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strong_mono_of_regular_mono (f : X ⟶ Y) [regular_mono f] : strong_mono f | strong_mono.mk' begin
introsI A B z hz u v sq,
have : v ≫ (regular_mono.left : Y ⟶ regular_mono.Z f) = v ≫ regular_mono.right,
{ apply (cancel_epi z).1,
simp only [regular_mono.w, ← reassoc_of sq.w] },
obtain ⟨t, ht⟩ := regular_mono.lift' _ _ this,
refine comm_sq.has_lift.mk' ⟨t, (cancel_mono f).1 _, ht⟩,... | instance | category_theory.strong_mono_of_regular_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/regular_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_of_regular_mono_of_epi (f : X ⟶ Y) [regular_mono f] [e : epi f] : is_iso f | is_iso_of_epi_of_strong_mono _ | lemma | category_theory.is_iso_of_regular_mono_of_epi | category_theory.limits.shapes | src/category_theory/limits/shapes/regular_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi",
"category_theory.limits.shapes.equalizers"
] | [] | A regular monomorphism is an isomorphism if it is an epimorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
regular_mono_category | (regular_mono_of_mono : ∀ {X Y : C} (f : X ⟶ Y) [mono f], regular_mono f) | class | category_theory.regular_mono_category | category_theory.limits.shapes | src/category_theory/limits/shapes/regular_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi",
"category_theory.limits.shapes.equalizers"
] | [] | A regular mono category is a category in which every monomorphism is regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
regular_mono_of_mono [regular_mono_category C] (f : X ⟶ Y) [mono f] : regular_mono f | regular_mono_category.regular_mono_of_mono _ | def | category_theory.regular_mono_of_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/regular_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi",
"category_theory.limits.shapes.equalizers"
] | [] | In a category in which every monomorphism is regular, we can express every monomorphism as
an equalizer. This is not an instance because it would create an instance loop. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
regular_mono_category_of_split_mono_category [split_mono_category C] :
regular_mono_category C | { regular_mono_of_mono := λ _ _ f _,
by { haveI := by exactI is_split_mono_of_mono f, apply_instance } } | instance | category_theory.regular_mono_category_of_split_mono_category | category_theory.limits.shapes | src/category_theory/limits/shapes/regular_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strong_mono_category_of_regular_mono_category [regular_mono_category C] :
strong_mono_category C | { strong_mono_of_mono := λ _ _ f _,
by { haveI := by exactI regular_mono_of_mono f, apply_instance } } | instance | category_theory.strong_mono_category_of_regular_mono_category | category_theory.limits.shapes | src/category_theory/limits/shapes/regular_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
regular_epi (f : X ⟶ Y) | (W : C)
(left right : W ⟶ X)
(w : left ≫ f = right ≫ f)
(is_colimit : is_colimit (cofork.of_π f w)) | class | category_theory.regular_epi | category_theory.limits.shapes | src/category_theory/limits/shapes/regular_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi",
"category_theory.limits.shapes.equalizers"
] | [] | A regular epimorphism is a morphism which is the coequalizer of some parallel pair. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
regular_epi.epi (f : X ⟶ Y) [regular_epi f] : epi f | epi_of_is_colimit_cofork regular_epi.is_colimit | instance | category_theory.regular_epi.epi | category_theory.limits.shapes | src/category_theory/limits/shapes/regular_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi",
"category_theory.limits.shapes.equalizers"
] | [] | Every regular epimorphism is an epimorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coequalizer_regular (g h : X ⟶ Y) [has_colimit (parallel_pair g h)] :
regular_epi (coequalizer.π g h) | { W := X,
left := g,
right := h,
w := coequalizer.condition g h,
is_colimit := cofork.is_colimit.mk _ (λ s, colimit.desc _ s) (by simp)
(λ s m w, by { ext1, simp [←w] }) } | instance | category_theory.coequalizer_regular | category_theory.limits.shapes | src/category_theory/limits/shapes/regular_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
regular_epi.of_split_epi (f : X ⟶ Y) [is_split_epi f] : regular_epi f | { W := X,
left := 𝟙 X,
right := f ≫ section_ f,
w := by tidy,
is_colimit := is_split_epi_coequalizes f } | instance | category_theory.regular_epi.of_split_epi | category_theory.limits.shapes | src/category_theory/limits/shapes/regular_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi",
"category_theory.limits.shapes.equalizers"
] | [] | Every split epimorphism is a regular epimorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
regular_epi.desc' {W : C} (f : X ⟶ Y) [regular_epi f] (k : X ⟶ W)
(h : (regular_epi.left : regular_epi.W f ⟶ X) ≫ k = regular_epi.right ≫ k) :
{l : Y ⟶ W // f ≫ l = k} | cofork.is_colimit.desc' (regular_epi.is_colimit) _ h | def | category_theory.regular_epi.desc' | category_theory.limits.shapes | src/category_theory/limits/shapes/regular_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi",
"category_theory.limits.shapes.equalizers"
] | [] | If `f` is a regular epi, then every morphism `k : X ⟶ W` coequalizing `regular_epi.left` and
`regular_epi.right` induces `l : Y ⟶ W` such that `f ≫ l = k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
regular_of_is_pushout_snd_of_regular
{P Q R S : C} {f : P ⟶ Q} {g : P ⟶ R} {h : Q ⟶ S} {k : R ⟶ S}
[gr : regular_epi g] (comm : f ≫ h = g ≫ k) (t : is_colimit (pushout_cocone.mk _ _ comm)) :
regular_epi h | { W := gr.W,
left := gr.left ≫ f,
right := gr.right ≫ f,
w := by rw [category.assoc, category.assoc, comm, reassoc_of gr.w],
is_colimit :=
begin
apply cofork.is_colimit.mk' _ _,
intro s,
have l₁ : gr.left ≫ f ≫ s.π = gr.right ≫ f ≫ s.π,
rw [← category.assoc, ← category.assoc, s.condition],
... | def | category_theory.regular_of_is_pushout_snd_of_regular | category_theory.limits.shapes | src/category_theory/limits/shapes/regular_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi",
"category_theory.limits.shapes.equalizers"
] | [
"comm"
] | The second leg of a pushout cocone is a regular epimorphism if the right component is too.
