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has_coequalizer | has_colimit (parallel_pair f g) | abbreviation | category_theory.limits.has_coequalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | `has_coequalizer f g` represents a particular choice of colimiting cocone
for the parallel pair of morphisms `f` and `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coequalizer : C | colimit (parallel_pair f g) | abbreviation | category_theory.limits.coequalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | If a coequalizer of `f` and `g` exists, we can access an arbitrary choice of such by
saying `coequalizer f g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coequalizer.π : Y ⟶ coequalizer f g | colimit.ι (parallel_pair f g) one | abbreviation | category_theory.limits.coequalizer.π | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | If a coequalizer of `f` and `g` exists, we can access the corresponding projection by
saying `coequalizer.π f g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coequalizer.cofork : cofork f g | colimit.cocone (parallel_pair f g) | abbreviation | category_theory.limits.coequalizer.cofork | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | An arbitrary choice of coequalizer cocone for a parallel pair `f` and `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coequalizer.cofork_π :
(coequalizer.cofork f g).π = coequalizer.π f g | rfl | lemma | category_theory.limits.coequalizer.cofork_π | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coequalizer.cofork_ι_app_one :
(coequalizer.cofork f g).ι.app one = coequalizer.π f g | rfl | lemma | category_theory.limits.coequalizer.cofork_ι_app_one | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coequalizer.condition : f ≫ coequalizer.π f g = g ≫ coequalizer.π f g | cofork.condition $ colimit.cocone $ parallel_pair f g | lemma | category_theory.limits.coequalizer.condition | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coequalizer_is_coequalizer :
is_colimit (cofork.of_π (coequalizer.π f g) (coequalizer.condition f g)) | is_colimit.of_iso_colimit (colimit.is_colimit _) (cofork.ext (iso.refl _) (by tidy)) | def | category_theory.limits.coequalizer_is_coequalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | The cofork built from `coequalizer.π f g` is colimiting. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coequalizer.desc {W : C} (k : Y ⟶ W) (h : f ≫ k = g ≫ k) : coequalizer f g ⟶ W | colimit.desc (parallel_pair f g) (cofork.of_π k h) | abbreviation | category_theory.limits.coequalizer.desc | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | Any morphism `k : Y ⟶ W` satisfying `f ≫ k = g ≫ k` factors through the coequalizer of `f`
and `g` via `coequalizer.desc : coequalizer f g ⟶ W`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coequalizer.π_desc {W : C} (k : Y ⟶ W) (h : f ≫ k = g ≫ k) :
coequalizer.π f g ≫ coequalizer.desc k h = k | colimit.ι_desc _ _ | lemma | category_theory.limits.coequalizer.π_desc | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coequalizer.π_colim_map_desc {X' Y' Z : C} (f' g' : X' ⟶ Y') [has_coequalizer f' g']
(p : X ⟶ X') (q : Y ⟶ Y') (wf : f ≫ q = p ≫ f') (wg : g ≫ q = p ≫ g')
(h : Y' ⟶ Z) (wh : f' ≫ h = g' ≫ h) :
coequalizer.π f g ≫ colim_map (parallel_pair_hom f g f' g' p q wf wg) ≫ coequalizer.desc h wh =
q ≫ h | by rw [ι_colim_map_assoc, parallel_pair_hom_app_one, coequalizer.π_desc] | lemma | category_theory.limits.coequalizer.π_colim_map_desc | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coequalizer.desc' {W : C} (k : Y ⟶ W) (h : f ≫ k = g ≫ k) :
{l : coequalizer f g ⟶ W // coequalizer.π f g ≫ l = k} | ⟨coequalizer.desc k h, coequalizer.π_desc _ _⟩ | def | category_theory.limits.coequalizer.desc' | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | Any morphism `k : Y ⟶ W` satisfying `f ≫ k = g ≫ k` induces a morphism
`l : coequalizer f g ⟶ W` satisfying `coequalizer.π ≫ g = l`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coequalizer.hom_ext {W : C} {k l : coequalizer f g ⟶ W}
(h : coequalizer.π f g ≫ k = coequalizer.π f g ≫ l) : k = l | cofork.is_colimit.hom_ext (colimit.is_colimit _) h | lemma | category_theory.limits.coequalizer.hom_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | Two maps from a coequalizer are equal if they are equal when composed with the coequalizer
map | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coequalizer.exists_unique {W : C} (k : Y ⟶ W) (h : f ≫ k = g ≫ k) :
∃! (d : coequalizer f g ⟶ W), coequalizer.π f g ≫ d = k | cofork.is_colimit.exists_unique (colimit.is_colimit _) _ h | lemma | category_theory.limits.coequalizer.exists_unique | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coequalizer.π_epi : epi (coequalizer.π f g) | { left_cancellation := λ Z h k w, coequalizer.hom_ext w } | instance | category_theory.limits.coequalizer.π_epi | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | A coequalizer morphism is an epimorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
