statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
geom_series_mul_neg (x : R) (h : ‖x‖ < 1) :
(∑' i:ℕ, x ^ i) * (1 - x) = 1 | begin
have := ((normed_ring.summable_geometric_of_norm_lt_1 x h).has_sum.mul_right (1 - x)),
refine tendsto_nhds_unique this.tendsto_sum_nat _,
have : tendsto (λ (n : ℕ), 1 - x ^ n) at_top (𝓝 1),
{ simpa using tendsto_const_nhds.sub (tendsto_pow_at_top_nhds_0_of_norm_lt_1 h) },
convert ← this,
ext n,
rw ... | lemma | geom_series_mul_neg | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"finset.sum_mul",
"has_sum.mul_right",
"normed_ring.summable_geometric_of_norm_lt_1",
"tendsto_nhds_unique",
"tendsto_pow_at_top_nhds_0_of_norm_lt_1"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_neg_geom_series (x : R) (h : ‖x‖ < 1) :
(1 - x) * ∑' i:ℕ, x ^ i = 1 | begin
have := (normed_ring.summable_geometric_of_norm_lt_1 x h).has_sum.mul_left (1 - x),
refine tendsto_nhds_unique this.tendsto_sum_nat _,
have : tendsto (λ (n : ℕ), 1 - x ^ n) at_top (nhds 1),
{ simpa using tendsto_const_nhds.sub
(tendsto_pow_at_top_nhds_0_of_norm_lt_1 h) },
convert ← this,
ext n,
... | lemma | mul_neg_geom_series | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"finset.mul_sum",
"has_sum.mul_left",
"nhds",
"normed_ring.summable_geometric_of_norm_lt_1",
"tendsto_nhds_unique",
"tendsto_pow_at_top_nhds_0_of_norm_lt_1"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_of_ratio_norm_eventually_le {α : Type*} [seminormed_add_comm_group α]
[complete_space α] {f : ℕ → α} {r : ℝ} (hr₁ : r < 1)
(h : ∀ᶠ n in at_top, ‖f (n+1)‖ ≤ r * ‖f n‖) : summable f | begin
by_cases hr₀ : 0 ≤ r,
{ rw eventually_at_top at h,
rcases h with ⟨N, hN⟩,
rw ← @summable_nat_add_iff α _ _ _ _ N,
refine summable_of_norm_bounded (λ n, ‖f N‖ * r^n)
(summable.mul_left _ $ summable_geometric_of_lt_1 hr₀ hr₁) (λ n, _),
conv_rhs {rw [mul_comm, ← zero_add N]},
refine le_... | lemma | summable_of_ratio_norm_eventually_le | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"complete_space",
"le_geom",
"mul_comm",
"mul_neg_of_neg_of_pos",
"nat.cofinite_eq_at_top",
"seminormed_add_comm_group",
"summable",
"summable.mul_left",
"summable_geometric_of_lt_1",
"summable_nat_add_iff",
"summable_of_norm_bounded",
"summable_of_norm_bounded_eventually",
"summable_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_of_ratio_test_tendsto_lt_one {α : Type*} [normed_add_comm_group α] [complete_space α]
{f : ℕ → α} {l : ℝ} (hl₁ : l < 1) (hf : ∀ᶠ n in at_top, f n ≠ 0)
(h : tendsto (λ n, ‖f (n+1)‖/‖f n‖) at_top (𝓝 l)) : summable f | begin
rcases exists_between hl₁ with ⟨r, hr₀, hr₁⟩,
refine summable_of_ratio_norm_eventually_le hr₁ _,
filter_upwards [eventually_le_of_tendsto_lt hr₀ h, hf] with _ _ h₁,
rwa ← div_le_iff (norm_pos_iff.mpr h₁),
end | lemma | summable_of_ratio_test_tendsto_lt_one | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"complete_space",
"div_le_iff",
"eventually_le_of_tendsto_lt",
"exists_between",
"normed_add_comm_group",
"summable",
"summable_of_ratio_norm_eventually_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_summable_of_ratio_norm_eventually_ge {α : Type*} [seminormed_add_comm_group α]
{f : ℕ → α} {r : ℝ} (hr : 1 < r) (hf : ∃ᶠ n in at_top, ‖f n‖ ≠ 0)
(h : ∀ᶠ n in at_top, r * ‖f n‖ ≤ ‖f (n+1)‖) : ¬ summable f | begin
rw eventually_at_top at h,
rcases h with ⟨N₀, hN₀⟩,
rw frequently_at_top at hf,
rcases hf N₀ with ⟨N, hNN₀ : N₀ ≤ N, hN⟩,
rw ← @summable_nat_add_iff α _ _ _ _ N,
refine mt summable.tendsto_at_top_zero
(λ h', not_tendsto_at_top_of_tendsto_nhds (tendsto_norm_zero.comp h') _),
convert tendsto_at_to... | lemma | not_summable_of_ratio_norm_eventually_ge | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"not_tendsto_at_top_of_tendsto_nhds",
"seminormed_add_comm_group",
"summable",
"summable.tendsto_at_top_zero",
"summable_nat_add_iff",
"tendsto_at_top_of_geom_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_summable_of_ratio_test_tendsto_gt_one {α : Type*} [seminormed_add_comm_group α]
{f : ℕ → α} {l : ℝ} (hl : 1 < l)
(h : tendsto (λ n, ‖f (n+1)‖/‖f n‖) at_top (𝓝 l)) : ¬ summable f | begin
have key : ∀ᶠ n in at_top, ‖f n‖ ≠ 0,
{ filter_upwards [eventually_ge_of_tendsto_gt hl h] with _ hn hc,
rw [hc, div_zero] at hn,
linarith },
rcases exists_between hl with ⟨r, hr₀, hr₁⟩,
refine not_summable_of_ratio_norm_eventually_ge hr₀ key.frequently _,
filter_upwards [eventually_ge_of_tendsto... | lemma | not_summable_of_ratio_test_tendsto_gt_one | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"div_zero",
"eventually_ge_of_tendsto_gt",
"exists_between",
"le_div_iff",
"not_summable_of_ratio_norm_eventually_ge",
"seminormed_add_comm_group",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.cauchy_seq_series_mul_of_tendsto_zero_of_bounded
(hfa : monotone f) (hf0 : tendsto f at_top (𝓝 0)) (hgb : ∀ n, ‖∑ i in range n, z i‖ ≤ b) :
cauchy_seq (λ n, ∑ i in range (n + 1), (f i) • z i) | begin
simp_rw [finset.sum_range_by_parts _ _ (nat.succ _), sub_eq_add_neg,
nat.succ_sub_succ_eq_sub, tsub_zero],
apply (normed_field.tendsto_zero_smul_of_tendsto_zero_of_bounded hf0
⟨b, eventually_map.mpr $ eventually_of_forall $ λ n, hgb $ n+1⟩).cauchy_seq.add,
refine (cauchy_seq_range_of_norm_bou... | theorem | monotone.cauchy_seq_series_mul_of_tendsto_zero_of_bounded | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"abs_nonneg",
"abs_of_nonneg",
"cauchy_seq",
"cauchy_seq_range_of_norm_bounded",
"finset.sum_range_by_parts",
"monotone",
"mul_comm",
"mul_le_mul_of_nonneg_right",
"norm_smul",
"normed_field.tendsto_zero_smul_of_tendsto_zero_of_bounded",