See also `pushout.snd_of_epi` for the basic epimorphism version, and
`regular_of_is_pushout_fst_of_regular` for the flipped version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
regular_of_is_pushout_fst_of_regular
{P Q R S : C} {f : P ⟶ Q} {g : P ⟶ R} {h : Q ⟶ S} {k : R ⟶ S}
[fr : regular_epi f] (comm : f ≫ h = g ≫ k) (t : is_colimit (pushout_cocone.mk _ _ comm)) :
regular_epi k | regular_of_is_pushout_snd_of_regular comm.symm (pushout_cocone.flip_is_colimit t) | def | category_theory.regular_of_is_pushout_fst_of_regular | category_theory.limits.shapes | src/category_theory/limits/shapes/regular_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi",
"category_theory.limits.shapes.equalizers"
] | [
"comm"
] | The first leg of a pushout cocone is a regular epimorphism if the left component is too.
See also `pushout.fst_of_epi` for the basic epimorphism version, and
`regular_of_is_pushout_snd_of_regular` for the flipped version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strong_epi_of_regular_epi (f : X ⟶ Y) [regular_epi f] : strong_epi f | strong_epi.mk' begin
introsI A B z hz u v sq,
have : (regular_epi.left : regular_epi.W f ⟶ X) ≫ u = regular_epi.right ≫ u,
{ apply (cancel_mono z).1,
simp only [category.assoc, sq.w, regular_epi.w_assoc] },
obtain ⟨t, ht⟩ := regular_epi.desc' f u this,
exact comm_sq.has_lift.mk' ⟨t, ht, (cancel_epi f).1
... | instance | category_theory.strong_epi_of_regular_epi | category_theory.limits.shapes | src/category_theory/limits/shapes/regular_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_of_regular_epi_of_mono (f : X ⟶ Y) [regular_epi f] [m : mono f] : is_iso f | is_iso_of_mono_of_strong_epi _ | lemma | category_theory.is_iso_of_regular_epi_of_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/regular_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi",
"category_theory.limits.shapes.equalizers"
] | [] | A regular epimorphism is an isomorphism if it is a monomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
regular_epi_category | (regular_epi_of_epi : ∀ {X Y : C} (f : X ⟶ Y) [epi f], regular_epi f) | class | category_theory.regular_epi_category | category_theory.limits.shapes | src/category_theory/limits/shapes/regular_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi",
"category_theory.limits.shapes.equalizers"
] | [] | A regular epi category is a category in which every epimorphism is regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
regular_epi_of_epi [regular_epi_category C] (f : X ⟶ Y) [epi f] : regular_epi f | regular_epi_category.regular_epi_of_epi _ | def | category_theory.regular_epi_of_epi | category_theory.limits.shapes | src/category_theory/limits/shapes/regular_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi",
"category_theory.limits.shapes.equalizers"
] | [] | In a category in which every epimorphism is regular, we can express every epimorphism as
a coequalizer. This is not an instance because it would create an instance loop. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
regular_epi_category_of_split_epi_category [split_epi_category C] :
regular_epi_category C | { regular_epi_of_epi := λ _ _ f _, by { haveI := by exactI is_split_epi_of_epi f, apply_instance } } | instance | category_theory.regular_epi_category_of_split_epi_category | category_theory.limits.shapes | src/category_theory/limits/shapes/regular_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strong_epi_category_of_regular_epi_category [regular_epi_category C] :
strong_epi_category C | { strong_epi_of_epi := λ _ _ f _, by { haveI := by exactI regular_epi_of_epi f, apply_instance } } | instance | category_theory.strong_epi_category_of_regular_epi_category | category_theory.limits.shapes | src/category_theory/limits/shapes/regular_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_split_coequalizer {Z : C} (π : Y ⟶ Z) | (right_section : Z ⟶ Y)
(left_section : Y ⟶ X)
(condition : f ≫ π = g ≫ π)
(right_section_π : right_section ≫ π = 𝟙 Z)
(left_section_bottom : left_section ≫ g = 𝟙 Y)
(left_section_top : left_section ≫ f = π ≫ right_section) | structure | category_theory.is_split_coequalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/split_coequalizer.lean | [
"category_theory.limits.shapes.equalizers"
] | [] | A split coequalizer diagram consists of morphisms
f π
X ⇉ Y → Z
g
satisfying `f ≫ π = g ≫ π` together with morphisms
t s
X ← Y ← Z
satisfying `s ≫ π = 𝟙 Z`, `t ≫ g = 𝟙 Y` and `t ≫ f = π ≫ s`.