epi_of_is_colimit_cofork {c : cofork f g} (i : is_colimit c) : epi c.π | { left_cancellation := λ Z h k w, cofork.is_colimit.hom_ext i w } | lemma | category_theory.limits.epi_of_is_colimit_cofork | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | The coequalizer morphism in any colimit cocone is an epimorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_cofork (h : f = g) : cofork f g | cofork.of_π (𝟙 Y) $ h ▸ rfl | def | category_theory.limits.id_cofork | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | The identity determines a cocone on the coequalizer diagram of `f` and `g`, if `f = g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_id_cofork (h : f = g) : is_colimit (id_cofork h) | cofork.is_colimit.mk _
(λ s, cofork.π s)
(λ s, category.id_comp _)
(λ s m h, by { convert h, exact (category.id_comp _).symm }) | def | category_theory.limits.is_colimit_id_cofork | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | The identity on `Y` is a coequalizer of `(f, g)`, where `f = g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_colimit_cocone_parallel_pair_of_eq (h₀ : f = g) {c : cofork f g} (h : is_colimit c) :
is_iso c.π | is_iso.of_iso $ is_colimit.cocone_point_unique_up_to_iso (is_colimit_id_cofork h₀) h | lemma | category_theory.limits.is_iso_colimit_cocone_parallel_pair_of_eq | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | Every coequalizer of `(f, g)`, where `f = g`, is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coequalizer.π_of_eq [has_coequalizer f g] (h : f = g) : is_iso (coequalizer.π f g) | is_iso_colimit_cocone_parallel_pair_of_eq h $ colimit.is_colimit _ | lemma | category_theory.limits.coequalizer.π_of_eq | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | The coequalizer of `(f, g)`, where `f = g`, is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_colimit_cocone_parallel_pair_of_self {c : cofork f f} (h : is_colimit c) :
is_iso c.π | is_iso_colimit_cocone_parallel_pair_of_eq rfl h | lemma | category_theory.limits.is_iso_colimit_cocone_parallel_pair_of_self | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | Every coequalizer of `(f, f)` is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_limit_cocone_parallel_pair_of_epi {c : cofork f g}
(h : is_colimit c) [mono c.π] : is_iso c.π | is_iso_colimit_cocone_parallel_pair_of_eq ((cancel_mono _).1 (cofork.condition c)) h | lemma | category_theory.limits.is_iso_limit_cocone_parallel_pair_of_epi | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | A coequalizer that is a monomorphism is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_mono_cofork_π (t : cofork f g) [mono (cofork.π t)] : f = g | (cancel_mono (cofork.π t)).1 $ cofork.condition t | lemma | category_theory.limits.eq_of_mono_cofork_π | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | Two morphisms are equal if there is a cofork whose projection is mono. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_mono_coequalizer [has_coequalizer f g] [mono (coequalizer.π f g)] : f = g | (cancel_mono (coequalizer.π f g)).1 $ coequalizer.condition _ _ | lemma | category_theory.limits.eq_of_mono_coequalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | If the coequalizer of two morphisms is a monomorphism, then the two morphisms are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coequalizer_of_self : has_coequalizer f f | has_colimit.mk
{ cocone := id_cofork rfl,
is_colimit := is_colimit_id_cofork rfl } | instance | category_theory.limits.has_coequalizer_of_self | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coequalizer.π_of_self : is_iso (coequalizer.π f f) | coequalizer.π_of_eq rfl | instance | category_theory.limits.coequalizer.π_of_self | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | The coequalizer projection for `(f, f)` is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coequalizer.iso_target_of_self : coequalizer f f ≅ Y | (as_iso (coequalizer.π f f)).symm | def | category_theory.limits.coequalizer.iso_target_of_self | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | The coequalizer of a morphism with itself is isomorphic to the target. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coequalizer.iso_target_of_self_hom :
(coequalizer.iso_target_of_self f).hom = coequalizer.desc (𝟙 Y) (by simp) | by { ext, simp [coequalizer.iso_target_of_self], } | lemma | category_theory.limits.coequalizer.iso_target_of_self_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coequalizer.iso_target_of_self_inv :
(coequalizer.iso_target_of_self f).inv = coequalizer.π f f | rfl | lemma | category_theory.limits.coequalizer.iso_target_of_self_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equalizer_comparison [has_equalizer f g] [has_equalizer (G.map f) (G.map g)] :
G.obj (equalizer f g) ⟶ equalizer (G.map f) (G.map g) | equalizer.lift (G.map (equalizer.ι _ _)) (by simp only [←G.map_comp, equalizer.condition]) | def | category_theory.limits.equalizer_comparison | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | The comparison morphism for the equalizer of `f,g`.
