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vanishing_ideal (t : set (projective_spectrum 𝒜)) : homogeneous_ideal 𝒜 | ⨅ (x : projective_spectrum 𝒜) (h : x ∈ t), x.as_homogeneous_ideal | def | projective_spectrum.vanishing_ideal | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"homogeneous_ideal",
"projective_spectrum"
] | The vanishing ideal of a set `t` of points of the projective spectrum of a commutative ring `R`
is the intersection of all the relevant homogeneous prime ideals in the set `t`.
An element `f` of `A` can be thought of as a dependent function on the projective spectrum of `𝒜`.
At a point `x` (a homogeneous prime ideal)... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_vanishing_ideal (t : set (projective_spectrum 𝒜)) :
(vanishing_ideal t : set A) =
{f | ∀ x : projective_spectrum 𝒜, x ∈ t → f ∈ x.as_homogeneous_ideal} | begin
ext f,
rw [vanishing_ideal, set_like.mem_coe, ← homogeneous_ideal.mem_iff,
homogeneous_ideal.to_ideal_infi, submodule.mem_infi],
apply forall_congr (λ x, _),
rw [homogeneous_ideal.to_ideal_infi, submodule.mem_infi, homogeneous_ideal.mem_iff],
end | lemma | projective_spectrum.coe_vanishing_ideal | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"homogeneous_ideal.mem_iff",
"homogeneous_ideal.to_ideal_infi",
"projective_spectrum",
"set_like.mem_coe",
"submodule.mem_infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_vanishing_ideal (t : set (projective_spectrum 𝒜)) (f : A) :
f ∈ vanishing_ideal t ↔
∀ x : projective_spectrum 𝒜, x ∈ t → f ∈ x.as_homogeneous_ideal | by rw [← set_like.mem_coe, coe_vanishing_ideal, set.mem_set_of_eq] | lemma | projective_spectrum.mem_vanishing_ideal | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"projective_spectrum",
"set_like.mem_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vanishing_ideal_singleton (x : projective_spectrum 𝒜) :
vanishing_ideal ({x} : set (projective_spectrum 𝒜)) = x.as_homogeneous_ideal | by simp [vanishing_ideal] | lemma | projective_spectrum.vanishing_ideal_singleton | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"projective_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_zero_locus_iff_le_vanishing_ideal (t : set (projective_spectrum 𝒜))
(I : ideal A) :
t ⊆ zero_locus 𝒜 I ↔ I ≤ (vanishing_ideal t).to_ideal | ⟨λ h f k, (mem_vanishing_ideal _ _).mpr (λ x j, (mem_zero_locus _ _ _).mpr (h j) k), λ h,
λ x j, (mem_zero_locus _ _ _).mpr (le_trans h (λ f h, ((mem_vanishing_ideal _ _).mp h) x j))⟩ | lemma | projective_spectrum.subset_zero_locus_iff_le_vanishing_ideal | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"ideal",
"projective_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gc_ideal : @galois_connection
(ideal A) (set (projective_spectrum 𝒜))ᵒᵈ _ _
(λ I, zero_locus 𝒜 I) (λ t, (vanishing_ideal t).to_ideal) | λ I t, subset_zero_locus_iff_le_vanishing_ideal t I | lemma | projective_spectrum.gc_ideal | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"galois_connection",
"ideal",
"projective_spectrum"
] | `zero_locus` and `vanishing_ideal` form a galois connection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gc_set : @galois_connection
(set A) (set (projective_spectrum 𝒜))ᵒᵈ _ _
(λ s, zero_locus 𝒜 s) (λ t, vanishing_ideal t) | have ideal_gc : galois_connection (ideal.span) coe := (submodule.gi A _).gc,
by simpa [zero_locus_span, function.comp] using galois_connection.compose ideal_gc (gc_ideal 𝒜) | lemma | projective_spectrum.gc_set | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"galois_connection",
"galois_connection.compose",
"ideal.span",
"projective_spectrum",
"submodule.gi"
] | `zero_locus` and `vanishing_ideal` form a galois connection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gc_homogeneous_ideal : @galois_connection
(homogeneous_ideal 𝒜) (set (projective_spectrum 𝒜))ᵒᵈ _ _
(λ I, zero_locus 𝒜 I) (λ t, (vanishing_ideal t)) | λ I t, by simpa [show I.to_ideal ≤ (vanishing_ideal t).to_ideal ↔ I ≤ (vanishing_ideal t),
from iff.rfl] using subset_zero_locus_iff_le_vanishing_ideal t I.to_ideal | lemma | projective_spectrum.gc_homogeneous_ideal | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"galois_connection",
"homogeneous_ideal",
"projective_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_zero_locus_iff_subset_vanishing_ideal (t : set (projective_spectrum 𝒜))
(s : set A) :
t ⊆ zero_locus 𝒜 s ↔ s ⊆ vanishing_ideal t | (gc_set _) s t | lemma | projective_spectrum.subset_zero_locus_iff_subset_vanishing_ideal | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"projective_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_vanishing_ideal_zero_locus (s : set A) :
s ⊆ vanishing_ideal (zero_locus 𝒜 s) | (gc_set _).le_u_l s | lemma | projective_spectrum.subset_vanishing_ideal_zero_locus | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal_le_vanishing_ideal_zero_locus (I : ideal A) :
I ≤ (vanishing_ideal (zero_locus 𝒜 I)).to_ideal | (gc_ideal _).le_u_l I | lemma | projective_spectrum.ideal_le_vanishing_ideal_zero_locus | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homogeneous_ideal_le_vanishing_ideal_zero_locus (I : homogeneous_ideal 𝒜) :
I ≤ vanishing_ideal (zero_locus 𝒜 I) | (gc_homogeneous_ideal _).le_u_l I | lemma | projective_spectrum.homogeneous_ideal_le_vanishing_ideal_zero_locus | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"homogeneous_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_zero_locus_vanishing_ideal (t : set (projective_spectrum 𝒜)) :
t ⊆ zero_locus 𝒜 (vanishing_ideal t) | (gc_ideal _).l_u_le t | lemma | projective_spectrum.subset_zero_locus_vanishing_ideal | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"projective_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_anti_mono {s t : set A} (h : s ⊆ t) : zero_locus 𝒜 t ⊆ zero_locus 𝒜 s | (gc_set _).monotone_l h | lemma | projective_spectrum.zero_locus_anti_mono | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_anti_mono_ideal {s t : ideal A} (h : s ≤ t) :