The name "coequalizer" is appropriate, since any split coequalizer is a coequalizer, see
`c... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_split_coequalizer.map {Z : C} {π : Y ⟶ Z} (q : is_split_coequalizer f g π) (F : C ⥤ D) :
is_split_coequalizer (F.map f) (F.map g) (F.map π) | { right_section := F.map q.right_section,
left_section := F.map q.left_section,
condition := by rw [←F.map_comp, q.condition, F.map_comp],
right_section_π := by rw [←F.map_comp, q.right_section_π, F.map_id],
left_section_bottom := by rw [←F.map_comp, q.left_section_bottom, F.map_id],
left_section_top := by rw... | def | category_theory.is_split_coequalizer.map | category_theory.limits.shapes | src/category_theory/limits/shapes/split_coequalizer.lean | [
"category_theory.limits.shapes.equalizers"
] | [] | Split coequalizers are absolute: they are preserved by any functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_split_coequalizer.as_cofork {Z : C} {h : Y ⟶ Z} (t : is_split_coequalizer f g h) :
cofork f g | cofork.of_π h t.condition | def | category_theory.is_split_coequalizer.as_cofork | category_theory.limits.shapes | src/category_theory/limits/shapes/split_coequalizer.lean | [
"category_theory.limits.shapes.equalizers"
] | [] | A split coequalizer clearly induces a cofork. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_split_coequalizer.as_cofork_π {Z : C} {h : Y ⟶ Z}
(t : is_split_coequalizer f g h) : t.as_cofork.π = h | rfl | lemma | category_theory.is_split_coequalizer.as_cofork_π | category_theory.limits.shapes | src/category_theory/limits/shapes/split_coequalizer.lean | [
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_split_coequalizer.is_coequalizer
{Z : C} {h : Y ⟶ Z} (t : is_split_coequalizer f g h) :
is_colimit t.as_cofork | cofork.is_colimit.mk' _ $ λ s,
⟨t.right_section ≫ s.π,
by { dsimp, rw [← t.left_section_top_assoc, s.condition, t.left_section_bottom_assoc] },
λ m hm, by { simp [←hm] }⟩ | def | category_theory.is_split_coequalizer.is_coequalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/split_coequalizer.lean | [
"category_theory.limits.shapes.equalizers"
] | [] | The cofork induced by a split coequalizer is a coequalizer, justifying the name. In some cases it
is more convenient to show a given cofork is a coequalizer by showing it is split. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_split_coequalizer : Prop | (splittable [] : ∃ {Z : C} (h : Y ⟶ Z), nonempty (is_split_coequalizer f g h)) | class | category_theory.has_split_coequalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/split_coequalizer.lean | [
"category_theory.limits.shapes.equalizers"
] | [] | The pair `f,g` is a split pair if there is a `h : Y ⟶ Z` so that `f, g, h` forms a split coequalizer
in `C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor.is_split_pair : Prop | has_split_coequalizer (G.map f) (G.map g) | abbreviation | category_theory.functor.is_split_pair | category_theory.limits.shapes | src/category_theory/limits/shapes/split_coequalizer.lean | [
"category_theory.limits.shapes.equalizers"
] | [] | The pair `f,g` is a `G`-split pair if there is a `h : G Y ⟶ Z` so that `G f, G g, h` forms a split
coequalizer in `D`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_split_coequalizer.coequalizer_of_split [has_split_coequalizer f g] : C | (has_split_coequalizer.splittable f g).some | def | category_theory.has_split_coequalizer.coequalizer_of_split | category_theory.limits.shapes | src/category_theory/limits/shapes/split_coequalizer.lean | [
"category_theory.limits.shapes.equalizers"
] | [] | Get the coequalizer object from the typeclass `is_split_pair`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_split_coequalizer.coequalizer_π [has_split_coequalizer f g] :
Y ⟶ has_split_coequalizer.coequalizer_of_split f g | (has_split_coequalizer.splittable f g).some_spec.some | def | category_theory.has_split_coequalizer.coequalizer_π | category_theory.limits.shapes | src/category_theory/limits/shapes/split_coequalizer.lean | [
"category_theory.limits.shapes.equalizers"
] | [] | Get the coequalizer morphism from the typeclass `is_split_pair`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_split_coequalizer.is_split_coequalizer [has_split_coequalizer f g] :
is_split_coequalizer f g (has_split_coequalizer.coequalizer_π f g) | classical.choice (has_split_coequalizer.splittable f g).some_spec.some_spec | def | category_theory.has_split_coequalizer.is_split_coequalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/split_coequalizer.lean | [
"category_theory.limits.shapes.equalizers"
] | [] | The coequalizer morphism `coequalizer_ι` gives a split coequalizer on `f,g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_is_split_pair [has_split_coequalizer f g] :
has_split_coequalizer (G.map f) (G.map g) | { splittable :=
⟨_, _, ⟨is_split_coequalizer.map (has_split_coequalizer.is_split_coequalizer f g) _⟩⟩ } | instance | category_theory.map_is_split_pair | category_theory.limits.shapes | src/category_theory/limits/shapes/split_coequalizer.lean | [
"category_theory.limits.shapes.equalizers"
] | [] | If `f, g` is split, then `G f, G g` is split. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coequalizer_of_has_split_coequalizer [has_split_coequalizer f g] :
has_coequalizer f g | has_colimit.mk ⟨_, (has_split_coequalizer.is_split_coequalizer f g).is_coequalizer⟩ | instance | category_theory.limits.has_coequalizer_of_has_split_coequalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/split_coequalizer.lean | [
"category_theory.limits.shapes.equalizers"
] | [] | If a pair has a split coequalizer, it has a coequalizer. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_strict_initial_objects : Prop | (out : ∀ {I A : C} (f : A ⟶ I), is_initial I → is_iso f) | class | category_theory.limits.has_strict_initial_objects | category_theory.limits.shapes | src/category_theory/limits/shapes/strict_initial.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.binary_products"
] | [] | We say `C` has strict initial objects if every initial object is strict, ie given any morphism
`f : A ⟶ I` where `I` is initial, then `f` is an isomorphism.
