statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
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is_image (F : mono_factorisation f) | (lift : Π (F' : mono_factorisation f), F.I ⟶ F'.I)
(lift_fac' : Π (F' : mono_factorisation f), lift F' ≫ F'.m = F.m . obviously) | structure | category_theory.limits.is_image | 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"
] | [
"lift"
] | Data exhibiting that a given factorisation through a mono is initial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fac_lift {F : mono_factorisation f} (hF : is_image F)
(F' : mono_factorisation f) : F.e ≫ hF.lift F' = F'.e | (cancel_mono F'.m).1 $ by simp | lemma | category_theory.limits.is_image.fac_lift | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self [mono f] : is_image (mono_factorisation.self f) | { lift := λ F', F'.e } | def | category_theory.limits.is_image.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"
] | [
"lift"
] | The trivial factorisation of a monomorphism satisfies the universal property. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_ext {F F' : mono_factorisation f} (hF : is_image F) (hF' : is_image F') : F.I ≅ F'.I | { hom := hF.lift F',
inv := hF'.lift F,
hom_inv_id' := (cancel_mono F.m).1 (by simp),
inv_hom_id' := (cancel_mono F'.m).1 (by simp) } | def | category_theory.limits.is_image.iso_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_ext_hom_m : (iso_ext hF hF').hom ≫ F'.m = F.m | by simp | lemma | category_theory.limits.is_image.iso_ext_hom_m | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_ext_inv_m : (iso_ext hF hF').inv ≫ F.m = F'.m | by simp | lemma | category_theory.limits.is_image.iso_ext_inv_m | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
e_iso_ext_hom : F.e ≫ (iso_ext hF hF').hom = F'.e | by simp | lemma | category_theory.limits.is_image.e_iso_ext_hom | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
e_iso_ext_inv : F'.e ≫ (iso_ext hF hF').inv = F.e | by simp | lemma | category_theory.limits.is_image.e_iso_ext_inv | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_arrow_iso {f g : arrow C} {F : mono_factorisation f.hom} (hF : is_image F)
(sq : f ⟶ g) [is_iso sq] :
is_image (F.of_arrow_iso sq) | { lift := λ F', hF.lift (F'.of_arrow_iso (inv sq)),
lift_fac' := λ F', by simpa only [mono_factorisation.of_arrow_iso_m, arrow.inv_right,
← category.assoc, is_iso.comp_inv_eq] using hF.lift_fac (F'.of_arrow_iso (inv sq)) } | def | category_theory.limits.is_image.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"
] | [
"lift"
] | If `f` and `g` are isomorphic arrows, then a mono factorisation of `f` that is an image
gives a mono factorisation of `g` that is an image | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_factorisation (f : X ⟶ Y) | (F : mono_factorisation f)
(is_image : is_image F) | structure | category_theory.limits.image_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"
] | [] | Data exhibiting that a morphism `f` has an image. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_arrow_iso {f g : arrow C} (F : image_factorisation f.hom) (sq : f ⟶ g) [is_iso sq] :
image_factorisation g.hom | { F := F.F.of_arrow_iso sq,
is_image := F.is_image.of_arrow_iso sq } | def | category_theory.limits.image_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 an image factorisation of `f`
gives an image factorisation of `g` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_image (f : X ⟶ Y) : Prop | mk' :: (exists_image : nonempty (image_factorisation f)) | class | category_theory.limits.has_image | 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"
] | [
"mk'"
] | `has_image f` means that there exists an image factorisation of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_image.mk {f : X ⟶ Y} (F : image_factorisation f) : has_image f | ⟨nonempty.intro F⟩ | lemma | category_theory.limits.has_image.mk | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_image.of_arrow_iso {f g : arrow C} [h : has_image f.hom] (sq : f ⟶ g) [is_iso sq] :
has_image g.hom | ⟨⟨h.exists_image.some.of_arrow_iso sq⟩⟩ | lemma | category_theory.limits.has_image.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_has_image (f : X ⟶ Y) [mono f] : has_image f | has_image.mk ⟨_, is_image.self f⟩ | instance | category_theory.limits.mono_has_image | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image.mono_factorisation : mono_factorisation f | (classical.choice (has_image.exists_image)).F | def | category_theory.limits.image.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"
