statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
is_cospearator.is_codetector [balanced C] {G : C} : is_coseparator G → is_codetector G | is_coseparating.is_codetecting | lemma | category_theory.is_cospearator.is_codetector | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"balanced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_separator_def (G : C) :
is_separator G ↔ ∀ ⦃X Y : C⦄ (f g : X ⟶ Y), (∀ h : G ⟶ X, h ≫ f = h ≫ g) → f = g | ⟨λ hG X Y f g hfg, hG _ _ $ λ H hH h, by { obtain rfl := set.mem_singleton_iff.1 hH, exact hfg h },
λ hG X Y f g hfg, hG _ _ $ λ h, hfg _ (set.mem_singleton _) _⟩ | lemma | category_theory.is_separator_def | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"set.mem_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_separator.def {G : C} :
is_separator G → ∀ ⦃X Y : C⦄ (f g : X ⟶ Y), (∀ h : G ⟶ X, h ≫ f = h ≫ g) → f = g | (is_separator_def _).1 | lemma | category_theory.is_separator.def | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coseparator_def (G : C) :
is_coseparator G ↔ ∀ ⦃X Y : C⦄ (f g : X ⟶ Y), (∀ h : Y ⟶ G, f ≫ h = g ≫ h) → f = g | ⟨λ hG X Y f g hfg, hG _ _ $ λ H hH h, by { obtain rfl := set.mem_singleton_iff.1 hH, exact hfg h },
λ hG X Y f g hfg, hG _ _ $ λ h, hfg _ (set.mem_singleton _) _⟩ | lemma | category_theory.is_coseparator_def | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"set.mem_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coseparator.def {G : C} :
is_coseparator G → ∀ ⦃X Y : C⦄ (f g : X ⟶ Y), (∀ h : Y ⟶ G, f ≫ h = g ≫ h) → f = g | (is_coseparator_def _).1 | lemma | category_theory.is_coseparator.def | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_detector_def (G : C) :
is_detector G ↔ ∀ ⦃X Y : C⦄ (f : X ⟶ Y), (∀ h : G ⟶ Y, ∃! h', h' ≫ f = h) → is_iso f | ⟨λ hG X Y f hf, hG _ $ λ H hH h, by { obtain rfl := set.mem_singleton_iff.1 hH, exact hf h },
λ hG X Y f hf, hG _ $ λ h, hf _ (set.mem_singleton _) _⟩ | lemma | category_theory.is_detector_def | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"set.mem_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_detector.def {G : C} :
is_detector G → ∀ ⦃X Y : C⦄ (f : X ⟶ Y), (∀ h : G ⟶ Y, ∃! h', h' ≫ f = h) → is_iso f | (is_detector_def _).1 | lemma | category_theory.is_detector.def | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_codetector_def (G : C) :
is_codetector G ↔ ∀ ⦃X Y : C⦄ (f : X ⟶ Y), (∀ h : X ⟶ G, ∃! h', f ≫ h' = h) → is_iso f | ⟨λ hG X Y f hf, hG _ $ λ H hH h, by { obtain rfl := set.mem_singleton_iff.1 hH, exact hf h },
λ hG X Y f hf, hG _ $ λ h, hf _ (set.mem_singleton _) _⟩ | lemma | category_theory.is_codetector_def | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"set.mem_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_codetector.def {G : C} :
is_codetector G → ∀ ⦃X Y : C⦄ (f : X ⟶ Y), (∀ h : X ⟶ G, ∃! h', f ≫ h' = h) → is_iso f | (is_codetector_def _).1 | lemma | category_theory.is_codetector.def | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_separator_iff_faithful_coyoneda_obj (G : C) :
is_separator G ↔ faithful (coyoneda.obj (op G)) | ⟨λ hG, ⟨λ X Y f g hfg, hG.def _ _ (congr_fun hfg)⟩,
λ h, (is_separator_def _).2 $ λ X Y f g hfg,
by exactI (coyoneda.obj (op G)).map_injective (funext hfg)⟩ | lemma | category_theory.is_separator_iff_faithful_coyoneda_obj | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coseparator_iff_faithful_yoneda_obj (G : C) :
is_coseparator G ↔ faithful (yoneda.obj G) | ⟨λ hG, ⟨λ X Y f g hfg, quiver.hom.unop_inj (hG.def _ _ (congr_fun hfg))⟩,
λ h, (is_coseparator_def _).2 $ λ X Y f g hfg, quiver.hom.op_inj $
by exactI (yoneda.obj G).map_injective (funext hfg)⟩ | lemma | category_theory.is_coseparator_iff_faithful_yoneda_obj | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"quiver.hom.op_inj",
"quiver.hom.unop_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_separator_iff_epi (G : C) [Π A : C, has_coproduct (λ (f : G ⟶ A), G)] :
is_separator G ↔ ∀ (A : C), epi (sigma.desc (λ (f : G ⟶ A), f)) | begin
rw is_separator_def,
refine ⟨λ h A, ⟨λ Z u v huv, h _ _ (λ i, _)⟩, λ h X Y f g hh, _⟩,
{ simpa using (sigma.ι _ i) ≫= huv },
{ haveI := h X,
refine (cancel_epi (sigma.desc (λ (f : G ⟶ X), f))).1 (colimit.hom_ext (λ j, _)),
simpa using hh j.as }
