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map_biproduct_hom : (map_biproduct F f).hom = biproduct.lift (λ j, F.map (biproduct.π f j)) | rfl | lemma | category_theory.functor.map_biproduct_hom | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_biproduct_inv : (map_biproduct F f).inv = biproduct.desc (λ j, F.map (biproduct.ι f j)) | rfl | lemma | category_theory.functor.map_biproduct_inv | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biprod_comparison : F.obj (X ⊞ Y) ⟶ F.obj X ⊞ F.obj Y | biprod.lift (F.map biprod.fst) (F.map biprod.snd) | def | category_theory.functor.biprod_comparison | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | As for products, any functor between categories with binary biproducts gives rise to a
morphism `F.obj (X ⊞ Y) ⟶ F.obj X ⊞ F.obj Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biprod_comparison_fst :
biprod_comparison F X Y ≫ biprod.fst = F.map biprod.fst | biprod.lift_fst _ _ | lemma | category_theory.functor.biprod_comparison_fst | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biprod_comparison_snd :
biprod_comparison F X Y ≫ biprod.snd = F.map biprod.snd | biprod.lift_snd _ _ | lemma | category_theory.functor.biprod_comparison_snd | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biprod_comparison' : F.obj X ⊞ F.obj Y ⟶ F.obj (X ⊞ Y) | biprod.desc (F.map biprod.inl) (F.map biprod.inr) | def | category_theory.functor.biprod_comparison' | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | As for coproducts, any functor between categories with binary biproducts gives rise to a
morphism `F.obj X ⊞ F.obj Y ⟶ F.obj (X ⊞ Y)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inl_biprod_comparison' :
biprod.inl ≫ biprod_comparison' F X Y = F.map biprod.inl | biprod.inl_desc _ _ | lemma | category_theory.functor.inl_biprod_comparison' | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inr_biprod_comparison' :
biprod.inr ≫ biprod_comparison' F X Y = F.map biprod.inr | biprod.inr_desc _ _ | lemma | category_theory.functor.inr_biprod_comparison' | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biprod_comparison'_comp_biprod_comparison :
biprod_comparison' F X Y ≫ biprod_comparison F X Y = 𝟙 (F.obj X ⊞ F.obj Y) | by { ext; simp [← functor.map_comp] } | lemma | category_theory.functor.biprod_comparison'_comp_biprod_comparison | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | The composition in the opposite direction is equal to the identity if and only if `F` preserves
the biproduct, see `preserves_binary_biproduct_of_mono_biprod_comparison`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
split_epi_biprod_comparison : split_epi (biprod_comparison F X Y) | ⟨biprod_comparison' F X Y⟩ | def | category_theory.functor.split_epi_biprod_comparison | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | `biprod_comparison F X Y` is a split epi. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
split_mono_biprod_comparison' : split_mono (biprod_comparison' F X Y) | ⟨biprod_comparison F X Y⟩ | def | category_theory.functor.split_mono_biprod_comparison' | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | `biprod_comparison' F X Y` is a split mono. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_binary_biproduct_of_preserves : has_binary_biproduct (F.obj X) (F.obj Y) | has_binary_biproduct.mk
{ bicone := F.map_binary_bicone (binary_biproduct.bicone X Y),
is_bilimit := preserves_binary_biproduct.preserves (binary_biproduct.is_bilimit _ _) } | instance | category_theory.functor.has_binary_biproduct_of_preserves | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_biprod : F.obj (X ⊞ Y) ≅ F.obj X ⊞ F.obj Y | biprod.unique_up_to_iso _ _
(preserves_binary_biproduct.preserves (binary_biproduct.is_bilimit _ _)) | def | category_theory.functor.map_biprod | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | If `F` preserves a binary biproduct, we get a definitionally nice isomorphism
`F.obj (X ⊞ Y) ≅ F.obj X ⊞ F.obj Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_biprod_hom : (map_biprod F X Y).hom = biprod.lift (F.map biprod.fst) (F.map biprod.snd) | rfl | lemma | category_theory.functor.map_biprod_hom | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_biprod_inv : (map_biprod F X Y).inv = biprod.desc (F.map biprod.inl) (F.map biprod.inr) | rfl | lemma | category_theory.functor.map_biprod_inv | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.map_lift_map_biprod (g : Π j, W ⟶ f j) :
F.map (biproduct.lift g) ≫ (F.map_biproduct f).hom = biproduct.lift (λ j, F.map (g j)) | by { ext, simp [← F.map_comp] } | lemma | category_theory.limits.biproduct.map_lift_map_biprod | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.map_biproduct_inv_map_desc (g : Π j, f j ⟶ W) :
(F.map_biproduct f).inv ≫ F.map (biproduct.desc g) = biproduct.desc (λ j, F.map (g j)) | by { ext, simp [← F.map_comp] } | lemma | category_theory.limits.biproduct.map_biproduct_inv_map_desc | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.map_biproduct_hom_desc (g : Π j, f j ⟶ W) :
(F.map_biproduct f).hom ≫ biproduct.desc (λ j, F.map (g j)) = F.map (biproduct.desc g) | by rw [← biproduct.map_biproduct_inv_map_desc, iso.hom_inv_id_assoc] | lemma | category_theory.limits.biproduct.map_biproduct_hom_desc | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biprod.map_lift_map_biprod (f : W ⟶ X) (g : W ⟶ Y) :
F.map (biprod.lift f g) ≫ (F.map_biprod X Y).hom = biprod.lift (F.map f) (F.map g) | by ext; simp [← F.map_comp] | lemma | category_theory.limits.biprod.map_lift_map_biprod | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biprod.lift_map_biprod (f : W ⟶ X) (g : W ⟶ Y) :