] | [] | Some factorisation of `f` through a monomorphism (selected with choice). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image.is_image : is_image (image.mono_factorisation f) | (classical.choice (has_image.exists_image)).is_image | def | category_theory.limits.image.is_image | 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 witness of the universal property for the chosen factorisation of `f` through
a monomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image : C | (image.mono_factorisation f).I | def | category_theory.limits.image | 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 categorical image of a morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image.ι : image f ⟶ Y | (image.mono_factorisation f).m | def | category_theory.limits.image.ι | 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 inclusion of the image of a morphism into the target. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image.as_ι : (image.mono_factorisation f).m = image.ι f | rfl | lemma | category_theory.limits.image.as_ι | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
factor_thru_image : X ⟶ image f | (image.mono_factorisation f).e | def | category_theory.limits.factor_thru_image | 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 map from the source to the image of a morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
as_factor_thru_image : (image.mono_factorisation f).e = factor_thru_image f | rfl | lemma | category_theory.limits.as_factor_thru_image | 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"
] | [] | Rewrite in terms of the `factor_thru_image` interface. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image.fac : factor_thru_image f ≫ image.ι f = f | (image.mono_factorisation f).fac' | lemma | category_theory.limits.image.fac | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image.lift (F' : mono_factorisation f) : image f ⟶ F'.I | (image.is_image f).lift F' | def | category_theory.limits.image.lift | 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"
] | [
"lift"
] | Any other factorisation of the morphism `f` through a monomorphism receives a map from the
image. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image.lift_fac (F' : mono_factorisation f) : image.lift F' ≫ F'.m = image.ι f | (image.is_image f).lift_fac' F' | lemma | category_theory.limits.image.lift_fac | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image.fac_lift (F' : mono_factorisation f) : factor_thru_image f ≫ image.lift F' = F'.e | (image.is_image f).fac_lift F' | lemma | category_theory.limits.image.fac_lift | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image.is_image_lift (F : mono_factorisation f) :
(image.is_image f).lift F = image.lift F | rfl | lemma | category_theory.limits.image.is_image_lift | 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"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_image.lift_ι {F : mono_factorisation f} (hF : is_image F) :
hF.lift (image.mono_factorisation f) ≫ image.ι f = F.m | hF.lift_fac _ | lemma | category_theory.limits.is_image.lift_ι | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image.lift_mono (F' : mono_factorisation f) : mono (image.lift F') | by { apply mono_of_mono _ F'.m, simpa using mono_factorisation.m_mono _ } | instance | category_theory.limits.image.lift_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_image.uniq
(F' : mono_factorisation f) (l : image f ⟶ F'.I) (w : l ≫ F'.m = image.ι f) :
l = image.lift F' | (cancel_mono F'.m).1 (by simp [w]) | lemma | category_theory.limits.has_image.uniq | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_images : Prop | (has_image : Π {X Y : C} (f : X ⟶ Y), has_image f) | class | category_theory.limits.has_images | 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"
] | [] | `has_images` asserts that every morphism has an image. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_mono_iso_source [mono f] : image f ≅ X | is_image.iso_ext (image.is_image f) (is_image.self f) | def | category_theory.limits.image_mono_iso_source | 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 image of a monomorphism is isomorphic to the source. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_mono_iso_source_inv_ι [mono f] : (image_mono_iso_source f).inv ≫ image.ι f = f | by simp [image_mono_iso_source] | lemma | category_theory.limits.image_mono_iso_source_inv_ι | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_mono_iso_source_hom_self [mono f] : (image_mono_iso_source f).hom ≫ f = image.ι f | begin
