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walking_parallel_pair_op_right :
walking_parallel_pair_op.map right = @quiver.hom.op _ _ zero one right | rfl | lemma | category_theory.limits.walking_parallel_pair_op_right | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [
"quiver.hom.op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
walking_parallel_pair_op_equiv : walking_parallel_pair ≌ walking_parallel_pairᵒᵖ | { functor := walking_parallel_pair_op,
inverse := walking_parallel_pair_op.left_op,
unit_iso := nat_iso.of_components (λ j, eq_to_iso (by { cases j; refl }))
(by { rintros (_|_) (_|_) (_|_|_); refl }),
counit_iso := nat_iso.of_components (λ j, eq_to_iso
(by { induction j using opposite.rec, cases j; refl ... | def | category_theory.limits.walking_parallel_pair_op_equiv | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [
"opposite.rec"
] | The equivalence `walking_parallel_pair ⥤ walking_parallel_pairᵒᵖ` sending left to left and right to
right. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
walking_parallel_pair_op_equiv_unit_iso_zero :
walking_parallel_pair_op_equiv.unit_iso.app zero = iso.refl zero | rfl | lemma | category_theory.limits.walking_parallel_pair_op_equiv_unit_iso_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
walking_parallel_pair_op_equiv_unit_iso_one :
walking_parallel_pair_op_equiv.unit_iso.app one = iso.refl one | rfl | lemma | category_theory.limits.walking_parallel_pair_op_equiv_unit_iso_one | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
walking_parallel_pair_op_equiv_counit_iso_zero :
walking_parallel_pair_op_equiv.counit_iso.app (op zero) = iso.refl (op zero) | rfl | lemma | category_theory.limits.walking_parallel_pair_op_equiv_counit_iso_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
walking_parallel_pair_op_equiv_counit_iso_one :
walking_parallel_pair_op_equiv.counit_iso.app (op one) = iso.refl (op one) | rfl | lemma | category_theory.limits.walking_parallel_pair_op_equiv_counit_iso_one | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
parallel_pair (f g : X ⟶ Y) : walking_parallel_pair ⥤ C | { obj := λ x, match x with
| zero := X
| one := Y
end,
map := λ x y h, match x, y, h with
| _, _, (id _) := 𝟙 _
| _, _, left := f
| _, _, right := g
end,
-- `tidy` can cope with this, but it's too slow:
map_comp' := begin rintros (⟨⟩|⟨⟩) (⟨⟩|⟨⟩) (⟨⟩|⟨⟩) ⟨⟩⟨⟩; { unfold_aux, simp; refl }, end, }. | def | category_theory.limits.parallel_pair | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | `parallel_pair f g` is the diagram in `C` consisting of the two morphisms `f` and `g` with
common domain and codomain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
parallel_pair_obj_zero (f g : X ⟶ Y) : (parallel_pair f g).obj zero = X | rfl | lemma | category_theory.limits.parallel_pair_obj_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
parallel_pair_obj_one (f g : X ⟶ Y) : (parallel_pair f g).obj one = Y | rfl | lemma | category_theory.limits.parallel_pair_obj_one | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
parallel_pair_map_left (f g : X ⟶ Y) : (parallel_pair f g).map left = f | rfl | lemma | category_theory.limits.parallel_pair_map_left | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
parallel_pair_map_right (f g : X ⟶ Y) : (parallel_pair f g).map right = g | rfl | lemma | category_theory.limits.parallel_pair_map_right | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
parallel_pair_functor_obj
{F : walking_parallel_pair ⥤ C} (j : walking_parallel_pair) :
(parallel_pair (F.map left) (F.map right)).obj j = F.obj j | begin
cases j; refl
end | lemma | category_theory.limits.parallel_pair_functor_obj | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagram_iso_parallel_pair (F : walking_parallel_pair ⥤ C) :
F ≅ parallel_pair (F.map left) (F.map right) | nat_iso.of_components (λ j, eq_to_iso $ by cases j; tidy) $ by tidy | def | category_theory.limits.diagram_iso_parallel_pair | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | Every functor indexing a (co)equalizer is naturally isomorphic (actually, equal) to a
`parallel_pair` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
parallel_pair_hom {X' Y' : C} (f g : X ⟶ Y) (f' g' : X' ⟶ Y') (p : X ⟶ X') (q : Y ⟶ Y')
(wf : f ≫ q = p ≫ f') (wg : g ≫ q = p ≫ g') : parallel_pair f g ⟶ parallel_pair f' g' | { app := λ j, match j with
| zero := p
| one := q
end,
naturality' := begin
rintros (⟨⟩|⟨⟩) (⟨⟩|⟨⟩) ⟨⟩; { unfold_aux, simp [wf, wg], },
end } | def | category_theory.limits.parallel_pair_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | Construct a morphism between parallel pairs. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