biprod.lift (F.map f) (F.map g) ≫ (F.map_biprod X Y).inv = F.map (biprod.lift f g) | by rw [← biprod.map_lift_map_biprod, category.assoc, iso.hom_inv_id, category.comp_id] | lemma | category_theory.limits.biprod.lift_map_biprod | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biprod.map_biprod_inv_map_desc (f : X ⟶ W) (g : Y ⟶ W) :
(F.map_biprod X Y).inv ≫ F.map (biprod.desc f g) = biprod.desc (F.map f) (F.map g) | by ext; simp [← F.map_comp] | lemma | category_theory.limits.biprod.map_biprod_inv_map_desc | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biprod.map_biprod_hom_desc (f : X ⟶ W) (g : Y ⟶ W) :
(F.map_biprod X Y).hom ≫ biprod.desc (F.map f) (F.map g) = F.map (biprod.desc f g) | by rw [← biprod.map_biprod_inv_map_desc, iso.hom_inv_id_assoc] | lemma | category_theory.limits.biprod.map_biprod_hom_desc | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_limit_map_cone_fork_equiv :
is_limit (G.map_cone (fork.of_ι h w)) ≃
is_limit (fork.of_ι (G.map h) (by simp only [←G.map_comp, w]) : fork (G.map f) (G.map g)) | (is_limit.postcompose_hom_equiv (diagram_iso_parallel_pair _) _).symm.trans
(is_limit.equiv_iso_limit (fork.ext (iso.refl _) (by { simp [fork.ι] }))) | def | category_theory.limits.is_limit_map_cone_fork_equiv | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/equalizers.lean | [
"category_theory.limits.shapes.split_coequalizer",
"category_theory.limits.preserves.basic"
] | [] | The map of a fork is a limit iff the fork consisting of the mapped morphisms is a limit. This
essentially lets us commute `fork.of_ι` with `functor.map_cone`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_fork_map_of_is_limit [preserves_limit (parallel_pair f g) G]
(l : is_limit (fork.of_ι h w)) :
is_limit (fork.of_ι (G.map h) (by simp only [←G.map_comp, w]) : fork (G.map f) (G.map g)) | is_limit_map_cone_fork_equiv G w (preserves_limit.preserves l) | def | category_theory.limits.is_limit_fork_map_of_is_limit | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/equalizers.lean | [
"category_theory.limits.shapes.split_coequalizer",
"category_theory.limits.preserves.basic"
] | [] | The property of preserving equalizers expressed in terms of forks. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_of_is_limit_fork_map [reflects_limit (parallel_pair f g) G]
(l : is_limit (fork.of_ι (G.map h) (by simp only [←G.map_comp, w]) : fork (G.map f) (G.map g))) :
is_limit (fork.of_ι h w) | reflects_limit.reflects ((is_limit_map_cone_fork_equiv G w).symm l) | def | category_theory.limits.is_limit_of_is_limit_fork_map | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/equalizers.lean | [
"category_theory.limits.shapes.split_coequalizer",
"category_theory.limits.preserves.basic"
] | [] | The property of reflecting equalizers expressed in terms of forks. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_of_has_equalizer_of_preserves_limit
[preserves_limit (parallel_pair f g) G] :
is_limit (fork.of_ι (G.map (equalizer.ι f g))
(by simp only [←G.map_comp, equalizer.condition])) | is_limit_fork_map_of_is_limit G _ (equalizer_is_equalizer f g) | def | category_theory.limits.is_limit_of_has_equalizer_of_preserves_limit | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/equalizers.lean | [
"category_theory.limits.shapes.split_coequalizer",
"category_theory.limits.preserves.basic"
] | [] | If `G` preserves equalizers and `C` has them, then the fork constructed of the mapped morphisms of
a fork is a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_equalizer.of_iso_comparison [i : is_iso (equalizer_comparison f g G)] :
preserves_limit (parallel_pair f g) G | begin
apply preserves_limit_of_preserves_limit_cone (equalizer_is_equalizer f g),
apply (is_limit_map_cone_fork_equiv _ _).symm _,
apply is_limit.of_point_iso (limit.is_limit (parallel_pair (G.map f) (G.map g))),
apply i,
end | def | category_theory.limits.preserves_equalizer.of_iso_comparison | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/equalizers.lean | [
"category_theory.limits.shapes.split_coequalizer",
"category_theory.limits.preserves.basic"
] | [] | If the equalizer comparison map for `G` at `(f,g)` is an isomorphism, then `G` preserves the
equalizer of `(f,g)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_equalizer.iso :
G.obj (equalizer f g) ≅ equalizer (G.map f) (G.map g) | is_limit.cone_point_unique_up_to_iso
(is_limit_of_has_equalizer_of_preserves_limit G f g)
(limit.is_limit _) | def | category_theory.limits.preserves_equalizer.iso | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/equalizers.lean | [
"category_theory.limits.shapes.split_coequalizer",
"category_theory.limits.preserves.basic"
] | [] | If `G` preserves the equalizer of `(f,g)`, then the equalizer comparison map for `G` at `(f,g)` is
an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_equalizer.iso_hom :
(preserves_equalizer.iso G f g).hom = equalizer_comparison f g G | rfl | lemma | category_theory.limits.preserves_equalizer.iso_hom | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/equalizers.lean | [
"category_theory.limits.shapes.split_coequalizer",
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_colimit_map_cocone_cofork_equiv :
is_colimit (G.map_cocone (cofork.of_π h w)) ≃
is_colimit (cofork.of_π (G.map h) (by simp only [←G.map_comp, w]) : cofork (G.map f) (G.map g)) | (is_colimit.precompose_inv_equiv (diagram_iso_parallel_pair _) _).symm.trans $
is_colimit.equiv_iso_colimit $ cofork.ext (iso.refl _) $
begin
dsimp only [cofork.π, cofork.of_π_ι_app],
dsimp, rw [category.comp_id, category.id_comp]
end | def | category_theory.limits.is_colimit_map_cocone_cofork_equiv | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/equalizers.lean | [
"category_theory.limits.shapes.split_coequalizer",
"category_theory.limits.preserves.basic"
] | [] | The map of a cofork is a colimit iff the cofork consisting of the mapped morphisms is a colimit.