conv { to_lhs, congr, skip, rw ←image_mono_iso_source_inv_ι f, },
rw [←category.assoc, iso.hom_inv_id, category.id_comp],
end | lemma | category_theory.limits.image_mono_iso_source_hom_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image.ext [has_image f] {W : C} {g h : image f ⟶ W} [has_limit (parallel_pair g h)]
(w : factor_thru_image f ≫ g = factor_thru_image f ≫ h) :
g = h | begin
let q := equalizer.ι g h,
let e' := equalizer.lift _ w,
let F' : mono_factorisation f :=
{ I := equalizer g h,
m := q ≫ image.ι f,
m_mono := by apply mono_comp,
e := e' },
let v := image.lift F',
have t₀ : v ≫ q ≫ image.ι f = image.ι f := image.lift_fac F',
have t : v ≫ q = 𝟙 (image f) ... | lemma | category_theory.limits.image.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_image_of_epi {X Y : C} (f : X ⟶ Y) [has_image f] [E : epi f] : epi (image.ι f) | begin
rw ←image.fac f at E,
resetI,
exact epi_of_epi (factor_thru_image f) (image.ι f),
end | lemma | category_theory.limits.epi_image_of_epi | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_of_epi_image {X Y : C} (f : X ⟶ Y) [has_image f]
[epi (image.ι f)] [epi (factor_thru_image f)] : epi f | by { rw [←image.fac f], apply epi_comp, } | lemma | category_theory.limits.epi_of_epi_image | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image.eq_to_hom (h : f = f') : image f ⟶ image f' | image.lift
{ I := image f',
m := image.ι f',
e := factor_thru_image f', }. | def | category_theory.limits.image.eq_to_hom | 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"
] | [] | An equation between morphisms gives a comparison map between the images
(which momentarily we prove is an iso). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image.eq_to_iso (h : f = f') : image f ≅ image f' | as_iso (image.eq_to_hom h) | def | category_theory.limits.image.eq_to_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"
] | [] | An equation between morphisms gives an isomorphism between the images. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image.eq_fac [has_equalizers C] (h : f = f') :
image.ι f = (image.eq_to_iso h).hom ≫ image.ι f' | by { ext, simp [image.eq_to_iso, image.eq_to_hom], } | lemma | category_theory.limits.image.eq_fac | 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"
] | [] | As long as the category has equalizers,
the image inclusion maps commute with `image.eq_to_iso`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image.pre_comp [has_image g] [has_image (f ≫ g)] : image (f ≫ g) ⟶ image g | image.lift
{ I := image g,
m := image.ι g,
e := f ≫ factor_thru_image g } | def | category_theory.limits.image.pre_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"
] | [] | The comparison map `image (f ≫ g) ⟶ image g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image.pre_comp_ι [has_image g] [has_image (f ≫ g)] :
image.pre_comp f g ≫ image.ι g = image.ι (f ≫ g) | by simp [image.pre_comp] | lemma | category_theory.limits.image.pre_comp_ι | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image.factor_thru_image_pre_comp [has_image g] [has_image (f ≫ g)] :
factor_thru_image (f ≫ g) ≫ image.pre_comp f g = f ≫ factor_thru_image g | by simp [image.pre_comp] | lemma | category_theory.limits.image.factor_thru_image_pre_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image.pre_comp_mono [has_image g] [has_image (f ≫ g)] : mono (image.pre_comp f g) | begin
apply mono_of_mono _ (image.ι g),
simp only [image.pre_comp_ι],
apply_instance,
end | instance | category_theory.limits.image.pre_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"
] | [] | `image.pre_comp f g` is a monomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image.pre_comp_comp {W : C} (h : Z ⟶ W)
[has_image (g ≫ h)] [has_image (f ≫ g ≫ h)]
[has_image h] [has_image ((f ≫ g) ≫ h)] :
image.pre_comp f (g ≫ h) ≫ image.pre_comp g h =
image.eq_to_hom (category.assoc f g h).symm ≫ (image.pre_comp (f ≫ g) h) | begin
apply (cancel_mono (image.ι h)).1,
simp [image.pre_comp, image.eq_to_hom],
end | lemma | category_theory.limits.image.pre_comp_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"
