fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
sum_to_range :
Set.range (sum_to C ho) = GoodProducts (π C (ord I · < o)) ∪ MaxProducts C ho := by
have h : Set.range (sum_to C ho) = _ ∪ _ := Set.Sum.elim_range _ _; rw [h]; congr <;> ext l
· exact ⟨fun ⟨m,hm⟩ ↦ by rw [← hm]; exact m.prop, fun hl ↦ ⟨⟨l,hl⟩, rfl⟩⟩
· exact ⟨fun ⟨m,hm⟩ ↦ by rw [← hm]; exact m.p... | theorem | Topology | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/Successor.lean | sum_to_range | null |
noncomputable
sum_equiv (hsC : contained C (Order.succ o)) (ho : o < Ordinal.type (· < · : I → I → Prop)) :
GoodProducts (π C (ord I · < o)) ⊕ (MaxProducts C ho) ≃ GoodProducts C :=
calc _ ≃ Set.range (sum_to C ho) := Equiv.ofInjective (sum_to C ho) (injective_sum_to C ho)
_ ≃ _ := Equiv.setCongr <| by rw ... | def | Topology | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/Successor.lean | sum_equiv | The equivalence from the sum of `GoodProducts (π C (ord I · < o))` and
`(MaxProducts C ho)` to `GoodProducts C`. |
sum_equiv_comp_eval_eq_elim : eval C ∘ (sum_equiv C hsC ho).toFun =
(Sum.elim (fun (l : GoodProducts (π C (ord I · < o))) ↦ Products.eval C l.1)
(fun (l : MaxProducts C ho) ↦ Products.eval C l.1)) := by
ext ⟨_, _⟩ <;> [rfl; rfl] | theorem | Topology | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/Successor.lean | sum_equiv_comp_eval_eq_elim | null |
SumEval : GoodProducts (π C (ord I · < o)) ⊕ MaxProducts C ho →
LocallyConstant C ℤ :=
Sum.elim (fun l ↦ l.1.eval C) (fun l ↦ l.1.eval C)
include hsC in | def | Topology | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/Successor.lean | SumEval | Let
`N := LocallyConstant (π C (ord I · < o)) ℤ`
`M := LocallyConstant C ℤ`
`P := LocallyConstant (C' C ho) ℤ`
`ι := GoodProducts (π C (ord I · < o))`
`ι' := GoodProducts (C' C ho')`
`v : ι → N := GoodProducts.eval (π C (ord I · < o))`
Then `SumEval C ho` is the map `u` in the diagram below. It is linearly indep... |
linearIndependent_iff_sum :
LinearIndependent ℤ (eval C) ↔ LinearIndependent ℤ (SumEval C ho) := by
rw [← linearIndependent_equiv (sum_equiv C hsC ho), SumEval,
← sum_equiv_comp_eval_eq_elim C hsC ho]
exact Iff.rfl
include hsC in | theorem | Topology | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/Successor.lean | linearIndependent_iff_sum | null |
span_sum : Set.range (eval C) = Set.range (Sum.elim
(fun (l : GoodProducts (π C (ord I · < o))) ↦ Products.eval C l.1)
(fun (l : MaxProducts C ho) ↦ Products.eval C l.1)) := by
rw [← sum_equiv_comp_eval_eq_elim C hsC ho, Equiv.toFun_as_coe,
EquivLike.range_comp (e := sum_equiv C hsC ho)] | theorem | Topology | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/Successor.lean | span_sum | null |
square_commutes : SumEval C ho ∘ Sum.inl =
ModuleCat.ofHom (πs C o) ∘ eval (π C (ord I · < o)) := by
ext l
dsimp [SumEval]
rw [← Products.eval_πs C (Products.prop_of_isGood _ _ l.prop)]
simp [eval] | theorem | Topology | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/Successor.lean | square_commutes | null |
swapTrue_eq_true (x : I → Bool) : SwapTrue o x (term I ho) = true := by
simp only [SwapTrue, ord_term_aux, ite_true] | theorem | Topology | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/Successor.lean | swapTrue_eq_true | null |
mem_C'_eq_false : ∀ x, x ∈ C' C ho → x (term I ho) = false := by
rintro x ⟨_, y, _, rfl⟩
simp only [Proj, ord_term_aux, lt_self_iff_false, ite_false] | theorem | Topology | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/Successor.lean | mem_C'_eq_false | null |
Products.Tail (l : Products I) : Products I :=
⟨l.val.tail, List.IsChain.tail l.prop⟩ | def | Topology | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/Successor.lean | Products.Tail | `List.tail` as a `Products`. |
Products.max_eq_o_cons_tail [Inhabited I] (l : Products I) (hl : l.val ≠ [])
(hlh : l.val.head! = term I ho) : l.val = term I ho :: l.Tail.val := by
rw [← List.cons_head!_tail hl, hlh]
simp [Tail] | theorem | Topology | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/Successor.lean | Products.max_eq_o_cons_tail | null |
Products.max_eq_o_cons_tail' [Inhabited I] (l : Products I) (hl : l.val ≠ [])
(hlh : l.val.head! = term I ho) (hlc : List.IsChain (· > ·) (term I ho :: l.Tail.val)) :
l = ⟨term I ho :: l.Tail.val, hlc⟩ := by
simp_rw [← max_eq_o_cons_tail ho l hl hlh, Subtype.coe_eta]
include hsC in | theorem | Topology | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/Successor.lean | Products.max_eq_o_cons_tail' | null |
GoodProducts.head!_eq_o_of_maxProducts [Inhabited I] (l : ↑(MaxProducts C ho)) :
l.val.val.head! = term I ho := by
rw [eq_comm, ← ord_term ho]
have hm := l.prop.2
have := Products.prop_of_isGood_of_contained C _ l.prop.1 hsC l.val.val.head!