This essentially lets us commute `cofork.of_π` with `functor.map_cocone`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_cofork_map_of_is_colimit [preserves_colimit (parallel_pair f g) G]
(l : is_colimit (cofork.of_π h w)) :
is_colimit (cofork.of_π (G.map h) (by simp only [←G.map_comp, w]) : cofork (G.map f) (G.map g)) | is_colimit_map_cocone_cofork_equiv G w (preserves_colimit.preserves l) | def | category_theory.limits.is_colimit_cofork_map_of_is_colimit | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/equalizers.lean | [
"category_theory.limits.shapes.split_coequalizer",
"category_theory.limits.preserves.basic"
] | [] | The property of preserving coequalizers expressed in terms of coforks. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_of_is_colimit_cofork_map [reflects_colimit (parallel_pair f g) G]
(l : is_colimit (cofork.of_π (G.map h) (by simp only [←G.map_comp, w])
: cofork (G.map f) (G.map g))) :
is_colimit (cofork.of_π h w) | reflects_colimit.reflects ((is_colimit_map_cocone_cofork_equiv G w).symm l) | def | category_theory.limits.is_colimit_of_is_colimit_cofork_map | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/equalizers.lean | [
"category_theory.limits.shapes.split_coequalizer",
"category_theory.limits.preserves.basic"
] | [] | The property of reflecting coequalizers expressed in terms of coforks. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_of_has_coequalizer_of_preserves_colimit
[preserves_colimit (parallel_pair f g) G] :
is_colimit (cofork.of_π (G.map (coequalizer.π f g)) _) | is_colimit_cofork_map_of_is_colimit G _ (coequalizer_is_coequalizer f g) | def | category_theory.limits.is_colimit_of_has_coequalizer_of_preserves_colimit | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/equalizers.lean | [
"category_theory.limits.shapes.split_coequalizer",
"category_theory.limits.preserves.basic"
] | [] | If `G` preserves coequalizers and `C` has them, then the cofork constructed of the mapped morphisms
of a cofork is a colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_iso_comparison [i : is_iso (coequalizer_comparison f g G)] :
preserves_colimit (parallel_pair f g) G | begin
apply preserves_colimit_of_preserves_colimit_cocone (coequalizer_is_coequalizer f g),
apply (is_colimit_map_cocone_cofork_equiv _ _).symm _,
apply is_colimit.of_point_iso (colimit.is_colimit (parallel_pair (G.map f) (G.map g))),
apply i,
end | def | category_theory.limits.of_iso_comparison | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/equalizers.lean | [
"category_theory.limits.shapes.split_coequalizer",
"category_theory.limits.preserves.basic"
] | [] | If the coequalizer comparison map for `G` at `(f,g)` is an isomorphism, then `G` preserves the
coequalizer of `(f,g)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_coequalizer.iso :
coequalizer (G.map f) (G.map g) ≅ G.obj (coequalizer f g) | is_colimit.cocone_point_unique_up_to_iso
(colimit.is_colimit _)
(is_colimit_of_has_coequalizer_of_preserves_colimit G f g) | def | category_theory.limits.preserves_coequalizer.iso | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/equalizers.lean | [
"category_theory.limits.shapes.split_coequalizer",
"category_theory.limits.preserves.basic"
] | [] | If `G` preserves the coequalizer of `(f,g)`, then the coequalizer comparison map for `G` at `(f,g)`
is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_coequalizer.iso_hom :
(preserves_coequalizer.iso G f g).hom = coequalizer_comparison f g G | rfl | lemma | category_theory.limits.preserves_coequalizer.iso_hom | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/equalizers.lean | [
"category_theory.limits.shapes.split_coequalizer",
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_π_epi : epi (G.map (coequalizer.π f g)) | ⟨λ W h k, by { rw ←ι_comp_coequalizer_comparison, apply (cancel_epi _).1, apply epi_comp }⟩ | instance | category_theory.limits.map_π_epi | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/equalizers.lean | [
"category_theory.limits.shapes.split_coequalizer",
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_π_preserves_coequalizer_inv :
G.map (coequalizer.π f g) ≫ (preserves_coequalizer.iso G f g).inv =
coequalizer.π (G.map f) (G.map g) | begin
rw [←ι_comp_coequalizer_comparison_assoc, ←preserves_coequalizer.iso_hom,
iso.hom_inv_id, comp_id],
end | lemma | category_theory.limits.map_π_preserves_coequalizer_inv | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/equalizers.lean | [
"category_theory.limits.shapes.split_coequalizer",
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_π_preserves_coequalizer_inv_desc
{W : D} (k : G.obj Y ⟶ W) (wk : G.map f ≫ k = G.map g ≫ k) :
G.map (coequalizer.π f g) ≫ (preserves_coequalizer.iso G f g).inv ≫ coequalizer.desc k wk = k | by rw [←category.assoc, map_π_preserves_coequalizer_inv, coequalizer.π_desc] | lemma | category_theory.limits.map_π_preserves_coequalizer_inv_desc | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/equalizers.lean | [
"category_theory.limits.shapes.split_coequalizer",
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_π_preserves_coequalizer_inv_colim_map
{X' Y' : D} (f' g' : X' ⟶ Y') [has_coequalizer f' g'] (p : G.obj X ⟶ X') (q : G.obj Y ⟶ Y')
(wf : (G.map f) ≫ q = p ≫ f') (wg : (G.map g) ≫ q = p ≫ g') :
G.map (coequalizer.π f g) ≫ (preserves_coequalizer.iso G f g).inv ≫
colim_map (parallel_pair_hom (G.map f) (G.map g)... | by rw [←category.assoc, map_π_preserves_coequalizer_inv, ι_colim_map, parallel_pair_hom_app_one] | lemma | category_theory.limits.map_π_preserves_coequalizer_inv_colim_map | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/equalizers.lean | [
"category_theory.limits.shapes.split_coequalizer",
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_π_preserves_coequalizer_inv_colim_map_desc
{X' Y' : D} (f' g' : X' ⟶ Y') [has_coequalizer f' g'] (p : G.obj X ⟶ X') (q : G.obj Y ⟶ Y')
(wf : (G.map f) ≫ q = p ≫ f') (wg : (G.map g) ≫ q = p ≫ g')
{Z' : D} (h : Y' ⟶ Z') (wh : f' ≫ h = g' ≫ h) :
G.map (coequalizer.π f g) ≫ (preserves_coequalizer.iso G f g).inv... | begin
slice_lhs 1 3 { rw map_π_preserves_coequalizer_inv_colim_map },
slice_lhs 2 3 { rw coequalizer.π_desc },
end | lemma | category_theory.limits.map_π_preserves_coequalizer_inv_colim_map_desc | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/equalizers.lean | [
"category_theory.limits.shapes.split_coequalizer",
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_split_coequalizers (f g : X ⟶ Y) [has_split_coequalizer f g] :
preserves_colimit (parallel_pair f g) G | begin
apply preserves_colimit_of_preserves_colimit_cocone
((has_split_coequalizer.is_split_coequalizer f g).is_coequalizer),
apply (is_colimit_map_cocone_cofork_equiv G _).symm
((has_split_coequalizer.is_split_coequalizer f g).map G).is_coequalizer,
end | instance | category_theory.limits.preserves_split_coequalizers | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/equalizers.lean | [
"category_theory.limits.shapes.split_coequalizer",
"category_theory.limits.preserves.basic"
] | [] | Any functor preserves coequalizers of split pairs. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso {X Y : A} (f : X ⟶ Y) : image (L.map f) ≅ L.obj (image f) | let aux1 : strong_epi_mono_factorisation (L.map f) :=