] | [] | The two step comparison map
`image (f ≫ (g ≫ h)) ⟶ image (g ≫ h) ⟶ image h`
agrees with the one step comparison map
`image (f ≫ (g ≫ h)) ≅ image ((f ≫ g) ≫ h) ⟶ image h`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image.pre_comp_epi_of_epi [has_image g] [has_image (f ≫ g)] [epi f] :
epi (image.pre_comp f g) | begin
apply epi_of_epi_fac (image.factor_thru_image_pre_comp _ _),
exact epi_comp _ _
end | instance | category_theory.limits.image.pre_comp_epi_of_epi | 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"
] | [] | `image.pre_comp f g` is an epimorphism when `f` is an epimorphism
(we need `C` to have equalizers to prove this). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_image_iso_comp [is_iso f] [has_image g] : has_image (f ≫ g) | has_image.mk
{ F := (image.mono_factorisation g).iso_comp f,
is_image := { lift := λ F', image.lift (F'.of_iso_comp f) }, } | instance | category_theory.limits.has_image_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"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image.is_iso_precomp_iso (f : X ⟶ Y) [is_iso f] [has_image g] :
is_iso (image.pre_comp f g) | ⟨⟨image.lift
{ I := image (f ≫ g),
m := image.ι (f ≫ g),
e := inv f ≫ factor_thru_image (f ≫ g) },
⟨by { ext, simp [image.pre_comp], }, by { ext, simp [image.pre_comp], }⟩⟩⟩ | instance | category_theory.limits.image.is_iso_precomp_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"
] | [] | `image.pre_comp f g` is an isomorphism when `f` is an isomorphism
(we need `C` to have equalizers to prove this). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_image_comp_iso [has_image f] [is_iso g] : has_image (f ≫ g) | has_image.mk
{ F := (image.mono_factorisation f).comp_mono g,
is_image := { lift := λ F', image.lift F'.of_comp_iso }, } | instance | category_theory.limits.has_image_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"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image.comp_iso [has_image f] [is_iso g] :
image f ≅ image (f ≫ g) | { hom := image.lift (image.mono_factorisation (f ≫ g)).of_comp_iso,
inv := image.lift ((image.mono_factorisation f).comp_mono g) } | def | category_theory.limits.image.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"
] | [] | Postcomposing by an isomorphism induces an isomorphism on the image. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image.comp_iso_hom_comp_image_ι [has_image f] [is_iso g] :
(image.comp_iso f g).hom ≫ image.ι (f ≫ g) = image.ι f ≫ g | by { ext, simp [image.comp_iso] } | lemma | category_theory.limits.image.comp_iso_hom_comp_image_ι | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image.comp_iso_inv_comp_image_ι [has_image f] [is_iso g] :
(image.comp_iso f g).inv ≫ image.ι f = image.ι (f ≫ g) ≫ inv g | by { ext, simp [image.comp_iso] } | lemma | category_theory.limits.image.comp_iso_inv_comp_image_ι | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_map {f g : arrow C} [has_image f.hom] [has_image g.hom] (sq : f ⟶ g) | (map : image f.hom ⟶ image g.hom)
(map_ι' : map ≫ image.ι g.hom = image.ι f.hom ≫ sq.right . obviously) | structure | category_theory.limits.image_map | 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"
] | [] | An image map is a morphism `image f → image g` fitting into a commutative square and satisfying
the obvious commutativity conditions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inhabited_image_map {f : arrow C} [has_image f.hom] : inhabited (image_map (𝟙 f)) | ⟨⟨𝟙 _, by tidy⟩⟩ | instance | category_theory.limits.inhabited_image_map | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_map.factor_map {f g : arrow C} [has_image f.hom] [has_image g.hom] (sq : f ⟶ g)
(m : image_map sq) :
factor_thru_image f.hom ≫ m.map = sq.left ≫ factor_thru_image g.hom | (cancel_mono (image.ι g.hom)).1 $ by simp | lemma | category_theory.limits.image_map.factor_map | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_map.transport {f g : arrow C} [has_image f.hom] [has_image g.hom] (sq : f ⟶ g)
(F : mono_factorisation f.hom) {F' : mono_factorisation g.hom} (hF' : is_image F')
{map : F.I ⟶ F'.I} (map_ι : map ≫ F'.m = F.m ≫ sq.right) : image_map sq | { map := image.lift F ≫ map ≫ hF'.lift (image.mono_factorisation g.hom),
map_ι' := by simp [map_ι] } | def | category_theory.limits.image_map.transport | 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"