This is an isomorphism iff `G` preserves the equalizer of `f,g`; see
`category_theory/limits/preserves/shapes/equalizers.lean` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equalizer_comparison_comp_π [has_equalizer f g] [has_equalizer (G.map f) (G.map g)] :
equalizer_comparison f g G ≫ equalizer.ι (G.map f) (G.map g) = G.map (equalizer.ι f g) | equalizer.lift_ι _ _ | lemma | category_theory.limits.equalizer_comparison_comp_π | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_lift_equalizer_comparison [has_equalizer f g] [has_equalizer (G.map f) (G.map g)]
{Z : C} {h : Z ⟶ X} (w : h ≫ f = h ≫ g) :
G.map (equalizer.lift h w) ≫ equalizer_comparison f g G =
equalizer.lift (G.map h) (by simp only [←G.map_comp, w]) | by { ext, simp [← G.map_comp] } | lemma | category_theory.limits.map_lift_equalizer_comparison | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coequalizer_comparison [has_coequalizer f g] [has_coequalizer (G.map f) (G.map g)] :
coequalizer (G.map f) (G.map g) ⟶ G.obj (coequalizer f g) | coequalizer.desc (G.map (coequalizer.π _ _)) (by simp only [←G.map_comp, coequalizer.condition]) | def | category_theory.limits.coequalizer_comparison | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | The comparison morphism for the coequalizer of `f,g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_comp_coequalizer_comparison [has_coequalizer f g] [has_coequalizer (G.map f) (G.map g)] :
coequalizer.π _ _ ≫ coequalizer_comparison f g G = G.map (coequalizer.π _ _) | coequalizer.π_desc _ _ | lemma | category_theory.limits.ι_comp_coequalizer_comparison | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coequalizer_comparison_map_desc [has_coequalizer f g] [has_coequalizer (G.map f) (G.map g)]
{Z : C} {h : Y ⟶ Z} (w : f ≫ h = g ≫ h) :
coequalizer_comparison f g G ≫ G.map (coequalizer.desc h w) =
coequalizer.desc (G.map h) (by simp only [←G.map_comp, w]) | by { ext, simp [← G.map_comp] } | lemma | category_theory.limits.coequalizer_comparison_map_desc | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_equalizers | has_limits_of_shape walking_parallel_pair C | abbreviation | category_theory.limits.has_equalizers | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | `has_equalizers` represents a choice of equalizer for every pair of morphisms | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coequalizers | has_colimits_of_shape walking_parallel_pair C | abbreviation | category_theory.limits.has_coequalizers | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | `has_coequalizers` represents a choice of coequalizer for every pair of morphisms | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_equalizers_of_has_limit_parallel_pair
[Π {X Y : C} {f g : X ⟶ Y}, has_limit (parallel_pair f g)] : has_equalizers C | { has_limit := λ F, has_limit_of_iso (diagram_iso_parallel_pair F).symm } | lemma | category_theory.limits.has_equalizers_of_has_limit_parallel_pair | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | If `C` has all limits of diagrams `parallel_pair f g`, then it has all equalizers | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coequalizers_of_has_colimit_parallel_pair
[Π {X Y : C} {f g : X ⟶ Y}, has_colimit (parallel_pair f g)] : has_coequalizers C | { has_colimit := λ F, has_colimit_of_iso (diagram_iso_parallel_pair F) } | lemma | category_theory.limits.has_coequalizers_of_has_colimit_parallel_pair | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | If `C` has all colimits of diagrams `parallel_pair f g`, then it has all coequalizers | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_of_is_split_mono : fork (𝟙 Y) (retraction f ≫ f) | fork.of_ι f (by simp) | def | category_theory.limits.cone_of_is_split_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | A split mono `f` equalizes `(retraction f ≫ f)` and `(𝟙 Y)`.
Here we build the cone, and show in `is_split_mono_equalizes` that it is a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_of_is_split_mono_ι : (cone_of_is_split_mono f).ι = f | rfl | lemma | category_theory.limits.cone_of_is_split_mono_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_split_mono_equalizes {X Y : C} (f : X ⟶ Y) [is_split_mono f] :
is_limit (cone_of_is_split_mono f) | fork.is_limit.mk' _ $ λ s,
⟨s.ι ≫ retraction f,
by { dsimp, rw [category.assoc, ←s.condition], apply category.comp_id },
λ m hm, by simp [←hm]⟩ | def | category_theory.limits.is_split_mono_equalizes | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | A split mono `f` equalizes `(retraction f ≫ f)` and `(𝟙 Y)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
split_mono_of_equalizer {X Y : C} {f : X ⟶ Y} {r : Y ⟶ X} (hr : f ≫ r ≫ f = f)
(h : is_limit (fork.of_ι f (hr.trans (category.comp_id _).symm : f ≫ r ≫ f = f ≫ 𝟙 Y))) :
split_mono f | { retraction := r,
id' := fork.is_limit.hom_ext h
((category.assoc _ _ _).trans $ hr.trans (category.id_comp _).symm) } | def | category_theory.limits.split_mono_of_equalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | We show that the converse to `is_split_mono_equalizes` is true:
Whenever `f` equalizes `(r ≫ f)` and `(𝟙 Y)`, then `r` is a retraction of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_equalizer_comp_mono {c : fork f g} (i : is_limit c) {Z : C} (h : Y ⟶ Z) [hm : mono h] :
is_limit (fork.of_ι c.ι (by simp [reassoc_of c.condition]) : fork (f ≫ h) (g ≫ h)) | fork.is_limit.mk' _ $ λ s,
let s' : fork f g := fork.of_ι s.ι (by apply hm.right_cancellation; simp [s.condition]) in
let l := fork.is_limit.lift' i s'.ι s'.condition in
⟨l.1, l.2, λ m hm, by apply fork.is_limit.hom_ext i; rw fork.ι_of_ι at hm; rw hm; exact l.2.symm⟩ | def | category_theory.limits.is_equalizer_comp_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | The fork obtained by postcomposing an equalizer fork with a monomorphism is an equalizer. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_equalizer_comp_mono [has_equalizer f g] {Z : C} (h : Y ⟶ Z) [mono h] :
has_equalizer (f ≫ h) (g ≫ h) | ⟨⟨{ cone := _, is_limit := is_equalizer_comp_mono (limit.is_limit _) h }⟩⟩ | lemma | category_theory.limits.has_equalizer_comp_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_mono_of_idempotent_of_is_limit_fork {X : C} {f : X ⟶ X} (hf : f ≫ f = f)
{c : fork (𝟙 X) f} (i : is_limit c) : split_mono c.ι | { retraction := i.lift (fork.of_ι f (by simp [hf])),
id' :=
begin
letI := mono_of_is_limit_fork i,
rw [←cancel_mono_id c.ι, category.assoc, fork.is_limit.lift_ι, fork.ι_of_ι, ←c.condition],
exact category.comp_id c.ι
end } | def | category_theory.limits.split_mono_of_idempotent_of_is_limit_fork | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | An equalizer of an idempotent morphism and the identity is split mono. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
split_mono_of_idempotent_equalizer {X : C} {f : X ⟶ X} (hf : f ≫ f = f)
[has_equalizer (𝟙 X) f] : split_mono (equalizer.ι (𝟙 X) f) | split_mono_of_idempotent_of_is_limit_fork _ hf (limit.is_limit _) | def | category_theory.limits.split_mono_of_idempotent_equalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | The equalizer of an idempotent morphism and the identity is split mono. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_of_is_split_epi : cofork (𝟙 X) (f ≫ section_ f) | cofork.of_π f (by simp) | def | category_theory.limits.cocone_of_is_split_epi | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | A split epi `f` coequalizes `(f ≫ section_ f)` and `(𝟙 X)`.
