statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
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preserves_limits_of_size.zero.preserves_finite_limits (F : C ⥤ D)
[preserves_limits_of_size.{0 0} F] : preserves_finite_limits F | preserves_limits_of_size.preserves_finite_limits F | instance | category_theory.limits.preserves_limits_of_size.zero.preserves_finite_limits | category_theory.limits.preserves | src/category_theory/limits/preserves/finite.lean | [
"category_theory.limits.preserves.basic",
"category_theory.fin_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_limits.preserves_finite_limits (F : C ⥤ D)
[preserves_limits F] : preserves_finite_limits F | preserves_limits_of_size.preserves_finite_limits F | instance | category_theory.limits.preserves_limits.preserves_finite_limits | category_theory.limits.preserves | src/category_theory/limits/preserves/finite.lean | [
"category_theory.limits.preserves.basic",
"category_theory.fin_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_finite_limits_of_preserves_finite_limits_of_size (F : C ⥤ D)
(h : ∀ (J : Type w) {𝒥 : small_category J} (hJ : @fin_category J 𝒥),
by { resetI, exact preserves_limits_of_shape J F }) :
preserves_finite_limits F | ⟨λ J hJ hhJ,
begin
resetI,
letI : category.{w w} (ulift_hom.{w} (ulift.{w 0} J)),
{ apply ulift_hom.category.{0}, exact category_theory.ulift_category J },
haveI := h (ulift_hom.{w} (ulift.{w} J)) category_theory.fin_category_ulift,
exact preserves_limits_of_shape_of_equiv (ulift_hom_ulift_categor... | def | category_theory.limits.preserves_finite_limits_of_preserves_finite_limits_of_size | category_theory.limits.preserves | src/category_theory/limits/preserves/finite.lean | [
"category_theory.limits.preserves.basic",
"category_theory.fin_category"
] | [
"category_theory.fin_category_ulift",
"category_theory.ulift_category"
] | We can always derive `preserves_finite_limits C` by showing that we are preserving limits at an
arbitrary universe. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_preserves_finite_limits : preserves_finite_limits (𝟭 C) | {} | instance | category_theory.limits.id_preserves_finite_limits | category_theory.limits.preserves | src/category_theory/limits/preserves/finite.lean | [
"category_theory.limits.preserves.basic",
"category_theory.fin_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_preserves_finite_limits (F : C ⥤ D) (G : D ⥤ E)
[preserves_finite_limits F] [preserves_finite_limits G] : preserves_finite_limits (F ⋙ G) | ⟨λ _ _ _, by { resetI, apply_instance }⟩ | def | category_theory.limits.comp_preserves_finite_limits | category_theory.limits.preserves | src/category_theory/limits/preserves/finite.lean | [
"category_theory.limits.preserves.basic",
"category_theory.fin_category"
] | [] | The composition of two left exact functors is left exact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_finite_colimits (F : C ⥤ D) | (preserves_finite_colimits : Π (J : Type) [small_category J] [fin_category J],
preserves_colimits_of_shape J F . tactic.apply_instance) | class | category_theory.limits.preserves_finite_colimits | category_theory.limits.preserves | src/category_theory/limits/preserves/finite.lean | [
"category_theory.limits.preserves.basic",
"category_theory.fin_category"
] | [] | A functor is said to preserve finite colimits, if it preserves all colimits of
shape `J`, where `J : Type` is a finite category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimits_of_shape_of_preserves_finite_colimits (F : C ⥤ D)
[preserves_finite_colimits F] (J : Type w) [small_category J] [fin_category J] :
preserves_colimits_of_shape J F | by apply preserves_colimits_of_shape_of_equiv (fin_category.equiv_as_type J) | instance | category_theory.limits.preserves_colimits_of_shape_of_preserves_finite_colimits | category_theory.limits.preserves | src/category_theory/limits/preserves/finite.lean | [
"category_theory.limits.preserves.basic",
"category_theory.fin_category"
] | [] | Preserving finite limits also implies preserving limits over finite shapes in higher universes,
though through a noncomputable instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimits_of_size.preserves_finite_colimits (F : C ⥤ D)
[preserves_colimits_of_size.{w w₂} F] : preserves_finite_colimits F | ⟨λ J sJ fJ,
begin
haveI := preserves_smallest_colimits_of_preserves_colimits F,
exact preserves_colimits_of_shape_of_equiv (fin_category.equiv_as_type J) F,
end⟩ | def | category_theory.limits.preserves_colimits_of_size.preserves_finite_colimits | category_theory.limits.preserves | src/category_theory/limits/preserves/finite.lean | [
"category_theory.limits.preserves.basic",
"category_theory.fin_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_colimits_of_size.zero.preserves_finite_colimits (F : C ⥤ D)
[preserves_colimits_of_size.{0 0} F] : preserves_finite_colimits F | preserves_colimits_of_size.preserves_finite_colimits F | instance | category_theory.limits.preserves_colimits_of_size.zero.preserves_finite_colimits | category_theory.limits.preserves | src/category_theory/limits/preserves/finite.lean | [