] | [] | To give an image map for a commutative square with `f` at the top and `g` at the bottom, it
suffices to give a map between any mono factorisation of `f` and any image factorisation of
`g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_image_map {f g : arrow C} [has_image f.hom] [has_image g.hom] (sq : f ⟶ g) : Prop | mk' :: (has_image_map : nonempty (image_map sq)) | class | category_theory.limits.has_image_map | 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"
] | [
"mk'"
] | `has_image_map sq` means that there is an `image_map` for the square `sq`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_image_map.mk {f g : arrow C} [has_image f.hom] [has_image g.hom] {sq : f ⟶ g}
(m : image_map sq) : has_image_map sq | ⟨nonempty.intro m⟩ | lemma | category_theory.limits.has_image_map.mk | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_image_map.transport {f g : arrow C} [has_image f.hom] [has_image g.hom] (sq : f ⟶ g)
(F : mono_factorisation f.hom) {F' : mono_factorisation g.hom} (hF' : is_image F')
(map : F.I ⟶ F'.I) (map_ι : map ≫ F'.m = F.m ≫ sq.right) : has_image_map sq | has_image_map.mk $ image_map.transport sq F hF' map_ι | lemma | category_theory.limits.has_image_map.transport | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_image_map.image_map {f g : arrow C} [has_image f.hom] [has_image g.hom] (sq : f ⟶ g)
[has_image_map sq] : image_map sq | classical.choice $ @has_image_map.has_image_map _ _ _ _ _ _ sq _ | def | category_theory.limits.has_image_map.image_map | 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"
] | [] | Obtain an `image_map` from a `has_image_map` instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_image_map_of_is_iso {f g : arrow C} [has_image f.hom] [has_image g.hom]
(sq : f ⟶ g) [is_iso sq] :
has_image_map sq | has_image_map.mk
{ map := image.lift ((image.mono_factorisation g.hom).of_arrow_iso (inv sq)),
map_ι' := begin
erw [← cancel_mono (inv sq).right, category.assoc, ← mono_factorisation.of_arrow_iso_m,
image.lift_fac, category.assoc, ← comma.comp_right, is_iso.hom_inv_id,
comma.id_right, category.comp_id... | instance | category_theory.limits.has_image_map_of_is_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_image_map.comp {f g h : arrow C} [has_image f.hom] [has_image g.hom] [has_image h.hom]
(sq1 : f ⟶ g) (sq2 : g ⟶ h) [has_image_map sq1] [has_image_map sq2] :
has_image_map (sq1 ≫ sq2) | has_image_map.mk
{ map := (has_image_map.image_map sq1).map ≫ (has_image_map.image_map sq2).map,
map_ι' :=
by simp only [image_map.map_ι, image_map.map_ι_assoc, comma.comp_right, category.assoc] } | instance | category_theory.limits.has_image_map.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image.map : image f.hom ⟶ image g.hom | (has_image_map.image_map sq).map | abbreviation | category_theory.limits.image.map | 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 map on images induced by a commutative square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image.factor_map :
factor_thru_image f.hom ≫ image.map sq = sq.left ≫ factor_thru_image g.hom | by simp | lemma | category_theory.limits.image.factor_map | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image.map_ι : image.map sq ≫ image.ι g.hom = image.ι f.hom ≫ sq.right | by simp | lemma | category_theory.limits.image.map_ι | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image.map_hom_mk'_ι {X Y P Q : C} {k : X ⟶ Y} [has_image k] {l : P ⟶ Q} [has_image l]
{m : X ⟶ P} {n : Y ⟶ Q} (w : m ≫ l = k ≫ n) [has_image_map (arrow.hom_mk' w)] :
image.map (arrow.hom_mk' w) ≫ image.ι l = image.ι k ≫ n | image.map_ι _ | lemma | category_theory.limits.image.map_hom_mk'_ι | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_map_comp : image_map (sq ≫ sq') | { map := image.map sq ≫ image.map sq' } | def | category_theory.limits.image_map_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"
] | [] | Image maps for composable commutative squares induce an image map in the composite square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image.map_comp [has_image_map (sq ≫ sq')] :
image.map (sq ≫ sq') = image.map sq ≫ image.map sq' | show (has_image_map.image_map (sq ≫ sq')).map = (image_map_comp sq sq').map, by congr | lemma | category_theory.limits.image.map_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_map_id : image_map (𝟙 f) | { map := 𝟙 (image f.hom) } | def | category_theory.limits.image_map_id | 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 identity `image f ⟶ image f` fits into the commutative square represented by the identity