Here we build the cocone, and show in `is_split_epi_coequalizes` that it is a colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_of_is_split_epi_π : (cocone_of_is_split_epi f).π = f | rfl | lemma | category_theory.limits.cocone_of_is_split_epi_π | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_split_epi_coequalizes {X Y : C} (f : X ⟶ Y) [is_split_epi f] :
is_colimit (cocone_of_is_split_epi f) | cofork.is_colimit.mk' _ $ λ s,
⟨section_ f ≫ s.π,
by { dsimp, rw [← category.assoc, ← s.condition, category.id_comp] },
λ m hm, by simp [← hm]⟩ | def | category_theory.limits.is_split_epi_coequalizes | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | A split epi `f` coequalizes `(f ≫ section_ f)` and `(𝟙 X)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
split_epi_of_coequalizer {X Y : C} {f : X ⟶ Y} {s : Y ⟶ X} (hs : f ≫ s ≫ f = f)
(h : is_colimit (cofork.of_π f ((category.assoc _ _ _).trans $
hs.trans (category.id_comp f).symm : (f ≫ s) ≫ f = 𝟙 X ≫ f))) :
split_epi f | { section_ := s,
id' := cofork.is_colimit.hom_ext h (hs.trans (category.comp_id _).symm) } | def | category_theory.limits.split_epi_of_coequalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | We show that the converse to `is_split_epi_equalizes` is true:
Whenever `f` coequalizes `(f ≫ s)` and `(𝟙 X)`, then `s` is a section of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_coequalizer_epi_comp {c : cofork f g} (i : is_colimit c) {W : C} (h : W ⟶ X) [hm : epi h] :
is_colimit (cofork.of_π c.π (by simp) : cofork (h ≫ f) (h ≫ g)) | cofork.is_colimit.mk' _ $ λ s,
let s' : cofork f g := cofork.of_π s.π
(by apply hm.left_cancellation; simp_rw [←category.assoc, s.condition]) in
let l := cofork.is_colimit.desc' i s'.π s'.condition in
⟨l.1, l.2,
λ m hm,by apply cofork.is_colimit.hom_ext i; rw cofork.π_of_π at hm; rw hm; exact l.2.symm⟩ | def | category_theory.limits.is_coequalizer_epi_comp | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | The cofork obtained by precomposing a coequalizer cofork with an epimorphism is
a coequalizer. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coequalizer_epi_comp [has_coequalizer f g] {W : C} (h : W ⟶ X) [hm : epi h] :
has_coequalizer (h ≫ f) (h ≫ g) | ⟨⟨{ cocone := _, is_colimit := is_coequalizer_epi_comp (colimit.is_colimit _) h }⟩⟩ | lemma | category_theory.limits.has_coequalizer_epi_comp | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_epi_of_idempotent_of_is_colimit_cofork {X : C} {f : X ⟶ X} (hf : f ≫ f = f)
{c : cofork (𝟙 X) f} (i : is_colimit c) : split_epi c.π | { section_ := i.desc (cofork.of_π f (by simp [hf])),
id' :=
begin
letI := epi_of_is_colimit_cofork i,
rw [← cancel_epi_id c.π, ← category.assoc, cofork.is_colimit.π_desc,
cofork.π_of_π, ← c.condition],
exact category.id_comp _,
end } | def | category_theory.limits.split_epi_of_idempotent_of_is_colimit_cofork | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | A coequalizer of an idempotent morphism and the identity is split epi. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
split_epi_of_idempotent_coequalizer {X : C} {f : X ⟶ X} (hf : f ≫ f = f)
[has_coequalizer (𝟙 X) f] : split_epi (coequalizer.π (𝟙 X) f) | split_epi_of_idempotent_of_is_colimit_cofork _ hf (colimit.is_colimit _) | def | category_theory.limits.split_epi_of_idempotent_coequalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | The coequalizer of an idempotent morphism and the identity is split epi. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_initial_of_equivalence (e : D ⥤ C) [is_equivalence e] [has_initial C] : has_initial D | adjunction.has_colimits_of_shape_of_equivalence e | lemma | category_theory.has_initial_of_equivalence | category_theory.limits.shapes | src/category_theory/limits/shapes/equivalence.lean | [
"category_theory.adjunction.limits",
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equivalence.has_initial_iff (e : C ≌ D) : has_initial C ↔ has_initial D | ⟨λ h, by exactI has_initial_of_equivalence e.inverse,
λ h, by exactI has_initial_of_equivalence e.functor⟩ | lemma | category_theory.equivalence.has_initial_iff | category_theory.limits.shapes | src/category_theory/limits/shapes/equivalence.lean | [
"category_theory.adjunction.limits",