"category_theory.limits.preserves.basic",
"category_theory.fin_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_colimits.preserves_finite_colimits (F : C ⥤ D)
[preserves_colimits F] : preserves_finite_colimits F | preserves_colimits_of_size.preserves_finite_colimits F | instance | category_theory.limits.preserves_colimits.preserves_finite_colimits | category_theory.limits.preserves | src/category_theory/limits/preserves/finite.lean | [
"category_theory.limits.preserves.basic",
"category_theory.fin_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_finite_colimits_of_preserves_finite_colimits_of_size (F : C ⥤ D)
(h : ∀ (J : Type w) {𝒥 : small_category J} (hJ : @fin_category J 𝒥),
by { resetI, exact preserves_colimits_of_shape J F }) :
preserves_finite_colimits F | ⟨λ J hJ hhJ,
begin
resetI,
letI : category.{w w} (ulift_hom.{w} (ulift.{w 0} J)),
{ apply ulift_hom.category.{0}, exact category_theory.ulift_category J },
haveI := h (ulift_hom.{w} (ulift.{w} J)) category_theory.fin_category_ulift,
exact preserves_colimits_of_shape_of_equiv (ulift_hom_ulift_categ... | def | category_theory.limits.preserves_finite_colimits_of_preserves_finite_colimits_of_size | category_theory.limits.preserves | src/category_theory/limits/preserves/finite.lean | [
"category_theory.limits.preserves.basic",
"category_theory.fin_category"
] | [
"category_theory.fin_category_ulift",
"category_theory.ulift_category"
] | We can always derive `preserves_finite_limits C` by showing that we are preserving limits at an
arbitrary universe. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_preserves_finite_colimits : preserves_finite_colimits (𝟭 C) | {} | instance | category_theory.limits.id_preserves_finite_colimits | category_theory.limits.preserves | src/category_theory/limits/preserves/finite.lean | [
"category_theory.limits.preserves.basic",
"category_theory.fin_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_preserves_finite_colimits (F : C ⥤ D) (G : D ⥤ E)
[preserves_finite_colimits F] [preserves_finite_colimits G] :
preserves_finite_colimits (F ⋙ G) | ⟨λ _ _ _, by { resetI, apply_instance }⟩ | def | category_theory.limits.comp_preserves_finite_colimits | category_theory.limits.preserves | src/category_theory/limits/preserves/finite.lean | [
"category_theory.limits.preserves.basic",
"category_theory.fin_category"
] | [] | The composition of two right exact functors is right exact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor_category.prod_preserves_colimits [has_binary_products D] [has_colimits D]
[∀ (X : D), preserves_colimits (prod.functor.obj X)]
(F : C ⥤ D) :
preserves_colimits (prod.functor.obj F) | { preserves_colimits_of_shape := λ J 𝒥, by exactI
{ preserves_colimit := λ K,
{ preserves := λ c t,
begin
apply evaluation_jointly_reflects_colimits _ (λ k, _),
change is_colimit ((prod.functor.obj F ⋙ (evaluation _ _).obj k).map_cocone c),
let := is_colimit_of_preserves ((evaluatio... | def | category_theory.functor_category.prod_preserves_colimits | category_theory.limits.preserves | src/category_theory/limits/preserves/functor_category.lean | [
"category_theory.limits.functor_category",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.limits.yoneda",
"category_theory.limits.presheaf"
] | [] | If `X × -` preserves colimits in `D` for any `X : D`, then the product functor `F ⨯ -` for
`F : C ⥤ D` also preserves colimits.
Note this is (mathematically) a special case of the statement that
"if limits commute with colimits in `D`, then they do as well in `C ⥤ D`"
but the story in Lean is a bit more complex, and t... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whiskering_left_preserves_limits [has_limits D] (F : C ⥤ E) :
preserves_limits ((whiskering_left C E D).obj F) | ⟨λ J hJ, by exactI ⟨λ K, ⟨λ c hc,
begin
apply evaluation_jointly_reflects_limits,
intro Y,
change is_limit (((evaluation E D).obj (F.obj Y)).map_cone c),
exact preserves_limit.preserves hc,
end ⟩⟩⟩ | instance | category_theory.whiskering_left_preserves_limits | category_theory.limits.preserves | src/category_theory/limits/preserves/functor_category.lean | [
"category_theory.limits.functor_category",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.limits.yoneda",
"category_theory.limits.presheaf"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whiskering_right_preserves_limits_of_shape {C : Type u} [category C]
{D : Type*} [category.{u} D] {E : Type*} [category.{u} E]
{J : Type u} [small_category J] [has_limits_of_shape J D]
(F : D ⥤ E) [preserves_limits_of_shape J F] :
preserves_limits_of_shape J ((whiskering_right C D E).obj F) | ⟨λ K, ⟨λ c hc,
begin
apply evaluation_jointly_reflects_limits,
intro k,
change is_limit (((evaluation _ _).obj k ⋙ F).map_cone c),
exact preserves_limit.preserves hc,
end ⟩⟩ | instance | category_theory.whiskering_right_preserves_limits_of_shape | category_theory.limits.preserves | src/category_theory/limits/preserves/functor_category.lean | [
"category_theory.limits.functor_category",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.limits.yoneda",