Strictly speaking, this says that *any* initial object must be strict, rather than that strict
initial objects exist. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_initial.is_iso_to (hI : is_initial I) {A : C} (f : A ⟶ I) :
is_iso f | has_strict_initial_objects.out f hI | lemma | category_theory.limits.is_initial.is_iso_to | category_theory.limits.shapes | src/category_theory/limits/shapes/strict_initial.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_initial.strict_hom_ext (hI : is_initial I) {A : C} (f g : A ⟶ I) :
f = g | begin
haveI := hI.is_iso_to f,
haveI := hI.is_iso_to g,
exact eq_of_inv_eq_inv (hI.hom_ext (inv f) (inv g)),
end | lemma | category_theory.limits.is_initial.strict_hom_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/strict_initial.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_initial.subsingleton_to (hI : is_initial I) {A : C} :
subsingleton (A ⟶ I) | ⟨hI.strict_hom_ext⟩ | lemma | category_theory.limits.is_initial.subsingleton_to | category_theory.limits.shapes | src/category_theory/limits/shapes/strict_initial.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
initial_mono_of_strict_initial_objects : initial_mono_class C | { is_initial_mono_from := λ I A hI,
{ right_cancellation := λ B g h i, hI.strict_hom_ext _ _ } } | instance | category_theory.limits.initial_mono_of_strict_initial_objects | category_theory.limits.shapes | src/category_theory/limits/shapes/strict_initial.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_is_initial (X : C) [has_binary_product X I] (hI : is_initial I) :
X ⨯ I ≅ I | @@as_iso _ prod.snd (hI.is_iso_to _) | def | category_theory.limits.mul_is_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/strict_initial.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.binary_products"
] | [] | If `I` is initial, then `X ⨯ I` is isomorphic to it. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_is_initial_inv (X : C) [has_binary_product X I] (hI : is_initial I) :
(mul_is_initial X hI).inv = hI.to _ | hI.hom_ext _ _ | lemma | category_theory.limits.mul_is_initial_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/strict_initial.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_initial_mul (X : C) [has_binary_product I X] (hI : is_initial I) :
I ⨯ X ≅ I | @@as_iso _ prod.fst (hI.is_iso_to _) | def | category_theory.limits.is_initial_mul | category_theory.limits.shapes | src/category_theory/limits/shapes/strict_initial.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.binary_products"
] | [] | If `I` is initial, then `I ⨯ X` is isomorphic to it. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_initial_mul_inv (X : C) [has_binary_product I X] (hI : is_initial I) :
(is_initial_mul X hI).inv = hI.to _ | hI.hom_ext _ _ | lemma | category_theory.limits.is_initial_mul_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/strict_initial.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
initial_is_iso_to {A : C} (f : A ⟶ ⊥_ C) : is_iso f | initial_is_initial.is_iso_to _ | instance | category_theory.limits.initial_is_iso_to | category_theory.limits.shapes | src/category_theory/limits/shapes/strict_initial.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
initial.hom_ext {A : C} (f g : A ⟶ ⊥_ C) : f = g | initial_is_initial.strict_hom_ext _ _ | lemma | category_theory.limits.initial.hom_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/strict_initial.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
initial.subsingleton_to {A : C} : subsingleton (A ⟶ ⊥_ C) | initial_is_initial.subsingleton_to | lemma | category_theory.limits.initial.subsingleton_to | category_theory.limits.shapes | src/category_theory/limits/shapes/strict_initial.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_initial (X : C) [has_binary_product X ⊥_ C] :
X ⨯ ⊥_ C ≅ ⊥_ C | mul_is_initial _ initial_is_initial | def | category_theory.limits.mul_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/strict_initial.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.binary_products"
] | [] | The product of `X` with an initial object in a category with strict initial objects is itself
initial.
This is the generalisation of the fact that `X × empty ≃ empty` for types (or `n * 0 = 0`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_initial_inv (X : C) [has_binary_product X ⊥_ C] :
(mul_initial X).inv = initial.to _ | subsingleton.elim _ _ | lemma | category_theory.limits.mul_initial_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/strict_initial.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
initial_mul (X : C) [has_binary_product (⊥_ C) X] :
⊥_ C ⨯ X ≅ ⊥_ C | is_initial_mul _ initial_is_initial | def | category_theory.limits.initial_mul | category_theory.limits.shapes | src/category_theory/limits/shapes/strict_initial.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.binary_products"
] | [] | The product of `X` with an initial object in a category with strict initial objects is itself
initial.