end | lemma | category_theory.is_separator_iff_epi | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coseparator_iff_mono (G : C) [Π A : C, has_product (λ (f : A ⟶ G), G)] :
is_coseparator G ↔ ∀ (A : C), mono (pi.lift (λ (f : A ⟶ G), f)) | begin
rw is_coseparator_def,
refine ⟨λ h A, ⟨λ Z u v huv, h _ _ (λ i, _)⟩, λ h X Y f g hh, _⟩,
{ simpa using huv =≫ (pi.π _ i) },
{ haveI := h Y,
refine (cancel_mono (pi.lift (λ (f : Y ⟶ G), f))).1 (limit.hom_ext (λ j, _)),
simpa using hh j.as }
end | lemma | category_theory.is_coseparator_iff_mono | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_separator_coprod (G H : C) [has_binary_coproduct G H] :
is_separator (G ⨿ H) ↔ is_separating ({G, H} : set C) | begin
refine ⟨λ h X Y u v huv, _, λ h, (is_separator_def _).2 (λ X Y u v huv, h _ _ (λ Z hZ g, _))⟩,
{ refine h.def _ _ (λ g, coprod.hom_ext _ _),
{ simpa using huv G (by simp) (coprod.inl ≫ g) },
{ simpa using huv H (by simp) (coprod.inr ≫ g) } },
{ simp only [set.mem_insert_iff, set.mem_singleton_iff] a... | lemma | category_theory.is_separator_coprod | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"set.mem_insert_iff",
"set.mem_singleton_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_separator_coprod_of_is_separator_left (G H : C) [has_binary_coproduct G H]
(hG : is_separator G) : is_separator (G ⨿ H) | (is_separator_coprod _ _).2 $ is_separating.mono hG $ by simp | lemma | category_theory.is_separator_coprod_of_is_separator_left | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_separator_coprod_of_is_separator_right (G H : C) [has_binary_coproduct G H]
(hH : is_separator H) : is_separator (G ⨿ H) | (is_separator_coprod _ _).2 $ is_separating.mono hH $ by simp | lemma | category_theory.is_separator_coprod_of_is_separator_right | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_separator_sigma {β : Type w} (f : β → C) [has_coproduct f] :
is_separator (∐ f) ↔ is_separating (set.range f) | begin
refine ⟨λ h X Y u v huv, _, λ h, (is_separator_def _).2 (λ X Y u v huv, h _ _ (λ Z hZ g, _))⟩,
{ refine h.def _ _ (λ g, colimit.hom_ext (λ b, _)),
simpa using huv (f b.as) (by simp) (colimit.ι (discrete.functor f) _ ≫ g) },
{ obtain ⟨b, rfl⟩ := set.mem_range.1 hZ,
classical,
simpa using sigma.ι ... | lemma | category_theory.is_separator_sigma | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_separator_sigma_of_is_separator {β : Type w} (f : β → C) [has_coproduct f]
(b : β) (hb : is_separator (f b)) : is_separator (∐ f) | (is_separator_sigma _).2 $ is_separating.mono hb $ by simp | lemma | category_theory.is_separator_sigma_of_is_separator | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coseparator_prod (G H : C) [has_binary_product G H] :
is_coseparator (G ⨯ H) ↔ is_coseparating ({G, H} : set C) | begin
refine ⟨λ h X Y u v huv, _, λ h, (is_coseparator_def _).2 (λ X Y u v huv, h _ _ (λ Z hZ g, _))⟩,
{ refine h.def _ _ (λ g, prod.hom_ext _ _),
{ simpa using huv G (by simp) (g ≫ limits.prod.fst) },
{ simpa using huv H (by simp) (g ≫ limits.prod.snd) } },
{ simp only [set.mem_insert_iff, set.mem_single... | lemma | category_theory.is_coseparator_prod | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"set.mem_insert_iff",
"set.mem_singleton_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coseparator_prod_of_is_coseparator_left (G H : C) [has_binary_product G H]
(hG : is_coseparator G) : is_coseparator (G ⨯ H) | (is_coseparator_prod _ _).2 $ is_coseparating.mono hG $ by simp | lemma | category_theory.is_coseparator_prod_of_is_coseparator_left | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coseparator_prod_of_is_coseparator_right (G H : C) [has_binary_product G H]
(hH : is_coseparator H) : is_coseparator (G ⨯ H) | (is_coseparator_prod _ _).2 $ is_coseparating.mono hH $ by simp | lemma | category_theory.is_coseparator_prod_of_is_coseparator_right | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coseparator_pi {β : Type w} (f : β → C) [has_product f] :
is_coseparator (∏ f) ↔ is_coseparating (set.range f) | begin
refine ⟨λ h X Y u v huv, _, λ h, (is_coseparator_def _).2 (λ X Y u v huv, h _ _ (λ Z hZ g, _))⟩,
{ refine h.def _ _ (λ g, limit.hom_ext (λ b, _)),
simpa using huv (f b.as) (by simp) (g ≫ limit.π (discrete.functor f) _ ) },
{ obtain ⟨b, rfl⟩ := set.mem_range.1 hZ,
classical,