parallel_pair_hom_app_zero
{X' Y' : C} (f g : X ⟶ Y) (f' g' : X' ⟶ Y') (p : X ⟶ X') (q : Y ⟶ Y')
(wf : f ≫ q = p ≫ f') (wg : g ≫ q = p ≫ g') :
(parallel_pair_hom f g f' g' p q wf wg).app zero = p | rfl | lemma | category_theory.limits.parallel_pair_hom_app_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
parallel_pair_hom_app_one
{X' Y' : C} (f g : X ⟶ Y) (f' g' : X' ⟶ Y') (p : X ⟶ X') (q : Y ⟶ Y')
(wf : f ≫ q = p ≫ f') (wg : g ≫ q = p ≫ g') :
(parallel_pair_hom f g f' g' p q wf wg).app one = q | rfl | lemma | category_theory.limits.parallel_pair_hom_app_one | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
parallel_pair.ext {F G : walking_parallel_pair ⥤ C}
(zero : F.obj zero ≅ G.obj zero) (one : F.obj one ≅ G.obj one)
(left : F.map left ≫ one.hom = zero.hom ≫ G.map left)
(right : F.map right ≫ one.hom = zero.hom ≫ G.map right) : F ≅ G | nat_iso.of_components
(by { rintro ⟨j⟩, exacts [zero, one] })
(by { rintro ⟨j₁⟩ ⟨j₂⟩ ⟨f⟩; simp [left, right], }) | def | category_theory.limits.parallel_pair.ext | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | Construct a natural isomorphism between functors out of the walking parallel pair from
its components. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
parallel_pair.eq_of_hom_eq {f g f' g' : X ⟶ Y} (hf : f = f') (hg : g = g') :
parallel_pair f g ≅ parallel_pair f' g' | parallel_pair.ext (iso.refl _) (iso.refl _) (by simp [hf]) (by simp [hg]) | def | category_theory.limits.parallel_pair.eq_of_hom_eq | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | Construct a natural isomorphism between `parallel_pair f g` and `parallel_pair f' g'` given
equalities `f = f'` and `g = g'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fork (f g : X ⟶ Y) | cone (parallel_pair f g) | abbreviation | category_theory.limits.fork | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | A fork on `f` and `g` is just a `cone (parallel_pair f g)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cofork (f g : X ⟶ Y) | cocone (parallel_pair f g) | abbreviation | category_theory.limits.cofork | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | A cofork on `f` and `g` is just a `cocone (parallel_pair f g)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fork.ι (t : fork f g) | t.π.app zero | def | category_theory.limits.fork.ι | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | A fork `t` on the parallel pair `f g : X ⟶ Y` consists of two morphisms `t.π.app zero : t.X ⟶ X`
and `t.π.app one : t.X ⟶ Y`. Of these, only the first one is interesting, and we give it the
shorter name `fork.ι t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fork.app_zero_eq_ι (t : fork f g) : t.π.app zero = t.ι | rfl | lemma | category_theory.limits.fork.app_zero_eq_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cofork.π (t : cofork f g) | t.ι.app one | def | category_theory.limits.cofork.π | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | A cofork `t` on the parallel_pair `f g : X ⟶ Y` consists of two morphisms
`t.ι.app zero : X ⟶ t.X` and `t.ι.app one : Y ⟶ t.X`. Of these, only the second one is
interesting, and we give it the shorter name `cofork.π t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cofork.app_one_eq_π (t : cofork f g) : t.ι.app one = t.π | rfl | lemma | category_theory.limits.cofork.app_one_eq_π | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fork.app_one_eq_ι_comp_left (s : fork f g) : s.π.app one = s.ι ≫ f | by rw [←s.app_zero_eq_ι, ←s.w left, parallel_pair_map_left] | lemma | category_theory.limits.fork.app_one_eq_ι_comp_left | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fork.app_one_eq_ι_comp_right (s : fork f g) : s.π.app one = s.ι ≫ g | by rw [←s.app_zero_eq_ι, ←s.w right, parallel_pair_map_right] | lemma | category_theory.limits.fork.app_one_eq_ι_comp_right | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cofork.app_zero_eq_comp_π_left (s : cofork f g) : s.ι.app zero = f ≫ s.π | by rw [←s.app_one_eq_π, ←s.w left, parallel_pair_map_left] | lemma | category_theory.limits.cofork.app_zero_eq_comp_π_left | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cofork.app_zero_eq_comp_π_right (s : cofork f g) : s.ι.app zero = g ≫ s.π | by rw [←s.app_one_eq_π, ←s.w right, parallel_pair_map_right] | lemma | category_theory.limits.cofork.app_zero_eq_comp_π_right | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fork.of_ι {P : C} (ι : P ⟶ X) (w : ι ≫ f = ι ≫ g) : fork f g | { X := P,
π :=
{ app := λ X, begin cases X, exact ι, exact ι ≫ f, end,
naturality' := λ X Y f,
begin
cases X; cases Y; cases f; dsimp; simp,
{ dsimp, simp, }, -- See note [dsimp, simp].