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_terminal_of_equivalence (e : D ⥤ C) [is_equivalence e] [has_terminal C] :
has_terminal D | adjunction.has_limits_of_shape_of_equivalence e | lemma | category_theory.has_terminal_of_equivalence | category_theory.limits.shapes | src/category_theory/limits/shapes/equivalence.lean | [
"category_theory.adjunction.limits",
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equivalence.has_terminal_iff (e : C ≌ D) : has_terminal C ↔ has_terminal D | ⟨λ h, by exactI has_terminal_of_equivalence e.inverse,
λ h, by exactI has_terminal_of_equivalence e.functor⟩ | lemma | category_theory.equivalence.has_terminal_iff | category_theory.limits.shapes | src/category_theory/limits/shapes/equivalence.lean | [
"category_theory.adjunction.limits",
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_finite_limits : Prop | (out (J : Type) [𝒥 : small_category J] [@fin_category J 𝒥] : @has_limits_of_shape J 𝒥 C _) | class | category_theory.limits.has_finite_limits | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_limits.lean | [
"category_theory.fin_category",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.pullbacks",
"data.fintype.option"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limits_of_shape_of_has_finite_limits
(J : Type w) [small_category J] [fin_category J] [has_finite_limits C] :
has_limits_of_shape J C | begin
apply has_limits_of_shape_of_equivalence (fin_category.equiv_as_type J),
apply has_finite_limits.out
end | instance | category_theory.limits.has_limits_of_shape_of_has_finite_limits | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_limits.lean | [
"category_theory.fin_category",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.pullbacks",
"data.fintype.option"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_finite_limits_of_has_limits_of_size [has_limits_of_size.{v' u'} C] :
has_finite_limits C | ⟨λ J hJ hJ', by { haveI := has_limits_of_size_shrink.{0 0} C,
exact has_limits_of_shape_of_equivalence (fin_category.equiv_as_type J) }⟩ | instance | category_theory.limits.has_finite_limits_of_has_limits_of_size | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_limits.lean | [
"category_theory.fin_category",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.pullbacks",
"data.fintype.option"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_finite_limits_of_has_limits [has_limits C] : has_finite_limits C | infer_instance | instance | category_theory.limits.has_finite_limits_of_has_limits | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_limits.lean | [
"category_theory.fin_category",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.pullbacks",
"data.fintype.option"
] | [] | If `C` has all limits, it has finite limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_finite_limits_of_has_finite_limits_of_size
(h : ∀ (J : Type w) {𝒥 : small_category J} (hJ : @fin_category J 𝒥),
by { resetI, exact has_limits_of_shape J C }) :
has_finite_limits C | ⟨λ J hJ hhJ,
begin
resetI,
letI : category.{w w} (ulift_hom.{w} (ulift.{w 0} J)),
{ apply ulift_hom.category.{0}, exact category_theory.ulift_category J },
haveI := h (ulift_hom.{w} (ulift.{w} J)) category_theory.fin_category_ulift,
exact has_limits_of_shape_of_equivalence (ulift_hom_ulift_categor... | lemma | category_theory.limits.has_finite_limits_of_has_finite_limits_of_size | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_limits.lean | [
"category_theory.fin_category",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.pullbacks",
"data.fintype.option"
] | [
"category_theory.fin_category_ulift",
"category_theory.ulift_category"
] | We can always derive `has_finite_limits C` by providing limits at an
arbitrary universe. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_finite_colimits : Prop | (out (J : Type) [𝒥 : small_category J] [@fin_category J 𝒥] : @has_colimits_of_shape J 𝒥 C _) | class | category_theory.limits.has_finite_colimits | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_limits.lean | [
"category_theory.fin_category",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.pullbacks",
"data.fintype.option"
] | [] | A category has all finite colimits if every functor `J ⥤ C` with a `fin_category J`
instance and `J : Type` has a colimit.