{ I := L.obj (limits.image f),
m := L.map $ limits.image.ι _,
m_mono := preserves_mono_of_preserves_limit _ _,
e := L.map $ factor_thru_image _,
e_strong_epi := @@strong_epi_of_epi _ _ _ $ preserves_epi_of_preserves_colimit L _,
fac' := by rw [←L.map_comp,... | def | category_theory.preserves_image.iso | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/images.lean | [
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.epi_mono"
] | [
"iso"
] | If a functor preserves span and cospan, then it preserves images. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
factor_thru_image_comp_hom {X Y : A} (f : X ⟶ Y) :
factor_thru_image (L.map f) ≫ (iso L f).hom =
L.map (factor_thru_image f) | by simp | lemma | category_theory.preserves_image.factor_thru_image_comp_hom | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/images.lean | [
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.epi_mono"
] | [
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_comp_map_image_ι {X Y : A} (f : X ⟶ Y) :
(iso L f).hom ≫ L.map (image.ι f) = image.ι (L.map f) | by simp | lemma | category_theory.preserves_image.hom_comp_map_image_ι | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/images.lean | [
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.epi_mono"
] | [
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_comp_image_ι_map {X Y : A} (f : X ⟶ Y) :
(iso L f).inv ≫ image.ι (L.map f) = L.map (image.ι f) | by simp | lemma | category_theory.preserves_image.inv_comp_image_ι_map | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/images.lean | [
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.epi_mono"
] | [
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_limit_map_cone_fork_equiv' :
is_limit (G.map_cone (kernel_fork.of_ι h w)) ≃
is_limit (kernel_fork.of_ι (G.map h) (by simp only [←G.map_comp, w, functor.map_zero])
: fork (G.map f) 0) | begin
refine (is_limit.postcompose_hom_equiv _ _).symm.trans (is_limit.equiv_iso_limit _),
refine parallel_pair.ext (iso.refl _) (iso.refl _) _ _; simp,
refine fork.ext (iso.refl _) _,
simp [fork.ι]
end | def | category_theory.limits.is_limit_map_cone_fork_equiv' | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/kernels.lean | [
"category_theory.limits.shapes.kernels",
"category_theory.limits.preserves.shapes.zero"
] | [] | The map of a kernel fork is a limit iff
the kernel fork consisting of the mapped morphisms is a limit.
This essentially lets us commute `kernel_fork.of_ι` with `functor.map_cone`.
This is a variant of `is_limit_map_cone_fork_equiv` for equalizers,
which we can't use directly between `G.map 0 = 0` does not hold definit... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_fork_map_of_is_limit' [preserves_limit (parallel_pair f 0) G]
(l : is_limit (kernel_fork.of_ι h w)) :
is_limit (kernel_fork.of_ι (G.map h) (by simp only [←G.map_comp, w, functor.map_zero]) :
fork (G.map f) 0) | is_limit_map_cone_fork_equiv' G w (preserves_limit.preserves l) | def | category_theory.limits.is_limit_fork_map_of_is_limit' | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/kernels.lean | [
"category_theory.limits.shapes.kernels",
"category_theory.limits.preserves.shapes.zero"
] | [] | The property of preserving kernels expressed in terms of kernel forks.
This is a variant of `is_limit_fork_map_of_is_limit` for equalizers,
which we can't use directly between `G.map 0 = 0` does not hold definitionally. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_of_has_kernel_of_preserves_limit [preserves_limit (parallel_pair f 0) G] :
is_limit (fork.of_ι (G.map (kernel.ι f))
(by simp only [←G.map_comp, equalizer.condition, comp_zero, functor.map_zero])
: fork (G.map f) 0) | is_limit_fork_map_of_is_limit' G (kernel.condition f) (kernel_is_kernel f) | def | category_theory.limits.is_limit_of_has_kernel_of_preserves_limit | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/kernels.lean | [
"category_theory.limits.shapes.kernels",
"category_theory.limits.preserves.shapes.zero"
] | [] | If `G` preserves kernels and `C` has them, then the fork constructed of the mapped morphisms of
a kernel fork is a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_kernel.of_iso_comparison [i : is_iso (kernel_comparison f G)] :
preserves_limit (parallel_pair f 0) G | begin
apply preserves_limit_of_preserves_limit_cone (kernel_is_kernel f),
apply (is_limit_map_cone_fork_equiv' G (kernel.condition f)).symm _,
apply is_limit.of_point_iso (kernel_is_kernel (G.map f)),
exact i,
end | def | category_theory.limits.preserves_kernel.of_iso_comparison | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/kernels.lean | [
"category_theory.limits.shapes.kernels",
"category_theory.limits.preserves.shapes.zero"
] | [] | If the kernel comparison map for `G` at `f` is an isomorphism, then `G` preserves the
kernel of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_kernel.iso :
G.obj (kernel f) ≅ kernel (G.map f) | is_limit.cone_point_unique_up_to_iso
(is_limit_of_has_kernel_of_preserves_limit G f)
(kernel_is_kernel _) | def | category_theory.limits.preserves_kernel.iso | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/kernels.lean | [
"category_theory.limits.shapes.kernels",
"category_theory.limits.preserves.shapes.zero"
] | [] | If `G` preserves the kernel of `f`, then the kernel comparison map for `G` at `f` is
an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_kernel.iso_hom :
(preserves_kernel.iso G f).hom = kernel_comparison f G | rfl | lemma | category_theory.limits.preserves_kernel.iso_hom | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/kernels.lean | [
"category_theory.limits.shapes.kernels",
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kernel_map_comp_preserves_kernel_iso_inv {X' Y' : C} (g : X' ⟶ Y') [has_kernel g]
[has_kernel (G.map g)] [preserves_limit (parallel_pair g 0) G] (p : X ⟶ X') (q : Y ⟶ Y')
(hpq : f ≫ q = p ≫ g) :
kernel.map (G.map f) (G.map g) (G.map p) (G.map q)
(by rw [←G.map_comp, hpq, G.map_comp]) ≫ (preserves_kernel.iso G... | begin
rw [iso.comp_inv_eq, category.assoc, preserves_kernel.iso_hom, iso.eq_inv_comp],
exact kernel_comparison_comp_kernel_map _ _ _ _ _ _,
end | lemma | category_theory.limits.kernel_map_comp_preserves_kernel_iso_inv | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/kernels.lean | [
"category_theory.limits.shapes.kernels",
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_colimit_map_cocone_cofork_equiv' :
is_colimit (G.map_cocone (cokernel_cofork.of_π h w)) ≃
is_colimit (cokernel_cofork.of_π (G.map h) (by simp only [←G.map_comp, w, functor.map_zero])
: cofork (G.map f) 0) | begin
refine (is_colimit.precompose_hom_equiv _ _).symm.trans (is_colimit.equiv_iso_colimit _),
refine parallel_pair.ext (iso.refl _) (iso.refl _) _ _; simp,
refine cofork.ext (iso.refl _) _,
simp only [cofork.π, iso.refl_hom, id_comp, cocones.precompose_obj_ι,
nat_trans.comp_app, parallel_pair.ext_hom_app,... | def | category_theory.limits.is_colimit_map_cocone_cofork_equiv' | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/kernels.lean | [
"category_theory.limits.shapes.kernels",
"category_theory.limits.preserves.shapes.zero"
] | [] | The map of a cokernel cofork is a colimit iff
the cokernel cofork consisting of the mapped morphisms is a colimit.