{ exact w },
{ dsimp, simp, },
end } } | def | category_theory.limits.fork.of_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | A fork on `f g : X ⟶ Y` is determined by the morphism `ι : P ⟶ X` satisfying `ι ≫ f = ι ≫ g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cofork.of_π {P : C} (π : Y ⟶ P) (w : f ≫ π = g ≫ π) : cofork f g | { X := P,
ι :=
{ app := λ X, walking_parallel_pair.cases_on X (f ≫ π) π,
naturality' := λ i j f, by { cases f; dsimp; simp [w] } } } | def | category_theory.limits.cofork.of_π | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | A cofork on `f g : X ⟶ Y` is determined by the morphism `π : Y ⟶ P` satisfying
`f ≫ π = g ≫ π`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fork.ι_of_ι {P : C} (ι : P ⟶ X) (w : ι ≫ f = ι ≫ g) :
(fork.of_ι ι w).ι = ι | rfl | lemma | category_theory.limits.fork.ι_of_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cofork.π_of_π {P : C} (π : Y ⟶ P) (w : f ≫ π = g ≫ π) :
(cofork.of_π π w).π = π | rfl | lemma | category_theory.limits.cofork.π_of_π | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fork.condition (t : fork f g) : t.ι ≫ f = t.ι ≫ g | by rw [←t.app_one_eq_ι_comp_left, ←t.app_one_eq_ι_comp_right] | lemma | category_theory.limits.fork.condition | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cofork.condition (t : cofork f g) : f ≫ t.π = g ≫ t.π | by rw [←t.app_zero_eq_comp_π_left, ←t.app_zero_eq_comp_π_right] | lemma | category_theory.limits.cofork.condition | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fork.equalizer_ext (s : fork f g) {W : C} {k l : W ⟶ s.X} (h : k ≫ s.ι = l ≫ s.ι) :
∀ (j : walking_parallel_pair), k ≫ s.π.app j = l ≫ s.π.app j | | zero := h
| one := by rw [s.app_one_eq_ι_comp_left, reassoc_of h] | lemma | category_theory.limits.fork.equalizer_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | To check whether two maps are equalized by both maps of a fork, it suffices to check it for the
first map | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cofork.coequalizer_ext (s : cofork f g) {W : C} {k l : s.X ⟶ W}
(h : cofork.π s ≫ k = cofork.π s ≫ l) : ∀ (j : walking_parallel_pair),
s.ι.app j ≫ k = s.ι.app j ≫ l | | zero := by simp only [s.app_zero_eq_comp_π_left, category.assoc, h]
| one := h | lemma | category_theory.limits.cofork.coequalizer_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | To check whether two maps are coequalized by both maps of a cofork, it suffices to check it for
the second map | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fork.is_limit.hom_ext {s : fork f g} (hs : is_limit s) {W : C} {k l : W ⟶ s.X}
(h : k ≫ fork.ι s = l ≫ fork.ι s) : k = l | hs.hom_ext $ fork.equalizer_ext _ h | lemma | category_theory.limits.fork.is_limit.hom_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cofork.is_colimit.hom_ext {s : cofork f g} (hs : is_colimit s) {W : C} {k l : s.X ⟶ W}
(h : cofork.π s ≫ k = cofork.π s ≫ l) : k = l | hs.hom_ext $ cofork.coequalizer_ext _ h | lemma | category_theory.limits.cofork.is_colimit.hom_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fork.is_limit.lift_ι {s t : fork f g} (hs : is_limit s) :
hs.lift t ≫ s.ι = t.ι | hs.fac _ _ | lemma | category_theory.limits.fork.is_limit.lift_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cofork.is_colimit.π_desc {s t : cofork f g} (hs : is_colimit s) :
s.π ≫ hs.desc t = t.π | hs.fac _ _ | lemma | category_theory.limits.cofork.is_colimit.π_desc | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fork.is_limit.lift' {s : fork f g} (hs : is_limit s) {W : C} (k : W ⟶ X) (h : k ≫ f = k ≫ g) :
{l : W ⟶ s.X // l ≫ fork.ι s = k} | ⟨hs.lift $ fork.of_ι _ h, hs.fac _ _⟩ | def | category_theory.limits.fork.is_limit.lift' | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | If `s` is a limit fork over `f` and `g`, then a morphism `k : W ⟶ X` satisfying
`k ≫ f = k ≫ g` induces a morphism `l : W ⟶ s.X` such that `l ≫ fork.ι s = k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cofork.is_colimit.desc' {s : cofork f g} (hs : is_colimit s) {W : C} (k : Y ⟶ W)