zero_locus 𝒜 (t : set A) ⊆ zero_locus 𝒜 (s : set A) | (gc_ideal _).monotone_l h | lemma | projective_spectrum.zero_locus_anti_mono_ideal | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_anti_mono_homogeneous_ideal {s t : homogeneous_ideal 𝒜} (h : s ≤ t) :
zero_locus 𝒜 (t : set A) ⊆ zero_locus 𝒜 (s : set A) | (gc_homogeneous_ideal _).monotone_l h | lemma | projective_spectrum.zero_locus_anti_mono_homogeneous_ideal | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"homogeneous_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vanishing_ideal_anti_mono {s t : set (projective_spectrum 𝒜)} (h : s ⊆ t) :
vanishing_ideal t ≤ vanishing_ideal s | (gc_ideal _).monotone_u h | lemma | projective_spectrum.vanishing_ideal_anti_mono | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"projective_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_bot :
zero_locus 𝒜 ((⊥ : ideal A) : set A) = set.univ | (gc_ideal 𝒜).l_bot | lemma | projective_spectrum.zero_locus_bot | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_singleton_zero :
zero_locus 𝒜 ({0} : set A) = set.univ | zero_locus_bot _ | lemma | projective_spectrum.zero_locus_singleton_zero | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_empty :
zero_locus 𝒜 (∅ : set A) = set.univ | (gc_set 𝒜).l_bot | lemma | projective_spectrum.zero_locus_empty | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vanishing_ideal_univ :
vanishing_ideal (∅ : set (projective_spectrum 𝒜)) = ⊤ | by simpa using (gc_ideal _).u_top | lemma | projective_spectrum.vanishing_ideal_univ | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"projective_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_empty_of_one_mem {s : set A} (h : (1:A) ∈ s) :
zero_locus 𝒜 s = ∅ | set.eq_empty_iff_forall_not_mem.mpr $ λ x hx,
(infer_instance : x.as_homogeneous_ideal.to_ideal.is_prime).ne_top $
x.as_homogeneous_ideal.to_ideal.eq_top_iff_one.mpr $ hx h | lemma | projective_spectrum.zero_locus_empty_of_one_mem | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_singleton_one :
zero_locus 𝒜 ({1} : set A) = ∅ | zero_locus_empty_of_one_mem 𝒜 (set.mem_singleton (1 : A)) | lemma | projective_spectrum.zero_locus_singleton_one | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"set.mem_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_univ :
zero_locus 𝒜 (set.univ : set A) = ∅ | zero_locus_empty_of_one_mem _ (set.mem_univ 1) | lemma | projective_spectrum.zero_locus_univ | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"set.mem_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_sup_ideal (I J : ideal A) :
zero_locus 𝒜 ((I ⊔ J : ideal A) : set A) = zero_locus _ I ∩ zero_locus _ J | (gc_ideal 𝒜).l_sup | lemma | projective_spectrum.zero_locus_sup_ideal | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_sup_homogeneous_ideal (I J : homogeneous_ideal 𝒜) :
zero_locus 𝒜 ((I ⊔ J : homogeneous_ideal 𝒜) : set A) = zero_locus _ I ∩ zero_locus _ J | (gc_homogeneous_ideal 𝒜).l_sup | lemma | projective_spectrum.zero_locus_sup_homogeneous_ideal | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"homogeneous_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_union (s s' : set A) :
zero_locus 𝒜 (s ∪ s') = zero_locus _ s ∩ zero_locus _ s' | (gc_set 𝒜).l_sup | lemma | projective_spectrum.zero_locus_union | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vanishing_ideal_union (t t' : set (projective_spectrum 𝒜)) :
vanishing_ideal (t ∪ t') = vanishing_ideal t ⊓ vanishing_ideal t' | by ext1; convert (gc_ideal 𝒜).u_inf | lemma | projective_spectrum.vanishing_ideal_union | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"projective_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_supr_ideal {γ : Sort*} (I : γ → ideal A) :
zero_locus _ ((⨆ i, I i : ideal A) : set A) = (⋂ i, zero_locus 𝒜 (I i)) | (gc_ideal 𝒜).l_supr | lemma | projective_spectrum.zero_locus_supr_ideal | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_supr_homogeneous_ideal {γ : Sort*} (I : γ → homogeneous_ideal 𝒜) :
zero_locus _ ((⨆ i, I i : homogeneous_ideal 𝒜) : set A) = (⋂ i, zero_locus 𝒜 (I i)) | (gc_homogeneous_ideal 𝒜).l_supr | lemma | projective_spectrum.zero_locus_supr_homogeneous_ideal | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"homogeneous_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_Union {γ : Sort*} (s : γ → set A) :
zero_locus 𝒜 (⋃ i, s i) = (⋂ i, zero_locus 𝒜 (s i)) | (gc_set 𝒜).l_supr | lemma | projective_spectrum.zero_locus_Union | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_bUnion (s : set (set A)) :
zero_locus 𝒜 (⋃ s' ∈ s, s' : set A) = ⋂ s' ∈ s, zero_locus 𝒜 s' | by simp only [zero_locus_Union] | lemma | projective_spectrum.zero_locus_bUnion | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vanishing_ideal_Union {γ : Sort*} (t : γ → set (projective_spectrum 𝒜)) :
vanishing_ideal (⋃ i, t i) = (⨅ i, vanishing_ideal (t i)) | homogeneous_ideal.to_ideal_injective $
by convert (gc_ideal 𝒜).u_infi; exact homogeneous_ideal.to_ideal_infi _ | lemma | projective_spectrum.vanishing_ideal_Union | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"homogeneous_ideal.to_ideal_infi",
"homogeneous_ideal.to_ideal_injective",
"projective_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_inf (I J : ideal A) :
zero_locus 𝒜 ((I ⊓ J : ideal A) : set A) = zero_locus 𝒜 I ∪ zero_locus 𝒜 J | set.ext $ λ x, x.is_prime.inf_le | lemma | projective_spectrum.zero_locus_inf | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"ideal",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
union_zero_locus (s s' : set A) :
zero_locus 𝒜 s ∪ zero_locus 𝒜 s' = zero_locus 𝒜 ((ideal.span s) ⊓ (ideal.span s'): ideal A) | by { rw zero_locus_inf, simp } | lemma | projective_spectrum.union_zero_locus | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"ideal",
"ideal.span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_mul_ideal (I J : ideal A) :