(List.head!_mem_self (List.ne_nil_of_mem hm))
simp only [Order.... | theorem | Topology | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/Successor.lean | GoodProducts.head | null |
GoodProducts.max_eq_o_cons_tail (l : MaxProducts C ho) :
l.val.val = (term I ho) :: l.val.Tail.val :=
have : Inhabited I := ⟨term I ho⟩
Products.max_eq_o_cons_tail ho l.val (List.ne_nil_of_mem l.prop.2)
(head!_eq_o_of_maxProducts _ hsC ho l) | theorem | Topology | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/Successor.lean | GoodProducts.max_eq_o_cons_tail | null |
Products.evalCons {I} [LinearOrder I] {C : Set (I → Bool)} {l : List I} {a : I}
(hla : (a::l).IsChain (· > ·)) : Products.eval C ⟨a::l,hla⟩ =
(e C a) * Products.eval C ⟨l,List.IsChain.sublist hla (List.tail_sublist (a::l))⟩ := by
simp only [eval.eq_1, List.map, List.prod_cons] | theorem | Topology | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/Successor.lean | Products.evalCons | null |
Products.max_eq_eval [Inhabited I] (l : Products I) (hl : l.val ≠ [])
(hlh : l.val.head! = term I ho) :
Linear_CC' C hsC ho (l.eval C) = l.Tail.eval (C' C ho) := by
have hlc : ((term I ho) :: l.Tail.val).IsChain (· > ·) := by
rw [← max_eq_o_cons_tail ho l hl hlh]; exact l.prop
rw [max_eq_o_cons_tail' ho... | theorem | Topology | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/Successor.lean | Products.max_eq_eval | null |
max_eq_eval (l : MaxProducts C ho) :
Linear_CC' C hsC ho (l.val.eval C) = l.val.Tail.eval (C' C ho) :=
have : Inhabited I := ⟨term I ho⟩
Products.max_eq_eval _ _ _ _ (List.ne_nil_of_mem l.prop.2)
(head!_eq_o_of_maxProducts _ hsC ho l) | theorem | Topology | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/Successor.lean | max_eq_eval | null |
max_eq_eval_unapply :
(Linear_CC' C hsC ho) ∘ (fun (l : MaxProducts C ho) ↦ Products.eval C l.val) =
(fun l ↦ l.val.Tail.eval (C' C ho)) := by
ext1 l
exact max_eq_eval _ _ _ _
include hsC in | theorem | Topology | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/Successor.lean | max_eq_eval_unapply | null |
isChain_cons_of_lt (l : MaxProducts C ho)
(q : Products I) (hq : q < l.val.Tail) :
List.IsChain (fun x x_1 ↦ x > x_1) (term I ho :: q.val) := by
have : Inhabited I := ⟨term I ho⟩
rw [List.isChain_iff_pairwise]
simp only [gt_iff_lt, List.pairwise_cons]
refine ⟨fun a ha ↦ lt_of_le_of_lt (Products.rel_head... | theorem | Topology | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/Successor.lean | isChain_cons_of_lt | null |
good_lt_maxProducts (q : GoodProducts (π C (ord I · < o)))
(l : MaxProducts C ho) : List.Lex (· < ·) q.val.val l.val.val := by
have : Inhabited I := ⟨term I ho⟩
by_cases h : q.val.val = []
· rw [h, max_eq_o_cons_tail C hsC ho l]
exact List.Lex.nil
· rw [← List.cons_head!_tail h, max_eq_o_cons_tail C hsC... | theorem | Topology | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/Successor.lean | good_lt_maxProducts | null |
maxTail_isGood (l : MaxProducts C ho)
(h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C (ord I · < o))))) :
l.val.Tail.isGood (C' C ho) := by
have : Inhabited I := ⟨term I ho⟩
intro h
rw [Finsupp.mem_span_image_iff_linearCombination, ← max_eq_eval C hsC ho] at h
obtain ⟨m, ⟨hmmem, hmsum⟩⟩ := h
rw [Fins... | theorem | Topology | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/Successor.lean | maxTail_isGood | Removing the leading `o` from a term of `MaxProducts C` yields a list which `isGood` with respect to
`C'`. |
noncomputable
MaxToGood
(h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C (ord I · < o))))) :
MaxProducts C ho → GoodProducts (C' C ho) :=
fun l ↦ ⟨l.val.Tail, maxTail_isGood C hC hsC ho l h₁⟩ | def | Topology | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/Successor.lean | MaxToGood | Given `l : MaxProducts C ho`, its `Tail` is a `GoodProducts (C' C ho)`. |
maxToGood_injective
(h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C (ord I · < o))))) :
(MaxToGood C hC hsC ho h₁).Injective := by
intro m n h
apply Subtype.ext ∘ Subtype.ext
rw [Subtype.ext_iff] at h
dsimp [MaxToGood] at h
rw [max_eq_o_cons_tail C hsC ho m, max_eq_o_cons_tail C hsC ho n, h]
include ... | theorem | Topology | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/Successor.lean | maxToGood_injective | null |
linearIndependent_comp_of_eval
(h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C (ord I · < o))))) :
LinearIndependent ℤ (eval (C' C ho)) →
LinearIndependent ℤ (ModuleCat.ofHom (Linear_CC' C hsC ho) ∘ SumEval C ho ∘ Sum.inr) := by
dsimp [SumEval, ModuleCat.ofHom]
rw [max_eq_eval_unapply C hsC ho]
intro... | theorem | Topology | [
"Mathlib.Algebra.Homology.ShortComplex.ModuleCat",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/Successor.lean | linearIndependent_comp_of_eval | null |
GoodProducts.linearIndependentEmpty {I} [LinearOrder I] :
LinearIndependent ℤ (eval (∅ : Set (I → Bool))) := linearIndependent_empty_type | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearIndependent.Basic",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/ZeroLimit.lean | GoodProducts.linearIndependentEmpty | null |
Products.nil : Products I := ⟨[], by simp only [List.isChain_nil]⟩ | def | Topology | [
"Mathlib.LinearAlgebra.LinearIndependent.Basic",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/ZeroLimit.lean | Products.nil | The empty list as a `Products` |
Products.lt_nil_empty {I} [LinearOrder I] : { m : Products I | m < Products.nil } = ∅ := by
ext ⟨m, hm⟩
refine ⟨fun h ↦ ?_, by tauto⟩
simp only [Set.mem_setOf_eq, lt_iff_lex_lt, nil, List.not_lex_nil] at h | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearIndependent.Basic",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/ZeroLimit.lean | Products.lt_nil_empty | null |
Products.isGood_nil {I} [LinearOrder I] :
Products.isGood ({fun _ ↦ false} : Set (I → Bool)) Products.nil := by
intro h
simp [Products.eval, Products.nil] at h | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearIndependent.Basic",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/ZeroLimit.lean | Products.isGood_nil | null |
Products.span_nil_eq_top {I} [LinearOrder I] :
Submodule.span ℤ (eval ({fun _ ↦ false} : Set (I → Bool)) '' {nil}) = ⊤ := by
rw [Set.image_singleton, eq_top_iff]
intro f _
rw [Submodule.mem_span_singleton]