This essentially lets us commute `cokernel_cofork.of_π` with `functor.map_cocone`.
This is a variant of `is_colimit_map_cocone_cofork_equiv` for equalizers,
which we can't use directly between `G.map 0 = ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_cofork_map_of_is_colimit' [preserves_colimit (parallel_pair f 0) G]
(l : is_colimit (cokernel_cofork.of_π h w)) :
is_colimit (cokernel_cofork.of_π (G.map h) (by simp only [←G.map_comp, w, functor.map_zero]) :
cofork (G.map f) 0) | is_colimit_map_cocone_cofork_equiv' G w (preserves_colimit.preserves l) | def | category_theory.limits.is_colimit_cofork_map_of_is_colimit' | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/kernels.lean | [
"category_theory.limits.shapes.kernels",
"category_theory.limits.preserves.shapes.zero"
] | [] | The property of preserving cokernels expressed in terms of cokernel coforks.
This is a variant of `is_colimit_cofork_map_of_is_colimit` for equalizers,
which we can't use directly between `G.map 0 = 0` does not hold definitionally. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_of_has_cokernel_of_preserves_colimit [preserves_colimit (parallel_pair f 0) G] :
is_colimit (cofork.of_π (G.map (cokernel.π f))
(by simp only [←G.map_comp, coequalizer.condition, zero_comp, functor.map_zero])
: cofork (G.map f) 0) | is_colimit_cofork_map_of_is_colimit' G (cokernel.condition f) (cokernel_is_cokernel f) | def | category_theory.limits.is_colimit_of_has_cokernel_of_preserves_colimit | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/kernels.lean | [
"category_theory.limits.shapes.kernels",
"category_theory.limits.preserves.shapes.zero"
] | [] | If `G` preserves cokernels and `C` has them, then the cofork constructed of the mapped morphisms of
a cokernel cofork is a colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_cokernel.of_iso_comparison [i : is_iso (cokernel_comparison f G)] :
preserves_colimit (parallel_pair f 0) G | begin
apply preserves_colimit_of_preserves_colimit_cocone (cokernel_is_cokernel f),
apply (is_colimit_map_cocone_cofork_equiv' G (cokernel.condition f)).symm _,
apply is_colimit.of_point_iso (cokernel_is_cokernel (G.map f)),
exact i,
end | def | category_theory.limits.preserves_cokernel.of_iso_comparison | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/kernels.lean | [
"category_theory.limits.shapes.kernels",
"category_theory.limits.preserves.shapes.zero"
] | [] | If the cokernel comparison map for `G` at `f` is an isomorphism, then `G` preserves the
cokernel of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_cokernel.iso :
G.obj (cokernel f) ≅ cokernel (G.map f) | is_colimit.cocone_point_unique_up_to_iso
(is_colimit_of_has_cokernel_of_preserves_colimit G f)
(cokernel_is_cokernel _) | def | category_theory.limits.preserves_cokernel.iso | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/kernels.lean | [
"category_theory.limits.shapes.kernels",
"category_theory.limits.preserves.shapes.zero"
] | [] | If `G` preserves the cokernel of `f`, then the cokernel comparison map for `G` at `f` is
an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_cokernel.iso_inv :
(preserves_cokernel.iso G f).inv = cokernel_comparison f G | rfl | lemma | category_theory.limits.preserves_cokernel.iso_inv | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/kernels.lean | [
"category_theory.limits.shapes.kernels",
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_cokernel_iso_comp_cokernel_map {X' Y' : C} (g : X' ⟶ Y')
[has_cokernel g] [has_cokernel (G.map g)] [preserves_colimit (parallel_pair g 0) G]
(p : X ⟶ X') (q : Y ⟶ Y') (hpq : f ≫ q = p ≫ g) :
(preserves_cokernel.iso G _).hom ≫ cokernel.map (G.map f) (G.map g) (G.map p) (G.map q)
(by rw [←G.map_comp, ... | begin
rw [←iso.comp_inv_eq, category.assoc, ←iso.eq_inv_comp],
exact cokernel_map_comp_cokernel_comparison _ _ _ _ _ _,
end | lemma | category_theory.limits.preserves_cokernel_iso_comp_cokernel_map | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/kernels.lean | [
"category_theory.limits.shapes.kernels",
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_limit_map_cone_fan_mk_equiv {P : C} (g : Π j, P ⟶ f j) :
is_limit (G.map_cone (fan.mk P g)) ≃
is_limit (fan.mk _ (λ j, G.map (g j)) : fan (λ j, G.obj (f j))) | begin
refine (is_limit.postcompose_hom_equiv _ _).symm.trans (is_limit.equiv_iso_limit _),
refine discrete.nat_iso (λ j, iso.refl (G.obj (f j.as))),
refine cones.ext (iso.refl _) (λ j, by { discrete_cases, dsimp, simp }),
end | def | category_theory.limits.is_limit_map_cone_fan_mk_equiv | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/products.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.preserves.basic"
] | [] | The map of a fan is a limit iff the fan consisting of the mapped morphisms is a limit. This
essentially lets us commute `fan.mk` with `functor.map_cone`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_fan_mk_obj_of_is_limit [preserves_limit (discrete.functor f) G]
{P : C} (g : Π j, P ⟶ f j) (t : is_limit (fan.mk _ g)) :
is_limit (fan.mk (G.obj P) (λ j, G.map (g j)) : fan (λ j, G.obj (f j))) | is_limit_map_cone_fan_mk_equiv _ _ _ (preserves_limit.preserves t) | def | category_theory.limits.is_limit_fan_mk_obj_of_is_limit | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/products.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.preserves.basic"
] | [] | The property of preserving products expressed in terms of fans. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_of_is_limit_fan_mk_obj [reflects_limit (discrete.functor f) G]
{P : C} (g : Π j, P ⟶ f j) (t : is_limit (fan.mk _ (λ j, G.map (g j)) : fan (λ j, G.obj (f j)))) :
is_limit (fan.mk P g) | reflects_limit.reflects ((is_limit_map_cone_fan_mk_equiv _ _ _).symm t) | def | category_theory.limits.is_limit_of_is_limit_fan_mk_obj | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/products.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.preserves.basic"
] | [] | The property of reflecting products expressed in terms of fans. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_of_has_product_of_preserves_limit [preserves_limit (discrete.functor f) G] :
is_limit (fan.mk _ (λ (j : J), G.map (pi.π f j)) : fan (λ j, G.obj (f j))) | is_limit_fan_mk_obj_of_is_limit G f _ (product_is_product _) | def | category_theory.limits.is_limit_of_has_product_of_preserves_limit | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/products.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.preserves.basic"