simpa using huv (pi.l... | lemma | category_theory.is_coseparator_pi | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coseparator_pi_of_is_coseparator {β : Type w} (f : β → C) [has_product f]
(b : β) (hb : is_coseparator (f b)) : is_coseparator (∏ f) | (is_coseparator_pi _).2 $ is_coseparating.mono hb $ by simp | lemma | category_theory.is_coseparator_pi_of_is_coseparator | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_detector_iff_reflects_isomorphisms_coyoneda_obj (G : C) :
is_detector G ↔ reflects_isomorphisms (coyoneda.obj (op G)) | begin
refine ⟨λ hG, ⟨λ X Y f hf, hG.def _ (λ h, _)⟩, λ h, (is_detector_def _).2 (λ X Y f hf, _)⟩,
{ rw [is_iso_iff_bijective, function.bijective_iff_exists_unique] at hf,
exact hf h },
{ suffices : is_iso ((coyoneda.obj (op G)).map f),
{ exactI @is_iso_of_reflects_iso _ _ _ _ _ _ _ (coyoneda.obj (op G)) _... | lemma | category_theory.is_detector_iff_reflects_isomorphisms_coyoneda_obj | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"function.bijective_iff_exists_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_codetector_iff_reflects_isomorphisms_yoneda_obj (G : C) :
is_codetector G ↔ reflects_isomorphisms (yoneda.obj G) | begin
refine ⟨λ hG, ⟨λ X Y f hf, _ ⟩, λ h, (is_codetector_def _).2 (λ X Y f hf, _)⟩,
{ refine (is_iso_unop_iff _).1 (hG.def _ _),
rwa [is_iso_iff_bijective, function.bijective_iff_exists_unique] at hf },
{ rw ← is_iso_op_iff,
suffices : is_iso ((yoneda.obj G).map f.op),
{ exactI @is_iso_of_reflects_is... | lemma | category_theory.is_codetector_iff_reflects_isomorphisms_yoneda_obj | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [
"function.bijective_iff_exists_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
well_powered_of_is_detector [has_pullbacks C] (G : C) (hG : is_detector G) :
well_powered C | well_powered_of_is_detecting hG | lemma | category_theory.well_powered_of_is_detector | category_theory | src/category_theory/generator.lean | [
"category_theory.balanced",
"category_theory.limits.essentially_small",
"category_theory.limits.opposites",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.subobject.lattice",
"category_theory.subobject.well_powered",
"data.set.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
glue_data | (J : Type v)
(U : J → C)
(V : J × J → C)
(f : Π i j, V (i, j) ⟶ U i)
(f_mono : ∀ i j, mono (f i j) . tactic.apply_instance)
(f_has_pullback : ∀ i j k, has_pullback (f i j) (f i k) . tactic.apply_instance)
(f_id : ∀ i, is_iso (f i i) . tactic.apply_instance)
(t : Π i j, V (i, j) ⟶ V (j, i))
(t_id : ∀ i, t i i = 𝟙 _)
(t... | structure | category_theory.glue_data | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | A gluing datum consists of
1. An index type `J`
2. An object `U i` for each `i : J`.
3. An object `V i j` for each `i j : J`.
4. A monomorphism `f i j : V i j ⟶ U i` for each `i j : J`.
5. A transition map `t i j : V i j ⟶ V j i` for each `i j : J`.
such that
6. `f i i` is an isomorphism.
7. `t i i` is the identity.
8.... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
t'_iij (i j : D.J) : D.t' i i j = (pullback_symmetry _ _).hom | begin
have eq₁ := D.t_fac i i j,
have eq₂ := (is_iso.eq_comp_inv (D.f i i)).mpr (@pullback.condition _ _ _ _ _ _ (D.f i j) _),
rw [D.t_id, category.comp_id, eq₂] at eq₁,
have eq₃ := (is_iso.eq_comp_inv (D.f i i)).mp eq₁,
rw [category.assoc, ←pullback.condition, ←category.assoc] at eq₃,
exact mono.right_canc... | lemma | category_theory.glue_data.t'_iij | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t'_jii (i j : D.J) : D.t' j i i = pullback.fst ≫ D.t j i ≫ inv pullback.snd | by { rw [←category.assoc, ←D.t_fac], simp } | lemma | category_theory.glue_data.t'_jii | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t'_iji (i j : D.J) : D.t' i j i = pullback.fst ≫ D.t i j ≫ inv pullback.snd | by { rw [←category.assoc, ←D.t_fac], simp } | lemma | category_theory.glue_data.t'_iji | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t_inv (i j : D.J) :
D.t i j ≫ D.t j i = 𝟙 _ | begin
have eq : (pullback_symmetry (D.f i i) (D.f i j)).hom = pullback.snd ≫ inv pullback.fst,
{ simp },
have := D.cocycle i j i,
rw [D.t'_iij, D.t'_jii, D.t'_iji, fst_eq_snd_of_mono_eq, eq] at this,
simp only [category.assoc, is_iso.inv_hom_id_assoc] at this,