(h : f ≫ k = g ≫ k) : {l : s.X ⟶ W // cofork.π s ≫ l = k} | ⟨hs.desc $ cofork.of_π _ h, hs.fac _ _⟩ | def | category_theory.limits.cofork.is_colimit.desc' | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | If `s` is a colimit cofork over `f` and `g`, then a morphism `k : Y ⟶ W` satisfying
`f ≫ k = g ≫ k` induces a morphism `l : s.X ⟶ W` such that `cofork.π s ≫ l = k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fork.is_limit.exists_unique {s : fork f g} (hs : is_limit s) {W : C} (k : W ⟶ X)
(h : k ≫ f = k ≫ g) : ∃! (l : W ⟶ s.X), l ≫ fork.ι s = k | ⟨hs.lift $ fork.of_ι _ h, hs.fac _ _, λ m hm, fork.is_limit.hom_ext hs $
hm.symm ▸ (hs.fac (fork.of_ι _ h) walking_parallel_pair.zero).symm⟩ | lemma | category_theory.limits.fork.is_limit.exists_unique | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cofork.is_colimit.exists_unique {s : cofork f g} (hs : is_colimit s) {W : C} (k : Y ⟶ W)
(h : f ≫ k = g ≫ k) : ∃! (d : s.X ⟶ W), cofork.π s ≫ d = k | ⟨hs.desc $ cofork.of_π _ h, hs.fac _ _, λ m hm, cofork.is_colimit.hom_ext hs $
hm.symm ▸ (hs.fac (cofork.of_π _ h) walking_parallel_pair.one).symm⟩ | lemma | category_theory.limits.cofork.is_colimit.exists_unique | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fork.is_limit.mk (t : fork f g)
(lift : Π (s : fork f g), s.X ⟶ t.X)
(fac : ∀ (s : fork f g), lift s ≫ fork.ι t = fork.ι s)
(uniq : ∀ (s : fork f g) (m : s.X ⟶ t.X) (w : m ≫ t.ι = s.ι), m = lift s) :
is_limit t | { lift := lift,
fac' := λ s j, walking_parallel_pair.cases_on j (fac s) $
by erw [←s.w left, ←t.w left, ←category.assoc, fac]; refl,
uniq' := λ s m j, by tidy } | def | category_theory.limits.fork.is_limit.mk | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [
"lift"
] | This is a slightly more convenient method to verify that a fork is a limit cone. It
only asks for a proof of facts that carry any mathematical content | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fork.is_limit.mk' {X Y : C} {f g : X ⟶ Y} (t : fork f g)
(create : Π (s : fork f g), {l // l ≫ t.ι = s.ι ∧ ∀ {m}, m ≫ t.ι = s.ι → m = l}) :
is_limit t | fork.is_limit.mk t
(λ s, (create s).1)
(λ s, (create s).2.1)
(λ s m w, (create s).2.2 w) | def | category_theory.limits.fork.is_limit.mk' | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | This is another convenient method to verify that a fork is a limit cone. It
only asks for a proof of facts that carry any mathematical content, and allows access to the
same `s` for all parts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cofork.is_colimit.mk (t : cofork f g)
(desc : Π (s : cofork f g), t.X ⟶ s.X)
(fac : ∀ (s : cofork f g), cofork.π t ≫ desc s = cofork.π s)
(uniq : ∀ (s : cofork f g) (m : t.X ⟶ s.X) (w : t.π ≫ m = s.π), m = desc s) :
is_colimit t | { desc := desc,
fac' := λ s j, walking_parallel_pair.cases_on j
(by erw [←s.w left, ←t.w left, category.assoc, fac]; refl) (fac s),
uniq' := by tidy } | def | category_theory.limits.cofork.is_colimit.mk | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | This is a slightly more convenient method to verify that a cofork is a colimit cocone. It
only asks for a proof of facts that carry any mathematical content | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cofork.is_colimit.mk' {X Y : C} {f g : X ⟶ Y} (t : cofork f g)
(create : Π (s : cofork f g), {l : t.X ⟶ s.X // t.π ≫ l = s.π ∧ ∀ {m}, t.π ≫ m = s.π → m = l}) :
is_colimit t | cofork.is_colimit.mk t
(λ s, (create s).1)
(λ s, (create s).2.1)
(λ s m w, (create s).2.2 w) | def | category_theory.limits.cofork.is_colimit.mk' | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | This is another convenient method to verify that a fork is a limit cone. It
only asks for a proof of facts that carry any mathematical content, and allows access to the
same `s` for all parts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fork.is_limit.of_exists_unique {t : fork f g}
(hs : ∀ (s : fork f g), ∃! l : s.X ⟶ t.X, l ≫ fork.ι t = fork.ι s) : is_limit t | by { choose d hd hd' using hs, exact fork.is_limit.mk _ d hd (λ s m hm, hd' _ _ hm) } | def | category_theory.limits.fork.is_limit.of_exists_unique | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | Noncomputably make a limit cone from the existence of unique factorizations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cofork.is_colimit.of_exists_unique {t : cofork f g}