"tsub_zero"
] | **Dirichlet's Test** for monotone sequences. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone.cauchy_seq_series_mul_of_tendsto_zero_of_bounded
(hfa : antitone f) (hf0 : tendsto f at_top (𝓝 0)) (hzb : ∀ n, ‖∑ i in range n, z i‖ ≤ b) :
cauchy_seq (λ n, ∑ i in range (n+1), (f i) • z i) | begin
have hfa': monotone (λ n, -f n) := λ _ _ hab, neg_le_neg $ hfa hab,
have hf0': tendsto (λ n, -f n) at_top (𝓝 0) := by { convert hf0.neg, norm_num },
convert (hfa'.cauchy_seq_series_mul_of_tendsto_zero_of_bounded hf0' hzb).neg,
funext,
simp
end | theorem | antitone.cauchy_seq_series_mul_of_tendsto_zero_of_bounded | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"antitone",
"cauchy_seq",
"monotone"
] | **Dirichlet's test** for antitone sequences. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_sum_neg_one_pow_le (n : ℕ) : ‖∑ i in range n, (-1 : ℝ) ^ i‖ ≤ 1 | by { rw [neg_one_geom_sum], split_ifs; norm_num } | lemma | norm_sum_neg_one_pow_le | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"neg_one_geom_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.cauchy_seq_alternating_series_of_tendsto_zero
(hfa : monotone f) (hf0 : tendsto f at_top (𝓝 0)) :
cauchy_seq (λ n, ∑ i in range (n+1), (-1) ^ i * f i) | begin
simp_rw [mul_comm],
exact hfa.cauchy_seq_series_mul_of_tendsto_zero_of_bounded hf0 norm_sum_neg_one_pow_le
end | theorem | monotone.cauchy_seq_alternating_series_of_tendsto_zero | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"cauchy_seq",
"monotone",
"mul_comm",
"norm_sum_neg_one_pow_le"
] | The **alternating series test** for monotone sequences.
See also `tendsto_alternating_series_of_monotone_tendsto_zero`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.tendsto_alternating_series_of_tendsto_zero
(hfa : monotone f) (hf0 : tendsto f at_top (𝓝 0)) :
∃ l, tendsto (λ n, ∑ i in range (n+1), (-1) ^ i * f i) at_top (𝓝 l) | cauchy_seq_tendsto_of_complete $ hfa.cauchy_seq_alternating_series_of_tendsto_zero hf0 | theorem | monotone.tendsto_alternating_series_of_tendsto_zero | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"cauchy_seq_tendsto_of_complete",
"monotone"
] | The **alternating series test** for monotone sequences. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone.cauchy_seq_alternating_series_of_tendsto_zero
(hfa : antitone f) (hf0 : tendsto f at_top (𝓝 0)) :
cauchy_seq (λ n, ∑ i in range (n+1), (-1) ^ i * f i) | begin
simp_rw [mul_comm],
exact
hfa.cauchy_seq_series_mul_of_tendsto_zero_of_bounded hf0 norm_sum_neg_one_pow_le
end | theorem | antitone.cauchy_seq_alternating_series_of_tendsto_zero | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"antitone",
"cauchy_seq",
"mul_comm",
"norm_sum_neg_one_pow_le"
] | The **alternating series test** for antitone sequences.
See also `tendsto_alternating_series_of_antitone_tendsto_zero`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone.tendsto_alternating_series_of_tendsto_zero
(hfa : antitone f) (hf0 : tendsto f at_top (𝓝 0)) :
∃ l, tendsto (λ n, ∑ i in range (n+1), (-1) ^ i * f i) at_top (𝓝 l) | cauchy_seq_tendsto_of_complete $ hfa.cauchy_seq_alternating_series_of_tendsto_zero hf0 | theorem | antitone.tendsto_alternating_series_of_tendsto_zero | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"antitone",
"cauchy_seq_tendsto_of_complete"
] | The **alternating series test** for antitone sequences. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real.summable_pow_div_factorial (x : ℝ) :
summable (λ n, x ^ n / n! : ℕ → ℝ) | begin
-- We start with trivial extimates
have A : (0 : ℝ) < ⌊‖x‖⌋₊ + 1, from zero_lt_one.trans_le (by simp),
have B : ‖x‖ / (⌊‖x‖⌋₊ + 1) < 1, from (div_lt_one A).2 (nat.lt_floor_add_one _),
-- Then we apply the ratio test. The estimate works for `n ≥ ⌊‖x‖⌋₊`.
suffices : ∀ n ≥ ⌊‖x‖⌋₊, ‖x ^ (n + 1) / (n + 1)!‖ ... | lemma | real.summable_pow_div_factorial | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"div_lt_one",
"div_mul_div_comm",
"nat.cast_mul",
"nat.cast_succ",
"nat.factorial_succ",
"nat.lt_floor_add_one",
"norm_div",
"norm_mul",
"pow_succ",
"real.norm_coe_nat",
"summable",
"summable_of_ratio_norm_eventually_le"
] | The series `∑' n, x ^ n / n!` is summable of any `x : ℝ`. See also `exp_series_div_summable`
for a version that also works in `ℂ`, and `exp_series_summable'` for a version that works in
any normed algebra over `ℝ` or `ℂ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real.tendsto_pow_div_factorial_at_top (x : ℝ) :
tendsto (λ n, x ^ n / n! : ℕ → ℝ) at_top (𝓝 0) | (real.summable_pow_div_factorial x).tendsto_at_top_zero | lemma | real.tendsto_pow_div_factorial_at_top | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"real.summable_pow_div_factorial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wstar_algebra (M : Type u) [normed_ring M] [star_ring M] [cstar_ring M]
[module ℂ M] [normed_algebra ℂ M] [star_module ℂ M] | (exists_predual : ∃ (X : Type u) [normed_add_comm_group X] [normed_space ℂ X] [complete_space X],
nonempty (normed_space.dual ℂ X ≃ₗᵢ⋆[ℂ] M)) | class | wstar_algebra | analysis.von_neumann_algebra | src/analysis/von_neumann_algebra/basic.lean | [
"analysis.normed_space.dual",
"analysis.normed_space.star.basic",
"analysis.complex.basic",
"analysis.inner_product_space.adjoint",
"algebra.star.subalgebra"
] | [
"complete_space",
"cstar_ring",
"module",
"normed_add_comm_group",
"normed_algebra",
"normed_ring",
"normed_space",
"normed_space.dual",
"star_module",
"star_ring"
] | Sakai's definition of a von Neumann algebra as a C^* algebra with a Banach space predual.