refine ⟨f default, ?_⟩
simp only [eval, List.map, List.prod_nil, zsmul_eq_mul, mul_one, Products.nil... | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearIndependent.Basic",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/ZeroLimit.lean | Products.span_nil_eq_top | null |
smaller (o : Ordinal) : Set (LocallyConstant C ℤ) :=
(πs C o) '' (range (π C (ord I · < o))) | def | Topology | [
"Mathlib.LinearAlgebra.LinearIndependent.Basic",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/ZeroLimit.lean | smaller | There is a unique `GoodProducts` for the singleton `{fun _ ↦ false}`. -/
noncomputable
instance : Unique { l // Products.isGood ({fun _ ↦ false} : Set (I → Bool)) l } where
default := ⟨Products.nil, Products.isGood_nil⟩
uniq := by
intro ⟨⟨l, hl⟩, hll⟩
ext
apply Subtype.ext
apply (List.lex_nil_or_eq_... |
noncomputable
range_equiv_smaller_toFun (o : Ordinal) (x : range (π C (ord I · < o))) : smaller C o :=
⟨πs C o ↑x, x.val, x.property, rfl⟩ | def | Topology | [
"Mathlib.LinearAlgebra.LinearIndependent.Basic",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/ZeroLimit.lean | range_equiv_smaller_toFun | The map from the image of the `GoodProducts` in `LocallyConstant (π C (ord I · < o)) ℤ` to
`smaller C o` |
range_equiv_smaller_toFun_bijective (o : Ordinal) :
Function.Bijective (range_equiv_smaller_toFun C o) := by
dsimp +unfoldPartialApp [range_equiv_smaller_toFun]
refine ⟨fun a b hab ↦ ?_, fun ⟨a, b, hb⟩ ↦ ?_⟩
· ext1
simp only [Subtype.mk.injEq] at hab
exact injective_πs C o hab
· use ⟨b, hb.1⟩
si... | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearIndependent.Basic",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/ZeroLimit.lean | range_equiv_smaller_toFun_bijective | null |
noncomputable
range_equiv_smaller (o : Ordinal) : range (π C (ord I · < o)) ≃ smaller C o :=
Equiv.ofBijective (range_equiv_smaller_toFun C o) (range_equiv_smaller_toFun_bijective C o) | def | Topology | [
"Mathlib.LinearAlgebra.LinearIndependent.Basic",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/ZeroLimit.lean | range_equiv_smaller | The equivalence from the image of the `GoodProducts` in `LocallyConstant (π C (ord I · < o)) ℤ` to
`smaller C o` |
smaller_factorization (o : Ordinal) :
(fun (p : smaller C o) ↦ p.1) ∘ (range_equiv_smaller C o).toFun =
(πs C o) ∘ (fun (p : range (π C (ord I · < o))) ↦ p.1) := by rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearIndependent.Basic",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/ZeroLimit.lean | smaller_factorization | null |
linearIndependent_iff_smaller (o : Ordinal) :
LinearIndependent ℤ (GoodProducts.eval (π C (ord I · < o))) ↔
LinearIndependent ℤ (fun (p : smaller C o) ↦ p.1) := by
rw [GoodProducts.linearIndependent_iff_range,
← LinearMap.linearIndependent_iff (πs C o)
(LinearMap.ker_eq_bot_of_injective (injective_πs ... | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearIndependent.Basic",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/ZeroLimit.lean | linearIndependent_iff_smaller | null |
smaller_mono {o₁ o₂ : Ordinal} (h : o₁ ≤ o₂) : smaller C o₁ ⊆ smaller C o₂ := by
rintro f ⟨g, hg, rfl⟩
simp only [smaller, Set.mem_image]
use πs' C h g
obtain ⟨⟨l, gl⟩, rfl⟩ := hg
refine ⟨?_, ?_⟩
· use ⟨l, Products.isGood_mono C h gl⟩
ext x
rw [eval, ← Products.eval_πs' _ h (Products.prop_of_isGood ... | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearIndependent.Basic",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/ZeroLimit.lean | smaller_mono | null |
Products.limitOrdinal (l : Products I) : l.isGood (π C (ord I · < o)) ↔
∃ (o' : Ordinal), o' < o ∧ l.isGood (π C (ord I · < o')) := by
refine ⟨fun h ↦ ?_, fun ⟨o', ⟨ho', hl⟩⟩ ↦ isGood_mono C (le_of_lt ho') hl⟩
use Finset.sup l.val.toFinset (fun a ↦ Order.succ (ord I a))
have hslt : Finset.sup l.val.toFinset (... | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearIndependent.Basic",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/ZeroLimit.lean | Products.limitOrdinal | null |
GoodProducts.union : range C = ⋃ (e : {o' // o' < o}), (smaller C e.val) := by
ext p
simp only [smaller, range, Set.mem_iUnion, Set.mem_image, Set.mem_range, Subtype.exists]
refine ⟨fun hp ↦ ?_, fun hp ↦ ?_⟩
· obtain ⟨l, hl, rfl⟩ := hp
rw [contained_eq_proj C o hsC, Products.limitOrdinal C ho] at hl
obt... | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearIndependent.Basic",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/ZeroLimit.lean | GoodProducts.union | null |
GoodProducts.range_equiv : range C ≃ ⋃ (e : {o' // o' < o}), (smaller C e.val) :=
Equiv.setCongr (union C ho hsC) | def | Topology | [
"Mathlib.LinearAlgebra.LinearIndependent.Basic",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/ZeroLimit.lean | GoodProducts.range_equiv | The image of the `GoodProducts` in `C` is equivalent to the union of `smaller C o'` over all
ordinals `o' < o`. |
GoodProducts.range_equiv_factorization :
(fun (p : ⋃ (e : {o' // o' < o}), (smaller C e.val)) ↦ p.1) ∘ (range_equiv C ho hsC).toFun =
(fun (p : range C) ↦ (p.1 : LocallyConstant C ℤ)) := rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearIndependent.Basic",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/ZeroLimit.lean | GoodProducts.range_equiv_factorization | null |
GoodProducts.linearIndependent_iff_union_smaller :
LinearIndependent ℤ (GoodProducts.eval C) ↔
LinearIndependent ℤ (fun (p : ⋃ (e : {o' // o' < o}), (smaller C e.val)) ↦ p.1) := by
rw [GoodProducts.linearIndependent_iff_range, ← range_equiv_factorization C ho hsC]
exact linearIndependent_equiv (range_equi... | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearIndependent.Basic",
"Mathlib.Topology.Category.Profinite.Nobeling.Basic"
] | Mathlib/Topology/Category/Profinite/Nobeling/ZeroLimit.lean | GoodProducts.linearIndependent_iff_union_smaller | null |
limitCone (F : J ⥤ TopCat.{max v u}) : Cone F where
pt := TopCat.of { u : ∀ j : J, F.obj j | ∀ {i j : J} (f : i ⟶ j), F.map f (u i) = u j }
π :=
{ app := fun j => ofHom
{ toFun := fun u => u.val j
continuous_toFun := Continuous.comp (continuous_apply _) (continuous_subtype_val) }
natural... | def | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | limitCone | A choice of limit cone for a functor `F : J ⥤ TopCat`.