zero_locus 𝒜 ((I * J : ideal A) : set A) = zero_locus 𝒜 I ∪ zero_locus 𝒜 J | set.ext $ λ x, x.is_prime.mul_le | lemma | projective_spectrum.zero_locus_mul_ideal | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"ideal",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_mul_homogeneous_ideal (I J : homogeneous_ideal 𝒜) :
zero_locus 𝒜 ((I * J : homogeneous_ideal 𝒜) : set A) = zero_locus 𝒜 I ∪ zero_locus 𝒜 J | set.ext $ λ x, x.is_prime.mul_le | lemma | projective_spectrum.zero_locus_mul_homogeneous_ideal | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"homogeneous_ideal",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_singleton_mul (f g : A) :
zero_locus 𝒜 ({f * g} : set A) = zero_locus 𝒜 {f} ∪ zero_locus 𝒜 {g} | set.ext $ λ x, by simpa using x.is_prime.mul_mem_iff_mem_or_mem | lemma | projective_spectrum.zero_locus_singleton_mul | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_singleton_pow (f : A) (n : ℕ) (hn : 0 < n) :
zero_locus 𝒜 ({f ^ n} : set A) = zero_locus 𝒜 {f} | set.ext $ λ x, by simpa using x.is_prime.pow_mem_iff_mem n hn | lemma | projective_spectrum.zero_locus_singleton_pow | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_vanishing_ideal_le (t t' : set (projective_spectrum 𝒜)) :
vanishing_ideal t ⊔ vanishing_ideal t' ≤ vanishing_ideal (t ∩ t') | begin
intros r,
rw [← homogeneous_ideal.mem_iff, homogeneous_ideal.to_ideal_sup, mem_vanishing_ideal,
submodule.mem_sup],
rintro ⟨f, hf, g, hg, rfl⟩ x ⟨hxt, hxt'⟩,
erw mem_vanishing_ideal at hf hg,
apply submodule.add_mem; solve_by_elim
end | lemma | projective_spectrum.sup_vanishing_ideal_le | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"homogeneous_ideal.mem_iff",
"homogeneous_ideal.to_ideal_sup",
"projective_spectrum",
"submodule.add_mem",
"submodule.mem_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_compl_zero_locus_iff_not_mem {f : A} {I : projective_spectrum 𝒜} :
I ∈ (zero_locus 𝒜 {f} : set (projective_spectrum 𝒜))ᶜ ↔ f ∉ I.as_homogeneous_ideal | by rw [set.mem_compl_iff, mem_zero_locus, set.singleton_subset_iff]; refl | lemma | projective_spectrum.mem_compl_zero_locus_iff_not_mem | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"projective_spectrum",
"set.mem_compl_iff",
"set.singleton_subset_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zariski_topology : topological_space (projective_spectrum 𝒜) | topological_space.of_closed (set.range (projective_spectrum.zero_locus 𝒜))
(⟨set.univ, by simp⟩)
begin
intros Zs h,
rw set.sInter_eq_Inter,
let f : Zs → set _ := λ i, classical.some (h i.2),
have hf : ∀ i : Zs, ↑i = zero_locus 𝒜 (f i) := λ i, (classical.some_spec (h i.2)).symm,
simp only [hf],... | instance | projective_spectrum.zariski_topology | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"projective_spectrum",
"projective_spectrum.zero_locus",
"set.range",
"set.sInter_eq_Inter",
"topological_space",
"topological_space.of_closed"
] | The Zariski topology on the prime spectrum of a commutative ring is defined via the closed sets
of the topology: they are exactly those sets that are the zero locus of a subset of the ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Top : Top | Top.of (projective_spectrum 𝒜) | def | projective_spectrum.Top | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"Top",
"Top.of",
"projective_spectrum"
] | The underlying topology of `Proj` is the projective spectrum of graded ring `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_iff (U : set (projective_spectrum 𝒜)) :
is_open U ↔ ∃ s, Uᶜ = zero_locus 𝒜 s | by simp only [@eq_comm _ Uᶜ]; refl | lemma | projective_spectrum.is_open_iff | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"is_open",
"projective_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_iff_zero_locus (Z : set (projective_spectrum 𝒜)) :
is_closed Z ↔ ∃ s, Z = zero_locus 𝒜 s | by rw [← is_open_compl_iff, is_open_iff, compl_compl] | lemma | projective_spectrum.is_closed_iff_zero_locus | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"compl_compl",
"is_closed",
"is_open_compl_iff",
"projective_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_zero_locus (s : set A) :
is_closed (zero_locus 𝒜 s) | by { rw [is_closed_iff_zero_locus], exact ⟨s, rfl⟩ } | lemma | projective_spectrum.is_closed_zero_locus | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_vanishing_ideal_eq_closure (t : set (projective_spectrum 𝒜)) :
zero_locus 𝒜 (vanishing_ideal t : set A) = closure t | begin
apply set.subset.antisymm,
{ rintro x hx t' ⟨ht', ht⟩,
obtain ⟨fs, rfl⟩ : ∃ s, t' = zero_locus 𝒜 s,
by rwa [is_closed_iff_zero_locus] at ht',
rw [subset_zero_locus_iff_subset_vanishing_ideal] at ht,
exact set.subset.trans ht hx },
{ rw (is_closed_zero_locus _ _).closure_subset_iff,
exac... | lemma | projective_spectrum.zero_locus_vanishing_ideal_eq_closure | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"closure",
"projective_spectrum",
"set.subset.antisymm",
"set.subset.trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vanishing_ideal_closure (t : set (projective_spectrum 𝒜)) :
vanishing_ideal (closure t) = vanishing_ideal t | begin
have := (gc_ideal 𝒜).u_l_u_eq_u t,
dsimp only at this,
ext1,
erw zero_locus_vanishing_ideal_eq_closure 𝒜 t at this,
exact this,
end | lemma | projective_spectrum.vanishing_ideal_closure | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"closure",
"projective_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open (r : A) : topological_space.opens (projective_spectrum 𝒜) | { carrier := { x | r ∉ x.as_homogeneous_ideal },
is_open' := ⟨{r}, set.ext $ λ x, set.singleton_subset_iff.trans $ not_not.symm⟩ } | def | projective_spectrum.basic_open | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"projective_spectrum",
"set.ext",
"topological_space.opens"
] | `basic_open r` is the open subset containing all prime ideals not containing `r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_basic_open (f : A) (x : projective_spectrum 𝒜) :