This is the generalisation of the fact that `empty × X ≃ empty` for types (or `0 * n = 0`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
initial_mul_inv (X : C) [has_binary_product (⊥_ C) X] :
(initial_mul X).inv = initial.to _ | subsingleton.elim _ _ | lemma | category_theory.limits.initial_mul_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/strict_initial.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_initial_objects_of_initial_is_strict [has_initial C]
(h : ∀ A (f : A ⟶ ⊥_ C), is_iso f) :
has_strict_initial_objects C | { out := λ I A f hI,
begin
haveI := h A (f ≫ hI.to _),
exact ⟨⟨hI.to _ ≫ inv (f ≫ hI.to ⊥_ C), by rw [←assoc, is_iso.hom_inv_id], hI.hom_ext _ _⟩⟩,
end } | lemma | category_theory.limits.has_strict_initial_objects_of_initial_is_strict | category_theory.limits.shapes | src/category_theory/limits/shapes/strict_initial.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.binary_products"
] | [] | If `C` has an initial object such that every morphism *to* it is an isomorphism, then `C`
has strict initial objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_strict_terminal_objects : Prop | (out : ∀ {I A : C} (f : I ⟶ A), is_terminal I → is_iso f) | class | category_theory.limits.has_strict_terminal_objects | category_theory.limits.shapes | src/category_theory/limits/shapes/strict_initial.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.binary_products"
] | [] | We say `C` has strict terminal objects if every terminal object is strict, ie given any morphism
`f : I ⟶ A` where `I` is terminal, then `f` is an isomorphism.
Strictly speaking, this says that *any* terminal object must be strict, rather than that strict
terminal objects exist. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_terminal.is_iso_from (hI : is_terminal I) {A : C} (f : I ⟶ A) :
is_iso f | has_strict_terminal_objects.out f hI | lemma | category_theory.limits.is_terminal.is_iso_from | category_theory.limits.shapes | src/category_theory/limits/shapes/strict_initial.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_terminal.strict_hom_ext (hI : is_terminal I) {A : C} (f g : I ⟶ A) :
f = g | begin
haveI := hI.is_iso_from f,
haveI := hI.is_iso_from g,
exact eq_of_inv_eq_inv (hI.hom_ext (inv f) (inv g)),
end | lemma | category_theory.limits.is_terminal.strict_hom_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/strict_initial.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_terminal.subsingleton_to (hI : is_terminal I) {A : C} :
subsingleton (I ⟶ A) | ⟨hI.strict_hom_ext⟩ | lemma | category_theory.limits.is_terminal.subsingleton_to | category_theory.limits.shapes | src/category_theory/limits/shapes/strict_initial.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_π_is_iso_of_is_strict_terminal (F : J ⥤ C) [has_limit F] (i : J)
(H : ∀ j ≠ i, is_terminal (F.obj j)) [subsingleton (i ⟶ i)] :
is_iso (limit.π F i) | begin
classical,
refine ⟨⟨limit.lift _ ⟨_,⟨_,_⟩⟩,_,_⟩⟩,
{ exact λ j, dite (j = i) (λ h, eq_to_hom (by { cases h, refl })) (λ h, (H _ h).from _) },
{ intros j k f,
split_ifs,
{ cases h, cases h_1, obtain rfl : f = 𝟙 _ := subsingleton.elim _ _, simpa },
{ cases h, erw category.comp_id,
haveI : ... | lemma | category_theory.limits.limit_π_is_iso_of_is_strict_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/strict_initial.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.binary_products"
] | [
"hom_ext"
] | If all but one object in a diagram is strict terminal, the the limit is isomorphic to the
said object via `limit.π`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
terminal_is_iso_from {A : C} (f : ⊤_ C ⟶ A) : is_iso f | terminal_is_terminal.is_iso_from _ | instance | category_theory.limits.terminal_is_iso_from | category_theory.limits.shapes | src/category_theory/limits/shapes/strict_initial.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
terminal.hom_ext {A : C} (f g : ⊤_ C ⟶ A) : f = g | terminal_is_terminal.strict_hom_ext _ _ | lemma | category_theory.limits.terminal.hom_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/strict_initial.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
terminal.subsingleton_to {A : C} : subsingleton (⊤_ C ⟶ A) | terminal_is_terminal.subsingleton_to | lemma | category_theory.limits.terminal.subsingleton_to | category_theory.limits.shapes | src/category_theory/limits/shapes/strict_initial.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_terminal_objects_of_terminal_is_strict (I : C) (h : ∀ A (f : I ⟶ A), is_iso f) :
has_strict_terminal_objects C | { out := λ I' A f hI',
begin
haveI := h A (hI'.from _ ≫ f),
exact ⟨⟨inv (hI'.from I ≫ f) ≫ hI'.from I,
hI'.hom_ext _ _, by rw [assoc, is_iso.inv_hom_id]⟩⟩,
end } | lemma | category_theory.limits.has_strict_terminal_objects_of_terminal_is_strict | category_theory.limits.shapes | src/category_theory/limits/shapes/strict_initial.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.binary_products"
] | [] | If `C` has an object such that every morphism *from* it is an isomorphism, then `C`
has strict terminal objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strong_epi (f : P ⟶ Q) : Prop | (epi : epi f)
(llp : ∀ ⦃X Y : C⦄ (z : X ⟶ Y) [mono z], has_lifting_property f z) | class | category_theory.strong_epi | category_theory.limits.shapes | src/category_theory/limits/shapes/strong_epi.lean | [
"category_theory.balanced",
"category_theory.lifting_properties.basic"
] | [] | A strong epimorphism `f` is an epimorphism which has the left lifting property
with respect to monomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strong_epi.mk' {f : P ⟶ Q} [epi f]