rw [←is_iso.eq_inv_comp, ←category.assoc, is_is... | lemma | category_theory.glue_data.t_inv | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t'_inv (i j k : D.J) : D.t' i j k ≫ (pullback_symmetry _ _).hom ≫
D.t' j i k ≫ (pullback_symmetry _ _).hom = 𝟙 _ | begin
rw ← cancel_mono (pullback.fst : pullback (D.f i j) (D.f i k) ⟶ _),
simp [t_fac, t_fac_assoc]
end | lemma | category_theory.glue_data.t'_inv | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t_is_iso (i j : D.J) : is_iso (D.t i j) | ⟨⟨D.t j i, D.t_inv _ _, D.t_inv _ _⟩⟩ | instance | category_theory.glue_data.t_is_iso | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t'_is_iso (i j k : D.J) : is_iso (D.t' i j k) | ⟨⟨D.t' j k i ≫ D.t' k i j, D.cocycle _ _ _, (by simpa using D.cocycle _ _ _)⟩⟩ | instance | category_theory.glue_data.t'_is_iso | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t'_comp_eq_pullback_symmetry (i j k : D.J) :
D.t' j k i ≫ D.t' k i j = (pullback_symmetry _ _).hom ≫
D.t' j i k ≫ (pullback_symmetry _ _).hom | begin
transitivity inv (D.t' i j k),
{ exact is_iso.eq_inv_of_hom_inv_id (D.cocycle _ _ _) },
{ rw ← cancel_mono (pullback.fst : pullback (D.f i j) (D.f i k) ⟶ _),
simp [t_fac, t_fac_assoc] }
end | lemma | category_theory.glue_data.t'_comp_eq_pullback_symmetry | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sigma_opens [has_coproduct D.U] : C | ∐ D.U | def | category_theory.glue_data.sigma_opens | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | (Implementation) The disjoint union of `U i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagram : multispan_index C | { L := D.J × D.J, R := D.J,
fst_from := _root_.prod.fst, snd_from := _root_.prod.snd,
left := D.V, right := D.U,
fst := λ ⟨i, j⟩, D.f i j,
snd := λ ⟨i, j⟩, D.t i j ≫ D.f j i } | def | category_theory.glue_data.diagram | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | (Implementation) The diagram to take colimit of. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagram_L : D.diagram.L = (D.J × D.J) | rfl | lemma | category_theory.glue_data.diagram_L | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagram_R : D.diagram.R = D.J | rfl | lemma | category_theory.glue_data.diagram_R | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagram_fst_from (i j : D.J) : D.diagram.fst_from ⟨i, j⟩ = i | rfl | lemma | category_theory.glue_data.diagram_fst_from | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagram_snd_from (i j : D.J) : D.diagram.snd_from ⟨i, j⟩ = j | rfl | lemma | category_theory.glue_data.diagram_snd_from | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagram_fst (i j : D.J) : D.diagram.fst ⟨i, j⟩ = D.f i j | rfl | lemma | category_theory.glue_data.diagram_fst | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagram_snd (i j : D.J) : D.diagram.snd ⟨i, j⟩ = D.t i j ≫ D.f j i | rfl | lemma | category_theory.glue_data.diagram_snd | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagram_left : D.diagram.left = D.V | rfl | lemma | category_theory.glue_data.diagram_left | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagram_right : D.diagram.right = D.U | rfl | lemma | category_theory.glue_data.diagram_right | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
glued : C | multicoequalizer D.diagram | def | category_theory.glue_data.glued | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | The glued object given a family of gluing data. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι (i : D.J) : D.U i ⟶ D.glued | multicoequalizer.π D.diagram i | def | category_theory.glue_data.ι | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | The map `D.U i ⟶ D.glued` for each `i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
glue_condition (i j : D.J) :
D.t i j ≫ D.f j i ≫ D.ι j = D.f i j ≫ D.ι i | (category.assoc _ _ _).symm.trans (multicoequalizer.condition D.diagram ⟨i, j⟩).symm | lemma | category_theory.glue_data.glue_condition | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
V_pullback_cone (i j : D.J) : pullback_cone (D.ι i) (D.ι j) | pullback_cone.mk (D.f i j) (D.t i j ≫ D.f j i) (by simp) | def | category_theory.glue_data.V_pullback_cone | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | The pullback cone spanned by `V i j ⟶ U i` and `V i j ⟶ U j`.