So that we can unambiguously talk about these "abstract" von Neumann algebras
in parallel with the "concrete" ones (weakly closed *-subalgebras of B(H)),
we name this definition `wstar_algebra`.
Note that for now we only assert ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
von_neumann_algebra (H : Type u)
[normed_add_comm_group H] [inner_product_space ℂ H] [complete_space H] extends
star_subalgebra ℂ (H →L[ℂ] H) | (centralizer_centralizer' :
set.centralizer (set.centralizer carrier) = carrier) | structure | von_neumann_algebra | analysis.von_neumann_algebra | src/analysis/von_neumann_algebra/basic.lean | [
"analysis.normed_space.dual",
"analysis.normed_space.star.basic",
"analysis.complex.basic",
"analysis.inner_product_space.adjoint",
"algebra.star.subalgebra"
] | [
"complete_space",
"inner_product_space",
"normed_add_comm_group",
"set.centralizer",
"star_subalgebra"
] | The double commutant definition of a von Neumann algebra,
as a *-closed subalgebra of bounded operators on a Hilbert space,
which is equal to its double commutant.
Note that this definition is parameterised by the Hilbert space
on which the algebra faithfully acts, as is standard in the literature.
See `wstar_algebra`... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_carrier {S : von_neumann_algebra H} {x : H →L[ℂ] H}:
x ∈ S.carrier ↔ x ∈ (S : set (H →L[ℂ] H)) | iff.rfl | lemma | von_neumann_algebra.mem_carrier | analysis.von_neumann_algebra | src/analysis/von_neumann_algebra/basic.lean | [
"analysis.normed_space.dual",
"analysis.normed_space.star.basic",
"analysis.complex.basic",
"analysis.inner_product_space.adjoint",
"algebra.star.subalgebra"
] | [
"von_neumann_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {S T : von_neumann_algebra H} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T | set_like.ext h | theorem | von_neumann_algebra.ext | analysis.von_neumann_algebra | src/analysis/von_neumann_algebra/basic.lean | [
"analysis.normed_space.dual",
"analysis.normed_space.star.basic",
"analysis.complex.basic",
"analysis.inner_product_space.adjoint",
"algebra.star.subalgebra"
] | [
"set_like.ext",
"von_neumann_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
centralizer_centralizer (S : von_neumann_algebra H) :
set.centralizer (set.centralizer (S : set (H →L[ℂ] H))) = S | S.centralizer_centralizer' | lemma | von_neumann_algebra.centralizer_centralizer | analysis.von_neumann_algebra | src/analysis/von_neumann_algebra/basic.lean | [
"analysis.normed_space.dual",
"analysis.normed_space.star.basic",
"analysis.complex.basic",
"analysis.inner_product_space.adjoint",
"algebra.star.subalgebra"
] | [
"set.centralizer",
"von_neumann_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commutant (S : von_neumann_algebra H) : von_neumann_algebra H | { carrier := set.centralizer (S : set (H →L[ℂ] H)),
centralizer_centralizer' := by rw S.centralizer_centralizer,
.. star_subalgebra.centralizer ℂ (S : set (H →L[ℂ] H)) (λ a (ha : a ∈ S), (star_mem ha : _)) } | def | von_neumann_algebra.commutant | analysis.von_neumann_algebra | src/analysis/von_neumann_algebra/basic.lean | [
"analysis.normed_space.dual",
"analysis.normed_space.star.basic",
"analysis.complex.basic",
"analysis.inner_product_space.adjoint",
"algebra.star.subalgebra"
] | [
"set.centralizer",
"star_subalgebra.centralizer",
"von_neumann_algebra"
] | The centralizer of a `von_neumann_algebra`, as a `von_neumann_algebra`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_commutant (S : von_neumann_algebra H) :
↑S.commutant = set.centralizer (S : set (H →L[ℂ] H)) | rfl | lemma | von_neumann_algebra.coe_commutant | analysis.von_neumann_algebra | src/analysis/von_neumann_algebra/basic.lean | [
"analysis.normed_space.dual",
"analysis.normed_space.star.basic",
"analysis.complex.basic",
"analysis.inner_product_space.adjoint",
"algebra.star.subalgebra"
] | [
"set.centralizer",
"von_neumann_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_commutant_iff {S : von_neumann_algebra H} {z : H →L[ℂ] H} :
z ∈ S.commutant ↔ ∀ g ∈ S, g * z = z * g | iff.rfl | lemma | von_neumann_algebra.mem_commutant_iff | analysis.von_neumann_algebra | src/analysis/von_neumann_algebra/basic.lean | [
"analysis.normed_space.dual",
"analysis.normed_space.star.basic",
"analysis.complex.basic",
"analysis.inner_product_space.adjoint",
"algebra.star.subalgebra"
] | [
"von_neumann_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commutant_commutant (S : von_neumann_algebra H) :
S.commutant.commutant = S | set_like.coe_injective S.centralizer_centralizer' | lemma | von_neumann_algebra.commutant_commutant | analysis.von_neumann_algebra | src/analysis/von_neumann_algebra/basic.lean | [
"analysis.normed_space.dual",
"analysis.normed_space.star.basic",
"analysis.complex.basic",
"analysis.inner_product_space.adjoint",
"algebra.star.subalgebra"
] | [
"set_like.coe_injective",
"von_neumann_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
action_as_functor : single_obj M ⥤ Type u | { obj := λ _, X,
map := λ _ _, (•),
map_id' := λ _, funext $ mul_action.one_smul,
map_comp' := λ _ _ _ f g, funext $ λ x, (smul_smul g f x).symm } | def | category_theory.action_as_functor | category_theory | src/category_theory/action.lean | [
"category_theory.elements",
"category_theory.is_connected",
"category_theory.single_obj",
"group_theory.group_action.quotient",
"group_theory.semidirect_product"
] | [
"smul_smul"
] | A multiplicative action M ↻ X viewed as a functor mapping the single object of M to X
and an element `m : M` to the map `X → X` given by multiplication by `m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
action_category | (action_as_functor M X).elements | def | category_theory.action_category | category_theory | src/category_theory/action.lean | [
"category_theory.elements",
"category_theory.is_connected",
"category_theory.single_obj",
"group_theory.group_action.quotient",
"group_theory.semidirect_product"
] | [] | A multiplicative action M ↻ X induces a category strucure on X, where a morphism
from x to y is a scalar taking x to y. Due to implementation details, the object type