"category_theory.limits.presheaf"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whiskering_right_preserves_limits {C : Type u} [category C]
{D : Type*} [category.{u} D] {E : Type*} [category.{u} E] (F : D ⥤ E)
[has_limits D] [preserves_limits F] : preserves_limits ((whiskering_right C D E).obj F) | ⟨⟩ | instance | category_theory.whiskering_right_preserves_limits | category_theory.limits.preserves | src/category_theory/limits/preserves/functor_category.lean | [
"category_theory.limits.functor_category",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.limits.yoneda",
"category_theory.limits.presheaf"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_limit_of_Lan_preserves_limit {C D : Type u} [small_category C] [small_category D]
(F : C ⥤ D) (J : Type u) [small_category J]
[preserves_limits_of_shape J (Lan F.op : _ ⥤ (Dᵒᵖ ⥤ Type u))] :
preserves_limits_of_shape J F | begin
apply preserves_limits_of_shape_of_reflects_of_preserves F yoneda,
exact preserves_limits_of_shape_of_nat_iso (comp_yoneda_iso_yoneda_comp_Lan F).symm,
apply_instance
end | def | category_theory.preserves_limit_of_Lan_preserves_limit | category_theory.limits.preserves | src/category_theory/limits/preserves/functor_category.lean | [
"category_theory.limits.functor_category",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.limits.yoneda",
"category_theory.limits.presheaf"
] | [] | If `Lan F.op : (Cᵒᵖ ⥤ Type*) ⥤ (Dᵒᵖ ⥤ Type*)` preserves limits of shape `J`, so will `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_lift_map_cone (c₁ c₂ : cone F) (t : is_limit c₁) :
(preserves_limit.preserves t).lift (G.map_cone c₂) = G.map (t.lift c₂) | ((preserves_limit.preserves t).uniq (G.map_cone c₂) _ (by simp [← G.map_comp])).symm | lemma | category_theory.preserves_lift_map_cone | category_theory.limits.preserves | src/category_theory/limits/preserves/limits.lean | [
"category_theory.limits.preserves.basic"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_limit_iso : G.obj (limit F) ≅ limit (F ⋙ G) | (preserves_limit.preserves (limit.is_limit _)).cone_point_unique_up_to_iso (limit.is_limit _) | def | category_theory.preserves_limit_iso | category_theory.limits.preserves | src/category_theory/limits/preserves/limits.lean | [
"category_theory.limits.preserves.basic"
] | [] | If `G` preserves limits, we have an isomorphism from the image of the limit of a functor `F`
to the limit of the functor `F ⋙ G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limits_iso_hom_π (j) :
(preserves_limit_iso G F).hom ≫ limit.π _ j = G.map (limit.π F j) | is_limit.cone_point_unique_up_to_iso_hom_comp _ _ j | lemma | category_theory.preserves_limits_iso_hom_π | category_theory.limits.preserves | src/category_theory/limits/preserves/limits.lean | [
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_limits_iso_inv_π (j) :
(preserves_limit_iso G F).inv ≫ G.map (limit.π F j) = limit.π _ j | is_limit.cone_point_unique_up_to_iso_inv_comp _ _ j | lemma | category_theory.preserves_limits_iso_inv_π | category_theory.limits.preserves | src/category_theory/limits/preserves/limits.lean | [
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_comp_preserves_limits_iso_hom (t : cone F) :
G.map (limit.lift _ t) ≫ (preserves_limit_iso G F).hom = limit.lift (F ⋙ G) (G.map_cone _) | by { ext, simp [← G.map_comp] } | lemma | category_theory.lift_comp_preserves_limits_iso_hom | category_theory.limits.preserves | src/category_theory/limits/preserves/limits.lean | [
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_limit_nat_iso : lim ⋙ G ≅ (whiskering_right J C D).obj G ⋙ lim | nat_iso.of_components (λ F, preserves_limit_iso G F)
begin
intros _ _ f,
ext,
dsimp,
simp only [preserves_limits_iso_hom_π, whisker_right_app, lim_map_π, category.assoc,
preserves_limits_iso_hom_π_assoc, ← G.map_comp]
end | def | category_theory.preserves_limit_nat_iso | category_theory.limits.preserves | src/category_theory/limits/preserves/limits.lean | [
"category_theory.limits.preserves.basic"
] | [
"lim"
] | If `C, D` has all limits of shape `J`, and `G` preserves them, then `preserves_limit_iso` is
functorial wrt `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_desc_map_cocone (c₁ c₂ : cocone F) (t : is_colimit c₁) :
(preserves_colimit.preserves t).desc (G.map_cocone _) = G.map (t.desc c₂) | ((preserves_colimit.preserves t).uniq (G.map_cocone _) _ (by simp [← G.map_comp])).symm | lemma | category_theory.preserves_desc_map_cocone | category_theory.limits.preserves | src/category_theory/limits/preserves/limits.lean | [
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_colimit_iso : G.obj (colimit F) ≅ colimit (F ⋙ G) | (preserves_colimit.preserves (colimit.is_colimit _)).cocone_point_unique_up_to_iso
(colimit.is_colimit _) | def | category_theory.preserves_colimit_iso | category_theory.limits.preserves | src/category_theory/limits/preserves/limits.lean | [
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι_preserves_colimits_iso_inv (j : J) :
colimit.ι _ j ≫ (preserves_colimit_iso G F).inv = G.map (colimit.ι F j) | is_colimit.comp_cocone_point_unique_up_to_iso_inv _ (colimit.is_colimit (F ⋙ G)) j | lemma | category_theory.ι_preserves_colimits_iso_inv | category_theory.limits.preserves | src/category_theory/limits/preserves/limits.lean | [