] | [] | If `G` preserves products and `C` has them, then the fan constructed of the mapped projection of a
product is a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_product.of_iso_comparison [i : is_iso (pi_comparison G f)] :
preserves_limit (discrete.functor f) G | begin
apply preserves_limit_of_preserves_limit_cone (product_is_product f),
apply (is_limit_map_cone_fan_mk_equiv _ _ _).symm _,
apply is_limit.of_point_iso (limit.is_limit (discrete.functor (λ (j : J), G.obj (f j)))),
apply i,
end | def | category_theory.limits.preserves_product.of_iso_comparison | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/products.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.preserves.basic"
] | [] | If `pi_comparison G f` is an isomorphism, then `G` preserves the limit of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_product.iso : G.obj (∏ f) ≅ ∏ (λ j, G.obj (f j)) | is_limit.cone_point_unique_up_to_iso
(is_limit_of_has_product_of_preserves_limit G f)
(limit.is_limit _) | def | category_theory.limits.preserves_product.iso | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/products.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.preserves.basic"
] | [] | If `G` preserves limits, we have an isomorphism from the image of a product to the product of the
images. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_product.iso_hom : (preserves_product.iso G f).hom = pi_comparison G f | rfl | lemma | category_theory.limits.preserves_product.iso_hom | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/products.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_colimit_map_cocone_cofan_mk_equiv {P : C} (g : Π j, f j ⟶ P) :
is_colimit (G.map_cocone (cofan.mk P g)) ≃
is_colimit (cofan.mk _ (λ j, G.map (g j)) : cofan (λ j, G.obj (f j))) | begin
refine (is_colimit.precompose_hom_equiv _ _).symm.trans (is_colimit.equiv_iso_colimit _),
refine discrete.nat_iso (λ j, iso.refl (G.obj (f j.as))),
refine cocones.ext (iso.refl _) (λ j, by { discrete_cases, dsimp, simp }),
end | def | category_theory.limits.is_colimit_map_cocone_cofan_mk_equiv | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/products.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.preserves.basic"
] | [] | The map of a cofan is a colimit iff the cofan consisting of the mapped morphisms is a colimit.
This essentially lets us commute `cofan.mk` with `functor.map_cocone`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_cofan_mk_obj_of_is_colimit [preserves_colimit (discrete.functor f) G]
{P : C} (g : Π j, f j ⟶ P) (t : is_colimit (cofan.mk _ g)) :
is_colimit (cofan.mk (G.obj P) (λ j, G.map (g j)) : cofan (λ j, G.obj (f j))) | is_colimit_map_cocone_cofan_mk_equiv _ _ _ (preserves_colimit.preserves t) | def | category_theory.limits.is_colimit_cofan_mk_obj_of_is_colimit | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/products.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.preserves.basic"
] | [] | The property of preserving coproducts expressed in terms of cofans. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_of_is_colimit_cofan_mk_obj [reflects_colimit (discrete.functor f) G]
{P : C} (g : Π j, f j ⟶ P)
(t : is_colimit (cofan.mk _ (λ j, G.map (g j)) : cofan (λ j, G.obj (f j)))) :
is_colimit (cofan.mk P g) | reflects_colimit.reflects ((is_colimit_map_cocone_cofan_mk_equiv _ _ _).symm t) | def | category_theory.limits.is_colimit_of_is_colimit_cofan_mk_obj | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/products.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.preserves.basic"
] | [] | The property of reflecting coproducts expressed in terms of cofans. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_of_has_coproduct_of_preserves_colimit [preserves_colimit (discrete.functor f) G] :
is_colimit (cofan.mk _ (λ (j : J), G.map (sigma.ι f j)) : cofan (λ j, G.obj (f j))) | is_colimit_cofan_mk_obj_of_is_colimit G f _ (coproduct_is_coproduct _) | def | category_theory.limits.is_colimit_of_has_coproduct_of_preserves_colimit | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/products.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.preserves.basic"
] | [] | If `G` preserves coproducts and `C` has them,
then the cofan constructed of the mapped inclusion of a coproduct is a colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_coproduct.of_iso_comparison [i : is_iso (sigma_comparison G f)] :
preserves_colimit (discrete.functor f) G | begin
apply preserves_colimit_of_preserves_colimit_cocone (coproduct_is_coproduct f),
apply (is_colimit_map_cocone_cofan_mk_equiv _ _ _).symm _,
apply is_colimit.of_point_iso (colimit.is_colimit (discrete.functor (λ (j : J), G.obj (f j)))),
apply i,
end | def | category_theory.limits.preserves_coproduct.of_iso_comparison | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/products.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.preserves.basic"
] | [] | If `sigma_comparison G f` is an isomorphism, then `G` preserves the colimit of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_coproduct.iso : G.obj (∐ f) ≅ ∐ (λ j, G.obj (f j)) | is_colimit.cocone_point_unique_up_to_iso
(is_colimit_of_has_coproduct_of_preserves_colimit G f)
(colimit.is_colimit _) | def | category_theory.limits.preserves_coproduct.iso | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/products.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.preserves.basic"
] | [] | If `G` preserves colimits,
we have an isomorphism from the image of a coproduct to the coproduct of the images. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_coproduct.inv_hom : (preserves_coproduct.iso G f).inv = sigma_comparison G f | rfl | lemma | category_theory.limits.preserves_coproduct.inv_hom | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/products.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_limit_map_cone_pullback_cone_equiv :
is_limit (G.map_cone (pullback_cone.mk h k comm)) ≃
is_limit (pullback_cone.mk (G.map h) (G.map k) (by simp only [← G.map_comp, comm])
: pullback_cone (G.map f) (G.map g)) | (is_limit.postcompose_hom_equiv (diagram_iso_cospan.{v₂} _) _).symm.trans $
is_limit.equiv_iso_limit $ cones.ext (iso.refl _) $