(hs : ∀ (s : cofork f g), ∃! d : t.X ⟶ s.X, cofork.π t ≫ d = cofork.π s) : is_colimit t | by { choose d hd hd' using hs, exact cofork.is_colimit.mk _ d hd (λ s m hm, hd' _ _ hm) } | def | category_theory.limits.cofork.is_colimit.of_exists_unique | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | Noncomputably make a colimit cocone from the existence of unique factorizations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fork.is_limit.hom_iso {X Y : C} {f g : X ⟶ Y} {t : fork f g} (ht : is_limit t) (Z : C) :
(Z ⟶ t.X) ≃ {h : Z ⟶ X // h ≫ f = h ≫ g} | { to_fun := λ k, ⟨k ≫ t.ι, by simp only [category.assoc, t.condition]⟩,
inv_fun := λ h, (fork.is_limit.lift' ht _ h.prop).1,
left_inv := λ k, fork.is_limit.hom_ext ht (fork.is_limit.lift' _ _ _).prop,
right_inv := λ h, subtype.ext (fork.is_limit.lift' ht _ _).prop } | def | category_theory.limits.fork.is_limit.hom_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [
"inv_fun",
"subtype.ext"
] | Given a limit cone for the pair `f g : X ⟶ Y`, for any `Z`, morphisms from `Z` to its point are in
bijection with morphisms `h : Z ⟶ X` such that `h ≫ f = h ≫ g`.
Further, this bijection is natural in `Z`: see `fork.is_limit.hom_iso_natural`.
This is a special case of `is_limit.hom_iso'`, often useful to construct adju... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fork.is_limit.hom_iso_natural {X Y : C} {f g : X ⟶ Y} {t : fork f g} (ht : is_limit t)
{Z Z' : C} (q : Z' ⟶ Z) (k : Z ⟶ t.X) :
(fork.is_limit.hom_iso ht _ (q ≫ k) : Z' ⟶ X) = q ≫ (fork.is_limit.hom_iso ht _ k : Z ⟶ X) | category.assoc _ _ _ | lemma | category_theory.limits.fork.is_limit.hom_iso_natural | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | The bijection of `fork.is_limit.hom_iso` is natural in `Z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cofork.is_colimit.hom_iso {X Y : C} {f g : X ⟶ Y} {t : cofork f g} (ht : is_colimit t) (Z : C) :
(t.X ⟶ Z) ≃ {h : Y ⟶ Z // f ≫ h = g ≫ h} | { to_fun := λ k, ⟨t.π ≫ k, by simp only [←category.assoc, t.condition]⟩,
inv_fun := λ h, (cofork.is_colimit.desc' ht _ h.prop).1,
left_inv := λ k, cofork.is_colimit.hom_ext ht (cofork.is_colimit.desc' _ _ _).prop,
right_inv := λ h, subtype.ext (cofork.is_colimit.desc' ht _ _).prop } | def | category_theory.limits.cofork.is_colimit.hom_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [
"inv_fun",
"subtype.ext"
] | Given a colimit cocone for the pair `f g : X ⟶ Y`, for any `Z`, morphisms from the cocone point
to `Z` are in bijection with morphisms `h : Y ⟶ Z` such that `f ≫ h = g ≫ h`.
Further, this bijection is natural in `Z`: see `cofork.is_colimit.hom_iso_natural`.
This is a special case of `is_colimit.hom_iso'`, often useful ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cofork.is_colimit.hom_iso_natural {X Y : C} {f g : X ⟶ Y} {t : cofork f g} {Z Z' : C}
(q : Z ⟶ Z') (ht : is_colimit t) (k : t.X ⟶ Z) :
(cofork.is_colimit.hom_iso ht _ (k ≫ q) : Y ⟶ Z') =
(cofork.is_colimit.hom_iso ht _ k : Y ⟶ Z) ≫ q | (category.assoc _ _ _).symm | lemma | category_theory.limits.cofork.is_colimit.hom_iso_natural | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | The bijection of `cofork.is_colimit.hom_iso` is natural in `Z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone.of_fork
{F : walking_parallel_pair ⥤ C} (t : fork (F.map left) (F.map right)) : cone F | { X := t.X,
π :=
{ app := λ X, t.π.app X ≫ eq_to_hom (by tidy),
naturality' := λ j j' g, by { cases j; cases j'; cases g; dsimp; simp } } } | def | category_theory.limits.cone.of_fork | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | This is a helper construction that can be useful when verifying that a category has all
equalizers. Given `F : walking_parallel_pair ⥤ C`, which is really the same as
`parallel_pair (F.map left) (F.map right)`, and a fork on `F.map left` and `F.map right`,
we get a cone on `F`.