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι_preserves_colimits_iso_hom (j : J) :
G.map (colimit.ι F j) ≫ (preserves_colimit_iso G F).hom = colimit.ι (F ⋙ G) j | (preserves_colimit.preserves (colimit.is_colimit _)).comp_cocone_point_unique_up_to_iso_hom _ j | lemma | category_theory.ι_preserves_colimits_iso_hom | category_theory.limits.preserves | src/category_theory/limits/preserves/limits.lean | [
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_colimits_iso_inv_comp_desc (t : cocone F) :
(preserves_colimit_iso G F).inv ≫ G.map (colimit.desc _ t) = colimit.desc _ (G.map_cocone t) | by { ext, simp [← G.map_comp] } | lemma | category_theory.preserves_colimits_iso_inv_comp_desc | category_theory.limits.preserves | src/category_theory/limits/preserves/limits.lean | [
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_colimit_nat_iso : colim ⋙ G ≅ (whiskering_right J C D).obj G ⋙ colim | nat_iso.of_components (λ F, preserves_colimit_iso G F)
begin
intros _ _ f,
rw [← iso.inv_comp_eq, ← category.assoc, ← iso.eq_comp_inv],
ext,
dsimp,
erw ι_colim_map_assoc,
simp only [ι_preserves_colimits_iso_inv, whisker_right_app, category.assoc,
ι_preserves_colimits_iso_inv_assoc, ← G.map_comp],
erw ... | def | category_theory.preserves_colimit_nat_iso | category_theory.limits.preserves | src/category_theory/limits/preserves/limits.lean | [
"category_theory.limits.preserves.basic"
] | [] | If `C, D` has all colimits of shape `J`, and `G` preserves them, then `preserves_colimit_iso`
is functorial wrt `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limit_op (K : J ⥤ Cᵒᵖ) (F : C ⥤ D) [preserves_colimit K.left_op F] :
preserves_limit K F.op | { preserves := λ c hc, is_limit_cone_right_op_of_cocone _
(is_colimit_of_preserves F (is_colimit_cocone_left_op_of_cone _ hc)) } | def | category_theory.limits.preserves_limit_op | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : C ⥤ D` preserves colimits of `K.left_op : Jᵒᵖ ⥤ C`, then `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves
limits of `K : J ⥤ Cᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limit_left_op (K : J ⥤ Cᵒᵖ) (F : C ⥤ Dᵒᵖ) [preserves_colimit K.left_op F] :
preserves_limit K F.left_op | { preserves := λ c hc, is_limit_cone_unop_of_cocone _
(is_colimit_of_preserves F (is_colimit_cocone_left_op_of_cone _ hc)) } | def | category_theory.limits.preserves_limit_left_op | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : C ⥤ Dᵒᵖ` preserves colimits of `K.left_op : Jᵒᵖ ⥤ C`, then `F.left_op : Cᵒᵖ ⥤ D`
preserves limits of `K : J ⥤ Cᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limit_right_op (K : J ⥤ C) (F : Cᵒᵖ ⥤ D) [preserves_colimit K.op F] :
preserves_limit K F.right_op | { preserves := λ c hc, is_limit_cone_right_op_of_cocone _
(is_colimit_of_preserves F (is_colimit_cone_op _ hc)) } | def | category_theory.limits.preserves_limit_right_op | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : Cᵒᵖ ⥤ D` preserves colimits of `K.op : Jᵒᵖ ⥤ Cᵒᵖ`, then `F.right_op : C ⥤ Dᵒᵖ` preserves
limits of `K : J ⥤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limit_unop (K : J ⥤ C) (F : Cᵒᵖ ⥤ Dᵒᵖ) [preserves_colimit K.op F] :
preserves_limit K F.unop | { preserves := λ c hc, is_limit_cone_unop_of_cocone _
(is_colimit_of_preserves F (is_colimit_cone_op _ hc)) } | def | category_theory.limits.preserves_limit_unop | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves colimits of `K.op : Jᵒᵖ ⥤ Cᵒᵖ`, then `F.unop : C ⥤ D` preserves
limits of `K : J ⥤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimit_op (K : J ⥤ Cᵒᵖ) (F : C ⥤ D) [preserves_limit K.left_op F] :
preserves_colimit K F.op | { preserves := λ c hc, is_colimit_cocone_right_op_of_cone _
(is_limit_of_preserves F (is_limit_cone_left_op_of_cocone _ hc)) } | def | category_theory.limits.preserves_colimit_op | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : C ⥤ D` preserves limits of `K.left_op : Jᵒᵖ ⥤ C`, then `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves
colimits of `K : J ⥤ Cᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimit_left_op (K : J ⥤ Cᵒᵖ) (F : C ⥤ Dᵒᵖ) [preserves_limit K.left_op F] :
preserves_colimit K F.left_op | { preserves := λ c hc, is_colimit_cocone_unop_of_cone _
(is_limit_of_preserves F (is_limit_cone_left_op_of_cocone _ hc)) } | def | category_theory.limits.preserves_colimit_left_op | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : C ⥤ Dᵒᵖ` preserves limits of `K.left_op : Jᵒᵖ ⥤ C`, then `F.left_op : Cᵒᵖ ⥤ D` preserves
colimits of `K : J ⥤ Cᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimit_right_op (K : J ⥤ C) (F : Cᵒᵖ ⥤ D) [preserves_limit K.op F] :
preserves_colimit K F.right_op | { preserves := λ c hc, is_colimit_cocone_right_op_of_cone _
(is_limit_of_preserves F (is_limit_cocone_op _ hc)) } | def | category_theory.limits.preserves_colimit_right_op | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : Cᵒᵖ ⥤ D` preserves limits of `K.op : Jᵒᵖ ⥤ Cᵒᵖ`, then `F.right_op : C ⥤ Dᵒᵖ` preserves