This is often called 'finitely cocomplete'. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimits_of_shape_of_has_finite_colimits
(J : Type w) [small_category J] [fin_category J] [has_finite_colimits C] :
has_colimits_of_shape J C | begin
apply has_colimits_of_shape_of_equivalence (fin_category.equiv_as_type J),
apply has_finite_colimits.out
end | instance | category_theory.limits.has_colimits_of_shape_of_has_finite_colimits | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_limits.lean | [
"category_theory.fin_category",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.pullbacks",
"data.fintype.option"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_finite_colimits_of_has_colimits_of_size [has_colimits_of_size.{v' u'} C] :
has_finite_colimits C | ⟨λ J hJ hJ', by { haveI := has_colimits_of_size_shrink.{0 0} C,
exact has_colimits_of_shape_of_equivalence (fin_category.equiv_as_type J) }⟩ | instance | category_theory.limits.has_finite_colimits_of_has_colimits_of_size | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_limits.lean | [
"category_theory.fin_category",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.pullbacks",
"data.fintype.option"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_finite_colimits_of_has_finite_colimits_of_size
(h : ∀ (J : Type w) {𝒥 : small_category J} (hJ : @fin_category J 𝒥),
by { resetI, exact has_colimits_of_shape J C }) :
has_finite_colimits C | ⟨λ J hJ hhJ,
begin
resetI,
letI : category.{w w} (ulift_hom.{w} (ulift.{w 0} J)),
{ apply ulift_hom.category.{0}, exact category_theory.ulift_category J },
haveI := h (ulift_hom.{w} (ulift.{w} J)) category_theory.fin_category_ulift,
exact has_colimits_of_shape_of_equivalence (ulift_hom_ulift_categ... | lemma | category_theory.limits.has_finite_colimits_of_has_finite_colimits_of_size | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_limits.lean | [
"category_theory.fin_category",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.pullbacks",
"data.fintype.option"
] | [
"category_theory.fin_category_ulift",
"category_theory.ulift_category"
] | We can always derive `has_finite_colimits C` by providing colimits at an
arbitrary universe. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fintype_walking_parallel_pair : fintype walking_parallel_pair | { elems := [walking_parallel_pair.zero, walking_parallel_pair.one].to_finset,
complete := λ x, by { cases x; simp } } | instance | category_theory.limits.fintype_walking_parallel_pair | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_limits.lean | [
"category_theory.fin_category",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.pullbacks",
"data.fintype.option"
] | [
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fintype_obj [fintype J] : fintype (wide_pullback_shape J) | by { rw wide_pullback_shape, apply_instance } | instance | category_theory.limits.wide_pullback_shape.fintype_obj | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_limits.lean | [
"category_theory.fin_category",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.pullbacks",
"data.fintype.option"
] | [
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fintype_hom (j j' : wide_pullback_shape J) : fintype (j ⟶ j') | { elems :=
begin
cases j',
{ cases j,
{ exact {hom.id none} },
{ exact {hom.term j} } },
{ by_cases some j' = j,
{ rw h,
exact {hom.id j} },
{ exact ∅ } }
end,
complete := by tidy } | instance | category_theory.limits.wide_pullback_shape.fintype_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_limits.lean | [
"category_theory.fin_category",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.pullbacks",
"data.fintype.option"
] | [
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fintype_obj [fintype J] : fintype (wide_pushout_shape J) | by { rw wide_pushout_shape, apply_instance } | instance | category_theory.limits.wide_pushout_shape.fintype_obj | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_limits.lean | [
"category_theory.fin_category",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.pullbacks",
"data.fintype.option"
] | [
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fintype_hom (j j' : wide_pushout_shape J) : fintype (j ⟶ j') | { elems :=
begin
cases j,
{ cases j',
{ exact {hom.id none} },
{ exact {hom.init j'} } },
{ by_cases some j = j',
{ rw h,
exact {hom.id j'} },
{ exact ∅ } }
end,
complete := by tidy } | instance | category_theory.limits.wide_pushout_shape.fintype_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_limits.lean | [
"category_theory.fin_category",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.pullbacks",
"data.fintype.option"
] | [
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fin_category_wide_pullback [fintype J] : fin_category (wide_pullback_shape J) | { fintype_hom := wide_pullback_shape.fintype_hom } | instance | category_theory.limits.fin_category_wide_pullback | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_limits.lean | [
"category_theory.fin_category",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.pullbacks",
"data.fintype.option"
] | [
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fin_category_wide_pushout [fintype J] :
fin_category (wide_pushout_shape J) | { fintype_hom := wide_pushout_shape.fintype_hom } | instance | category_theory.limits.fin_category_wide_pushout | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_limits.lean | [
"category_theory.fin_category",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.pullbacks",
"data.fintype.option"
] | [
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_finite_wide_pullbacks : Prop | (out (J : Type) [fintype J] : has_limits_of_shape (wide_pullback_shape J) C) | class | category_theory.limits.has_finite_wide_pullbacks | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_limits.lean | [
"category_theory.fin_category",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.pullbacks",
"data.fintype.option"
] | [
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limits_of_shape_wide_pullback_shape