If you're thinking about... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone.of_cofork
{F : walking_parallel_pair ⥤ C} (t : cofork (F.map left) (F.map right)) : cocone F | { X := t.X,
ι :=
{ app := λ X, eq_to_hom (by tidy) ≫ t.ι.app X,
naturality' := λ j j' g, by { cases j; cases j'; cases g; dsimp; simp } } } | def | category_theory.limits.cocone.of_cofork | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | This is a helper construction that can be useful when verifying that a category has all
coequalizers. Given `F : walking_parallel_pair ⥤ C`, which is really the same as
`parallel_pair (F.map left) (F.map right)`, and a cofork on `F.map left` and `F.map right`,
we get a cocone on `F`.
If you're thinking... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone.of_fork_π
{F : walking_parallel_pair ⥤ C} (t : fork (F.map left) (F.map right)) (j) :
(cone.of_fork t).π.app j = t.π.app j ≫ eq_to_hom (by tidy) | rfl | lemma | category_theory.limits.cone.of_fork_π | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocone.of_cofork_ι
{F : walking_parallel_pair ⥤ C} (t : cofork (F.map left) (F.map right)) (j) :
(cocone.of_cofork t).ι.app j = eq_to_hom (by tidy) ≫ t.ι.app j | rfl | lemma | category_theory.limits.cocone.of_cofork_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fork.of_cone
{F : walking_parallel_pair ⥤ C} (t : cone F) : fork (F.map left) (F.map right) | { X := t.X,
π := { app := λ X, t.π.app X ≫ eq_to_hom (by tidy) } } | def | category_theory.limits.fork.of_cone | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | Given `F : walking_parallel_pair ⥤ C`, which is really the same as
`parallel_pair (F.map left) (F.map right)` and a cone on `F`, we get a fork on
`F.map left` and `F.map right`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cofork.of_cocone
{F : walking_parallel_pair ⥤ C} (t : cocone F) : cofork (F.map left) (F.map right) | { X := t.X,
ι := { app := λ X, eq_to_hom (by tidy) ≫ t.ι.app X } } | def | category_theory.limits.cofork.of_cocone | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | Given `F : walking_parallel_pair ⥤ C`, which is really the same as
`parallel_pair (F.map left) (F.map right)` and a cocone on `F`, we get a cofork on
`F.map left` and `F.map right`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fork.of_cone_π {F : walking_parallel_pair ⥤ C} (t : cone F) (j) :
(fork.of_cone t).π.app j = t.π.app j ≫ eq_to_hom (by tidy) | rfl | lemma | category_theory.limits.fork.of_cone_π | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cofork.of_cocone_ι {F : walking_parallel_pair ⥤ C} (t : cocone F) (j) :
(cofork.of_cocone t).ι.app j = eq_to_hom (by tidy) ≫ t.ι.app j | rfl | lemma | category_theory.limits.cofork.of_cocone_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fork.ι_postcompose {f' g' : X ⟶ Y} {α : parallel_pair f g ⟶ parallel_pair f' g'}
{c : fork f g} : fork.ι ((cones.postcompose α).obj c) = c.ι ≫ α.app _ | rfl | lemma | category_theory.limits.fork.ι_postcompose | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cofork.π_precompose {f' g' : X ⟶ Y} {α : parallel_pair f g ⟶ parallel_pair f' g'}
{c : cofork f' g'} : cofork.π ((cocones.precompose α).obj c) = α.app _ ≫ c.π | rfl | lemma | category_theory.limits.cofork.π_precompose | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fork.mk_hom {s t : fork f g} (k : s.X ⟶ t.X) (w : k ≫ t.ι = s.ι) : s ⟶ t | { hom := k,
w' :=
begin
rintro ⟨_|_⟩,
{ exact w },
{ simp only [fork.app_one_eq_ι_comp_left, reassoc_of w] },
end } | def | category_theory.limits.fork.mk_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | Helper function for constructing morphisms between equalizer forks. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fork.ext {s t : fork f g} (i : s.X ≅ t.X) (w : i.hom ≫ t.ι = s.ι) : s ≅ t | { hom := fork.mk_hom i.hom w,
inv := fork.mk_hom i.inv (by rw [← w, iso.inv_hom_id_assoc]) } | def | category_theory.limits.fork.ext | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | To construct an isomorphism between forks,
it suffices to give an isomorphism between the cone points
and check that it commutes with the `ι` morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fork.iso_fork_of_ι (c : fork f g) : c ≅ fork.of_ι c.ι c.condition | fork.ext (by simp only [fork.of_ι_X, functor.const_obj_obj]) (by simp) | def | category_theory.limits.fork.iso_fork_of_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | Every fork is isomorphic to one of the form `fork.of_ι _ _`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cofork.mk_hom {s t : cofork f g} (k : s.X ⟶ t.X) (w : s.π ≫ k = t.π) : s ⟶ t | { hom := k,
w' :=
begin
rintro ⟨_|_⟩,
{ simp [cofork.app_zero_eq_comp_π_left, w] },
{ exact w }