Generally you should just use `limit.cone F`, unless you need the actual definition
(which is in terms of `Types.limitCone`). |
limitConeIsLimit (F : J ⥤ TopCat.{max v u}) : IsLimit (limitCone.{v,u} F) where
lift S := ofHom
{ toFun := fun x =>
⟨fun _ => S.π.app _ x, fun f => by
dsimp
rw [← S.w f]
rfl⟩
continuous_toFun :=
Continuous.subtype_mk (continuous_pi fun j => (S.π.app j).hom.2) fu... | def | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | limitConeIsLimit | The chosen cone `TopCat.limitCone F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`). |
conePtOfConeForget : Type _ := c.pt | def | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | conePtOfConeForget | Given a functor `F : J ⥤ TopCat` and a cone `c : Cone (F ⋙ forget)`
of the underlying functor to types, this is the type `c.pt`
with the infimum of the induced topologies by the maps `c.π.app j`. |
topologicalSpaceConePtOfConeForget :
TopologicalSpace (conePtOfConeForget c) :=
(⨅ j, (F.obj j).str.induced (c.π.app j)) | instance | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | topologicalSpaceConePtOfConeForget | null |
@[simps pt π_app]
coneOfConeForget : Cone F where
pt := of (conePtOfConeForget c)
π :=
{ app j := ofHom (ContinuousMap.mk (c.π.app j) (by
rw [continuous_iff_le_induced]
exact iInf_le (fun j ↦ (F.obj j).str.induced (c.π.app j)) j))
naturality j j' φ := by
ext
apply congr_fun... | def | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | coneOfConeForget | Given a functor `F : J ⥤ TopCat` and a cone `c : Cone (F ⋙ forget)`
of the underlying functor to types, this is a cone for `F` whose point is
`c.pt` with the infimum of the induced topologies by the maps `c.π.app j`. |
isLimitConeOfForget (c : Cone (F ⋙ forget)) (hc : IsLimit c) :
IsLimit (coneOfConeForget c) := by
refine IsLimit.ofFaithful forget (ht := hc)
(fun s ↦ ofHom (ContinuousMap.mk (hc.lift ((forget).mapCone s)) ?_)) (fun _ ↦ rfl)
rw [continuous_iff_coinduced_le]
dsimp [topologicalSpaceConePtOfConeForget]
rw ... | def | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | isLimitConeOfForget | Given a functor `F : J ⥤ TopCat` and a cone `c : Cone (F ⋙ forget)`
of the underlying functor to types, the limit of `F` is `c.pt` equipped
with the infimum of the induced topologies by the maps `c.π.app j`. |
induced_of_isLimit :
c.pt.str = ⨅ j, (F.obj j).str.induced (c.π.app j) := by
let c' := coneOfConeForget ((forget).mapCone c)
let hc' : IsLimit c' := isLimitConeOfForget _ (isLimitOfPreserves forget hc)
let e := IsLimit.conePointUniqueUpToIso hc' hc
have he (j : J) : e.inv ≫ c'.π.app j = c.π.app j :=
Is... | theorem | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | induced_of_isLimit | null |
limit_topology [HasLimit F] :
(limit F).str = ⨅ j, (F.obj j).str.induced (limit.π F j) :=
induced_of_isLimit _ (limit.isLimit _) | theorem | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | limit_topology | null |
hasLimit_iff_small_sections :
HasLimit F ↔ Small.{u} ((F ⋙ forget).sections) := by
rw [← Types.hasLimit_iff_small_sections]
constructor <;> intro
· infer_instance
· exact ⟨⟨_, isLimitConeOfForget _ (limit.isLimit _)⟩⟩ | lemma | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | hasLimit_iff_small_sections | null |
topCat_hasLimitsOfShape (J : Type v) [Category J] [Small.{u} J] :
HasLimitsOfShape J TopCat.{u} where
has_limit := fun F => by
rw [hasLimit_iff_small_sections]
infer_instance | instance | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | topCat_hasLimitsOfShape | null |
topCat_hasLimitsOfSize [UnivLE.{v, u}] : HasLimitsOfSize.{w, v} TopCat.{u} where | instance | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | topCat_hasLimitsOfSize | null |
topCat_hasLimits : HasLimits TopCat.{u} :=
TopCat.topCat_hasLimitsOfSize.{u, u} | instance | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | topCat_hasLimits | null |
forget_preservesLimitsOfSize :
PreservesLimitsOfSize.{w, v} (forget : TopCat.{u} ⥤ _) where | instance | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | forget_preservesLimitsOfSize | null |
forget_preservesLimits : PreservesLimits (forget : TopCat.{u} ⥤ _) where | instance | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | forget_preservesLimits | null |
coconePtOfCoconeForget : Type _ := c.pt | def | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | coconePtOfCoconeForget | Given a functor `F : J ⥤ TopCat` and a cocone `c : Cocone (F ⋙ forget)`
of the underlying cocone of types, this is the type `c.pt`
with the supremum of the topologies that are coinduced by the maps `c.ι.app j`. |
topologicalSpaceCoconePtOfCoconeForget :
TopologicalSpace (coconePtOfCoconeForget c) :=
(⨆ j, (F.obj j).str.coinduced (c.ι.app j)) | instance | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | topologicalSpaceCoconePtOfCoconeForget | null |
@[simps pt ι_app]
coconeOfCoconeForget : Cocone F where
pt := of (coconePtOfCoconeForget c)
ι :=