morphism `𝟙 f` in the arrow category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image.map_id [has_image_map (𝟙 f)] : image.map (𝟙 f) = 𝟙 (image f.hom) | show (has_image_map.image_map (𝟙 f)).map = (image_map_id f).map, by congr | lemma | category_theory.limits.image.map_id | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_image_maps | (has_image_map : Π {f g : arrow C} (st : f ⟶ g), has_image_map st) | class | category_theory.limits.has_image_maps | 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 a category `has_image_maps`, then all commutative squares induce morphisms on images. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
im : arrow C ⥤ C | { obj := λ f, image f.hom,
map := λ _ _ st, image.map st } | def | category_theory.limits.im | 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 functor from the arrow category of `C` to `C` itself that maps a morphism to its image
and a commutative square to the induced morphism on images. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strong_epi_mono_factorisation {X Y : C} (f : X ⟶ Y) extends mono_factorisation f | [e_strong_epi : strong_epi e] | structure | category_theory.limits.strong_epi_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 strong epi-mono factorisation is a decomposition `f = e ≫ m` with `e` a strong epimorphism
and `m` a monomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strong_epi_mono_factorisation_inhabited {X Y : C} (f : X ⟶ Y) [strong_epi f] :
inhabited (strong_epi_mono_factorisation f) | ⟨⟨⟨Y, 𝟙 Y, f, by simp⟩⟩⟩ | instance | category_theory.limits.strong_epi_mono_factorisation_inhabited | 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"
] | [] | Satisfying the inhabited linter | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strong_epi_mono_factorisation.to_mono_is_image {X Y : C} {f : X ⟶ Y}
(F : strong_epi_mono_factorisation f) : is_image F.to_mono_factorisation | { lift := λ G, (comm_sq.mk (show G.e ≫ G.m = F.e ≫ F.m,
by rw [F.to_mono_factorisation.fac, G.fac])).lift, } | def | category_theory.limits.strong_epi_mono_factorisation.to_mono_is_image | 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"
] | [
"lift"
] | A mono factorisation coming from a strong epi-mono factorisation always has the universal
property of the image. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_strong_epi_mono_factorisations : Prop | mk' :: (has_fac : Π {X Y : C} (f : X ⟶ Y), nonempty (strong_epi_mono_factorisation f)) | class | category_theory.limits.has_strong_epi_mono_factorisations | 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"
] | [
"mk'"
] | A category has strong epi-mono factorisations if every morphism admits a strong epi-mono
factorisation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_strong_epi_mono_factorisations.mk
(d : Π {X Y : C} (f : X ⟶ Y), strong_epi_mono_factorisation f) :
has_strong_epi_mono_factorisations C | ⟨λ X Y f, nonempty.intro $ d f⟩ | lemma | category_theory.limits.has_strong_epi_mono_factorisations.mk | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_images_of_has_strong_epi_mono_factorisations
[has_strong_epi_mono_factorisations C] : has_images C | { has_image := λ X Y f,
let F' := classical.choice (has_strong_epi_mono_factorisations.has_fac f) in
has_image.mk { F := F'.to_mono_factorisation,
is_image := F'.to_mono_is_image } } | instance | category_theory.limits.has_images_of_has_strong_epi_mono_factorisations | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strong_epi_images : Prop | (strong_factor_thru_image : Π {X Y : C} (f : X ⟶ Y), strong_epi (factor_thru_image f)) | class | category_theory.limits.has_strong_epi_images | 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 category has strong epi images if it has all images and `factor_thru_image f` is a strong
epimorphism for all `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strong_epi_of_strong_epi_mono_factorisation {X Y : C} {f : X ⟶ Y}
(F : strong_epi_mono_factorisation f) {F' : mono_factorisation f} (hF' : is_image F') :
strong_epi F'.e | by { rw ←is_image.e_iso_ext_hom F.to_mono_is_image hF', apply strong_epi_comp } | lemma | category_theory.limits.strong_epi_of_strong_epi_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"