of this category is not equal to X, but is in bijection with X. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
π : action_category M X ⥤ single_obj M | category_of_elements.π _ | def | category_theory.action_category.π | category_theory | src/category_theory/action.lean | [
"category_theory.elements",
"category_theory.is_connected",
"category_theory.single_obj",
"group_theory.group_action.quotient",
"group_theory.semidirect_product"
] | [] | The projection from the action category to the monoid, mapping a morphism to its
label. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
π_map (p q : action_category M X) (f : p ⟶ q) : (π M X).map f = f.val | rfl | lemma | category_theory.action_category.π_map | category_theory | src/category_theory/action.lean | [
"category_theory.elements",
"category_theory.is_connected",
"category_theory.single_obj",
"group_theory.group_action.quotient",
"group_theory.semidirect_product"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
π_obj (p : action_category M X) : (π M X).obj p = single_obj.star M | unit.ext | lemma | category_theory.action_category.π_obj | category_theory | src/category_theory/action.lean | [
"category_theory.elements",
"category_theory.is_connected",
"category_theory.single_obj",
"group_theory.group_action.quotient",
"group_theory.semidirect_product"
] | [
"unit.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
back : action_category M X → X | λ x, x.snd | def | category_theory.action_category.back | category_theory | src/category_theory/action.lean | [
"category_theory.elements",
"category_theory.is_connected",
"category_theory.single_obj",
"group_theory.group_action.quotient",
"group_theory.semidirect_product"
] | [] | The canonical map `action_category M X → X`. It is given by `λ x, x.snd`, but
has a more explicit type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_back (x : X) : (↑x : action_category M X).back = x | rfl | lemma | category_theory.action_category.coe_back | category_theory | src/category_theory/action.lean | [
"category_theory.elements",
"category_theory.is_connected",
"category_theory.single_obj",
"group_theory.group_action.quotient",
"group_theory.semidirect_product"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
back_coe (x : action_category M X) : ↑(x.back) = x | by ext; refl | lemma | category_theory.action_category.back_coe | category_theory | src/category_theory/action.lean | [
"category_theory.elements",
"category_theory.is_connected",
"category_theory.single_obj",
"group_theory.group_action.quotient",
"group_theory.semidirect_product"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
obj_equiv : X ≃ action_category M X | { to_fun := coe,
inv_fun := λ x, x.back,
left_inv := coe_back,
right_inv := back_coe } | def | category_theory.action_category.obj_equiv | category_theory | src/category_theory/action.lean | [
"category_theory.elements",
"category_theory.is_connected",
"category_theory.single_obj",
"group_theory.group_action.quotient",
"group_theory.semidirect_product"
] | [
"inv_fun"
] | An object of the action category given by M ↻ X corresponds to an element of X. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_as_subtype (p q : action_category M X) :
(p ⟶ q) = { m : M // m • p.back = q.back } | rfl | lemma | category_theory.action_category.hom_as_subtype | category_theory | src/category_theory/action.lean | [
"category_theory.elements",
"category_theory.is_connected",
"category_theory.single_obj",
"group_theory.group_action.quotient",
"group_theory.semidirect_product"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stabilizer_iso_End : stabilizer.submonoid M x ≃* End (↑x : action_category M X) | mul_equiv.refl _ | def | category_theory.action_category.stabilizer_iso_End | category_theory | src/category_theory/action.lean | [
"category_theory.elements",
"category_theory.is_connected",
"category_theory.single_obj",
"group_theory.group_action.quotient",
"group_theory.semidirect_product"
] | [
"mul_equiv.refl"
] | The stabilizer of a point is isomorphic to the endomorphism monoid at the
corresponding point. In fact they are definitionally equivalent. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stabilizer_iso_End_apply (f : stabilizer.submonoid M x) :
(stabilizer_iso_End M x).to_fun f = f | rfl | lemma | category_theory.action_category.stabilizer_iso_End_apply | category_theory | src/category_theory/action.lean | [
"category_theory.elements",
"category_theory.is_connected",
"category_theory.single_obj",
"group_theory.group_action.quotient",
"group_theory.semidirect_product"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stabilizer_iso_End_symm_apply (f : End _) :
(stabilizer_iso_End M x).inv_fun f = f | rfl | lemma | category_theory.action_category.stabilizer_iso_End_symm_apply | category_theory | src/category_theory/action.lean | [
"category_theory.elements",
"category_theory.is_connected",
"category_theory.single_obj",
"group_theory.group_action.quotient",
"group_theory.semidirect_product"
] | [
"inv_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_val (x : action_category M X) : subtype.val (𝟙 x) = 1 | rfl | lemma | category_theory.action_category.id_val | category_theory | src/category_theory/action.lean | [
"category_theory.elements",
"category_theory.is_connected",
"category_theory.single_obj",
"group_theory.group_action.quotient",
"group_theory.semidirect_product"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_val {x y z : action_category M X}
(f : x ⟶ y) (g : y ⟶ z) : (f ≫ g).val = g.val * f.val | rfl | lemma | category_theory.action_category.comp_val | category_theory | src/category_theory/action.lean | [
"category_theory.elements",
"category_theory.is_connected",
"category_theory.single_obj",
"group_theory.group_action.quotient",
"group_theory.semidirect_product"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
End_mul_equiv_subgroup (H : subgroup G) :
End (obj_equiv G (G ⧸ H) ↑(1 : G)) ≃* H | mul_equiv.trans
(stabilizer_iso_End G ((1 : G) : G ⧸ H)).symm
(mul_equiv.subgroup_congr $ stabilizer_quotient H) | def | category_theory.action_category.End_mul_equiv_subgroup | category_theory | src/category_theory/action.lean | [