{ app j := ofHom (ContinuousMap.mk (c.ι.app j) (by
rw [continuous_iff_coinduced_le]
exact le_iSup (fun j ↦ (F.obj j).str.coinduced (c.ι.app j)) j))
naturality j j' φ := by
ext
a... | def | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | coconeOfCoconeForget | Given a functor `F : J ⥤ TopCat` and a cocone `c : Cocone (F ⋙ forget)`
of the underlying cocone of types, this is a cocone for `F` whose point is
`c.pt` with the supremum of the coinduced topologies by the maps `c.ι.app j`. |
isColimitCoconeOfForget (c : Cocone (F ⋙ forget)) (hc : IsColimit c) :
IsColimit (coconeOfCoconeForget c) := by
refine IsColimit.ofFaithful forget (ht := hc)
(fun s ↦ ofHom (ContinuousMap.mk (hc.desc ((forget).mapCocone s)) ?_)) (fun _ ↦ rfl)
rw [continuous_iff_le_induced]
dsimp [topologicalSpaceCoconePtO... | def | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | isColimitCoconeOfForget | Given a functor `F : J ⥤ TopCat` and a cocone `c : Cocone (F ⋙ forget)`
of the underlying cocone of types, the colimit of `F` is `c.pt` equipped
with the supremum of the coinduced topologies by the maps `c.ι.app j`. |
coinduced_of_isColimit :
c.pt.str = ⨆ j, (F.obj j).str.coinduced (c.ι.app j) := by
let c' := coconeOfCoconeForget ((forget).mapCocone c)
let hc' : IsColimit c' := isColimitCoconeOfForget _ (isColimitOfPreserves forget hc)
let e := IsColimit.coconePointUniqueUpToIso hc' hc
have he (j : J) : c'.ι.app j ≫ e.ho... | theorem | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | coinduced_of_isColimit | null |
isOpen_iff_of_isColimit (X : Set c.pt) :
IsOpen X ↔ ∀ (j : J), IsOpen (c.ι.app j ⁻¹' X) := by
trans (⨆ (j : J), (F.obj j).str.coinduced (c.ι.app j)).IsOpen X
· rw [← coinduced_of_isColimit c hc, isOpen_fold]
· simp only [← isOpen_coinduced]
apply isOpen_iSup_iff | lemma | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | isOpen_iff_of_isColimit | null |
isClosed_iff_of_isColimit (X : Set c.pt) :
IsClosed X ↔ ∀ (j : J), IsClosed (c.ι.app j ⁻¹' X) := by
simp only [← isOpen_compl_iff, isOpen_iff_of_isColimit _ hc,
Functor.const_obj_obj, Set.preimage_compl] | lemma | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | isClosed_iff_of_isColimit | null |
continuous_iff_of_isColimit {X : Type u'} [TopologicalSpace X] (f : c.pt → X) :
Continuous f ↔ ∀ (j : J), Continuous (f ∘ c.ι.app j) := by
simp only [continuous_def, isOpen_iff_of_isColimit _ hc]
tauto | lemma | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | continuous_iff_of_isColimit | null |
colimit_topology (F : J ⥤ TopCat.{u}) [HasColimit F] :
(colimit F).str = ⨆ j, (F.obj j).str.coinduced (colimit.ι F j) :=
coinduced_of_isColimit _ (colimit.isColimit _) | theorem | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | colimit_topology | null |
colimit_isOpen_iff (F : J ⥤ TopCat.{u}) [HasColimit F]
(U : Set ((colimit F : _) : Type u)) :
IsOpen U ↔ ∀ j, IsOpen (colimit.ι F j ⁻¹' U) := by
apply isOpen_iff_of_isColimit _ (colimit.isColimit _) | theorem | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | colimit_isOpen_iff | null |
hasColimit_iff_small_colimitType :
HasColimit F ↔ Small.{u} (F ⋙ forget).ColimitType := by
rw [← Types.hasColimit_iff_small_colimitType]
constructor <;> intro
· infer_instance
· exact ⟨⟨_, isColimitCoconeOfForget _ (colimit.isColimit _)⟩⟩
@[deprecated (since := "2025-04-01")] alias hasColimit_iff_small_quot... | lemma | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | hasColimit_iff_small_colimitType | null |
topCat_hasColimitsOfShape (J : Type v) [Category J] [Small.{u} J] :
HasColimitsOfShape J TopCat.{u} where
has_colimit := fun F => by
rw [hasColimit_iff_small_colimitType]
infer_instance | instance | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | topCat_hasColimitsOfShape | null |
topCat_hasColimitsOfSize [UnivLE.{v, u}] : HasColimitsOfSize.{w, v} TopCat.{u} where | instance | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | topCat_hasColimitsOfSize | null |
topCat_hasColimits : HasColimits TopCat.{u} :=
TopCat.topCat_hasColimitsOfSize.{u, u} | instance | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | topCat_hasColimits | null |
forget_preservesColimitsOfSize :
PreservesColimitsOfSize.{w, v} (forget : TopCat.{u} ⥤ _) where | instance | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | forget_preservesColimitsOfSize | null |
forget_preservesColimits : PreservesColimits (forget : TopCat.{u} ⥤ Type u) where | instance | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | forget_preservesColimits | null |
isTerminalPUnit : IsTerminal (TopCat.of PUnit.{u + 1}) :=
haveI : ∀ X, Unique (X ⟶ TopCat.of PUnit.{u + 1}) := fun X =>
⟨⟨ofHom ⟨fun _ => PUnit.unit, continuous_const⟩⟩, fun f => by ext⟩
Limits.IsTerminal.ofUnique _ | def | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | isTerminalPUnit | The terminal object of `Top` is `PUnit`. |
terminalIsoPUnit : ⊤_ TopCat.{u} ≅ TopCat.of PUnit :=