x ∈ basic_open 𝒜 f ↔ f ∉ x.as_homogeneous_ideal | iff.rfl | lemma | projective_spectrum.mem_basic_open | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"projective_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_coe_basic_open (f : A) (x : projective_spectrum 𝒜) :
x ∈ (↑(basic_open 𝒜 f): set (projective_spectrum 𝒜)) ↔ f ∉ x.as_homogeneous_ideal | iff.rfl | lemma | projective_spectrum.mem_coe_basic_open | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"projective_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_basic_open {a : A} : is_open ((basic_open 𝒜 a) :
set (projective_spectrum 𝒜)) | (basic_open 𝒜 a).is_open | lemma | projective_spectrum.is_open_basic_open | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"is_open",
"projective_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_eq_zero_locus_compl (r : A) :
(basic_open 𝒜 r : set (projective_spectrum 𝒜)) = (zero_locus 𝒜 {r})ᶜ | set.ext $ λ x, by simpa only [set.mem_compl_iff, mem_zero_locus, set.singleton_subset_iff] | lemma | projective_spectrum.basic_open_eq_zero_locus_compl | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"projective_spectrum",
"set.ext",
"set.mem_compl_iff",
"set.singleton_subset_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_one : basic_open 𝒜 (1 : A) = ⊤ | topological_space.opens.ext $ by simp | lemma | projective_spectrum.basic_open_one | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"topological_space.opens.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_zero : basic_open 𝒜 (0 : A) = ⊥ | topological_space.opens.ext $ by simp | lemma | projective_spectrum.basic_open_zero | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"topological_space.opens.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_mul (f g : A) : basic_open 𝒜 (f * g) = basic_open 𝒜 f ⊓ basic_open 𝒜 g | topological_space.opens.ext $ by {simp [zero_locus_singleton_mul]} | lemma | projective_spectrum.basic_open_mul | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"topological_space.opens.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_mul_le_left (f g : A) : basic_open 𝒜 (f * g) ≤ basic_open 𝒜 f | by { rw basic_open_mul 𝒜 f g, exact inf_le_left } | lemma | projective_spectrum.basic_open_mul_le_left | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"inf_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_mul_le_right (f g : A) : basic_open 𝒜 (f * g) ≤ basic_open 𝒜 g | by { rw basic_open_mul 𝒜 f g, exact inf_le_right } | lemma | projective_spectrum.basic_open_mul_le_right | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"inf_le_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_pow (f : A) (n : ℕ) (hn : 0 < n) :
basic_open 𝒜 (f ^ n) = basic_open 𝒜 f | topological_space.opens.ext $ by simpa using zero_locus_singleton_pow 𝒜 f n hn | lemma | projective_spectrum.basic_open_pow | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"topological_space.opens.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_eq_union_of_projection (f : A) :
basic_open 𝒜 f = ⨆ (i : ℕ), basic_open 𝒜 (graded_algebra.proj 𝒜 i f) | topological_space.opens.ext $ set.ext $ λ z, begin
erw [mem_coe_basic_open, topological_space.opens.mem_Sup],
split; intros hz,
{ rcases show ∃ i, graded_algebra.proj 𝒜 i f ∉ z.as_homogeneous_ideal, begin
contrapose! hz with H,
classical,
rw ←direct_sum.sum_support_decompose 𝒜 f,
apply i... | lemma | projective_spectrum.basic_open_eq_union_of_projection | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"graded_algebra.proj",
"ideal.sum_mem",
"set.ext",
"topological_space.opens.ext",
"topological_space.opens.mem_Sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_topological_basis_basic_opens : topological_space.is_topological_basis
(set.range (λ (r : A), (basic_open 𝒜 r : set (projective_spectrum 𝒜)))) | begin
apply topological_space.is_topological_basis_of_open_of_nhds,
{ rintros _ ⟨r, rfl⟩,
exact is_open_basic_open 𝒜 },
{ rintros p U hp ⟨s, hs⟩,
rw [← compl_compl U, set.mem_compl_iff, ← hs, mem_zero_locus, set.not_subset] at hp,
obtain ⟨f, hfs, hfp⟩ := hp,
refine ⟨basic_open 𝒜 f, ⟨f, rfl⟩, hfp... | lemma | is_topological_basis_basic_opens | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"compl_compl",
"projective_spectrum",
"set.compl_subset_compl",
"set.mem_compl_iff",
"set.not_subset",
"set.range",
"topological_space.is_topological_basis",
"topological_space.is_topological_basis_of_open_of_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
as_ideal_le_as_ideal (x y : projective_spectrum 𝒜) :
x.as_homogeneous_ideal ≤ y.as_homogeneous_ideal ↔ x ≤ y | iff.rfl | lemma | as_ideal_le_as_ideal | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"projective_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
as_ideal_lt_as_ideal (x y : projective_spectrum 𝒜) :
x.as_homogeneous_ideal < y.as_homogeneous_ideal ↔ x < y | iff.rfl | lemma | as_ideal_lt_as_ideal | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"projective_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_iff_mem_closure (x y : projective_spectrum 𝒜) :
x ≤ y ↔ y ∈ closure ({x} : set (projective_spectrum 𝒜)) | begin
rw [← as_ideal_le_as_ideal, ← zero_locus_vanishing_ideal_eq_closure,
mem_zero_locus, vanishing_ideal_singleton],
simp only [coe_subset_coe, subtype.coe_le_coe, coe_coe],
end | lemma | le_iff_mem_closure | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"as_ideal_le_as_ideal",
"closure",
"coe_coe",
"projective_spectrum",
"subtype.coe_le_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
obj_d (n : ℕ) : X _[n+1] ⟶ X _[n] | ∑ (i : fin (n+2)), (-1 : ℤ)^(i : ℕ) • X.δ i | def | algebraic_topology.alternating_face_map_complex.obj_d | algebraic_topology | src/algebraic_topology/alternating_face_map_complex.lean | [
"algebra.homology.additive",
"algebraic_topology.Moore_complex",
"algebra.big_operators.fin",
"category_theory.preadditive.opposite",
"category_theory.idempotents.functor_categories",
"tactic.equiv_rw"
] | [] | The differential on the alternating face map complex is the alternate