"category_theory.elements",
"category_theory.is_connected",
"category_theory.single_obj",
"group_theory.group_action.quotient",
"group_theory.semidirect_product"
] | [
"mul_equiv.subgroup_congr",
"mul_equiv.trans",
"subgroup"
] | Any subgroup of `G` is a vertex group in its action groupoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_of_pair (t : X) (g : G) : ↑(g⁻¹ • t) ⟶ (t : action_category G X) | subtype.mk g (smul_inv_smul g t) | def | category_theory.action_category.hom_of_pair | category_theory | src/category_theory/action.lean | [
"category_theory.elements",
"category_theory.is_connected",
"category_theory.single_obj",
"group_theory.group_action.quotient",
"group_theory.semidirect_product"
] | [
"smul_inv_smul"
] | A target vertex `t` and a scalar `g` determine a morphism in the action groupoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_of_pair.val (t : X) (g : G) : (hom_of_pair t g).val = g | rfl | lemma | category_theory.action_category.hom_of_pair.val | category_theory | src/category_theory/action.lean | [
"category_theory.elements",
"category_theory.is_connected",
"category_theory.single_obj",
"group_theory.group_action.quotient",
"group_theory.semidirect_product"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cases {P : Π ⦃a b : action_category G X⦄, (a ⟶ b) → Sort*}
(hyp : ∀ t g, P (hom_of_pair t g)) ⦃a b⦄ (f : a ⟶ b) : P f | begin
refine cast _ (hyp b.back f.val),
rcases a with ⟨⟨⟩, a : X⟩,
rcases b with ⟨⟨⟩, b : X⟩,
rcases f with ⟨g : G, h : g • a = b⟩,
cases (inv_smul_eq_iff.mpr h.symm),
refl
end | def | category_theory.action_category.cases | category_theory | src/category_theory/action.lean | [
"category_theory.elements",
"category_theory.is_connected",
"category_theory.single_obj",
"group_theory.group_action.quotient",
"group_theory.semidirect_product"
] | [] | Any morphism in the action groupoid is given by some pair. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
curry (F : action_category G X ⥤ single_obj H) :
G →* (X → H) ⋊[mul_aut_arrow] G | have F_map_eq : ∀ {a b} {f : a ⟶ b}, F.map f = (F.map (hom_of_pair b.back f.val) : H) :=
action_category.cases (λ _ _, rfl),
{ to_fun := λ g, ⟨λ b, F.map (hom_of_pair b g), g⟩,
map_one' := by { congr, funext, exact F_map_eq.symm.trans (F.map_id b) },
map_mul' := begin
intros g h,
congr, funext,
exact ... | def | category_theory.action_category.curry | category_theory | src/category_theory/action.lean | [
"category_theory.elements",
"category_theory.is_connected",
"category_theory.single_obj",
"group_theory.group_action.quotient",
"group_theory.semidirect_product"
] | [
"mul_aut_arrow"
] | Given `G` acting on `X`, a functor from the corresponding action groupoid to a group `H`
can be curried to a group homomorphism `G →* (X → H) ⋊ G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uncurry (F : G →* (X → H) ⋊[mul_aut_arrow] G) (sane : ∀ g, (F g).right = g) :
action_category G X ⥤ single_obj H | { obj := λ _, (),
map := λ a b f, ((F f.val).left b.back),
map_id' := by { intro x, rw [action_category.id_val, F.map_one], refl },
map_comp' := begin
intros x y z f g, revert y z g,
refine action_category.cases _,
simp [single_obj.comp_as_mul, sane],
end } | def | category_theory.action_category.uncurry | category_theory | src/category_theory/action.lean | [
"category_theory.elements",
"category_theory.is_connected",
"category_theory.single_obj",
"group_theory.group_action.quotient",
"group_theory.semidirect_product"
] | [
"mul_aut_arrow"
] | Given `G` acting on `X`, a group homomorphism `φ : G →* (X → H) ⋊ G` can be uncurried to
a functor from the action groupoid to `H`, provided that `φ g = (_, g)` for all `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_pushout.is_van_kampen (H : is_pushout f g h i) : Prop | ∀ ⦃W' X' Y' Z' : C⦄ (f' : W' ⟶ X') (g' : W' ⟶ Y') (h' : X' ⟶ Z') (i' : Y' ⟶ Z')
(αW : W' ⟶ W) (αX : X' ⟶ X) (αY : Y' ⟶ Y) (αZ : Z' ⟶ Z)
(hf : is_pullback f' αW αX f) (hg : is_pullback g' αW αY g)
(hh : comm_sq h' αX αZ h) (hi : comm_sq i' αY αZ i)
(w : comm_sq f' g' h' i'),
is_pushout f' g' h' i' ↔ is_pullbac... | def | category_theory.is_pushout.is_van_kampen | category_theory | src/category_theory/adhesive.lean | [
"category_theory.extensive",
"category_theory.limits.shapes.kernel_pair"
] | [] | A convenience formulation for a pushout being a van Kampen colimit.
See `is_pushout.is_van_kampen_iff` below. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_pushout.is_van_kampen.flip {H : is_pushout f g h i} (H' : H.is_van_kampen) :
H.flip.is_van_kampen | begin
introv W' hf hg hh hi w,
simpa only [is_pushout.flip_iff, is_pullback.flip_iff, and_comm] using
H' g' f' i' h' αW αY αX αZ hg hf hi hh w.flip,
end | lemma | category_theory.is_pushout.is_van_kampen.flip | category_theory | src/category_theory/adhesive.lean | [
"category_theory.extensive",
"category_theory.limits.shapes.kernel_pair"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_pushout.is_van_kampen_iff (H : is_pushout f g h i) :
H.is_van_kampen ↔ is_van_kampen_colimit (pushout_cocone.mk h i H.w) | begin
split,
{ intros H F' c' α fα eα hα,
refine iff.trans _ ((H (F'.map walking_span.hom.fst) (F'.map walking_span.hom.snd)
(c'.ι.app _) (c'.ι.app _) (α.app _) (α.app _) (α.app _) fα
(by convert hα walking_span.hom.fst) (by convert hα walking_span.hom.snd)
_ _ _).trans _),
{ have : F'.map... | lemma | category_theory.is_pushout.is_van_kampen_iff | category_theory | src/category_theory/adhesive.lean | [
"category_theory.extensive",
"category_theory.limits.shapes.kernel_pair"
] | [
"functor.map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprod_iff_is_pushout {X E Y YE : C} (c : binary_cofan X E)
(hc : is_colimit c) {f : X ⟶ Y} {iY : Y ⟶ YE} {fE : c.X ⟶ YE}
(H : comm_sq f c.inl iY fE) :
nonempty (is_colimit (binary_cofan.mk (c.inr ≫ fE) iY)) ↔ is_pushout f c.inl iY fE | begin
split,
{ rintro ⟨h⟩,
refine ⟨H, ⟨limits.pushout_cocone.is_colimit_aux' _ _⟩⟩,
intro s,
dsimp,
refine ⟨h.desc (binary_cofan.mk (c.inr ≫ s.inr) s.inl), h.fac _ ⟨walking_pair.right⟩, _, _⟩,
{ apply binary_cofan.is_colimit.hom_ext hc,
