end_path hfg).symm | by { rw functor.conj_eq_to_hom_iff_heq, exact heq_path_of_eq_image hfg } | lemma | continuous_map.homotopy.eq_path_of_eq_image | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/induced_maps.lean | [
"topology.homotopy.equiv",
"category_theory.equivalence",
"algebraic_topology.fundamental_groupoid.product"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ulift_map : C(Top.of (ulift.{u} I × X), Y) | ⟨λ x, H (x.1.down, x.2),
H.continuous.comp ((continuous_induced_dom.comp continuous_fst).prod_mk continuous_snd)⟩ | def | continuous_map.homotopy.ulift_map | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/induced_maps.lean | [
"topology.homotopy.equiv",
"category_theory.equivalence",
"algebraic_topology.fundamental_groupoid.product"
] | [
"Top.of",
"continuous_fst",
"continuous_snd"
] | Interpret a homotopy `H : C(I × X, Y) as a map C(ulift I × X, Y) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ulift_apply (i : ulift.{u} I) (x : X) : H.ulift_map (i, x) = H (i.down, x) | rfl | lemma | continuous_map.homotopy.ulift_apply | algebraic_topology.fundamental_groupoid | src/algebraic_topology/fundamental_groupoid/induced_maps.lean | [
"topology.homotopy.equiv",
"category_theory.equivalence",
"algebraic_topology.fundamental_groupoid.product"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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