terminalIsTerminal.uniqueUpToIso isTerminalPUnit | def | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | terminalIsoPUnit | The terminal object of `Top` is `PUnit`. |
isInitialPEmpty : IsInitial (TopCat.of PEmpty.{u + 1}) :=
haveI : ∀ X, Unique (TopCat.of PEmpty.{u + 1} ⟶ X) := fun X =>
⟨⟨ofHom ⟨fun x => x.elim, by continuity⟩⟩, fun f => by ext ⟨⟩⟩
Limits.IsInitial.ofUnique _ | def | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | isInitialPEmpty | The initial object of `Top` is `PEmpty`. |
initialIsoPEmpty : ⊥_ TopCat.{u} ≅ TopCat.of PEmpty :=
initialIsInitial.uniqueUpToIso isInitialPEmpty | def | Topology | [
"Mathlib.Topology.Category.TopCat.Adjunctions",
"Mathlib.CategoryTheory.Limits.Types.Limits",
"Mathlib.CategoryTheory.Limits.Types.Colimits",
"Mathlib.CategoryTheory.Limits.Shapes.Terminal",
"Mathlib.CategoryTheory.Adjunction.Limits"
] | Mathlib/Topology/Category/TopCat/Limits/Basic.lean | initialIsoPEmpty | The initial object of `Top` is `PEmpty`. |
isTopologicalBasis_cofiltered_limit (hC : IsLimit C) (T : ∀ j, Set (Set (F.obj j)))
(hT : ∀ j, IsTopologicalBasis (T j)) (univ : ∀ i : J, Set.univ ∈ T i)
(inter : ∀ (i) (U1 U2 : Set (F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i)
(compat : ∀ (i j : J) (f : i ⟶ j) (V : Set (F.obj j)) (_hV : V ∈ T j), F.map ... | theorem | Topology | [
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.CategoryTheory.Filtered.Basic"
] | Mathlib/Topology/Category/TopCat/Limits/Cofiltered.lean | isTopologicalBasis_cofiltered_limit | Given a *compatible* collection of topological bases for the factors in a cofiltered limit
which contain `Set.univ` and are closed under intersections, the induced *naive* collection
of sets in the limit is, in fact, a topological basis. |
private FiniteDiagramArrow {J : Type u} [SmallCategory J] (G : Finset J) :=
Σ' (X Y : J) (_ : X ∈ G) (_ : Y ∈ G), X ⟶ Y | abbrev | Topology | [
"Mathlib.CategoryTheory.Filtered.Basic",
"Mathlib.Topology.Category.TopCat.Limits.Basic"
] | Mathlib/Topology/Category/TopCat/Limits/Konig.lean | FiniteDiagramArrow | null |
private FiniteDiagram (J : Type u) [SmallCategory J] :=
Σ G : Finset J, Finset (FiniteDiagramArrow G) | abbrev | Topology | [
"Mathlib.CategoryTheory.Filtered.Basic",
"Mathlib.Topology.Category.TopCat.Limits.Basic"
] | Mathlib/Topology/Category/TopCat/Limits/Konig.lean | FiniteDiagram | null |
partialSections {J : Type u} [SmallCategory J] (F : J ⥤ TopCat.{v}) {G : Finset J}
(H : Finset (FiniteDiagramArrow G)) : Set (∀ j, F.obj j) :=
{u | ∀ {f : FiniteDiagramArrow G} (_ : f ∈ H), F.map f.2.2.2.2 (u f.1) = u f.2.1} | def | Topology | [
"Mathlib.CategoryTheory.Filtered.Basic",
"Mathlib.Topology.Category.TopCat.Limits.Basic"
] | Mathlib/Topology/Category/TopCat/Limits/Konig.lean | partialSections | Partial sections of a cofiltered limit are sections when restricted to
a finite subset of objects and morphisms of `J`. |
partialSections.nonempty [IsCofilteredOrEmpty J] [h : ∀ j : J, Nonempty (F.obj j)]
{G : Finset J} (H : Finset (FiniteDiagramArrow G)) : (partialSections F H).Nonempty := by
classical
cases isEmpty_or_nonempty J
· exact ⟨isEmptyElim, fun {j} => IsEmpty.elim' inferInstance j.1⟩
haveI : IsCofiltered J := ⟨⟩
... | theorem | Topology | [
"Mathlib.CategoryTheory.Filtered.Basic",
"Mathlib.Topology.Category.TopCat.Limits.Basic"
] | Mathlib/Topology/Category/TopCat/Limits/Konig.lean | partialSections.nonempty | null |
partialSections.directed :
Directed Superset fun G : FiniteDiagram J => partialSections F G.2 := by
classical
intro A B
let ιA : FiniteDiagramArrow A.1 → FiniteDiagramArrow (A.1 ⊔ B.1) := fun f =>
⟨f.1, f.2.1, Finset.mem_union_left _ f.2.2.1, Finset.mem_union_left _ f.2.2.2.1, f.2.2.2.2⟩
let ιB : Finite... | theorem | Topology | [
"Mathlib.CategoryTheory.Filtered.Basic",
"Mathlib.Topology.Category.TopCat.Limits.Basic"
] | Mathlib/Topology/Category/TopCat/Limits/Konig.lean | partialSections.directed | null |
partialSections.closed [∀ j : J, T2Space (F.obj j)] {G : Finset J}
(H : Finset (FiniteDiagramArrow G)) : IsClosed (partialSections F H) := by
have :
partialSections F H =
⋂ (f : FiniteDiagramArrow G) (_ : f ∈ H), {u | F.map f.2.2.2.2 (u f.1) = u f.2.1} := by
ext1
simp only [Set.mem_iInter, Set.m... | theorem | Topology | [
"Mathlib.CategoryTheory.Filtered.Basic",
"Mathlib.Topology.Category.TopCat.Limits.Basic"
] | Mathlib/Topology/Category/TopCat/Limits/Konig.lean | partialSections.closed | null |
nonempty_limitCone_of_compact_t2_cofiltered_system (F : J ⥤ TopCat.{max v u})
[IsCofilteredOrEmpty J]
[∀ j : J, Nonempty (F.obj j)] [∀ j : J, CompactSpace (F.obj j)] [∀ j : J, T2Space (F.obj j)] :
Nonempty (TopCat.limitCone F).pt := by
classical
obtain ⟨u, hu⟩ :=
IsCompact.nonempty_iInter_of_directe... | theorem | Topology | [
"Mathlib.CategoryTheory.Filtered.Basic",
"Mathlib.Topology.Category.TopCat.Limits.Basic"
] | Mathlib/Topology/Category/TopCat/Limits/Konig.lean | nonempty_limitCone_of_compact_t2_cofiltered_system | Cofiltered limits of nonempty compact Hausdorff spaces are nonempty topological spaces. |