] | [] | If there is a single strong epi-mono factorisation of `f`, then every image factorisation is a
strong epi-mono factorisation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strong_epi_factor_thru_image_of_strong_epi_mono_factorisation {X Y : C} {f : X ⟶ Y}
[has_image f] (F : strong_epi_mono_factorisation f) : strong_epi (factor_thru_image f) | strong_epi_of_strong_epi_mono_factorisation F $ image.is_image f | lemma | category_theory.limits.strong_epi_factor_thru_image_of_strong_epi_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strong_epi_images_of_has_strong_epi_mono_factorisations
[has_strong_epi_mono_factorisations C] : has_strong_epi_images C | { strong_factor_thru_image := λ X Y f,
strong_epi_factor_thru_image_of_strong_epi_mono_factorisation $
classical.choice $ has_strong_epi_mono_factorisations.has_fac f } | instance | category_theory.limits.has_strong_epi_images_of_has_strong_epi_mono_factorisations | 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 we constructed our images from strong epi-mono factorisations, then these images are
strong epi images. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_image_maps_of_has_strong_epi_images [has_strong_epi_images C] :
has_image_maps C | { has_image_map := λ f g st, has_image_map.mk
{ map := (comm_sq.mk (show (st.left ≫ factor_thru_image g.hom) ≫ image.ι g.hom =
factor_thru_image f.hom ≫ (image.ι f.hom ≫ st.right), by simp)).lift, }, } | instance | category_theory.limits.has_image_maps_of_has_strong_epi_images | 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"
] | [
"lift"
] | A category with strong epi images has image maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_strong_epi_images_of_has_pullbacks_of_has_equalizers [has_pullbacks C]
[has_equalizers C] : has_strong_epi_images C | { strong_factor_thru_image := λ X Y f, strong_epi.mk'
(λ A B h h_mono x y sq, comm_sq.has_lift.mk'
{ l := image.lift
{ I := pullback h y,
m := pullback.snd ≫ image.ι f,
m_mono := by exactI mono_comp _ _,
e := pullback.lift _ _ sq.w } ≫ pullback.fst,
fac_left' := by simp... | instance | category_theory.limits.has_strong_epi_images_of_has_pullbacks_of_has_equalizers | 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 a category has images, equalizers and pullbacks, then images are automatically strong epi
images. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image.iso_strong_epi_mono {I' : C} (e : X ⟶ I') (m : I' ⟶ Y) (comm : e ≫ m = f) [strong_epi e]
[mono m] :
I' ≅ image f | is_image.iso_ext {strong_epi_mono_factorisation . I := I', m := m, e := e}.to_mono_is_image $
image.is_image f | def | category_theory.limits.image.iso_strong_epi_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"
] | [
"comm"
] | If `C` has strong epi mono factorisations, then the image is unique up to isomorphism, in that if
`f` factors as a strong epi followed by a mono, this factorisation is essentially the image
factorisation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image.iso_strong_epi_mono_hom_comp_ι {I' : C} (e : X ⟶ I') (m : I' ⟶ Y) (comm : e ≫ m = f)
[strong_epi e] [mono m] :
(image.iso_strong_epi_mono e m comm).hom ≫ image.ι f = m | is_image.lift_fac _ _ | lemma | category_theory.limits.image.iso_strong_epi_mono_hom_comp_ι | 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"
] | [
"comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image.iso_strong_epi_mono_inv_comp_mono {I' : C} (e : X ⟶ I') (m : I' ⟶ Y) (comm : e ≫ m = f)
[strong_epi e] [mono m] :
(image.iso_strong_epi_mono e m comm).inv ≫ m = image.ι f | image.lift_fac _ | lemma | category_theory.limits.image.iso_strong_epi_mono_inv_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"
] | [
"comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strong_epi_mono_factorisations_imp_of_is_equivalence (F : C ⥤ D) [is_equivalence F]
[h : has_strong_epi_mono_factorisations C] :
has_strong_epi_mono_factorisations D | ⟨λ X Y f, begin
let em : strong_epi_mono_factorisation (F.inv.map f) :=
(has_strong_epi_mono_factorisations.has_fac (F.inv.map f)).some,
haveI : mono (F.map em.m ≫ F.as_equivalence.counit_iso.hom.app Y) := mono_comp _ _,
haveI : strong_epi (F.as_equivalence.counit_iso.inv.app X ≫ F.map em.e) := strong_epi_com... | lemma | category_theory.functor.has_strong_epi_mono_factorisations_imp_of_is_equivalence | 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"