This will often be a pullback diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
π : D.sigma_opens ⟶ D.glued | multicoequalizer.sigma_π D.diagram | def | category_theory.glue_data.π | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | The projection `∐ D.U ⟶ D.glued` given by the colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
π_epi : epi D.π | by { unfold π, apply_instance } | instance | category_theory.glue_data.π_epi | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
types_π_surjective (D : glue_data Type*) :
function.surjective D.π | (epi_iff_surjective _).mp infer_instance | lemma | category_theory.glue_data.types_π_surjective | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
types_ι_jointly_surjective (D : glue_data Type*) (x : D.glued) :
∃ i (y : D.U i), D.ι i y = x | begin
delta category_theory.glue_data.ι,
simp_rw ← multicoequalizer.ι_sigma_π D.diagram,
rcases D.types_π_surjective x with ⟨x', rfl⟩,
have := colimit.iso_colimit_cocone (types.coproduct_colimit_cocone _),
rw ← (show (colimit.iso_colimit_cocone (types.coproduct_colimit_cocone _)).inv _ = x',
from concrete... | lemma | category_theory.glue_data.types_ι_jointly_surjective | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [
"category_theory.glue_data.ι"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_glue_data :
glue_data C' | { J := D.J,
U := λ i, F.obj (D.U i),
V := λ i, F.obj (D.V i),
f := λ i j, F.map (D.f i j),
f_mono := λ i j, preserves_mono_of_preserves_limit _ _,
f_id := λ i, infer_instance,
t := λ i j, F.map (D.t i j),
t_id := λ i, by { rw D.t_id i, simp },
t' := λ i j k, (preserves_pullback.iso F (D.f i j) (D.f i k)... | def | category_theory.glue_data.map_glue_data | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | A functor that preserves the pullbacks of `f i j` and `f i k` can map a family of glue data. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagram_iso : D.diagram.multispan ⋙ F ≅ (D.map_glue_data F).diagram.multispan | nat_iso.of_components
(λ x, match x with
| walking_multispan.left a := iso.refl _
| walking_multispan.right b := iso.refl _
end)
(begin
rintros (⟨_,_⟩|_) _ (_|_|_),
{ erw [category.comp_id, category.id_comp, functor.map_id], refl },
{ erw [category.comp_id, category.id_comp], refl },
{ e... | def | category_theory.glue_data.diagram_iso | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [
"functor.map_id"
] | The diagram of the image of a `glue_data` under a functor `F` is naturally isomorphic to the
original diagram of the `glue_data` via `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagram_iso_app_left (i : D.J × D.J) :
(D.diagram_iso F).app (walking_multispan.left i) = iso.refl _ | rfl | lemma | category_theory.glue_data.diagram_iso_app_left | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagram_iso_app_right (i : D.J) :
(D.diagram_iso F).app (walking_multispan.right i) = iso.refl _ | rfl | lemma | category_theory.glue_data.diagram_iso_app_right | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagram_iso_hom_app_left (i : D.J × D.J) :
(D.diagram_iso F).hom.app (walking_multispan.left i) = 𝟙 _ | rfl | lemma | category_theory.glue_data.diagram_iso_hom_app_left | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagram_iso_hom_app_right (i : D.J) :
(D.diagram_iso F).hom.app (walking_multispan.right i) = 𝟙 _ | rfl | lemma | category_theory.glue_data.diagram_iso_hom_app_right | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagram_iso_inv_app_left (i : D.J × D.J) :
(D.diagram_iso F).inv.app (walking_multispan.left i) = 𝟙 _ | rfl | lemma | category_theory.glue_data.diagram_iso_inv_app_left | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagram_iso_inv_app_right (i : D.J) :
(D.diagram_iso F).inv.app (walking_multispan.right i) = 𝟙 _ | rfl | lemma | category_theory.glue_data.diagram_iso_inv_app_right | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimit_multispan_comp : has_colimit (D.diagram.multispan ⋙ F) | ⟨⟨⟨_,preserves_colimit.preserves (colimit.is_colimit _)⟩⟩⟩ | lemma | category_theory.glue_data.has_colimit_multispan_comp | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimit_map_glue_data_diagram : has_multicoequalizer (D.map_glue_data F).diagram | has_colimit_of_iso (D.diagram_iso F).symm | lemma | category_theory.glue_data.has_colimit_map_glue_data_diagram | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
glued_iso : F.obj D.glued ≅ (D.map_glue_data F).glued | preserves_colimit_iso F D.diagram.multispan ≪≫
(limits.has_colimit.iso_of_nat_iso (D.diagram_iso F)) | def | category_theory.glue_data.glued_iso | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | If `F` preserves the gluing, we obtain an iso between the glued objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_glued_iso_hom (i : D.J) :
F.map (D.ι i) ≫ (D.glued_iso F).hom = (D.map_glue_data F).ι i | by { erw ι_preserves_colimits_iso_hom_assoc, rw has_colimit.iso_of_nat_iso_ι_hom,
erw category.id_comp, refl } | lemma | category_theory.glue_data.ι_glued_iso_hom | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι_glued_iso_inv (i : D.J) :
(D.map_glue_data F).ι i ≫ (D.glued_iso F).inv = F.map (D.ι i) | by rw [iso.comp_inv_eq, ι_glued_iso_hom] | lemma | category_theory.glue_data.ι_glued_iso_inv | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
V_pullback_cone_is_limit_of_map (i j : D.J) [reflects_limit (cospan (D.ι i) (D.ι j)) F]
(hc : is_limit ((D.map_glue_data F).V_pullback_cone i j)) :
is_limit (D.V_pullback_cone i j) | begin
apply is_limit_of_reflects F,
apply (is_limit_map_cone_pullback_cone_equiv _ _).symm _,