piπ {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι) : TopCat.of (∀ i, α i) ⟶ α i :=
ofHom ⟨fun f => f i, continuous_apply i⟩ | abbrev | Topology | [
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.CategoryTheory.Limits.Shapes.Products",
"Mathlib.CategoryTheory.Limits.ConcreteCategory.Basic",
"Mathlib.Data.Set.Subsingleton",
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.Topology.Homeomorph... | Mathlib/Topology/Category/TopCat/Limits/Products.lean | piπ | The projection from the product as a bundled continuous map. |
@[simps! pt π_app]
piFan {ι : Type v} (α : ι → TopCat.{max v u}) : Fan α :=
Fan.mk (TopCat.of (∀ i, α i)) (piπ.{v,u} α) | def | Topology | [
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.CategoryTheory.Limits.Shapes.Products",
"Mathlib.CategoryTheory.Limits.ConcreteCategory.Basic",
"Mathlib.Data.Set.Subsingleton",
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.Topology.Homeomorph... | Mathlib/Topology/Category/TopCat/Limits/Products.lean | piFan | The explicit fan of a family of topological spaces given by the pi type. |
piFanIsLimit {ι : Type v} (α : ι → TopCat.{max v u}) : IsLimit (piFan α) where
lift S := ofHom
{ toFun := fun s i => S.π.app ⟨i⟩ s
continuous_toFun := continuous_pi (fun i => (S.π.app ⟨i⟩).hom.2) }
uniq := by
intro S m h
ext x
funext i
simp [ContinuousMap.coe_mk, ← h ⟨i⟩]
fac _ _ := rfl | def | Topology | [
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.CategoryTheory.Limits.Shapes.Products",
"Mathlib.CategoryTheory.Limits.ConcreteCategory.Basic",
"Mathlib.Data.Set.Subsingleton",
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.Topology.Homeomorph... | Mathlib/Topology/Category/TopCat/Limits/Products.lean | piFanIsLimit | The constructed fan is indeed a limit |
piIsoPi {ι : Type v} (α : ι → TopCat.{max v u}) : ∏ᶜ α ≅ TopCat.of (∀ i, α i) :=
(limit.isLimit _).conePointUniqueUpToIso (piFanIsLimit.{v, u} α)
@[reassoc (attr := simp)] | def | Topology | [
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.CategoryTheory.Limits.Shapes.Products",
"Mathlib.CategoryTheory.Limits.ConcreteCategory.Basic",
"Mathlib.Data.Set.Subsingleton",
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.Topology.Homeomorph... | Mathlib/Topology/Category/TopCat/Limits/Products.lean | piIsoPi | The product is homeomorphic to the product of the underlying spaces,
equipped with the product topology. |
piIsoPi_inv_π {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι) :
(piIsoPi α).inv ≫ Pi.π α i = piπ α i := by simp [piIsoPi] | theorem | Topology | [
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.CategoryTheory.Limits.Shapes.Products",
"Mathlib.CategoryTheory.Limits.ConcreteCategory.Basic",
"Mathlib.Data.Set.Subsingleton",
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.Topology.Homeomorph... | Mathlib/Topology/Category/TopCat/Limits/Products.lean | piIsoPi_inv_π | null |
piIsoPi_inv_π_apply {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι) (x : ∀ i, α i) :
(Pi.π α i :) ((piIsoPi α).inv x) = x i :=
ConcreteCategory.congr_hom (piIsoPi_inv_π α i) x | theorem | Topology | [
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.CategoryTheory.Limits.Shapes.Products",
"Mathlib.CategoryTheory.Limits.ConcreteCategory.Basic",
"Mathlib.Data.Set.Subsingleton",
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.Topology.Homeomorph... | Mathlib/Topology/Category/TopCat/Limits/Products.lean | piIsoPi_inv_π_apply | null |
piIsoPi_hom_apply {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι)
(x : (∏ᶜ α : TopCat.{max v u})) : (piIsoPi α).hom x i = (Pi.π α i :) x := by
have := piIsoPi_inv_π α i
rw [Iso.inv_comp_eq] at this
exact ConcreteCategory.congr_hom this x | theorem | Topology | [
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.CategoryTheory.Limits.Shapes.Products",
"Mathlib.CategoryTheory.Limits.ConcreteCategory.Basic",
"Mathlib.Data.Set.Subsingleton",
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.Topology.Homeomorph... | Mathlib/Topology/Category/TopCat/Limits/Products.lean | piIsoPi_hom_apply | null |
sigmaι {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι) : α i ⟶ TopCat.of (Σ i, α i) := by
refine ofHom (ContinuousMap.mk ?_ ?_)
· dsimp
apply Sigma.mk i
· dsimp; continuity | abbrev | Topology | [
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.CategoryTheory.Limits.Shapes.Products",
"Mathlib.CategoryTheory.Limits.ConcreteCategory.Basic",
"Mathlib.Data.Set.Subsingleton",
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.Topology.Homeomorph... | Mathlib/Topology/Category/TopCat/Limits/Products.lean | sigmaι | The inclusion to the coproduct as a bundled continuous map. |
@[simps! pt ι_app]
sigmaCofan {ι : Type v} (α : ι → TopCat.{max v u}) : Cofan α :=
Cofan.mk (TopCat.of (Σ i, α i)) (sigmaι α) | def | Topology | [
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.CategoryTheory.Limits.Shapes.Products",