{ rw ← H.w_assoc, erw h.fac _ ⟨walking_pair.right⟩, e... | lemma | category_theory.is_coprod_iff_is_pushout | category_theory | src/category_theory/adhesive.lean | [
"category_theory.extensive",
"category_theory.limits.shapes.kernel_pair"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_pushout.is_van_kampen_inl {W E X Z : C} (c : binary_cofan W E)
[finitary_extensive C]
[has_pullbacks C]
(hc : is_colimit c) (f : W ⟶ X) (h : X ⟶ Z) (i : c.X ⟶ Z)
(H : is_pushout f c.inl h i) : H.is_van_kampen | begin
obtain ⟨hc₁⟩ := (is_coprod_iff_is_pushout c hc H.1).mpr H,
introv W' hf hg hh hi w,
obtain ⟨hc₂⟩ := ((binary_cofan.is_van_kampen_iff _).mp (finitary_extensive.van_kampen c hc)
(binary_cofan.mk _ pullback.fst) _ _ _ hg.w.symm pullback.condition.symm).mpr
⟨hg, is_pullback.of_has_pullback αY c.inr⟩,
... | lemma | category_theory.is_pushout.is_van_kampen_inl | category_theory | src/category_theory/adhesive.lean | [
"category_theory.extensive",
"category_theory.limits.shapes.kernel_pair"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_pushout.is_van_kampen.is_pullback_of_mono_left [mono f]
{H : is_pushout f g h i} (H' : H.is_van_kampen) :
is_pullback f g h i | ((H' (𝟙 _) g g (𝟙 Y) (𝟙 _) f (𝟙 _) i
(is_kernel_pair.id_of_mono f) (is_pullback.of_vert_is_iso ⟨by simp⟩) H.1.flip ⟨rfl⟩
⟨by simp⟩).mp (is_pushout.of_horiz_is_iso ⟨by simp⟩)).1.flip | lemma | category_theory.is_pushout.is_van_kampen.is_pullback_of_mono_left | category_theory | src/category_theory/adhesive.lean | [
"category_theory.extensive",
"category_theory.limits.shapes.kernel_pair"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_pushout.is_van_kampen.is_pullback_of_mono_right [mono g]
{H : is_pushout f g h i} (H' : H.is_van_kampen) :
is_pullback f g h i | ((H' f (𝟙 _) (𝟙 _) f (𝟙 _) (𝟙 _) g h (is_pullback.of_vert_is_iso ⟨by simp⟩)
(is_kernel_pair.id_of_mono g) ⟨rfl⟩ H.1
⟨by simp⟩).mp (is_pushout.of_vert_is_iso ⟨by simp⟩)).2 | lemma | category_theory.is_pushout.is_van_kampen.is_pullback_of_mono_right | category_theory | src/category_theory/adhesive.lean | [
"category_theory.extensive",
"category_theory.limits.shapes.kernel_pair"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_pushout.is_van_kampen.mono_of_mono_left [mono f]
{H : is_pushout f g h i} (H' : H.is_van_kampen) :
mono i | is_kernel_pair.mono_of_is_iso_fst
(((H' (𝟙 _) g g (𝟙 Y) (𝟙 _) f (𝟙 _) i
(is_kernel_pair.id_of_mono f) (is_pullback.of_vert_is_iso ⟨by simp⟩) H.1.flip ⟨rfl⟩
⟨by simp⟩).mp (is_pushout.of_horiz_is_iso ⟨by simp⟩)).2) | lemma | category_theory.is_pushout.is_van_kampen.mono_of_mono_left | category_theory | src/category_theory/adhesive.lean | [
"category_theory.extensive",
"category_theory.limits.shapes.kernel_pair"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_pushout.is_van_kampen.mono_of_mono_right [mono g]
{H : is_pushout f g h i} (H' : H.is_van_kampen) :
mono h | is_kernel_pair.mono_of_is_iso_fst
((H' f (𝟙 _) (𝟙 _) f (𝟙 _) (𝟙 _) g h (is_pullback.of_vert_is_iso ⟨by simp⟩)
(is_kernel_pair.id_of_mono g) ⟨rfl⟩ H.1
⟨by simp⟩).mp (is_pushout.of_vert_is_iso ⟨by simp⟩)).1 | lemma | category_theory.is_pushout.is_van_kampen.mono_of_mono_right | category_theory | src/category_theory/adhesive.lean | [
"category_theory.extensive",
"category_theory.limits.shapes.kernel_pair"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adhesive (C : Type u) [category.{v} C] : Prop | [has_pullback_of_mono_left : ∀ {X Y S : C} (f : X ⟶ S) (g : Y ⟶ S) [mono f], has_pullback f g]
[has_pushout_of_mono_left : ∀ {X Y S : C} (f : S ⟶ X) (g : S ⟶ Y) [mono f], has_pushout f g]
(van_kampen : ∀ {W X Y Z : C} {f : W ⟶ X} {g : W ⟶ Y} {h : X ⟶ Z} {i : Y ⟶ Z} [mono f]
(H : is_pushout f g h i), H.is_van_kampen) | class | category_theory.adhesive | category_theory | src/category_theory/adhesive.lean | [
"category_theory.extensive",
"category_theory.limits.shapes.kernel_pair"
] | [] | A category is adhesive if it has pushouts and pullbacks along monomorphisms,
and such pushouts are van Kampen. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adhesive.van_kampen' [adhesive C] [mono g] (H : is_pushout f g h i) : H.is_van_kampen | (adhesive.van_kampen H.flip).flip | lemma | category_theory.adhesive.van_kampen' | category_theory | src/category_theory/adhesive.lean | [
"category_theory.extensive",
"category_theory.limits.shapes.kernel_pair"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adhesive.is_pullback_of_is_pushout_of_mono_left [adhesive C]
(H : is_pushout f g h i) [mono f] : is_pullback f g h i | (adhesive.van_kampen H).is_pullback_of_mono_left | lemma | category_theory.adhesive.is_pullback_of_is_pushout_of_mono_left | category_theory | src/category_theory/adhesive.lean | [
"category_theory.extensive",
"category_theory.limits.shapes.kernel_pair"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adhesive.is_pullback_of_is_pushout_of_mono_right [adhesive C]
(H : is_pushout f g h i) [mono g] : is_pullback f g h i | (adhesive.van_kampen' H).is_pullback_of_mono_right | lemma | category_theory.adhesive.is_pullback_of_is_pushout_of_mono_right | category_theory | src/category_theory/adhesive.lean | [
"category_theory.extensive",
"category_theory.limits.shapes.kernel_pair"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adhesive.mono_of_is_pushout_of_mono_left [adhesive C]
(H : is_pushout f g h i) [mono f] : mono i | (adhesive.van_kampen H).mono_of_mono_left | lemma | category_theory.adhesive.mono_of_is_pushout_of_mono_left | category_theory | src/category_theory/adhesive.lean | [
"category_theory.extensive",
"category_theory.limits.shapes.kernel_pair"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adhesive.mono_of_is_pushout_of_mono_right [adhesive C]
(H : is_pushout f g h i) [mono g] : mono h | (adhesive.van_kampen' H).mono_of_mono_right | lemma | category_theory.adhesive.mono_of_is_pushout_of_mono_right | category_theory | src/category_theory/adhesive.lean | [
"category_theory.extensive",
"category_theory.limits.shapes.kernel_pair"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
type.adhesive : adhesive (Type u) | begin
constructor,
intros,
exactI (is_pushout.is_van_kampen_inl _ (types.is_coprod_of_mono f) _ _ _ H.flip).flip