sum of the face maps | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
d_squared (n : ℕ) : obj_d X (n+1) ≫ obj_d X n = 0 | begin
/- we start by expanding d ≫ d as a double sum -/
dsimp,
rw comp_sum,
let d_l := λ (j : fin (n+3)), (-1 : ℤ)^(j : ℕ) • X.δ j,
let d_r := λ (i : fin (n+2)), (-1 : ℤ)^(i : ℕ) • X.δ i,
rw [show (λ i , (∑ j : fin (n+3), d_l j) ≫ d_r i) =
(λ i, ∑ j : fin (n+3), (d_l j ≫ d_r i)), by { ext i, rw sum_comp... | lemma | algebraic_topology.alternating_face_map_complex.d_squared | algebraic_topology | src/algebraic_topology/alternating_face_map_complex.lean | [
"algebra.homology.additive",
"algebraic_topology.Moore_complex",
"algebra.big_operators.fin",
"category_theory.preadditive.opposite",
"category_theory.idempotents.functor_categories",
"tactic.equiv_rw"
] | [
"category_theory.simplicial_object.δ_comp_δ",
"fin.cast_lt",
"fin.cast_lt_cast_succ",
"fin.cast_succ",
"fin.cast_succ_cast_lt",
"fin.coe_cast_lt",
"fin.coe_cast_succ",
"fin.coe_fin_le",
"fin.coe_pred",
"fin.coe_succ",
"fin.coe_zero",
"fin.is_lt",
"fin.succ_pred",
"finset.compl_filter",
"... | ## The chain complex relation `d ≫ d` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
obj : chain_complex C ℕ | chain_complex.of (λ n, X _[n]) (obj_d X) (d_squared X) | def | algebraic_topology.alternating_face_map_complex.obj | algebraic_topology | src/algebraic_topology/alternating_face_map_complex.lean | [
"algebra.homology.additive",
"algebraic_topology.Moore_complex",
"algebra.big_operators.fin",
"category_theory.preadditive.opposite",
"category_theory.idempotents.functor_categories",
"tactic.equiv_rw"
] | [
"chain_complex",
"chain_complex.of"
] | The alternating face map complex, on objects | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
obj_X (X : simplicial_object C) (n : ℕ) :
(alternating_face_map_complex.obj X).X n = X _[n] | rfl | lemma | algebraic_topology.alternating_face_map_complex.obj_X | algebraic_topology | src/algebraic_topology/alternating_face_map_complex.lean | [
"algebra.homology.additive",
"algebraic_topology.Moore_complex",
"algebra.big_operators.fin",
"category_theory.preadditive.opposite",
"category_theory.idempotents.functor_categories",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
obj_d_eq (X : simplicial_object C) (n : ℕ) :
(alternating_face_map_complex.obj X).d (n+1) n =
∑ (i : fin (n+2)), (-1 : ℤ)^(i : ℕ) • X.δ i | by apply chain_complex.of_d | lemma | algebraic_topology.alternating_face_map_complex.obj_d_eq | algebraic_topology | src/algebraic_topology/alternating_face_map_complex.lean | [
"algebra.homology.additive",
"algebraic_topology.Moore_complex",
"algebra.big_operators.fin",
"category_theory.preadditive.opposite",
"category_theory.idempotents.functor_categories",
"tactic.equiv_rw"
] | [
"chain_complex.of_d"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (f : X ⟶ Y) : obj X ⟶ obj Y | chain_complex.of_hom _ _ _ _ _ _
(λ n, f.app (op [n]))
(λ n,
begin
dsimp,
rw [comp_sum, sum_comp],
apply finset.sum_congr rfl (λ x h, _),
rw [comp_zsmul, zsmul_comp],
congr' 1,
symmetry,
apply f.naturality,
end) | def | algebraic_topology.alternating_face_map_complex.map | algebraic_topology | src/algebraic_topology/alternating_face_map_complex.lean | [
"algebra.homology.additive",
"algebraic_topology.Moore_complex",
"algebra.big_operators.fin",
"category_theory.preadditive.opposite",
"category_theory.idempotents.functor_categories",
"tactic.equiv_rw"
] | [
"chain_complex.of_hom"
] | The alternating face map complex, on morphisms | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_f (f : X ⟶ Y) (n : ℕ) : (map f).f n = f.app (op [n]) | rfl | lemma | algebraic_topology.alternating_face_map_complex.map_f | algebraic_topology | src/algebraic_topology/alternating_face_map_complex.lean | [
"algebra.homology.additive",
"algebraic_topology.Moore_complex",
"algebra.big_operators.fin",
"category_theory.preadditive.opposite",
"category_theory.idempotents.functor_categories",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
alternating_face_map_complex : simplicial_object C ⥤ chain_complex C ℕ | { obj := alternating_face_map_complex.obj,
map := λ X Y f, alternating_face_map_complex.map f } | def | algebraic_topology.alternating_face_map_complex | algebraic_topology | src/algebraic_topology/alternating_face_map_complex.lean | [
"algebra.homology.additive",
"algebraic_topology.Moore_complex",
"algebra.big_operators.fin",
"category_theory.preadditive.opposite",
"category_theory.idempotents.functor_categories",
"tactic.equiv_rw"
] | [
"chain_complex"
] | The alternating face map complex, as a functor | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
alternating_face_map_complex_obj_X (X : simplicial_object C) (n : ℕ) :
((alternating_face_map_complex C).obj X).X n = X _[n] | rfl | lemma | algebraic_topology.alternating_face_map_complex_obj_X | algebraic_topology | src/algebraic_topology/alternating_face_map_complex.lean | [
"algebra.homology.additive",
"algebraic_topology.Moore_complex",
"algebra.big_operators.fin",
"category_theory.preadditive.opposite",
"category_theory.idempotents.functor_categories",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
alternating_face_map_complex_obj_d (X : simplicial_object C) (n : ℕ) :
((alternating_face_map_complex C).obj X).d (n+1) n =
alternating_face_map_complex.obj_d X n | by apply chain_complex.of_d | lemma | algebraic_topology.alternating_face_map_complex_obj_d | algebraic_topology | src/algebraic_topology/alternating_face_map_complex.lean | [
"algebra.homology.additive",
"algebraic_topology.Moore_complex",
"algebra.big_operators.fin",
"category_theory.preadditive.opposite",
"category_theory.idempotents.functor_categories",
"tactic.equiv_rw"
] | [
"chain_complex.of_d"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
alternating_face_map_complex_map_f {X Y : simplicial_object C} (f : X ⟶ Y) (n : ℕ) :