(hf : ∀ (X Y : C) (z : X ⟶ Y) (hz : mono z) (u : P ⟶ X) (v : Q ⟶ Y)
(sq : comm_sq u f z v), sq.has_lift) : strong_epi f | { epi := infer_instance,
llp := λ X Y z hz, ⟨λ u v sq, hf X Y z hz u v sq⟩, } | lemma | category_theory.strong_epi.mk' | category_theory.limits.shapes | src/category_theory/limits/shapes/strong_epi.lean | [
"category_theory.balanced",
"category_theory.lifting_properties.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strong_mono (f : P ⟶ Q) : Prop | (mono : mono f)
(rlp : ∀ ⦃X Y : C⦄ (z : X ⟶ Y) [epi z], has_lifting_property z f) | class | category_theory.strong_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/strong_epi.lean | [
"category_theory.balanced",
"category_theory.lifting_properties.basic"
] | [] | A strong monomorphism `f` is a monomorphism which has the right lifting property
with respect to epimorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strong_mono.mk' {f : P ⟶ Q} [mono f]
(hf : ∀ (X Y : C) (z : X ⟶ Y) (hz : epi z) (u : X ⟶ P) (v : Y ⟶ Q)
(sq : comm_sq u z f v), sq.has_lift) : strong_mono f | { mono := infer_instance,
rlp := λ X Y z hz, ⟨λ u v sq, hf X Y z hz u v sq⟩, } | lemma | category_theory.strong_mono.mk' | category_theory.limits.shapes | src/category_theory/limits/shapes/strong_epi.lean | [
"category_theory.balanced",
"category_theory.lifting_properties.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_of_strong_epi (f : P ⟶ Q) [strong_epi f] : epi f | strong_epi.epi | instance | category_theory.epi_of_strong_epi | category_theory.limits.shapes | src/category_theory/limits/shapes/strong_epi.lean | [
"category_theory.balanced",
"category_theory.lifting_properties.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_of_strong_mono (f : P ⟶ Q) [strong_mono f] : mono f | strong_mono.mono | instance | category_theory.mono_of_strong_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/strong_epi.lean | [
"category_theory.balanced",
"category_theory.lifting_properties.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strong_epi_comp [strong_epi f] [strong_epi g] : strong_epi (f ≫ g) | { epi := epi_comp _ _,
llp := by { introsI, apply_instance, }, } | lemma | category_theory.strong_epi_comp | category_theory.limits.shapes | src/category_theory/limits/shapes/strong_epi.lean | [
"category_theory.balanced",
"category_theory.lifting_properties.basic"
] | [] | The composition of two strong epimorphisms is a strong epimorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strong_mono_comp [strong_mono f] [strong_mono g] : strong_mono (f ≫ g) | { mono := mono_comp _ _,
rlp := by { introsI, apply_instance, }, } | lemma | category_theory.strong_mono_comp | category_theory.limits.shapes | src/category_theory/limits/shapes/strong_epi.lean | [
"category_theory.balanced",
"category_theory.lifting_properties.basic"
] | [] | The composition of two strong monomorphisms is a strong monomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strong_epi_of_strong_epi [strong_epi (f ≫ g)] : strong_epi g | { epi := epi_of_epi f g,
llp := begin
introsI,
constructor,
intros u v sq,
have h₀ : (f ≫ u) ≫ z = (f ≫ g) ≫ v, by simp only [category.assoc, sq.w],
exact comm_sq.has_lift.mk' ⟨(comm_sq.mk h₀).lift,
by simp only [← cancel_mono z, category.assoc, comm_sq.fac_right, sq.w], by simp⟩,
end, } | lemma | category_theory.strong_epi_of_strong_epi | category_theory.limits.shapes | src/category_theory/limits/shapes/strong_epi.lean | [
"category_theory.balanced",
"category_theory.lifting_properties.basic"
] | [
"lift"
] | If `f ≫ g` is a strong epimorphism, then so is `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strong_mono_of_strong_mono [strong_mono (f ≫ g)] : strong_mono f | { mono := mono_of_mono f g,
rlp := begin
introsI,
constructor,
intros u v sq,
have h₀ : u ≫ f ≫ g = z ≫ v ≫ g, by rw reassoc_of sq.w,
exact comm_sq.has_lift.mk' ⟨(comm_sq.mk h₀).lift, by simp, by simp [← cancel_epi z, sq.w]⟩,
end, } | lemma | category_theory.strong_mono_of_strong_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/strong_epi.lean | [
"category_theory.balanced",
"category_theory.lifting_properties.basic"
] | [
"lift"
] | If `f ≫ g` is a strong monomorphism, then so is `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strong_epi_of_is_iso [is_iso f] : strong_epi f | { epi := by apply_instance,
llp := λ X Y z hz, has_lifting_property.of_left_iso _ _, } | instance | category_theory.strong_epi_of_is_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/strong_epi.lean | [
"category_theory.balanced",
"category_theory.lifting_properties.basic"
] | [] | An isomorphism is in particular a strong epimorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strong_mono_of_is_iso [is_iso f] : strong_mono f | { mono := by apply_instance,
rlp := λ X Y z hz, has_lifting_property.of_right_iso _ _, } | instance | category_theory.strong_mono_of_is_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/strong_epi.lean | [
"category_theory.balanced",
"category_theory.lifting_properties.basic"
] | [] | An isomorphism is in particular a strong monomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strong_epi.of_arrow_iso {A B A' B' : C} {f : A ⟶ B} {g : A' ⟶ B'}
(e : arrow.mk f ≅ arrow.mk g) [h : strong_epi f] : strong_epi g | { epi := begin
rw arrow.iso_w' e,
haveI := epi_comp f e.hom.right,
apply epi_comp,
end,
llp := λ X Y z, by { introI, apply has_lifting_property.of_arrow_iso_left e z, }, } | lemma | category_theory.strong_epi.of_arrow_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/strong_epi.lean | [
"category_theory.balanced",
"category_theory.lifting_properties.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strong_mono.of_arrow_iso {A B A' B' : C} {f : A ⟶ B} {g : A' ⟶ B'}