end | instance | category_theory.type.adhesive | category_theory | src/category_theory/adhesive.lean | [
"category_theory.extensive",
"category_theory.limits.shapes.kernel_pair"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adhesive.to_regular_mono_category [adhesive C] : regular_mono_category C | ⟨λ X Y f hf, by exactI
{ Z := pushout f f,
left := pushout.inl,
right := pushout.inr,
w := pushout.condition,
is_limit := (adhesive.is_pullback_of_is_pushout_of_mono_left
(is_pushout.of_has_pushout f f)).is_limit_fork }⟩ | instance | category_theory.adhesive.to_regular_mono_category | category_theory | src/category_theory/adhesive.lean | [
"category_theory.extensive",
"category_theory.limits.shapes.kernel_pair"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arrow | comma.{v v v} (𝟭 T) (𝟭 T) | def | category_theory.arrow | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | The arrow category of `T` has as objects all morphisms in `T` and as morphisms commutative
squares in `T`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
arrow.inhabited [inhabited T] : inhabited (arrow T) | { default := show comma (𝟭 T) (𝟭 T), from default } | instance | category_theory.arrow.inhabited | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_left (f : arrow T) : comma_morphism.left (𝟙 f) = 𝟙 (f.left) | rfl | lemma | category_theory.arrow.id_left | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_right (f : arrow T) : comma_morphism.right (𝟙 f) = 𝟙 (f.right) | rfl | lemma | category_theory.arrow.id_right | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk {X Y : T} (f : X ⟶ Y) : arrow T | { left := X,
right := Y,
hom := f } | def | category_theory.arrow.mk | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | An object in the arrow category is simply a morphism in `T`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_eq (f : arrow T) : arrow.mk f.hom = f | by { cases f, refl, } | lemma | category_theory.arrow.mk_eq | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_injective (A B : T) :
function.injective (arrow.mk : (A ⟶ B) → arrow T) | λ f g h, by { cases h, refl } | theorem | category_theory.arrow.mk_injective | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_inj (A B : T) {f g : A ⟶ B} : arrow.mk f = arrow.mk g ↔ f = g | (mk_injective A B).eq_iff | theorem | category_theory.arrow.mk_inj | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_mk {f g : arrow T} {u : f.left ⟶ g.left} {v : f.right ⟶ g.right}
(w : u ≫ g.hom = f.hom ≫ v) : f ⟶ g | { left := u,
right := v,
w' := w } | def | category_theory.arrow.hom_mk | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | A morphism in the arrow category is a commutative square connecting two objects of the arrow
category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_mk' {X Y : T} {f : X ⟶ Y} {P Q : T} {g : P ⟶ Q} {u : X ⟶ P} {v : Y ⟶ Q}
(w : u ≫ g = f ≫ v) : arrow.mk f ⟶ arrow.mk g | { left := u,
right := v,
w' := w } | def | category_theory.arrow.hom_mk' | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | We can also build a morphism in the arrow category out of any commutative square in `T`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
w {f g : arrow T} (sq : f ⟶ g) : sq.left ≫ g.hom = f.hom ≫ sq.right | sq.w | lemma | category_theory.arrow.w | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
w_mk_right {f : arrow T} {X Y : T} {g : X ⟶ Y} (sq : f ⟶ mk g) :
sq.left ≫ g = f.hom ≫ sq.right | sq.w | lemma | category_theory.arrow.w_mk_right | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_of_iso_left_of_is_iso_right
{f g : arrow T} (ff : f ⟶ g) [is_iso ff.left] [is_iso ff.right] : is_iso ff | { out := ⟨⟨inv ff.left, inv ff.right⟩,
by { ext; dsimp; simp only [is_iso.hom_inv_id] },
by { ext; dsimp; simp only [is_iso.inv_hom_id] }⟩ } | lemma | category_theory.arrow.is_iso_of_iso_left_of_is_iso_right | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_mk {f g : arrow T}
(l : f.left ≅ g.left) (r : f.right ≅ g.right) (h : l.hom ≫ g.hom = f.hom ≫ r.hom) :
f ≅ g | comma.iso_mk l r h | def | category_theory.arrow.iso_mk | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | Create an isomorphism between arrows,
by providing isomorphisms between the domains and codomains,
and a proof that the square commutes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_mk' {W X Y Z : T} (f : W ⟶ X) (g : Y ⟶ Z)
(e₁ : W ≅ Y) (e₂ : X ≅ Z) (h : e₁.hom ≫ g = f ≫ e₂.hom) : arrow.mk f ≅ arrow.mk g | arrow.iso_mk e₁ e₂ h | abbreviation | category_theory.arrow.iso_mk' | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | A variant of `arrow.iso_mk` that creates an iso between two `arrow.mk`s with a better type
signature. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom.congr_left {f g : arrow T} {φ₁ φ₂ : f ⟶ g} (h : φ₁ = φ₂) :
φ₁.left = φ₂.left | by rw h | lemma | category_theory.arrow.hom.congr_left | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom.congr_right {f g : arrow T} {φ₁ φ₂ : f ⟶ g} (h : φ₁ = φ₂) :
φ₁.right = φ₂.right | by rw h | lemma | category_theory.arrow.hom.congr_right | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_w {f g : arrow T} (e : f ≅ g) : g.hom = e.inv.left ≫ f.hom ≫ e.hom.right | begin
have eq := arrow.hom.congr_right e.inv_hom_id,
dsimp at eq,
erw [w_assoc, eq, category.comp_id],
end | lemma | category_theory.arrow.iso_w | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_w' {W X Y Z : T} {f : W ⟶ X} {g : Y ⟶ Z} (e : arrow.mk f ≅ arrow.mk g) :
g = e.inv.left ≫ f ≫ e.hom.right | iso_w e | lemma | category_theory.arrow.iso_w' | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_left [is_iso sq] : is_iso sq.left | { out := ⟨(inv sq).left, by simp only [← comma.comp_left, is_iso.hom_inv_id, is_iso.inv_hom_id,
arrow.id_left, eq_self_iff_true, and_self]⟩ } | instance | category_theory.arrow.is_iso_left | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_right [is_iso sq] : is_iso sq.right | { out := ⟨(inv sq).right, by simp only [← comma.comp_right, is_iso.hom_inv_id, is_iso.inv_hom_id,