let e : cospan (F.map (D.ι i)) (F.map (D.ι j)) ≅
cospan ((D.map_glue_data F).ι i) ((D.map_glue_data F).ι j),
exact nat_iso.of_components
(λ x, by { cases x, exacts [D.glued_iso F, iso.refl _] })
(by rintros... | def | category_theory.glue_data.V_pullback_cone_is_limit_of_map | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | If `F` preserves the gluing, and reflects the pullback of `U i ⟶ glued` and `U j ⟶ glued`,
then `F` reflects the fact that `V_pullback_cone` is a pullback. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_jointly_surjective (F : C ⥤ Type v) [preserves_colimit D.diagram.multispan F]
[Π (i j k : D.J), preserves_limit (cospan (D.f i j) (D.f i k)) F] (x : F.obj (D.glued)) :
∃ i (y : F.obj (D.U i)), F.map (D.ι i) y = x | begin
let e := D.glued_iso F,
obtain ⟨i, y, eq⟩ := (D.map_glue_data F).types_ι_jointly_surjective (e.hom x),
replace eq := congr_arg e.inv eq,
change ((D.map_glue_data F).ι i ≫ e.inv) y = (e.hom ≫ e.inv) x at eq,
rw [e.hom_inv_id, D.ι_glued_iso_inv] at eq,
exact ⟨i, y, eq⟩
end | lemma | category_theory.glue_data.ι_jointly_surjective | category_theory | src/category_theory/glue_data.lean | [
"tactic.elementwise",
"category_theory.limits.shapes.multiequalizer",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.preserves.limits",
"category_theory.limits.shapes.types"
] | [] | If there is a forgetful functor into `Type` that preserves enough (co)limits, then `D.ι` will
be jointly surjective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
graded_object (β : Type w) (C : Type u) : Type (max w u) | β → C | def | category_theory.graded_object | category_theory | src/category_theory/graded_object.lean | [
"algebra.group_power.lemmas",
"category_theory.pi.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | A type synonym for `β → C`, used for `β`-graded objects in a category `C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inhabited_graded_object (β : Type w) (C : Type u) [inhabited C] :
inhabited (graded_object β C) | ⟨λ b, inhabited.default⟩ | instance | category_theory.inhabited_graded_object | category_theory | src/category_theory/graded_object.lean | [
"algebra.group_power.lemmas",
"category_theory.pi.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
graded_object_with_shift {β : Type w} [add_comm_group β] (s : β) (C : Type u) :
Type (max w u) | graded_object β C | abbreviation | category_theory.graded_object_with_shift | category_theory | src/category_theory/graded_object.lean | [
"algebra.group_power.lemmas",
"category_theory.pi.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [
"add_comm_group"
] | A type synonym for `β → C`, used for `β`-graded objects in a category `C`
with a shift functor given by translation by `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
category_of_graded_objects (β : Type w) : category.{max w v} (graded_object β C) | category_theory.pi (λ _, C) | instance | category_theory.graded_object.category_of_graded_objects | category_theory | src/category_theory/graded_object.lean | [
"algebra.group_power.lemmas",
"category_theory.pi.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [
"category_theory.pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval {β : Type w} (b : β) : graded_object β C ⥤ C | { obj := λ X, X b,
map := λ X Y f, f b, } | def | category_theory.graded_object.eval | category_theory | src/category_theory/graded_object.lean | [
"algebra.group_power.lemmas",
"category_theory.pi.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | The projection of a graded object to its `i`-th component. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_eq {β γ : Type w} {f g : β → γ} (h : f = g) : comap (λ _, C) f ≅ comap (λ _, C) g | { hom := { app := λ X b, eq_to_hom begin dsimp [comap], subst h, end },
inv := { app := λ X b, eq_to_hom begin dsimp [comap], subst h, end }, } | def | category_theory.graded_object.comap_eq | category_theory | src/category_theory/graded_object.lean | [
"algebra.group_power.lemmas",
"category_theory.pi.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | The natural isomorphism comparing between
pulling back along two propositionally equal functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_eq_symm {β γ : Type w} {f g : β → γ} (h : f = g) :
comap_eq C h.symm = (comap_eq C h).symm | by tidy | lemma | category_theory.graded_object.comap_eq_symm | category_theory | src/category_theory/graded_object.lean | [
"algebra.group_power.lemmas",
"category_theory.pi.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_eq_trans {β γ : Type w} {f g h : β → γ} (k : f = g) (l : g = h) :
comap_eq C (k.trans l) = comap_eq C k ≪≫ comap_eq C l | begin
ext X b,
simp,
end | lemma | category_theory.graded_object.comap_eq_trans | category_theory | src/category_theory/graded_object.lean | [
"algebra.group_power.lemmas",
"category_theory.pi.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_to_hom_apply {β : Type w} {X Y : Π b : β, C} (h : X = Y) (b : β) :
(eq_to_hom h : X ⟶ Y) b = eq_to_hom (by subst h) | by { subst h, refl } | lemma | category_theory.graded_object.eq_to_hom_apply | category_theory | src/category_theory/graded_object.lean | [
"algebra.group_power.lemmas",
"category_theory.pi.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_equiv {β γ : Type w} (e : β ≃ γ) :
(graded_object β C) ≌ (graded_object γ C) | { functor := comap (λ _, C) (e.symm : γ → β),
inverse := comap (λ _, C) (e : β → γ),
counit_iso := (comap_comp (λ _, C) _ _).trans (comap_eq C (by { ext, simp } )),