(by rintro (_|_|_); dsimp; simp only [comp_id, id_comp, G.map_comp]) | def | category_theory.limits.is_limit_map_cone_pullback_cone_equiv | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/pullbacks.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.basic"
] | [
"comm"
] | The map of a pullback cone is a limit iff the fork consisting of the mapped morphisms is a
limit. This essentially lets us commute `pullback_cone.mk` with `functor.map_cone`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_pullback_cone_map_of_is_limit [preserves_limit (cospan f g) G]
(l : is_limit (pullback_cone.mk h k comm)) :
is_limit (pullback_cone.mk (G.map h) (G.map k) _) | is_limit_map_cone_pullback_cone_equiv G comm (preserves_limit.preserves l) | def | category_theory.limits.is_limit_pullback_cone_map_of_is_limit | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/pullbacks.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.basic"
] | [
"comm"
] | The property of preserving pullbacks expressed in terms of binary fans. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_of_is_limit_pullback_cone_map [reflects_limit (cospan f g) G]
(l : is_limit (pullback_cone.mk (G.map h) (G.map k) _)) :
is_limit (pullback_cone.mk h k comm) | reflects_limit.reflects ((is_limit_map_cone_pullback_cone_equiv G comm).symm l) | def | category_theory.limits.is_limit_of_is_limit_pullback_cone_map | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/pullbacks.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.basic"
] | [
"comm"
] | The property of reflecting pullbacks expressed in terms of binary fans. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_of_has_pullback_of_preserves_limit [has_pullback f g] :
is_limit (pullback_cone.mk (G.map pullback.fst) (G.map pullback.snd) _) | is_limit_pullback_cone_map_of_is_limit G _ (pullback_is_pullback f g) | def | category_theory.limits.is_limit_of_has_pullback_of_preserves_limit | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/pullbacks.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.basic"
] | [] | If `G` preserves pullbacks and `C` has them, then the pullback cone constructed of the mapped
morphisms of the pullback cone is a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_pullback_symmetry : preserves_limit (cospan g f) G | { preserves := λ c hc,
begin
apply (is_limit.postcompose_hom_equiv (diagram_iso_cospan.{v₂} _) _).to_fun,
apply is_limit.of_iso_limit _ (pullback_cone.iso_mk _).symm,
apply pullback_cone.flip_is_limit,
apply (is_limit_map_cone_pullback_cone_equiv _ _).to_fun,
{ apply_with preserves_limit.preserves... | def | category_theory.limits.preserves_pullback_symmetry | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/pullbacks.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.basic"
] | [
"inv_fun"
] | If `F` preserves the pullback of `f, g`, it also preserves the pullback of `g, f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_pullback_of_preserves_pullback [has_pullback f g] :
has_pullback (G.map f) (G.map g) | ⟨⟨⟨_, is_limit_pullback_cone_map_of_is_limit G _ (pullback_is_pullback _ _)⟩⟩⟩ | lemma | category_theory.limits.has_pullback_of_preserves_pullback | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/pullbacks.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_pullback.iso :
G.obj (pullback f g) ≅ pullback (G.map f) (G.map g) | is_limit.cone_point_unique_up_to_iso
(is_limit_of_has_pullback_of_preserves_limit G f g)
(limit.is_limit _) | def | category_theory.limits.preserves_pullback.iso | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/pullbacks.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.basic"
] | [] | If `G` preserves the pullback of `(f,g)`, then the pullback comparison map for `G` at `(f,g)` is
an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_pullback.iso_hom_fst :
(preserves_pullback.iso G f g).hom ≫ pullback.fst = G.map pullback.fst | by simp [preserves_pullback.iso] | lemma | category_theory.limits.preserves_pullback.iso_hom_fst | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/pullbacks.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_pullback.iso_hom_snd :
(preserves_pullback.iso G f g).hom ≫ pullback.snd = G.map pullback.snd | by simp [preserves_pullback.iso] | lemma | category_theory.limits.preserves_pullback.iso_hom_snd | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/pullbacks.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_pullback.iso_inv_fst :
(preserves_pullback.iso G f g).inv ≫ G.map pullback.fst = pullback.fst | by simp [preserves_pullback.iso, iso.inv_comp_eq] | lemma | category_theory.limits.preserves_pullback.iso_inv_fst | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/pullbacks.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_pullback.iso_inv_snd :
(preserves_pullback.iso G f g).inv ≫ G.map pullback.snd = pullback.snd | by simp [preserves_pullback.iso, iso.inv_comp_eq] | lemma | category_theory.limits.preserves_pullback.iso_inv_snd | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/pullbacks.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_colimit_map_cocone_pushout_cocone_equiv :
is_colimit (G.map_cocone (pushout_cocone.mk h k comm)) ≃
is_colimit (pushout_cocone.mk (G.map h) (G.map k) (by simp only [← G.map_comp, comm])
: pushout_cocone (G.map f) (G.map g)) | (is_colimit.precompose_hom_equiv (diagram_iso_span.{v₂} _).symm _).symm.trans $
is_colimit.equiv_iso_colimit $ cocones.ext (iso.refl _) $
(by rintro (_|_|_); dsimp; simp only [category.comp_id, category.id_comp, ← G.map_comp]) | def | category_theory.limits.is_colimit_map_cocone_pushout_cocone_equiv | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/pullbacks.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.basic"
] | [
"comm"
] | The map of a pushout cocone is a colimit iff the cofork consisting of the mapped morphisms is a
colimit. This essentially lets us commute `pushout_cocone.mk` with `functor.map_cocone`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_pushout_cocone_map_of_is_colimit [preserves_colimit (span f g) G]
(l : is_colimit (pushout_cocone.mk h k comm)) :
is_colimit (pushout_cocone.mk (G.map h) (G.map k) _) | is_colimit_map_cocone_pushout_cocone_equiv G comm (preserves_colimit.preserves l) | def | category_theory.limits.is_colimit_pushout_cocone_map_of_is_colimit | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/pullbacks.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.basic"