((alternating_face_map_complex C).map f).f n = f.app (op [n]) | rfl | lemma | algebraic_topology.alternating_face_map_complex_map_f | algebraic_topology | src/algebraic_topology/alternating_face_map_complex.lean | [
"algebra.homology.additive",
"algebraic_topology.Moore_complex",
"algebra.big_operators.fin",
"category_theory.preadditive.opposite",
"category_theory.idempotents.functor_categories",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_alternating_face_map_complex {D : Type*} [category D] [preadditive D]
(F : C ⥤ D) [F.additive] :
alternating_face_map_complex C ⋙ F.map_homological_complex _ =
(simplicial_object.whiskering C D).obj F ⋙ alternating_face_map_complex D | begin
apply category_theory.functor.ext,
{ intros X Y f,
ext n,
simp only [functor.comp_map, homological_complex.comp_f,
alternating_face_map_complex_map_f, functor.map_homological_complex_map_f,
homological_complex.eq_to_hom_f, eq_to_hom_refl, comp_id, id_comp,
simplicial_object.whiskerin... | lemma | algebraic_topology.map_alternating_face_map_complex | algebraic_topology | src/algebraic_topology/alternating_face_map_complex.lean | [
"algebra.homology.additive",
"algebraic_topology.Moore_complex",
"algebra.big_operators.fin",
"category_theory.preadditive.opposite",
"category_theory.idempotents.functor_categories",
"tactic.equiv_rw"
] | [
"category_theory.functor.ext",
"homological_complex.comp_f",
"homological_complex.eq_to_hom_f",
"homological_complex.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
karoubi_alternating_face_map_complex_d (P : karoubi (simplicial_object C)) (n : ℕ) :
(((alternating_face_map_complex.obj (karoubi_functor_category_embedding.obj P)).d (n+1) n).f) =
P.p.app (op [n+1]) ≫ ((alternating_face_map_complex.obj P.X).d (n+1) n) | begin
dsimp,
simpa only [alternating_face_map_complex.obj_d_eq, karoubi.sum_hom,
preadditive.comp_sum, karoubi.zsmul_hom, preadditive.comp_zsmul],
end | lemma | algebraic_topology.karoubi_alternating_face_map_complex_d | algebraic_topology | src/algebraic_topology/alternating_face_map_complex.lean | [
"algebra.homology.additive",
"algebraic_topology.Moore_complex",
"algebra.big_operators.fin",
"category_theory.preadditive.opposite",
"category_theory.idempotents.functor_categories",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ε [limits.has_zero_object C] :
simplicial_object.augmented.drop ⋙ algebraic_topology.alternating_face_map_complex C ⟶
simplicial_object.augmented.point ⋙ chain_complex.single₀ C | { app := λ X, begin
equiv_rw chain_complex.to_single₀_equiv _ _,
refine ⟨X.hom.app (op [0]), _⟩,
dsimp,
simp only [alternating_face_map_complex_obj_d, obj_d, fin.sum_univ_two,
fin.coe_zero, pow_zero, one_zsmul, fin.coe_one, pow_one, neg_smul, add_comp,
simplicial_object.δ_naturality, neg_com... | def | algebraic_topology.alternating_face_map_complex.ε | algebraic_topology | src/algebraic_topology/alternating_face_map_complex.lean | [
"algebra.homology.additive",
"algebraic_topology.Moore_complex",
"algebra.big_operators.fin",
"category_theory.preadditive.opposite",
"category_theory.idempotents.functor_categories",
"tactic.equiv_rw"
] | [
"algebraic_topology.alternating_face_map_complex",
"chain_complex.single₀",
"chain_complex.to_single₀_equiv",
"fin.coe_one",
"fin.coe_zero",
"neg_smul",
"pow_one",
"pow_zero"
] | The natural transformation which gives the augmentation of the alternating face map
complex attached to an augmented simplicial object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inclusion_of_Moore_complex_map (X : simplicial_object A) :
(normalized_Moore_complex A).obj X ⟶ (alternating_face_map_complex A).obj X | chain_complex.of_hom _ _ _ _ _ _
(λ n, (normalized_Moore_complex.obj_X X n).arrow)
(λ n,
begin
/- we have to show the compatibility of the differentials on the alternating
face map complex with those defined on the normalized Moore complex:
we first get rid of the terms of the alternatin... | def | algebraic_topology.inclusion_of_Moore_complex_map | algebraic_topology | src/algebraic_topology/alternating_face_map_complex.lean | [
"algebra.homology.additive",
"algebraic_topology.Moore_complex",
"algebra.big_operators.fin",
"category_theory.preadditive.opposite",
"category_theory.idempotents.functor_categories",
"tactic.equiv_rw"
] | [
"chain_complex.of_hom",
"finset.mem_univ",
"finset.univ",
"one_smul",
"ring"
] | The inclusion map of the Moore complex in the alternating face map complex | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inclusion_of_Moore_complex_map_f (X : simplicial_object A) (n : ℕ) :
(inclusion_of_Moore_complex_map X).f n = (normalized_Moore_complex.obj_X X n).arrow | chain_complex.of_hom_f _ _ _ _ _ _ _ _ n | lemma | algebraic_topology.inclusion_of_Moore_complex_map_f | algebraic_topology | src/algebraic_topology/alternating_face_map_complex.lean | [
"algebra.homology.additive",
"algebraic_topology.Moore_complex",
"algebra.big_operators.fin",
"category_theory.preadditive.opposite",
"category_theory.idempotents.functor_categories",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inclusion_of_Moore_complex :
(normalized_Moore_complex A) ⟶ (alternating_face_map_complex A) | { app := inclusion_of_Moore_complex_map, } | def | algebraic_topology.inclusion_of_Moore_complex | algebraic_topology | src/algebraic_topology/alternating_face_map_complex.lean | [
"algebra.homology.additive",
"algebraic_topology.Moore_complex",
"algebra.big_operators.fin",
"category_theory.preadditive.opposite",
"category_theory.idempotents.functor_categories",
"tactic.equiv_rw"
] | [] | The inclusion map of the Moore complex in the alternating face map complex,
as a natural transformation | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
obj_d (n : ℕ) : X.obj [n] ⟶ X.obj [n+1] | ∑ (i : fin (n+2)), (-1 : ℤ)^(i : ℕ) • X.δ i | def | algebraic_topology.alternating_coface_map_complex.obj_d | algebraic_topology | src/algebraic_topology/alternating_face_map_complex.lean | [
"algebra.homology.additive",
"algebraic_topology.Moore_complex",
"algebra.big_operators.fin",