colimits of `K : J ⥤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimit_unop (K : J ⥤ C) (F : Cᵒᵖ ⥤ Dᵒᵖ) [preserves_limit K.op F] :
preserves_colimit K F.unop | { preserves := λ c hc, is_colimit_cocone_unop_of_cone _
(is_limit_of_preserves F (is_limit_cocone_op _ hc)) } | def | category_theory.limits.preserves_colimit_unop | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves limits of `K.op : Jᵒᵖ ⥤ Cᵒᵖ`, then `F.unop : C ⥤ D` preserves
colimits of `K : J ⥤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limits_of_shape_op (F : C ⥤ D) [preserves_colimits_of_shape Jᵒᵖ F] :
preserves_limits_of_shape J F.op | { preserves_limit := λ K, preserves_limit_op K F } | def | category_theory.limits.preserves_limits_of_shape_op | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : C ⥤ D` preserves colimits of shape `Jᵒᵖ`, then `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves limits of
shape `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limits_of_shape_left_op (F : C ⥤ Dᵒᵖ) [preserves_colimits_of_shape Jᵒᵖ F] :
preserves_limits_of_shape J F.left_op | { preserves_limit := λ K, preserves_limit_left_op K F } | def | category_theory.limits.preserves_limits_of_shape_left_op | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : C ⥤ Dᵒᵖ` preserves colimits of shape `Jᵒᵖ`, then `F.left_op : Cᵒᵖ ⥤ D` preserves limits
of shape `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limits_of_shape_right_op (F : Cᵒᵖ ⥤ D) [preserves_colimits_of_shape Jᵒᵖ F] :
preserves_limits_of_shape J F.right_op | { preserves_limit := λ K, preserves_limit_right_op K F } | def | category_theory.limits.preserves_limits_of_shape_right_op | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : Cᵒᵖ ⥤ D` preserves colimits of shape `Jᵒᵖ`, then `F.right_op : C ⥤ Dᵒᵖ` preserves limits
of shape `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limits_of_shape_unop (F : Cᵒᵖ ⥤ Dᵒᵖ) [preserves_colimits_of_shape Jᵒᵖ F] :
preserves_limits_of_shape J F.unop | { preserves_limit := λ K, preserves_limit_unop K F } | def | category_theory.limits.preserves_limits_of_shape_unop | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves colimits of shape `Jᵒᵖ`, then `F.unop : C ⥤ D` preserves limits of
shape `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimits_of_shape_op (F : C ⥤ D) [preserves_limits_of_shape Jᵒᵖ F] :
preserves_colimits_of_shape J F.op | { preserves_colimit := λ K, preserves_colimit_op K F } | def | category_theory.limits.preserves_colimits_of_shape_op | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : C ⥤ D` preserves limits of shape `Jᵒᵖ`, then `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves colimits of
shape `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimits_of_shape_left_op (F : C ⥤ Dᵒᵖ) [preserves_limits_of_shape Jᵒᵖ F] :
preserves_colimits_of_shape J F.left_op | { preserves_colimit := λ K, preserves_colimit_left_op K F } | def | category_theory.limits.preserves_colimits_of_shape_left_op | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : C ⥤ Dᵒᵖ` preserves limits of shape `Jᵒᵖ`, then `F.left_op : Cᵒᵖ ⥤ D` preserves colimits
of shape `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimits_of_shape_right_op (F : Cᵒᵖ ⥤ D) [preserves_limits_of_shape Jᵒᵖ F] :
preserves_colimits_of_shape J F.right_op | { preserves_colimit := λ K, preserves_colimit_right_op K F } | def | category_theory.limits.preserves_colimits_of_shape_right_op | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : Cᵒᵖ ⥤ D` preserves limits of shape `Jᵒᵖ`, then `F.right_op : C ⥤ Dᵒᵖ` preserves colimits
of shape `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimits_of_shape_unop (F : Cᵒᵖ ⥤ Dᵒᵖ) [preserves_limits_of_shape Jᵒᵖ F] :
preserves_colimits_of_shape J F.unop | { preserves_colimit := λ K, preserves_colimit_unop K F } | def | category_theory.limits.preserves_colimits_of_shape_unop | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves limits of shape `Jᵒᵖ`, then `F.unop : C ⥤ D` preserves colimits
of shape `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limits_op (F : C ⥤ D) [preserves_colimits F] : preserves_limits F.op | { preserves_limits_of_shape := λ J _, by exactI preserves_limits_of_shape_op J F } | def | category_theory.limits.preserves_limits_op | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : C ⥤ D` preserves colimits, then `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limits_left_op (F : C ⥤ Dᵒᵖ) [preserves_colimits F] : preserves_limits F.left_op | { preserves_limits_of_shape := λ J _, by exactI preserves_limits_of_shape_left_op J F } | def | category_theory.limits.preserves_limits_left_op | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : C ⥤ Dᵒᵖ` preserves colimits, then `F.left_op : Cᵒᵖ ⥤ D` preserves limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limits_right_op (F : Cᵒᵖ ⥤ D) [preserves_colimits F] : preserves_limits F.right_op | { preserves_limits_of_shape := λ J _, by exactI preserves_limits_of_shape_right_op J F } | def | category_theory.limits.preserves_limits_right_op | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : Cᵒᵖ ⥤ D` preserves colimits, then `F.right_op : C ⥤ Dᵒᵖ` preserves limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limits_unop (F : Cᵒᵖ ⥤ Dᵒᵖ) [preserves_colimits F] : preserves_limits F.unop | { preserves_limits_of_shape := λ J _, by exactI preserves_limits_of_shape_unop J F } | def | category_theory.limits.preserves_limits_unop | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves colimits, then `F.unop : C ⥤ D` preserves limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