unit_iso := (comap_eq C (by { ext, simp } )).trans (comap_comp _ _ _).symm,
functor_unit_iso_comp' := λ X, by { ext b, dsimp, simp, }, } | def | category_theory.graded_object.comap_equiv | category_theory | src/category_theory/graded_object.lean | [
"algebra.group_power.lemmas",
"category_theory.pi.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | The equivalence between β-graded objects and γ-graded objects,
given an equivalence between β and γ. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_shift {β : Type*} [add_comm_group β] (s : β) :
has_shift (graded_object_with_shift s C) ℤ | has_shift_mk _ _
{ F := λ n, comap (λ _, C) $ λ (b : β), b + n • s,
zero := comap_eq C (by { ext, simp }) ≪≫ comap_id β (λ _, C),
add := λ m n, comap_eq C (by { ext, simp [add_zsmul, add_comm], }) ≪≫
(comap_comp _ _ _).symm,
assoc_hom_app := λ m₁ m₂ m₃ X, by { ext, dsimp, simp, },
zero_add_hom_app := λ n X... | instance | category_theory.graded_object.has_shift | category_theory | src/category_theory/graded_object.lean | [
"algebra.group_power.lemmas",
"category_theory.pi.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [
"add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
shift_functor_obj_apply {β : Type*} [add_comm_group β]
(s : β) (X : β → C) (t : β) (n : ℤ) :
(shift_functor (graded_object_with_shift s C) n).obj X t = X (t + n • s) | rfl | lemma | category_theory.graded_object.shift_functor_obj_apply | category_theory | src/category_theory/graded_object.lean | [
"algebra.group_power.lemmas",
"category_theory.pi.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [
"add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
shift_functor_map_apply {β : Type*} [add_comm_group β] (s : β)
{X Y : graded_object_with_shift s C} (f : X ⟶ Y) (t : β) (n : ℤ) :
(shift_functor (graded_object_with_shift s C) n).map f t = f (t + n • s) | rfl | lemma | category_theory.graded_object.shift_functor_map_apply | category_theory | src/category_theory/graded_object.lean | [
"algebra.group_power.lemmas",
"category_theory.pi.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [
"add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_zero_morphisms [has_zero_morphisms C] (β : Type w) :
has_zero_morphisms.{max w v} (graded_object β C) | { has_zero := λ X Y,
{ zero := λ b, 0 } } | instance | category_theory.graded_object.has_zero_morphisms | category_theory | src/category_theory/graded_object.lean | [
"algebra.group_power.lemmas",
"category_theory.pi.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_apply [has_zero_morphisms C] (β : Type w) (X Y : graded_object β C) (b : β) :
(0 : X ⟶ Y) b = 0 | rfl | lemma | category_theory.graded_object.zero_apply | category_theory | src/category_theory/graded_object.lean | [
"algebra.group_power.lemmas",
"category_theory.pi.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_zero_object [has_zero_object C] [has_zero_morphisms C] (β : Type w) :
has_zero_object.{max w v} (graded_object β C) | by { refine ⟨⟨λ b, 0, λ X, ⟨⟨⟨λ b, 0⟩, λ f, _⟩⟩, λ X, ⟨⟨⟨λ b, 0⟩, λ f, _⟩⟩⟩⟩; ext, } | instance | category_theory.graded_object.has_zero_object | category_theory | src/category_theory/graded_object.lean | [
"algebra.group_power.lemmas",
"category_theory.pi.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
total : graded_object β C ⥤ C | { obj := λ X, ∐ (λ i : β, X i),
map := λ X Y f, limits.sigma.map (λ i, f i) }. | def | category_theory.graded_object.total | category_theory | src/category_theory/graded_object.lean | [
"algebra.group_power.lemmas",
"category_theory.pi.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | The total object of a graded object is the coproduct of the graded components. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
grothendieck | (base : C)
(fiber : F.obj base) | structure | category_theory.grothendieck | category_theory | src/category_theory/grothendieck.lean | [
"category_theory.category.Cat",
"category_theory.elements"
] | [] | The Grothendieck construction (often written as `∫ F` in mathematics) for a functor `F : C ⥤ Cat`
gives a category whose
* objects `X` consist of `X.base : C` and `X.fiber : F.obj base`
* morphisms `f : X ⟶ Y` consist of
`base : X.base ⟶ Y.base` and
`f.fiber : (F.map base).obj X.fiber ⟶ Y.fiber` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom (X Y : grothendieck F) | (base : X.base ⟶ Y.base)
(fiber : (F.map base).obj X.fiber ⟶ Y.fiber) | structure | category_theory.grothendieck.hom | category_theory | src/category_theory/grothendieck.lean | [
"category_theory.category.Cat",
"category_theory.elements"
] | [] | A morphism in the Grothendieck category `F : C ⥤ Cat` consists of
`base : X.base ⟶ Y.base` and `f.fiber : (F.map base).obj X.fiber ⟶ Y.fiber`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {X Y : grothendieck F} (f g : hom X Y)
(w_base : f.base = g.base) (w_fiber : eq_to_hom (by rw w_base) ≫ f.fiber = g.fiber) : f = g | begin
cases f; cases g,
congr,
dsimp at w_base,
induction w_base,
refl,
dsimp at w_base,
induction w_base,
simpa using w_fiber,
end | lemma | category_theory.grothendieck.ext | category_theory | src/category_theory/grothendieck.lean | [
"category_theory.category.Cat",
"category_theory.elements"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id (X : grothendieck F) : hom X X | { base := 𝟙 X.base,
fiber := eq_to_hom (by erw [category_theory.functor.map_id, functor.id_obj X.fiber]), } | def | category_theory.grothendieck.id | category_theory | src/category_theory/grothendieck.lean | [
"category_theory.category.Cat",
"category_theory.elements"