"category_theory.preadditive.opposite",
"category_theory.idempotents.functor_categories",
"tactic.equiv_rw"
] | [] | The differential on the alternating coface map complex is the alternate
sum of the coface maps | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
d_eq_unop_d (n : ℕ) :
obj_d X n = (alternating_face_map_complex.obj_d
((cosimplicial_simplicial_equiv C).functor.obj (op X)) n).unop | by simpa only [obj_d, alternating_face_map_complex.obj_d, unop_sum, unop_zsmul] | lemma | algebraic_topology.alternating_coface_map_complex.d_eq_unop_d | algebraic_topology | src/algebraic_topology/alternating_face_map_complex.lean | [
"algebra.homology.additive",
"algebraic_topology.Moore_complex",
"algebra.big_operators.fin",
"category_theory.preadditive.opposite",
"category_theory.idempotents.functor_categories",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
d_squared (n : ℕ) : obj_d X n ≫ obj_d X (n+1) = 0 | by simp only [d_eq_unop_d, ← unop_comp, alternating_face_map_complex.d_squared, unop_zero] | lemma | algebraic_topology.alternating_coface_map_complex.d_squared | algebraic_topology | src/algebraic_topology/alternating_face_map_complex.lean | [
"algebra.homology.additive",
"algebraic_topology.Moore_complex",
"algebra.big_operators.fin",
"category_theory.preadditive.opposite",
"category_theory.idempotents.functor_categories",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
obj : cochain_complex C ℕ | cochain_complex.of (λ n, X.obj [n]) (obj_d X) (d_squared X) | def | algebraic_topology.alternating_coface_map_complex.obj | algebraic_topology | src/algebraic_topology/alternating_face_map_complex.lean | [
"algebra.homology.additive",
"algebraic_topology.Moore_complex",
"algebra.big_operators.fin",
"category_theory.preadditive.opposite",
"category_theory.idempotents.functor_categories",
"tactic.equiv_rw"
] | [
"cochain_complex",
"cochain_complex.of"
] | The alternating coface map complex, on objects | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map (f : X ⟶ Y) : obj X ⟶ obj Y | cochain_complex.of_hom _ _ _ _ _ _
(λ n, f.app [n])
(λ n,
begin
dsimp,
rw [comp_sum, sum_comp],
apply finset.sum_congr rfl (λ x h, _),
rw [comp_zsmul, zsmul_comp],
congr' 1,
symmetry,
apply f.naturality,
end) | def | algebraic_topology.alternating_coface_map_complex.map | algebraic_topology | src/algebraic_topology/alternating_face_map_complex.lean | [
"algebra.homology.additive",
"algebraic_topology.Moore_complex",
"algebra.big_operators.fin",
"category_theory.preadditive.opposite",
"category_theory.idempotents.functor_categories",
"tactic.equiv_rw"
] | [
"cochain_complex.of_hom"
] | The alternating face map complex, on morphisms | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
alternating_coface_map_complex : cosimplicial_object C ⥤ cochain_complex C ℕ | { obj := alternating_coface_map_complex.obj,
map := λ X Y f, alternating_coface_map_complex.map f } | def | algebraic_topology.alternating_coface_map_complex | algebraic_topology | src/algebraic_topology/alternating_face_map_complex.lean | [
"algebra.homology.additive",
"algebraic_topology.Moore_complex",
"algebra.big_operators.fin",
"category_theory.preadditive.opposite",
"category_theory.idempotents.functor_categories",
"tactic.equiv_rw"
] | [
"cochain_complex"
] | The alternating coface map complex, as a functor | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cech_nerve : simplicial_object C | { obj := λ n, wide_pullback.{0} f.right
(λ i : fin (n.unop.len + 1), f.left) (λ i, f.hom),
map := λ m n g, wide_pullback.lift (wide_pullback.base _)
(λ i, wide_pullback.π (λ i, f.hom) $ g.unop.to_order_hom i) $ λ j, by simp,
map_id' := λ x, by { ext ⟨⟩, { simpa }, { simp } },
map_comp' := λ x y z f g, by ... | def | category_theory.arrow.cech_nerve | algebraic_topology | src/algebraic_topology/cech_nerve.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products",
"category_theory.arrow"
] | [] | The Čech nerve associated to an arrow. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_cech_nerve {f g : arrow C}
[∀ n : ℕ, has_wide_pullback f.right (λ i : fin (n+1), f.left) (λ i, f.hom)]
[∀ n : ℕ, has_wide_pullback g.right (λ i : fin (n+1), g.left) (λ i, g.hom)]
(F : f ⟶ g) : f.cech_nerve ⟶ g.cech_nerve | { app := λ n, wide_pullback.lift (wide_pullback.base _ ≫ F.right)
(λ i, wide_pullback.π _ i ≫ F.left) $ λ j, by simp,
naturality' := λ x y f, by { ext ⟨⟩, { simp }, { simp } } } | def | category_theory.arrow.map_cech_nerve | algebraic_topology | src/algebraic_topology/cech_nerve.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products",
"category_theory.arrow"
] | [] | The morphism between Čech nerves associated to a morphism of arrows. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
augmented_cech_nerve : simplicial_object.augmented C | { left := f.cech_nerve,
right := f.right,
hom :=
{ app := λ i, wide_pullback.base _,
naturality' := λ x y f, by { dsimp, simp } } } | def | category_theory.arrow.augmented_cech_nerve | algebraic_topology | src/algebraic_topology/cech_nerve.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products",
"category_theory.arrow"
] | [] | The augmented Čech nerve associated to an arrow. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_augmented_cech_nerve {f g : arrow C}
[∀ n : ℕ, has_wide_pullback f.right (λ i : fin (n+1), f.left) (λ i, f.hom)]
[∀ n : ℕ, has_wide_pullback g.right (λ i : fin (n+1), g.left) (λ i, g.hom)]
(F : f ⟶ g) : f.augmented_cech_nerve ⟶ g.augmented_cech_nerve | { left := map_cech_nerve F,
right := F.right,
w' := by { ext, simp } } | def | category_theory.arrow.map_augmented_cech_nerve | algebraic_topology | src/algebraic_topology/cech_nerve.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products",
"category_theory.arrow"
] | [] | The morphism between augmented Čech nerve associated to a morphism of arrows. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cech_nerve : arrow C ⥤ simplicial_object C | { obj := λ f, f.cech_nerve,