] | [
"em"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_kernel {X Y : C} (f : X ⟶ Y) : Prop | has_limit (parallel_pair f 0) | abbreviation | category_theory.limits.has_kernel | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | A morphism `f` has a kernel if the functor `parallel_pair f 0` has a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_cokernel {X Y : C} (f : X ⟶ Y) : Prop | has_colimit (parallel_pair f 0) | abbreviation | category_theory.limits.has_cokernel | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | A morphism `f` has a cokernel if the functor `parallel_pair f 0` has a colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel_fork | fork f 0 | abbreviation | category_theory.limits.kernel_fork | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | A kernel fork is just a fork where the second morphism is a zero morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel_fork.condition (s : kernel_fork f) : fork.ι s ≫ f = 0 | by erw [fork.condition, has_zero_morphisms.comp_zero] | lemma | category_theory.limits.kernel_fork.condition | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kernel_fork.app_one (s : kernel_fork f) : s.π.app one = 0 | by simp [fork.app_one_eq_ι_comp_right] | lemma | category_theory.limits.kernel_fork.app_one | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kernel_fork.of_ι {Z : C} (ι : Z ⟶ X) (w : ι ≫ f = 0) : kernel_fork f | fork.of_ι ι $ by rw [w, has_zero_morphisms.comp_zero] | abbreviation | category_theory.limits.kernel_fork.of_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | A morphism `ι` satisfying `ι ≫ f = 0` determines a kernel fork over `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel_fork.ι_of_ι {X Y P : C} (f : X ⟶ Y) (ι : P ⟶ X) (w : ι ≫ f = 0) :
fork.ι (kernel_fork.of_ι ι w) = ι | rfl | lemma | category_theory.limits.kernel_fork.ι_of_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_of_ι (s : fork f 0) : s ≅ fork.of_ι (fork.ι s) (fork.condition s) | cones.ext (iso.refl _) $ by tidy | def | category_theory.limits.iso_of_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | Every kernel fork `s` is isomorphic (actually, equal) to `fork.of_ι (fork.ι s) _`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_ι_congr {P : C} {ι ι' : P ⟶ X} {w : ι ≫ f = 0} (h : ι = ι') :
kernel_fork.of_ι ι w ≅ kernel_fork.of_ι ι' (by rw [←h, w]) | cones.ext (iso.refl _) $ by tidy | def | category_theory.limits.of_ι_congr | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | If `ι = ι'`, then `fork.of_ι ι _` and `fork.of_ι ι' _` are isomorphic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_nat_iso {D : Type u'} [category.{v} D] [has_zero_morphisms D] (F : C ⥤ D)
[is_equivalence F] : parallel_pair f 0 ⋙ F ≅ parallel_pair (F.map f) 0 | nat_iso.of_components (λ j, match j with
| zero := iso.refl _
| one := iso.refl _
end) $ by tidy | def | category_theory.limits.comp_nat_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | If `F` is an equivalence, then applying `F` to a diagram indexing a (co)kernel of `f` yields
the diagram indexing the (co)kernel of `F.map f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel_fork.is_limit.lift' {s : kernel_fork f} (hs : is_limit s) {W : C} (k : W ⟶ X)
(h : k ≫ f = 0) : {l : W ⟶ s.X // l ≫ fork.ι s = k} | ⟨hs.lift $ kernel_fork.of_ι _ h, hs.fac _ _⟩ | def | category_theory.limits.kernel_fork.is_limit.lift' | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | If `s` is a limit kernel fork and `k : W ⟶ X` satisfies ``k ≫ f = 0`, then there is some
`l : W ⟶ s.X` such that `l ≫ fork.ι s = k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_aux (t : kernel_fork f)
(lift : Π (s : kernel_fork f), s.X ⟶ t.X)
(fac : ∀ (s : kernel_fork f), lift s ≫ t.ι = s.ι)
(uniq : ∀ (s : kernel_fork f) (m : s.X ⟶ t.X) (w : m ≫ t.ι = s.ι), m = lift s) :
is_limit t | { lift := lift,
fac' := λ s j, by { cases j, { exact fac s, }, { simp, }, },
uniq' := λ s m w, uniq s m (w limits.walking_parallel_pair.zero), } | def | category_theory.limits.is_limit_aux | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [
"lift"
] | This is a slightly more convenient method to verify that a kernel fork is a limit cone. It
only asks for a proof of facts that carry any mathematical content | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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