(e : arrow.mk f ≅ arrow.mk g) [h : strong_mono f] : strong_mono g | { mono := begin
rw arrow.iso_w' e,
haveI := mono_comp f e.hom.right,
apply mono_comp,
end,
rlp := λ X Y z, by { introI, apply has_lifting_property.of_arrow_iso_right z e, }, } | lemma | category_theory.strong_mono.of_arrow_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/strong_epi.lean | [
"category_theory.balanced",
"category_theory.lifting_properties.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strong_epi.iff_of_arrow_iso {A B A' B' : C} {f : A ⟶ B} {g : A' ⟶ B'}
(e : arrow.mk f ≅ arrow.mk g) : strong_epi f ↔ strong_epi g | by { split; introI, exacts [strong_epi.of_arrow_iso e, strong_epi.of_arrow_iso e.symm], } | lemma | category_theory.strong_epi.iff_of_arrow_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/strong_epi.lean | [
"category_theory.balanced",
"category_theory.lifting_properties.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strong_mono.iff_of_arrow_iso {A B A' B' : C} {f : A ⟶ B} {g : A' ⟶ B'}
(e : arrow.mk f ≅ arrow.mk g) : strong_mono f ↔ strong_mono g | by { split; introI, exacts [strong_mono.of_arrow_iso e, strong_mono.of_arrow_iso e.symm], } | lemma | category_theory.strong_mono.iff_of_arrow_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/strong_epi.lean | [
"category_theory.balanced",
"category_theory.lifting_properties.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_of_mono_of_strong_epi (f : P ⟶ Q) [mono f] [strong_epi f] : is_iso f | ⟨⟨(comm_sq.mk (show 𝟙 P ≫ f = f ≫ 𝟙 Q, by simp)).lift, by tidy⟩⟩ | lemma | category_theory.is_iso_of_mono_of_strong_epi | category_theory.limits.shapes | src/category_theory/limits/shapes/strong_epi.lean | [
"category_theory.balanced",
"category_theory.lifting_properties.basic"
] | [
"lift"
] | A strong epimorphism that is a monomorphism is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_of_epi_of_strong_mono (f : P ⟶ Q) [epi f] [strong_mono f] : is_iso f | ⟨⟨(comm_sq.mk (show 𝟙 P ≫ f = f ≫ 𝟙 Q, by simp)).lift, by tidy⟩⟩ | lemma | category_theory.is_iso_of_epi_of_strong_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/strong_epi.lean | [
"category_theory.balanced",
"category_theory.lifting_properties.basic"
] | [
"lift"
] | A strong monomorphism that is an epimorphism is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strong_epi_category : Prop | (strong_epi_of_epi : ∀ {X Y : C} (f : X ⟶ Y) [epi f], strong_epi f) | class | category_theory.strong_epi_category | category_theory.limits.shapes | src/category_theory/limits/shapes/strong_epi.lean | [
"category_theory.balanced",
"category_theory.lifting_properties.basic"
] | [] | A strong epi category is a category in which every epimorphism is strong. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strong_mono_category : Prop | (strong_mono_of_mono : ∀ {X Y : C} (f : X ⟶ Y) [mono f], strong_mono f) | class | category_theory.strong_mono_category | category_theory.limits.shapes | src/category_theory/limits/shapes/strong_epi.lean | [
"category_theory.balanced",
"category_theory.lifting_properties.basic"
] | [] | A strong mono category is a category in which every monomorphism is strong. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strong_epi_of_epi [strong_epi_category C] (f : P ⟶ Q) [epi f] : strong_epi f | strong_epi_category.strong_epi_of_epi _ | lemma | category_theory.strong_epi_of_epi | category_theory.limits.shapes | src/category_theory/limits/shapes/strong_epi.lean | [
"category_theory.balanced",
"category_theory.lifting_properties.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strong_mono_of_mono [strong_mono_category C] (f : P ⟶ Q) [mono f] : strong_mono f | strong_mono_category.strong_mono_of_mono _ | lemma | category_theory.strong_mono_of_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/strong_epi.lean | [
"category_theory.balanced",
"category_theory.lifting_properties.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced_of_strong_epi_category [strong_epi_category C] : balanced C | { is_iso_of_mono_of_epi := λ _ _ _ _ _, by exactI is_iso_of_mono_of_strong_epi _ } | instance | category_theory.balanced_of_strong_epi_category | category_theory.limits.shapes | src/category_theory/limits/shapes/strong_epi.lean | [
"category_theory.balanced",
"category_theory.lifting_properties.basic"
] | [
"balanced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
balanced_of_strong_mono_category [strong_mono_category C] : balanced C | { is_iso_of_mono_of_epi := λ _ _ _ _ _, by exactI is_iso_of_epi_of_strong_mono _ } | instance | category_theory.balanced_of_strong_mono_category | category_theory.limits.shapes | src/category_theory/limits/shapes/strong_epi.lean | [
"category_theory.balanced",
"category_theory.lifting_properties.basic"
] | [
"balanced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
as_empty_cone (X : C) : cone (functor.empty.{0} C) | { X := X, π := by tidy } | def | category_theory.limits.as_empty_cone | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | Construct a cone for the empty diagram given an object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