end } | def | category_theory.limits.cofork.mk_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | Helper function for constructing morphisms between coequalizer coforks. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fork.hom_comp_ι {s t : fork f g} (f : s ⟶ t) : f.hom ≫ t.ι = s.ι | by tidy | lemma | category_theory.limits.fork.hom_comp_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fork.π_comp_hom {s t : cofork f g} (f : s ⟶ t) : s.π ≫ f.hom = t.π | by tidy | lemma | category_theory.limits.fork.π_comp_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cofork.ext {s t : cofork f g} (i : s.X ≅ t.X) (w : s.π ≫ i.hom = t.π) : s ≅ t | { hom := cofork.mk_hom i.hom w,
inv := cofork.mk_hom i.inv (by rw [iso.comp_inv_eq, w]) } | def | category_theory.limits.cofork.ext | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | To construct an isomorphism between coforks,
it suffices to give an isomorphism between the cocone points
and check that it commutes with the `π` morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cofork.iso_cofork_of_π (c : cofork f g) : c ≅ cofork.of_π c.π c.condition | cofork.ext (by simp only [cofork.of_π_X, functor.const_obj_obj]) (by dsimp; simp) | def | category_theory.limits.cofork.iso_cofork_of_π | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | Every cofork is isomorphic to one of the form `cofork.of_π _ _`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_equalizer | has_limit (parallel_pair f g) | abbreviation | category_theory.limits.has_equalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | `has_equalizer f g` represents a particular choice of limiting cone
for the parallel pair of morphisms `f` and `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equalizer : C | limit (parallel_pair f g) | abbreviation | category_theory.limits.equalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | If an equalizer of `f` and `g` exists, we can access an arbitrary choice of such by
saying `equalizer f g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equalizer.ι : equalizer f g ⟶ X | limit.π (parallel_pair f g) zero | abbreviation | category_theory.limits.equalizer.ι | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | If an equalizer of `f` and `g` exists, we can access the inclusion
`equalizer f g ⟶ X` by saying `equalizer.ι f g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equalizer.fork : fork f g | limit.cone (parallel_pair f g) | abbreviation | category_theory.limits.equalizer.fork | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | An equalizer cone for a parallel pair `f` and `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equalizer.fork_ι :
(equalizer.fork f g).ι = equalizer.ι f g | rfl | lemma | category_theory.limits.equalizer.fork_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equalizer.fork_π_app_zero :
(equalizer.fork f g).π.app zero = equalizer.ι f g | rfl | lemma | category_theory.limits.equalizer.fork_π_app_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equalizer.condition : equalizer.ι f g ≫ f = equalizer.ι f g ≫ g | fork.condition $ limit.cone $ parallel_pair f g | lemma | category_theory.limits.equalizer.condition | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equalizer_is_equalizer : is_limit (fork.of_ι (equalizer.ι f g) (equalizer.condition f g)) | is_limit.of_iso_limit (limit.is_limit _) (fork.ext (iso.refl _) (by tidy)) | def | category_theory.limits.equalizer_is_equalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | The equalizer built from `equalizer.ι f g` is limiting. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equalizer.lift {W : C} (k : W ⟶ X) (h : k ≫ f = k ≫ g) : W ⟶ equalizer f g | limit.lift (parallel_pair f g) (fork.of_ι k h) | abbreviation | category_theory.limits.equalizer.lift | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | A morphism `k : W ⟶ X` satisfying `k ≫ f = k ≫ g` factors through the equalizer of `f` and `g`
via `equalizer.lift : W ⟶ equalizer f g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equalizer.lift_ι {W : C} (k : W ⟶ X) (h : k ≫ f = k ≫ g) :
equalizer.lift k h ≫ equalizer.ι f g = k | limit.lift_π _ _ | lemma | category_theory.limits.equalizer.lift_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equalizer.lift' {W : C} (k : W ⟶ X) (h : k ≫ f = k ≫ g) :
{l : W ⟶ equalizer f g // l ≫ equalizer.ι f g = k} | ⟨equalizer.lift k h, equalizer.lift_ι _ _⟩ | def | category_theory.limits.equalizer.lift' | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | A morphism `k : W ⟶ X` satisfying `k ≫ f = k ≫ g` induces a morphism `l : W ⟶ equalizer f g`
satisfying `l ≫ equalizer.ι f g = k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equalizer.hom_ext {W : C} {k l : W ⟶ equalizer f g}
(h : k ≫ equalizer.ι f g = l ≫ equalizer.ι f g) : k = l | fork.is_limit.hom_ext (limit.is_limit _) h | lemma | category_theory.limits.equalizer.hom_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | Two maps into an equalizer are equal if they are are equal when composed with the equalizer
map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equalizer.exists_unique {W : C} (k : W ⟶ X) (h : k ≫ f = k ≫ g) :