arrow.id_right, eq_self_iff_true, and_self]⟩ } | instance | category_theory.arrow.is_iso_right | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_left [is_iso sq] : (inv sq).left = inv sq.left | is_iso.eq_inv_of_hom_inv_id $ by rw [← comma.comp_left, is_iso.hom_inv_id, id_left] | lemma | category_theory.arrow.inv_left | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_right [is_iso sq] : (inv sq).right = inv sq.right | is_iso.eq_inv_of_hom_inv_id $ by rw [← comma.comp_right, is_iso.hom_inv_id, id_right] | lemma | category_theory.arrow.inv_right | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_hom_inv_right [is_iso sq] : sq.left ≫ g.hom ≫ inv sq.right = f.hom | by simp only [← category.assoc, is_iso.comp_inv_eq, w] | lemma | category_theory.arrow.left_hom_inv_right | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_left_hom_right [is_iso sq] : inv sq.left ≫ f.hom ≫ sq.right = g.hom | by simp only [w, is_iso.inv_comp_eq] | lemma | category_theory.arrow.inv_left_hom_right | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_left [mono sq] : mono sq.left | { right_cancellation := λ Z φ ψ h, begin
let aux : (Z ⟶ f.left) → (arrow.mk (𝟙 Z) ⟶ f) := λ φ, { left := φ, right := φ ≫ f.hom },
show (aux φ).left = (aux ψ).left,
congr' 1,
rw ← cancel_mono sq,
ext,
{ exact h },
{ simp only [comma.comp_right, category.assoc, ← arrow.w],
simp only [← ... | instance | category_theory.arrow.mono_left | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [
"aux"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_right [epi sq] : epi sq.right | { left_cancellation := λ Z φ ψ h, begin
let aux : (g.right ⟶ Z) → (g ⟶ arrow.mk (𝟙 Z)) := λ φ, { right := φ, left := g.hom ≫ φ },
show (aux φ).right = (aux ψ).right,
congr' 1,
rw ← cancel_epi sq,
ext,
{ simp only [comma.comp_left, category.assoc, arrow.w_assoc, h], },
{ exact h },
end } | instance | category_theory.arrow.epi_right | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [
"aux"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
square_to_iso_invert (i : arrow T) {X Y : T} (p : X ≅ Y) (sq : i ⟶ arrow.mk p.hom) :
i.hom ≫ sq.right ≫ p.inv = sq.left | by simpa only [category.assoc] using (iso.comp_inv_eq p).mpr ((arrow.w_mk_right sq).symm) | lemma | category_theory.arrow.square_to_iso_invert | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | Given a square from an arrow `i` to an isomorphism `p`, express the source part of `sq`
in terms of the inverse of `p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
square_from_iso_invert {X Y : T} (i : X ≅ Y) (p : arrow T) (sq : arrow.mk i.hom ⟶ p) :
i.inv ≫ sq.left ≫ p.hom = sq.right | by simp only [iso.inv_hom_id_assoc, arrow.w, arrow.mk_hom] | lemma | category_theory.arrow.square_from_iso_invert | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | Given a square from an isomorphism `i` to an arrow `p`, express the target part of `sq`
in terms of the inverse of `i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
square_to_snd {X Y Z: C} {i : arrow C} {f : X ⟶ Y} {g : Y ⟶ Z}
(sq : i ⟶ arrow.mk (f ≫ g)) :
i ⟶ arrow.mk g | { left := sq.left ≫ f,
right := sq.right } | def | category_theory.arrow.square_to_snd | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | A helper construction: given a square between `i` and `f ≫ g`, produce a square between
`i` and `g`, whose top leg uses `f`:
A → X
↓f
↓i Y --> A → Y
↓g ↓i ↓g
B → Z B → Z | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_func : arrow C ⥤ C | comma.fst _ _ | def | category_theory.arrow.left_func | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | The functor sending an arrow to its source. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_func : arrow C ⥤ C | comma.snd _ _ | def | category_theory.arrow.right_func | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | The functor sending an arrow to its target. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_to_right : (left_func : arrow C ⥤ C) ⟶ right_func | { app := λ f, f.hom } | def | category_theory.arrow.left_to_right | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | The natural transformation from `left_func` to `right_func`, given by the arrow itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_arrow (F : C ⥤ D) : arrow C ⥤ arrow D | { obj := λ a,
{ left := F.obj a.left,
right := F.obj a.right,
hom := F.map a.hom, },
map := λ a b f,
{ left := F.map f.left,
right := F.map f.right,
w' := by { have w := f.w, simp only [id_map] at w, dsimp, simp only [←F.map_comp, w], } } } | def | category_theory.functor.map_arrow | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | A functor `C ⥤ D` induces a functor between the corresponding arrow categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
arrow.iso_of_nat_iso {C D : Type*} [category C] [category D]
{F G : C ⥤ D} (e : F ≅ G) (f : arrow C) :
F.map_arrow.obj f ≅ G.map_arrow.obj f | arrow.iso_mk (e.app f.left) (e.app f.right) (by simp) | def | category_theory.arrow.iso_of_nat_iso | category_theory | src/category_theory/arrow.lean | [
"category_theory.comma"
] | [] | The images of `f : arrow C` by two isomorphic functors `F : C ⥤ D` are
isomorphic arrows in `D`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
balanced : Prop | (is_iso_of_mono_of_epi : ∀ {X Y : C} (f : X ⟶ Y) [mono f] [epi f], is_iso f) | class | category_theory.balanced | category_theory | src/category_theory/balanced.lean | [
"category_theory.epi_mono"
] | [
"balanced"
] | A category is called balanced if any morphism that is both monic and epic is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_of_mono_of_epi [balanced C] {X Y : C} (f : X ⟶ Y) [mono f] [epi f] : is_iso f | balanced.is_iso_of_mono_of_epi _ | lemma | category_theory.is_iso_of_mono_of_epi | category_theory | src/category_theory/balanced.lean | [
"category_theory.epi_mono"
] | [
"balanced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_iff_mono_and_epi [balanced C] {X Y : C} (f : X ⟶ Y) : is_iso f ↔ mono f ∧ epi f | ⟨λ _, by exactI ⟨infer_instance, infer_instance⟩, λ ⟨_, _⟩, by exactI is_iso_of_mono_of_epi _⟩ | lemma | category_theory.is_iso_iff_mono_and_epi | category_theory | src/category_theory/balanced.lean | [
"category_theory.epi_mono"
] | [
"balanced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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