perserves_colimits_op (F : C ⥤ D) [preserves_limits F] : preserves_colimits F.op | { preserves_colimits_of_shape := λ J _, by exactI preserves_colimits_of_shape_op J F } | def | category_theory.limits.perserves_colimits_op | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : C ⥤ D` preserves limits, then `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimits_left_op (F : C ⥤ Dᵒᵖ) [preserves_limits F] : preserves_colimits F.left_op | { preserves_colimits_of_shape := λ J _, by exactI preserves_colimits_of_shape_left_op J F } | def | category_theory.limits.preserves_colimits_left_op | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : C ⥤ Dᵒᵖ` preserves limits, then `F.left_op : Cᵒᵖ ⥤ D` preserves colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimits_right_op (F : Cᵒᵖ ⥤ D) [preserves_limits F] :
preserves_colimits F.right_op | { preserves_colimits_of_shape := λ J _, by exactI preserves_colimits_of_shape_right_op J F } | def | category_theory.limits.preserves_colimits_right_op | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : Cᵒᵖ ⥤ D` preserves limits, then `F.right_op : C ⥤ Dᵒᵖ` preserves colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimits_unop (F : Cᵒᵖ ⥤ Dᵒᵖ) [preserves_limits F] : preserves_colimits F.unop | { preserves_colimits_of_shape := λ J _, by exactI preserves_colimits_of_shape_unop J F } | def | category_theory.limits.preserves_colimits_unop | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves limits, then `F.unop : C ⥤ D` preserves colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_finite_limits_op (F : C ⥤ D) [preserves_finite_colimits F] :
preserves_finite_limits F.op | { preserves_finite_limits := λ J _ _, by exactI preserves_limits_of_shape_op J F } | def | category_theory.limits.preserves_finite_limits_op | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : C ⥤ D` preserves finite colimits, then `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves finite
limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_finite_limits_left_op (F : C ⥤ Dᵒᵖ) [preserves_finite_colimits F] :
preserves_finite_limits F.left_op | { preserves_finite_limits := λ J _ _, by exactI preserves_limits_of_shape_left_op J F } | def | category_theory.limits.preserves_finite_limits_left_op | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : C ⥤ Dᵒᵖ` preserves finite colimits, then `F.left_op : Cᵒᵖ ⥤ D` preserves finite
limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_finite_limits_right_op (F : Cᵒᵖ ⥤ D) [preserves_finite_colimits F] :
preserves_finite_limits F.right_op | { preserves_finite_limits := λ J _ _, by exactI preserves_limits_of_shape_right_op J F } | def | category_theory.limits.preserves_finite_limits_right_op | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : Cᵒᵖ ⥤ D` preserves finite colimits, then `F.right_op : C ⥤ Dᵒᵖ` preserves finite
limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_finite_limits_unop (F : Cᵒᵖ ⥤ Dᵒᵖ) [preserves_finite_colimits F] :
preserves_finite_limits F.unop | { preserves_finite_limits := λ J _ _, by exactI preserves_limits_of_shape_unop J F } | def | category_theory.limits.preserves_finite_limits_unop | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves finite colimits, then `F.unop : C ⥤ D` preserves finite
limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_finite_colimits_op (F : C ⥤ D) [preserves_finite_limits F] :
preserves_finite_colimits F.op | { preserves_finite_colimits := λ J _ _, by exactI preserves_colimits_of_shape_op J F } | def | category_theory.limits.preserves_finite_colimits_op | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : C ⥤ D` preserves finite limits, then `F.op : Cᵒᵖ ⥤ Dᵒᵖ` preserves finite
colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_finite_colimits_left_op (F : C ⥤ Dᵒᵖ) [preserves_finite_limits F] :
preserves_finite_colimits F.left_op | { preserves_finite_colimits := λ J _ _, by exactI preserves_colimits_of_shape_left_op J F } | def | category_theory.limits.preserves_finite_colimits_left_op | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : C ⥤ Dᵒᵖ` preserves finite limits, then `F.left_op : Cᵒᵖ ⥤ D` preserves finite
colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_finite_colimits_right_op (F : Cᵒᵖ ⥤ D) [preserves_finite_limits F] :
preserves_finite_colimits F.right_op | { preserves_finite_colimits := λ J _ _, by exactI preserves_colimits_of_shape_right_op J F } | def | category_theory.limits.preserves_finite_colimits_right_op | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : Cᵒᵖ ⥤ D` preserves finite limits, then `F.right_op : C ⥤ Dᵒᵖ` preserves finite
colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_finite_colimits_unop (F : Cᵒᵖ ⥤ Dᵒᵖ) [preserves_finite_limits F] :
preserves_finite_colimits F.unop | { preserves_finite_colimits := λ J _ _, by exactI preserves_colimits_of_shape_unop J F } | def | category_theory.limits.preserves_finite_colimits_unop | category_theory.limits.preserves | src/category_theory/limits/preserves/opposites.lean | [
"category_theory.limits.opposites",
"category_theory.limits.preserves.finite"
] | [] | If `F : Cᵒᵖ ⥤ Dᵒᵖ` preserves finite limits, then `F.unop : C ⥤ D` preserves finite
colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_map_cone_binary_fan_equiv :
is_limit (G.map_cone (binary_fan.mk f g)) ≃ is_limit (binary_fan.mk (G.map f) (G.map g)) | (is_limit.postcompose_hom_equiv (diagram_iso_pair _) _).symm.trans
(is_limit.equiv_iso_limit (cones.ext (iso.refl _) (by { rintro (_ | _), tidy }))) | def | category_theory.limits.is_limit_map_cone_binary_fan_equiv | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/binary_products.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.basic"
] | [] | The map of a binary fan is a limit iff the fork consisting of the mapped morphisms is a limit. This
essentially lets us commute `binary_fan.mk` with `functor.map_cone`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_is_limit_of_preserves_of_is_limit [preserves_limit (pair X Y) G]
(l : is_limit (binary_fan.mk f g)) :
is_limit (binary_fan.mk (G.map f) (G.map g)) | is_limit_map_cone_binary_fan_equiv G f g (preserves_limit.preserves l) | def | category_theory.limits.map_is_limit_of_preserves_of_is_limit | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/binary_products.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.basic"
] | [] | The property of preserving products expressed in terms of binary fans. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_of_reflects_of_map_is_limit [reflects_limit (pair X Y) G]
(l : is_limit (binary_fan.mk (G.map f) (G.map g))) :
is_limit (binary_fan.mk f g) | reflects_limit.reflects ((is_limit_map_cone_binary_fan_equiv G f g).symm l) | def | category_theory.limits.is_limit_of_reflects_of_map_is_limit | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/binary_products.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.basic"
] | [] | The property of reflecting products expressed in terms of binary fans. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_of_has_binary_product_of_preserves_limit
[preserves_limit (pair X Y) G] :
is_limit (binary_fan.mk (G.map (limits.prod.fst : X ⨯ Y ⟶ X)) (G.map (limits.prod.snd))) | map_is_limit_of_preserves_of_is_limit G _ _ (prod_is_prod X Y) | def | category_theory.limits.is_limit_of_has_binary_product_of_preserves_limit | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/binary_products.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.basic"
] | [] | If `G` preserves binary products and `C` has them, then the binary fan constructed of the mapped
morphisms of the binary product cone is a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limit_pair.of_iso_prod_comparison [i : is_iso (prod_comparison G X Y)] :
preserves_limit (pair X Y) G | begin
apply preserves_limit_of_preserves_limit_cone (prod_is_prod X Y),
apply (is_limit_map_cone_binary_fan_equiv _ _ _).symm _,
apply is_limit.of_point_iso (limit.is_limit (pair (G.obj X) (G.obj Y))),
apply i,
end | def | category_theory.limits.preserves_limit_pair.of_iso_prod_comparison | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/binary_products.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.basic"
] | [] | If the product comparison map for `G` at `(X,Y)` is an isomorphism, then `G` preserves the
pair of `(X,Y)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limit_pair.iso :
G.obj (X ⨯ Y) ≅ G.obj X ⨯ G.obj Y | is_limit.cone_point_unique_up_to_iso
(is_limit_of_has_binary_product_of_preserves_limit G X Y)
(limit.is_limit _) | def | category_theory.limits.preserves_limit_pair.iso | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/binary_products.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.basic"
] | [] | If `G` preserves the product of `(X,Y)`, then the product comparison map for `G` at `(X,Y)` is
an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limit_pair.iso_hom : (preserves_limit_pair.iso G X Y).hom = prod_comparison G X Y | rfl | lemma | category_theory.limits.preserves_limit_pair.iso_hom | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/binary_products.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_colimit_map_cocone_binary_cofan_equiv :
is_colimit (G.map_cocone (binary_cofan.mk f g)) ≃
is_colimit (binary_cofan.mk (G.map f) (G.map g)) | (is_colimit.precompose_hom_equiv (diagram_iso_pair _).symm _).symm.trans
(is_colimit.equiv_iso_colimit (cocones.ext (iso.refl _) (by { rintro (_ | _), tidy, }))) | def | category_theory.limits.is_colimit_map_cocone_binary_cofan_equiv | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/binary_products.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.basic"
] | [] | The map of a binary cofan is a colimit iff
the cofork consisting of the mapped morphisms is a colimit.