∃! (l : W ⟶ equalizer f g), l ≫ equalizer.ι f g = k | fork.is_limit.exists_unique (limit.is_limit _) _ h | lemma | category_theory.limits.equalizer.exists_unique | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equalizer.ι_mono : mono (equalizer.ι f g) | { right_cancellation := λ Z h k w, equalizer.hom_ext w } | instance | category_theory.limits.equalizer.ι_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | An equalizer morphism is a monomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono_of_is_limit_fork {c : fork f g} (i : is_limit c) : mono (fork.ι c) | { right_cancellation := λ Z h k w, fork.is_limit.hom_ext i w } | lemma | category_theory.limits.mono_of_is_limit_fork | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | The equalizer morphism in any limit cone is a monomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_fork (h : f = g) : fork f g | fork.of_ι (𝟙 X) $ h ▸ rfl | def | category_theory.limits.id_fork | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | The identity determines a cone on the equalizer diagram of `f` and `g` if `f = g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_id_fork (h : f = g) : is_limit (id_fork h) | fork.is_limit.mk _
(λ s, fork.ι s)
(λ s, category.comp_id _)
(λ s m h, by { convert h, exact (category.comp_id _).symm }) | def | category_theory.limits.is_limit_id_fork | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | The identity on `X` is an equalizer of `(f, g)`, if `f = g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_limit_cone_parallel_pair_of_eq (h₀ : f = g) {c : fork f g}
(h : is_limit c) : is_iso c.ι | is_iso.of_iso $ is_limit.cone_point_unique_up_to_iso h $ is_limit_id_fork h₀ | lemma | category_theory.limits.is_iso_limit_cone_parallel_pair_of_eq | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | Every equalizer of `(f, g)`, where `f = g`, is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equalizer.ι_of_eq [has_equalizer f g] (h : f = g) : is_iso (equalizer.ι f g) | is_iso_limit_cone_parallel_pair_of_eq h $ limit.is_limit _ | lemma | category_theory.limits.equalizer.ι_of_eq | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | The equalizer of `(f, g)`, where `f = g`, is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_limit_cone_parallel_pair_of_self {c : fork f f} (h : is_limit c) : is_iso c.ι | is_iso_limit_cone_parallel_pair_of_eq rfl h | lemma | category_theory.limits.is_iso_limit_cone_parallel_pair_of_self | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | Every equalizer of `(f, f)` is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_limit_cone_parallel_pair_of_epi {c : fork f g}
(h : is_limit c) [epi (c.ι)] : is_iso c.ι | is_iso_limit_cone_parallel_pair_of_eq ((cancel_epi _).1 (fork.condition c)) h | lemma | category_theory.limits.is_iso_limit_cone_parallel_pair_of_epi | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | An equalizer that is an epimorphism is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_epi_fork_ι (t : fork f g) [epi (fork.ι t)] : f = g | (cancel_epi (fork.ι t)).1 $ fork.condition t | lemma | category_theory.limits.eq_of_epi_fork_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | Two morphisms are equal if there is a fork whose inclusion is epi. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_epi_equalizer [has_equalizer f g] [epi (equalizer.ι f g)] : f = g | (cancel_epi (equalizer.ι f g)).1 $ equalizer.condition _ _ | lemma | category_theory.limits.eq_of_epi_equalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | If the equalizer of two morphisms is an epimorphism, then the two morphisms are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_equalizer_of_self : has_equalizer f f | has_limit.mk
{ cone := id_fork rfl,
is_limit := is_limit_id_fork rfl } | instance | category_theory.limits.has_equalizer_of_self | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equalizer.ι_of_self : is_iso (equalizer.ι f f) | equalizer.ι_of_eq rfl | instance | category_theory.limits.equalizer.ι_of_self | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | The equalizer inclusion for `(f, f)` is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equalizer.iso_source_of_self : equalizer f f ≅ X | as_iso (equalizer.ι f f) | def | category_theory.limits.equalizer.iso_source_of_self | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | The equalizer of a morphism with itself is isomorphic to the source. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equalizer.iso_source_of_self_hom :
(equalizer.iso_source_of_self f).hom = equalizer.ι f f | rfl | lemma | category_theory.limits.equalizer.iso_source_of_self_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equalizer.iso_source_of_self_inv :
(equalizer.iso_source_of_self f).inv = equalizer.lift (𝟙 X) (by simp) | by { ext, simp [equalizer.iso_source_of_self], } | lemma | category_theory.limits.equalizer.iso_source_of_self_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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