(J : Type) [finite J] [has_finite_wide_pullbacks C] :
has_limits_of_shape (wide_pullback_shape J) C | by { casesI nonempty_fintype J, haveI := @has_finite_wide_pullbacks.out C _ _ J, apply_instance } | instance | category_theory.limits.has_limits_of_shape_wide_pullback_shape | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_limits.lean | [
"category_theory.fin_category",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.pullbacks",
"data.fintype.option"
] | [
"finite",
"nonempty_fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_finite_wide_pushouts : Prop | (out (J : Type) [fintype J] : has_colimits_of_shape (wide_pushout_shape J) C) | class | category_theory.limits.has_finite_wide_pushouts | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_limits.lean | [
"category_theory.fin_category",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.pullbacks",
"data.fintype.option"
] | [
"fintype"
] | `has_finite_wide_pushouts` represents a choice of wide pushout
for every finite collection of morphisms | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimits_of_shape_wide_pushout_shape
(J : Type) [finite J] [has_finite_wide_pushouts C] :
has_colimits_of_shape (wide_pushout_shape J) C | by { casesI nonempty_fintype J, haveI := @has_finite_wide_pushouts.out C _ _ J, apply_instance } | instance | category_theory.limits.has_colimits_of_shape_wide_pushout_shape | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_limits.lean | [
"category_theory.fin_category",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.pullbacks",
"data.fintype.option"
] | [
"finite",
"nonempty_fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_finite_wide_pullbacks_of_has_finite_limits [has_finite_limits C] :
has_finite_wide_pullbacks C | ⟨λ J _, by exactI has_finite_limits.out _⟩ | lemma | category_theory.limits.has_finite_wide_pullbacks_of_has_finite_limits | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_limits.lean | [
"category_theory.fin_category",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.pullbacks",
"data.fintype.option"
] | [] | Finite wide pullbacks are finite limits, so if `C` has all finite limits,
it also has finite wide pullbacks | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_finite_wide_pushouts_of_has_finite_limits [has_finite_colimits C] :
has_finite_wide_pushouts C | ⟨λ J _, by exactI has_finite_colimits.out _⟩ | lemma | category_theory.limits.has_finite_wide_pushouts_of_has_finite_limits | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_limits.lean | [
"category_theory.fin_category",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.pullbacks",
"data.fintype.option"
] | [] | Finite wide pushouts are finite colimits, so if `C` has all finite colimits,
it also has finite wide pushouts | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fintype_walking_pair : fintype walking_pair | { elems := {walking_pair.left, walking_pair.right},
complete := λ x, by { cases x; simp } } | instance | category_theory.limits.fintype_walking_pair | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_limits.lean | [
"category_theory.fin_category",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.pullbacks",
"data.fintype.option"
] | [
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_finite_products : Prop | (out [] (n : ℕ) : has_limits_of_shape (discrete (fin n)) C) | class | category_theory.limits.has_finite_products | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_products.lean | [
"category_theory.limits.shapes.finite_limits",
"category_theory.limits.shapes.products"
] | [] | A category has finite products if there is a chosen limit for every diagram
with shape `discrete J`, where we have `[finite J]`.
We require this condition only for `J = fin n` in the definition, then deduce a version for any
`J : Type*` as a corollary of this definition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_finite_products_of_has_finite_limits [has_finite_limits C] :
has_finite_products C | ⟨λ n, infer_instance⟩ | instance | category_theory.limits.has_finite_products_of_has_finite_limits | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_products.lean | [
"category_theory.limits.shapes.finite_limits",
"category_theory.limits.shapes.products"
] | [] | If `C` has finite limits then it has finite products. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits_of_shape_discrete [has_finite_products C] (ι : Type w) [finite ι] :
has_limits_of_shape (discrete ι) C | begin
rcases finite.exists_equiv_fin ι with ⟨n, ⟨e⟩⟩,
haveI := has_finite_products.out C n,
exact has_limits_of_shape_of_equivalence (discrete.equivalence e.symm)
end | instance | category_theory.limits.has_limits_of_shape_discrete | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_products.lean | [
"category_theory.limits.shapes.finite_limits",
"category_theory.limits.shapes.products"
] | [
"finite",
"finite.exists_equiv_fin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_finite_products_of_has_products [has_products.{w} C] : has_finite_products C | ⟨λ n, has_limits_of_shape_of_equivalence (discrete.equivalence equiv.ulift.{w})⟩ | lemma | category_theory.limits.has_finite_products_of_has_products | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_products.lean | [
"category_theory.limits.shapes.finite_limits",
"category_theory.limits.shapes.products"
] | [] | If a category has all products then in particular it has finite products. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_finite_coproducts : Prop | (out [] (n : ℕ) : has_colimits_of_shape (discrete (fin n)) C) | class | category_theory.limits.has_finite_coproducts | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_products.lean | [
"category_theory.limits.shapes.finite_limits",
"category_theory.limits.shapes.products"
] | [] | A category has finite coproducts if there is a chosen colimit for every diagram
with shape `discrete J`, where we have `[fintype J]`.