] | [] | The identity morphism in the Grothendieck category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp {X Y Z : grothendieck F} (f : hom X Y) (g : hom Y Z) : hom X Z | { base := f.base ≫ g.base,
fiber :=
eq_to_hom (by erw [functor.map_comp, functor.comp_obj]) ≫
(F.map g.base).map f.fiber ≫ g.fiber, } | def | category_theory.grothendieck.comp | category_theory | src/category_theory/grothendieck.lean | [
"category_theory.category.Cat",
"category_theory.elements"
] | [] | Composition of morphisms in the Grothendieck category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_fiber' (X : grothendieck F) :
hom.fiber (𝟙 X) = eq_to_hom (by erw [category_theory.functor.map_id, functor.id_obj X.fiber]) | id_fiber X | lemma | category_theory.grothendieck.id_fiber' | category_theory | src/category_theory/grothendieck.lean | [
"category_theory.category.Cat",
"category_theory.elements"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr {X Y : grothendieck F} {f g : X ⟶ Y} (h : f = g) :
f.fiber = eq_to_hom (by subst h) ≫ g.fiber | by { subst h, dsimp, simp, } | lemma | category_theory.grothendieck.congr | category_theory | src/category_theory/grothendieck.lean | [
"category_theory.category.Cat",
"category_theory.elements"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget : grothendieck F ⥤ C | { obj := λ X, X.1,
map := λ X Y f, f.1, } | def | category_theory.grothendieck.forget | category_theory | src/category_theory/grothendieck.lean | [
"category_theory.category.Cat",
"category_theory.elements"
] | [] | The forgetful functor from `grothendieck F` to the source category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
grothendieck_Type_to_Cat_functor : grothendieck (G ⋙ Type_to_Cat) ⥤ G.elements | { obj := λ X, ⟨X.1, X.2.as⟩,
map := λ X Y f, ⟨f.1, f.2.1.1⟩ } | def | category_theory.grothendieck.grothendieck_Type_to_Cat_functor | category_theory | src/category_theory/grothendieck.lean | [
"category_theory.category.Cat",
"category_theory.elements"
] | [] | Auxiliary definition for `grothendieck_Type_to_Cat`, to speed up elaboration. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
grothendieck_Type_to_Cat_inverse : G.elements ⥤ grothendieck (G ⋙ Type_to_Cat) | { obj := λ X, ⟨X.1, ⟨X.2⟩⟩,
map := λ X Y f, ⟨f.1, ⟨⟨f.2⟩⟩⟩ } | def | category_theory.grothendieck.grothendieck_Type_to_Cat_inverse | category_theory | src/category_theory/grothendieck.lean | [
"category_theory.category.Cat",
"category_theory.elements"
] | [] | Auxiliary definition for `grothendieck_Type_to_Cat`, to speed up elaboration. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
grothendieck_Type_to_Cat : grothendieck (G ⋙ Type_to_Cat) ≌ G.elements | { functor := grothendieck_Type_to_Cat_functor G,
inverse := grothendieck_Type_to_Cat_inverse G,
unit_iso := nat_iso.of_components (λ X, by { rcases X with ⟨_, ⟨⟩⟩, exact iso.refl _, })
(by { rintro ⟨_, ⟨⟩⟩ ⟨_, ⟨⟩⟩ ⟨base, ⟨⟨f⟩⟩⟩, dsimp at *, subst f, ext, simp, }),
counit_iso := nat_iso.of_components (λ X, by ... | def | category_theory.grothendieck.grothendieck_Type_to_Cat | category_theory | src/category_theory/grothendieck.lean | [
"category_theory.category.Cat",
"category_theory.elements"
] | [] | The Grothendieck construction applied to a functor to `Type`
(thought of as a functor to `Cat` by realising a type as a discrete category)
is the same as the 'category of elements' construction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
groupoid (obj : Type u) extends category.{v} obj : Type (max u (v+1)) | (inv : Π {X Y : obj}, (X ⟶ Y) → (Y ⟶ X))
(inv_comp' : ∀ {X Y : obj} (f : X ⟶ Y), comp (inv f) f = id Y . obviously)
(comp_inv' : ∀ {X Y : obj} (f : X ⟶ Y), comp f (inv f) = id X . obviously) | class | category_theory.groupoid | category_theory | src/category_theory/groupoid.lean | [
"category_theory.full_subcategory",
"category_theory.products.basic",
"category_theory.pi.basic",
"category_theory.category.basic",
"combinatorics.quiver.connected_component"
] | [] | A `groupoid` is a category such that all morphisms are isomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
large_groupoid (C : Type (u+1)) : Type (u+1) | groupoid.{u} C | abbreviation | category_theory.large_groupoid | category_theory | src/category_theory/groupoid.lean | [
"category_theory.full_subcategory",
"category_theory.products.basic",
"category_theory.pi.basic",
"category_theory.category.basic",
"combinatorics.quiver.connected_component"
] | [] | A `large_groupoid` is a groupoid
where the objects live in `Type (u+1)` while the morphisms live in `Type u`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
small_groupoid (C : Type u) : Type (u+1) | groupoid.{u} C | abbreviation | category_theory.small_groupoid | category_theory | src/category_theory/groupoid.lean | [
"category_theory.full_subcategory",
"category_theory.products.basic",
"category_theory.pi.basic",
"category_theory.category.basic",
"combinatorics.quiver.connected_component"
] | [] | A `small_groupoid` is a groupoid
where the objects and morphisms live in the same universe. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso.of_groupoid (f : X ⟶ Y) : is_iso f | ⟨⟨groupoid.inv f, groupoid.comp_inv f, groupoid.inv_comp f⟩⟩ | instance | category_theory.is_iso.of_groupoid | category_theory | src/category_theory/groupoid.lean | [
"category_theory.full_subcategory",
"category_theory.products.basic",
"category_theory.pi.basic",
"category_theory.category.basic",
"combinatorics.quiver.connected_component"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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