as_empty_cocone (X : C) : cocone (functor.empty.{0} C) | { X := X, ι := by tidy } | def | category_theory.limits.as_empty_cocone | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | Construct a cocone for the empty diagram given an object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_terminal (X : C) | is_limit (as_empty_cone X) | abbreviation | category_theory.limits.is_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | `X` is terminal if the cone it induces on the empty diagram is limiting. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_initial (X : C) | is_colimit (as_empty_cocone X) | abbreviation | category_theory.limits.is_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | `X` is initial if the cocone it induces on the empty diagram is colimiting. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_terminal_equiv_unique (F : discrete.{0} pempty.{1} ⥤ C) (Y : C) :
is_limit (⟨Y, by tidy⟩ : cone F) ≃ ∀ X : C, unique (X ⟶ Y) | { to_fun := λ t X, { default := t.lift ⟨X, by tidy⟩,
uniq := λ f, t.uniq ⟨X, by tidy⟩ f (by tidy) },
inv_fun := λ u, { lift := λ s, (u s.X).default, uniq' := λ s _ _, (u s.X).2 _ },
left_inv := by tidy,
right_inv := by tidy } | def | category_theory.limits.is_terminal_equiv_unique | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [
"inv_fun",
"lift",
"unique"
] | An object `Y` is terminal iff for every `X` there is a unique morphism `X ⟶ Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_terminal.of_unique (Y : C) [h : Π X : C, unique (X ⟶ Y)] : is_terminal Y | { lift := λ s, (h s.X).default } | def | category_theory.limits.is_terminal.of_unique | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [
"lift",
"unique"
] | An object `Y` is terminal if for every `X` there is a unique morphism `X ⟶ Y`
(as an instance). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_terminal_top {α : Type*} [preorder α] [order_top α] : is_terminal (⊤ : α) | is_terminal.of_unique _ | def | category_theory.limits.is_terminal_top | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [
"order_top"
] | If `α` is a preorder with top, then `⊤` is a terminal object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_terminal.of_iso {Y Z : C} (hY : is_terminal Y) (i : Y ≅ Z) : is_terminal Z | is_limit.of_iso_limit hY
{ hom := { hom := i.hom },
inv := { hom := i.inv } } | def | category_theory.limits.is_terminal.of_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | Transport a term of type `is_terminal` across an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_initial_equiv_unique (F : discrete.{0} pempty.{1} ⥤ C) (X : C) :
is_colimit (⟨X, by tidy⟩ : cocone F) ≃ ∀ Y : C, unique (X ⟶ Y) | { to_fun := λ t X, { default := t.desc ⟨X, by tidy⟩,
uniq := λ f, t.uniq ⟨X, by tidy⟩ f (by tidy) },
inv_fun := λ u, { desc := λ s, (u s.X).default, uniq' := λ s _ _, (u s.X).2 _ },
left_inv := by tidy,
right_inv := by tidy } | def | category_theory.limits.is_initial_equiv_unique | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [
"inv_fun",
"unique"
] | An object `X` is initial iff for every `Y` there is a unique morphism `X ⟶ Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_initial.of_unique (X : C) [h : Π Y : C, unique (X ⟶ Y)] : is_initial X | { desc := λ s, (h s.X).default } | def | category_theory.limits.is_initial.of_unique | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [
"unique"
] | An object `X` is initial if for every `Y` there is a unique morphism `X ⟶ Y`
(as an instance). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_initial_bot {α : Type*} [preorder α] [order_bot α] : is_initial (⊥ : α) | is_initial.of_unique _ | def | category_theory.limits.is_initial_bot | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [
"order_bot"
] | If `α` is a preorder with bot, then `⊥` is an initial object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_initial.of_iso {X Y : C} (hX : is_initial X) (i : X ≅ Y) : is_initial Y | is_colimit.of_iso_colimit hX
{ hom := { hom := i.hom },
inv := { hom := i.inv } } | def | category_theory.limits.is_initial.of_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | Transport a term of type `is_initial` across an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_terminal.from {X : C} (t : is_terminal X) (Y : C) : Y ⟶ X | t.lift (as_empty_cone Y) | def | category_theory.limits.is_terminal.from | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | Give the morphism to a terminal object from any other. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_terminal.hom_ext {X Y : C} (t : is_terminal X) (f g : Y ⟶ X) : f = g | t.hom_ext (by tidy) | lemma | category_theory.limits.is_terminal.hom_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | Any two morphisms to a terminal object are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_terminal.comp_from {Z : C} (t : is_terminal Z) {X Y : C} (f : X ⟶ Y) :
f ≫ t.from Y = t.from X | t.hom_ext _ _ | lemma | category_theory.limits.is_terminal.comp_from | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_terminal.from_self {X : C} (t : is_terminal X) : t.from X = 𝟙 X | t.hom_ext _ _ | lemma | category_theory.limits.is_terminal.from_self | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_initial.to {X : C} (t : is_initial X) (Y : C) : X ⟶ Y | t.desc (as_empty_cocone Y) | def | category_theory.limits.is_initial.to | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | Give the morphism from an initial object to any other. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_initial.hom_ext {X Y : C} (t : is_initial X) (f g : X ⟶ Y) : f = g | t.hom_ext (by tidy) | lemma | category_theory.limits.is_initial.hom_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | Any two morphisms from an initial object are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_initial.to_comp {X : C} (t : is_initial X) {Y Z : C} (f : Y ⟶ Z) :
t.to Y ≫ f = t.to Z | t.hom_ext _ _ | lemma | category_theory.limits.is_initial.to_comp | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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