] | [
"comm"
] | The property of preserving pushouts expressed in terms of binary cofans. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_of_is_colimit_pushout_cocone_map [reflects_colimit (span f g) G]
(l : is_colimit (pushout_cocone.mk (G.map h) (G.map k) _)) :
is_colimit (pushout_cocone.mk h k comm) | reflects_colimit.reflects ((is_colimit_map_cocone_pushout_cocone_equiv G comm).symm l) | def | category_theory.limits.is_colimit_of_is_colimit_pushout_cocone_map | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/pullbacks.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.basic"
] | [
"comm"
] | The property of reflecting pushouts expressed in terms of binary cofans. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_of_has_pushout_of_preserves_colimit [has_pushout f g] :
is_colimit (pushout_cocone.mk (G.map pushout.inl) (G.map pushout.inr) _) | is_colimit_pushout_cocone_map_of_is_colimit G _ (pushout_is_pushout f g) | def | category_theory.limits.is_colimit_of_has_pushout_of_preserves_colimit | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/pullbacks.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.basic"
] | [] | If `G` preserves pushouts and `C` has them, then the pushout cocone constructed of the mapped
morphisms of the pushout cocone is a colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_pushout_symmetry : preserves_colimit (span g f) G | { preserves := λ c hc,
begin
apply (is_colimit.precompose_hom_equiv (diagram_iso_span.{v₂} _).symm _).to_fun,
apply is_colimit.of_iso_colimit _ (pushout_cocone.iso_mk _).symm,
apply pushout_cocone.flip_is_colimit,
apply (is_colimit_map_cocone_pushout_cocone_equiv _ _).to_fun,
{ apply_with preserve... | def | category_theory.limits.preserves_pushout_symmetry | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/pullbacks.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.basic"
] | [
"inv_fun"
] | If `F` preserves the pushout of `f, g`, it also preserves the pushout of `g, f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_pushout_of_preserves_pushout [has_pushout f g] :
has_pushout (G.map f) (G.map g) | ⟨⟨⟨_, is_colimit_pushout_cocone_map_of_is_colimit G _ (pushout_is_pushout _ _)⟩⟩⟩ | lemma | category_theory.limits.has_pushout_of_preserves_pushout | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/pullbacks.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_pushout.iso :
pushout (G.map f) (G.map g) ≅ G.obj (pushout f g) | is_colimit.cocone_point_unique_up_to_iso
(colimit.is_colimit _)
(is_colimit_of_has_pushout_of_preserves_colimit G f g) | def | category_theory.limits.preserves_pushout.iso | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/pullbacks.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.basic"
] | [] | If `G` preserves the pushout of `(f,g)`, then the pushout comparison map for `G` at `(f,g)` is
an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_pushout.inl_iso_hom :
pushout.inl ≫ (preserves_pushout.iso G f g).hom = G.map pushout.inl | by { delta preserves_pushout.iso, simp } | lemma | category_theory.limits.preserves_pushout.inl_iso_hom | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/pullbacks.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_pushout.inr_iso_hom :
pushout.inr ≫ (preserves_pushout.iso G f g).hom = G.map pushout.inr | by { delta preserves_pushout.iso, simp } | lemma | category_theory.limits.preserves_pushout.inr_iso_hom | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/pullbacks.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_pushout.inl_iso_inv :
G.map pushout.inl ≫ (preserves_pushout.iso G f g).inv = pushout.inl | by simp [preserves_pushout.iso, iso.comp_inv_eq] | lemma | category_theory.limits.preserves_pushout.inl_iso_inv | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/pullbacks.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_pushout.inr_iso_inv :
G.map pushout.inr ≫ (preserves_pushout.iso G f g).inv = pushout.inr | by simp [preserves_pushout.iso, iso.comp_inv_eq] | lemma | category_theory.limits.preserves_pushout.inr_iso_inv | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/pullbacks.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_pullback.of_iso_comparison [i : is_iso (pullback_comparison G f g)] :
preserves_limit (cospan f g) G | begin
apply preserves_limit_of_preserves_limit_cone (pullback_is_pullback f g),
apply (is_limit_map_cone_pullback_cone_equiv _ _).symm _,
apply is_limit.of_point_iso (limit.is_limit (cospan (G.map f) (G.map g))),
apply i,
end | def | category_theory.limits.preserves_pullback.of_iso_comparison | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/pullbacks.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.basic"
] | [] | If the pullback comparison map for `G` at `(f,g)` is an isomorphism, then `G` preserves the
pullback of `(f,g)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_pullback.iso_hom :
(preserves_pullback.iso G f g).hom = pullback_comparison G f g | rfl | lemma | category_theory.limits.preserves_pullback.iso_hom | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/pullbacks.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_pushout.of_iso_comparison [i : is_iso (pushout_comparison G f g)] :
preserves_colimit (span f g) G | begin
apply preserves_colimit_of_preserves_colimit_cocone (pushout_is_pushout f g),
apply (is_colimit_map_cocone_pushout_cocone_equiv _ _).symm _,
apply is_colimit.of_point_iso (colimit.is_colimit (span (G.map f) (G.map g))),
apply i,
end | def | category_theory.limits.preserves_pushout.of_iso_comparison | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/pullbacks.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.basic"
] | [] | If the pushout comparison map for `G` at `(f,g)` is an isomorphism, then `G` preserves the
pushout of `(f,g)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_pushout.iso_hom :
(preserves_pushout.iso G f g).hom = pushout_comparison G f g | rfl | lemma | category_theory.limits.preserves_pushout.iso_hom | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/pullbacks.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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