We require this condition only for `J = fin n` in the definition, then deduce a version for any
`J : Type*` as a corollary of this definition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimits_of_shape_discrete [has_finite_coproducts C] (ι : Type w) [finite ι] :
has_colimits_of_shape (discrete ι) C | begin
rcases finite.exists_equiv_fin ι with ⟨n, ⟨e⟩⟩,
haveI := has_finite_coproducts.out C n,
exact has_colimits_of_shape_of_equivalence (discrete.equivalence e.symm)
end | instance | category_theory.limits.has_colimits_of_shape_discrete | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_products.lean | [
"category_theory.limits.shapes.finite_limits",
"category_theory.limits.shapes.products"
] | [
"finite",
"finite.exists_equiv_fin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_finite_coproducts_of_has_finite_colimits [has_finite_colimits C] :
has_finite_coproducts C | ⟨λ J, by apply_instance⟩ | instance | category_theory.limits.has_finite_coproducts_of_has_finite_colimits | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_products.lean | [
"category_theory.limits.shapes.finite_limits",
"category_theory.limits.shapes.products"
] | [] | If `C` has finite colimits then it has finite coproducts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_finite_coproducts_of_has_coproducts [has_coproducts.{w} C] : has_finite_coproducts C | ⟨λ J, has_colimits_of_shape_of_equivalence (discrete.equivalence (equiv.ulift.{w}))⟩ | lemma | category_theory.limits.has_finite_coproducts_of_has_coproducts | category_theory.limits.shapes | src/category_theory/limits/shapes/finite_products.lean | [
"category_theory.limits.shapes.finite_limits",
"category_theory.limits.shapes.products"
] | [] | If a category has all coproducts then in particular it has finite coproducts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor_category_has_finite_limits [has_finite_limits D] : has_finite_limits (C ⥤ D) | { out := λ J _ _, by exactI infer_instance, } | instance | category_theory.limits.functor_category_has_finite_limits | category_theory.limits.shapes | src/category_theory/limits/shapes/functor_category.lean | [
"category_theory.limits.shapes.finite_limits",
"category_theory.limits.functor_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_category_has_finite_colimits [has_finite_colimits D] :
has_finite_colimits (C ⥤ D) | { out := λ J _ _, by exactI infer_instance, } | instance | category_theory.limits.functor_category_has_finite_colimits | category_theory.limits.shapes | src/category_theory/limits/shapes/functor_category.lean | [
"category_theory.limits.shapes.finite_limits",
"category_theory.limits.functor_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_factorisation (f : X ⟶ Y) | (I : C)
(m : I ⟶ Y)
[m_mono : mono m]
(e : X ⟶ I)
(fac' : e ≫ m = f . obviously) | structure | category_theory.limits.mono_factorisation | category_theory.limits.shapes | src/category_theory/limits/shapes/images.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi"
] | [] | A factorisation of a morphism `f = e ≫ m`, with `m` monic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
self [mono f] : mono_factorisation f | { I := X,
m := f,
e := 𝟙 X } | def | category_theory.limits.mono_factorisation.self | category_theory.limits.shapes | src/category_theory/limits/shapes/images.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi"
] | [] | The obvious factorisation of a monomorphism through itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext
{F F' : mono_factorisation f} (hI : F.I = F'.I) (hm : F.m = (eq_to_hom hI) ≫ F'.m) : F = F' | begin
cases F, cases F',
cases hI,
simp at hm,
dsimp at F_fac' F'_fac',
congr,
{ assumption },
{ resetI, apply (cancel_mono F_m).1,
rw [F_fac', hm, F'_fac'], }
end | lemma | category_theory.limits.mono_factorisation.ext | category_theory.limits.shapes | src/category_theory/limits/shapes/images.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi"
] | [] | The morphism `m` in a factorisation `f = e ≫ m` through a monomorphism is uniquely
determined. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_mono (F : mono_factorisation f) {Y' : C} (g : Y ⟶ Y') [mono g] :
mono_factorisation (f ≫ g) | { I := F.I,
m := F.m ≫ g,
m_mono := mono_comp _ _,
e := F.e, } | def | category_theory.limits.mono_factorisation.comp_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/images.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi"
] | [] | Any mono factorisation of `f` gives a mono factorisation of `f ≫ g` when `g` is a mono. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_comp_iso {Y' : C} {g : Y ⟶ Y'} [is_iso g] (F : mono_factorisation (f ≫ g)) :
mono_factorisation f | { I := F.I,
m := F.m ≫ (inv g),
m_mono := mono_comp _ _,
e := F.e, } | def | category_theory.limits.mono_factorisation.of_comp_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/images.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi"
] | [] | A mono factorisation of `f ≫ g`, where `g` is an isomorphism,
gives a mono factorisation of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_comp (F : mono_factorisation f) {X' : C} (g : X' ⟶ X) :
mono_factorisation (g ≫ f) | { I := F.I,
m := F.m,
e := g ≫ F.e, } | def | category_theory.limits.mono_factorisation.iso_comp | category_theory.limits.shapes | src/category_theory/limits/shapes/images.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi"
] | [] | Any mono factorisation of `f` gives a mono factorisation of `g ≫ f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_iso_comp {X' : C} (g : X' ⟶ X) [is_iso g] (F : mono_factorisation (g ≫ f)) :
mono_factorisation f | { I := F.I,
m := F.m,
e := inv g ≫ F.e, } | def | category_theory.limits.mono_factorisation.of_iso_comp | category_theory.limits.shapes | src/category_theory/limits/shapes/images.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi"
] | [] | A mono factorisation of `g ≫ f`, where `g` is an isomorphism,
gives a mono factorisation of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_arrow_iso {f g : arrow C} (F : mono_factorisation f.hom) (sq : f ⟶ g) [is_iso sq] :
mono_factorisation g.hom | { I := F.I,
m := F.m ≫ sq.right,
e := inv sq.left ≫ F.e,
m_mono := mono_comp _ _,
fac' := by simp only [fac_assoc, arrow.w, is_iso.inv_comp_eq, category.assoc] } | def | category_theory.limits.mono_factorisation.of_arrow_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/images.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi"
] | [] | If `f` and `g` are isomorphic arrows, then a mono factorisation of `f`
gives a mono factorisation of `g` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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