"Mathlib.CategoryTheory.Limits.ConcreteCategory.Basic",
"Mathlib.Data.Set.Subsingleton",
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.Topology.Homeomorph... | Mathlib/Topology/Category/TopCat/Limits/Products.lean | sigmaCofan | The explicit cofan of a family of topological spaces given by the sigma type. |
sigmaCofanIsColimit {ι : Type v} (β : ι → TopCat.{max v u}) : IsColimit (sigmaCofan β) where
desc S := ofHom
{ toFun := fun (s : of (Σ i, β i)) => S.ι.app ⟨s.1⟩ s.2
continuous_toFun := by continuity }
uniq := by
intro S m h
ext ⟨i, x⟩
simp only [← h]
congr
fac s j := by
cases j
c... | def | Topology | [
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.CategoryTheory.Limits.Shapes.Products",
"Mathlib.CategoryTheory.Limits.ConcreteCategory.Basic",
"Mathlib.Data.Set.Subsingleton",
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.Topology.Homeomorph... | Mathlib/Topology/Category/TopCat/Limits/Products.lean | sigmaCofanIsColimit | The constructed cofan is indeed a colimit |
sigmaIsoSigma {ι : Type v} (α : ι → TopCat.{max v u}) : ∐ α ≅ TopCat.of (Σ i, α i) :=
(colimit.isColimit _).coconePointUniqueUpToIso (sigmaCofanIsColimit.{v, u} α)
@[reassoc (attr := simp)] | def | Topology | [
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.CategoryTheory.Limits.Shapes.Products",
"Mathlib.CategoryTheory.Limits.ConcreteCategory.Basic",
"Mathlib.Data.Set.Subsingleton",
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.Topology.Homeomorph... | Mathlib/Topology/Category/TopCat/Limits/Products.lean | sigmaIsoSigma | The coproduct is homeomorphic to the disjoint union of the topological spaces. |
sigmaIsoSigma_hom_ι {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι) :
Sigma.ι α i ≫ (sigmaIsoSigma α).hom = sigmaι α i := by simp [sigmaIsoSigma] | theorem | Topology | [
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.CategoryTheory.Limits.Shapes.Products",
"Mathlib.CategoryTheory.Limits.ConcreteCategory.Basic",
"Mathlib.Data.Set.Subsingleton",
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.Topology.Homeomorph... | Mathlib/Topology/Category/TopCat/Limits/Products.lean | sigmaIsoSigma_hom_ι | null |
sigmaIsoSigma_hom_ι_apply {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι) (x : α i) :
(sigmaIsoSigma α).hom ((Sigma.ι α i :) x) = Sigma.mk i x :=
ConcreteCategory.congr_hom (sigmaIsoSigma_hom_ι α i) x | theorem | Topology | [
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.CategoryTheory.Limits.Shapes.Products",
"Mathlib.CategoryTheory.Limits.ConcreteCategory.Basic",
"Mathlib.Data.Set.Subsingleton",
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.Topology.Homeomorph... | Mathlib/Topology/Category/TopCat/Limits/Products.lean | sigmaIsoSigma_hom_ι_apply | null |
sigmaIsoSigma_inv_apply {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι) (x : α i) :
(sigmaIsoSigma α).inv ⟨i, x⟩ = (Sigma.ι α i :) x := by
rw [← sigmaIsoSigma_hom_ι_apply, ← comp_app, ← comp_app, Iso.hom_inv_id,
Category.comp_id] | theorem | Topology | [
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.CategoryTheory.Limits.Shapes.Products",
"Mathlib.CategoryTheory.Limits.ConcreteCategory.Basic",
"Mathlib.Data.Set.Subsingleton",
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.Topology.Homeomorph... | Mathlib/Topology/Category/TopCat/Limits/Products.lean | sigmaIsoSigma_inv_apply | null |
prodFst {X Y : TopCat.{u}} : TopCat.of (X × Y) ⟶ X :=
ofHom { toFun := Prod.fst } | abbrev | Topology | [
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.CategoryTheory.Limits.Shapes.Products",
"Mathlib.CategoryTheory.Limits.ConcreteCategory.Basic",
"Mathlib.Data.Set.Subsingleton",
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.Topology.Homeomorph... | Mathlib/Topology/Category/TopCat/Limits/Products.lean | prodFst | The first projection from the product. |
prodSnd {X Y : TopCat.{u}} : TopCat.of (X × Y) ⟶ Y :=
ofHom { toFun := Prod.snd } | abbrev | Topology | [
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.CategoryTheory.Limits.Shapes.Products",
"Mathlib.CategoryTheory.Limits.ConcreteCategory.Basic",
"Mathlib.Data.Set.Subsingleton",
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.Topology.Homeomorph... | Mathlib/Topology/Category/TopCat/Limits/Products.lean | prodSnd | The second projection from the product. |
prodBinaryFan (X Y : TopCat.{u}) : BinaryFan X Y :=
BinaryFan.mk prodFst prodSnd | def | Topology | [
"Mathlib.Topology.Category.TopCat.EpiMono",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.CategoryTheory.Limits.Shapes.Products",
"Mathlib.CategoryTheory.Limits.ConcreteCategory.Basic",
"Mathlib.Data.Set.Subsingleton",
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.Topology.Homeomorph... | Mathlib/Topology/Category/TopCat/Limits/Products.lean | prodBinaryFan | The explicit binary cofan of `X, Y` given by `X × Y`. |
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