map := λ f g F, arrow.map_cech_nerve F,
map_id' := λ i, by { ext, { simp }, { simp } },
map_comp' := λ x y z f g, by { ext, { simp }, { simp } } } | def | category_theory.simplicial_object.cech_nerve | algebraic_topology | src/algebraic_topology/cech_nerve.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products",
"category_theory.arrow"
] | [] | The Čech nerve construction, as a functor from `arrow C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
augmented_cech_nerve : arrow C ⥤ simplicial_object.augmented C | { obj := λ f, f.augmented_cech_nerve,
map := λ f g F, arrow.map_augmented_cech_nerve F,
map_id' := λ x, by { ext, { simp }, { simp }, { refl } },
map_comp' := λ x y z f g, by { ext, { simp }, { simp }, { refl } } } | def | category_theory.simplicial_object.augmented_cech_nerve | algebraic_topology | src/algebraic_topology/cech_nerve.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products",
"category_theory.arrow"
] | [] | The augmented Čech nerve construction, as a functor from `arrow C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence_right_to_left (X : simplicial_object.augmented C) (F : arrow C)
(G : X ⟶ F.augmented_cech_nerve) : augmented.to_arrow.obj X ⟶ F | { left := G.left.app _ ≫ wide_pullback.π (λ i, F.hom) 0,
right := G.right,
w' := begin
have := G.w,
apply_fun (λ e, e.app (opposite.op $ simplex_category.mk 0)) at this,
simpa using this,
end } | def | category_theory.simplicial_object.equivalence_right_to_left | algebraic_topology | src/algebraic_topology/cech_nerve.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products",
"category_theory.arrow"
] | [
"opposite.op",
"simplex_category.mk"
] | A helper function used in defining the Čech adjunction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence_left_to_right (X : simplicial_object.augmented C) (F : arrow C)
(G : augmented.to_arrow.obj X ⟶ F) : X ⟶ F.augmented_cech_nerve | { left :=
{ app := λ x, limits.wide_pullback.lift (X.hom.app _ ≫ G.right)
(λ i, X.left.map (simplex_category.const x.unop i).op ≫ G.left)
(λ i, by { dsimp, erw [category.assoc, arrow.w,
augmented.to_arrow_obj_hom, nat_trans.naturality_assoc,
functor.const_obj_map, category.id_comp] } ),
... | def | category_theory.simplicial_object.equivalence_left_to_right | algebraic_topology | src/algebraic_topology/cech_nerve.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products",
"category_theory.arrow"
] | [
"simplex_category.const"
] | A helper function used in defining the Čech adjunction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cech_nerve_equiv (X : simplicial_object.augmented C) (F : arrow C) :
(augmented.to_arrow.obj X ⟶ F) ≃ (X ⟶ F.augmented_cech_nerve) | { to_fun := equivalence_left_to_right _ _,
inv_fun := equivalence_right_to_left _ _,
left_inv := begin
intro A,
dsimp,
ext,
{ dsimp,
erw wide_pullback.lift_π,
nth_rewrite 1 ← category.id_comp A.left,
congr' 1,
convert X.left.map_id _,
rw ← op_id,
congr' 1,
e... | def | category_theory.simplicial_object.cech_nerve_equiv | algebraic_topology | src/algebraic_topology/cech_nerve.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products",
"category_theory.arrow"
] | [
"inv_fun"
] | A helper function used in defining the Čech adjunction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cech_nerve_adjunction :
(augmented.to_arrow : _ ⥤ arrow C) ⊣ augmented_cech_nerve | adjunction.mk_of_hom_equiv
{ hom_equiv := cech_nerve_equiv,
hom_equiv_naturality_left_symm' := λ x y f g h, by { ext, { simp }, { simp } },
hom_equiv_naturality_right' := λ x y f g h, by { ext, { simp }, { simp }, { refl } } } | abbreviation | category_theory.simplicial_object.cech_nerve_adjunction | algebraic_topology | src/algebraic_topology/cech_nerve.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products",
"category_theory.arrow"
] | [] | The augmented Čech nerve construction is right adjoint to the `to_arrow` functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cech_conerve : cosimplicial_object C | { obj := λ n, wide_pushout f.left
(λ i : fin (n.len + 1), f.right) (λ i, f.hom),
map := λ m n g, wide_pushout.desc (wide_pushout.head _)
(λ i, wide_pushout.ι (λ i, f.hom) $ g.to_order_hom i) $
λ i, by { rw [wide_pushout.arrow_ι (λ i, f.hom)] },
map_id' := λ x, by { ext ⟨⟩, { simpa }, { simp } },
map_c... | def | category_theory.arrow.cech_conerve | algebraic_topology | src/algebraic_topology/cech_nerve.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products",
"category_theory.arrow"
] | [] | The Čech conerve associated to an arrow. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_cech_conerve {f g : arrow C}
[∀ n : ℕ, has_wide_pushout f.left (λ i : fin (n+1), f.right) (λ i, f.hom)]
[∀ n : ℕ, has_wide_pushout g.left (λ i : fin (n+1), g.right) (λ i, g.hom)]
(F : f ⟶ g) : f.cech_conerve ⟶ g.cech_conerve | { app := λ n, wide_pushout.desc (F.left ≫ wide_pushout.head _)
(λ i, F.right ≫ wide_pushout.ι _ i) $
λ i, by { rw [← arrow.w_assoc F, wide_pushout.arrow_ι (λ i, g.hom)] },
naturality' := λ x y f, by { ext, { simp }, { simp } } } | def | category_theory.arrow.map_cech_conerve | algebraic_topology | src/algebraic_topology/cech_nerve.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products",
"category_theory.arrow"
] | [] | The morphism between Čech conerves associated to a morphism of arrows. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
augmented_cech_conerve : cosimplicial_object.augmented C | { left := f.left,
right := f.cech_conerve,
hom :=
{ app := λ i, wide_pushout.head _,
naturality' := λ x y f, by { dsimp, simp } } } | def | category_theory.arrow.augmented_cech_conerve | algebraic_topology | src/algebraic_topology/cech_nerve.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products",
"category_theory.arrow"
] | [] | The augmented Čech conerve associated to an arrow. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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