This essentially lets us commute `binary_cofan.mk` with `functor.map_cocone`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_is_colimit_of_preserves_of_is_colimit [preserves_colimit (pair X Y) G]
(l : is_colimit (binary_cofan.mk f g)) :
is_colimit (binary_cofan.mk (G.map f) (G.map g)) | is_colimit_map_cocone_binary_cofan_equiv G f g (preserves_colimit.preserves l) | def | category_theory.limits.map_is_colimit_of_preserves_of_is_colimit | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/binary_products.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.basic"
] | [] | The property of preserving coproducts expressed in terms of binary cofans. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_of_reflects_of_map_is_colimit [reflects_colimit (pair X Y) G]
(l : is_colimit (binary_cofan.mk (G.map f) (G.map g))) :
is_colimit (binary_cofan.mk f g) | reflects_colimit.reflects ((is_colimit_map_cocone_binary_cofan_equiv G f g).symm l) | def | category_theory.limits.is_colimit_of_reflects_of_map_is_colimit | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/binary_products.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.basic"
] | [] | The property of reflecting coproducts expressed in terms of binary cofans. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_of_has_binary_coproduct_of_preserves_colimit
[preserves_colimit (pair X Y) G] :
is_colimit
(binary_cofan.mk (G.map (limits.coprod.inl : X ⟶ X ⨿ Y)) (G.map (limits.coprod.inr))) | map_is_colimit_of_preserves_of_is_colimit G _ _ (coprod_is_coprod X Y) | def | category_theory.limits.is_colimit_of_has_binary_coproduct_of_preserves_colimit | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/binary_products.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.basic"
] | [] | If `G` preserves binary coproducts and `C` has them, then the binary cofan constructed of the mapped
morphisms of the binary product cocone is a colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimit_pair.of_iso_coprod_comparison [i : is_iso (coprod_comparison G X Y)] :
preserves_colimit (pair X Y) G | begin
apply preserves_colimit_of_preserves_colimit_cocone (coprod_is_coprod X Y),
apply (is_colimit_map_cocone_binary_cofan_equiv _ _ _).symm _,
apply is_colimit.of_point_iso (colimit.is_colimit (pair (G.obj X) (G.obj Y))),
apply i,
end | def | category_theory.limits.preserves_colimit_pair.of_iso_coprod_comparison | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/binary_products.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.basic"
] | [] | If the coproduct comparison map for `G` at `(X,Y)` is an isomorphism, then `G` preserves the
pair of `(X,Y)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimit_pair.iso :
G.obj X ⨿ G.obj Y ≅ G.obj (X ⨿ Y) | is_colimit.cocone_point_unique_up_to_iso
(colimit.is_colimit _)
(is_colimit_of_has_binary_coproduct_of_preserves_colimit G X Y) | def | category_theory.limits.preserves_colimit_pair.iso | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/binary_products.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.basic"
] | [] | If `G` preserves the coproduct of `(X,Y)`, then the coproduct comparison map for `G` at `(X,Y)` is
an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimit_pair.iso_hom :
(preserves_colimit_pair.iso G X Y).hom = coprod_comparison G X Y | rfl | lemma | category_theory.limits.preserves_colimit_pair.iso_hom | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/binary_products.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_bicone {f : J → C} (b : bicone f) : bicone (F.obj ∘ f) | { X := F.obj b.X,
π := λ j, F.map (b.π j),
ι := λ j, F.map (b.ι j),
ι_π := λ j j',
begin
rw ← F.map_comp,
split_ifs,
{ subst h,
simp only [bicone_ι_π_self, category_theory.functor.map_id, eq_to_hom_refl] },
{ rw [bicone_ι_π_ne _ h, F.map_zero] }
end } | def | category_theory.functor.map_bicone | 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 image of a bicone under a functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_bicone_whisker {K : Type w₂} {g : K ≃ J} {f : J → C} (c : bicone f) :
F.map_bicone (c.whisker g) = (F.map_bicone c).whisker g | rfl | lemma | category_theory.functor.map_bicone_whisker | 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_binary_bicone {X Y : C} (b : binary_bicone X Y) : binary_bicone (F.obj X) (F.obj Y) | { X := F.obj b.X,
fst := F.map b.fst,
snd := F.map b.snd,
inl := F.map b.inl,
inr := F.map b.inr,
inl_fst' := by rw [← F.map_comp, b.inl_fst, F.map_id],
inl_snd' := by rw [← F.map_comp, b.inl_snd, F.map_zero],
inr_fst' := by rw [← F.map_comp, b.inr_fst, F.map_zero],
inr_snd' := by rw [← F.map_comp, b.in... | def | category_theory.functor.map_binary_bicone | 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 image of a binary bicone under a functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_biproduct (f : J → C) (F : C ⥤ D) [preserves_zero_morphisms F] | (preserves : Π {b : bicone f}, b.is_bilimit → (F.map_bicone b).is_bilimit) | class | category_theory.limits.preserves_biproduct | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | A functor `F` preserves biproducts of `f` if `F` maps every bilimit bicone over `f` to a
bilimit bicone over `F.obj ∘ f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_bilimit_of_preserves {f : J → C} (F : C ⥤ D) [preserves_zero_morphisms F]
[preserves_biproduct f F] {b : bicone f} (hb : b.is_bilimit) : (F.map_bicone b).is_bilimit | preserves_biproduct.preserves hb | def | category_theory.limits.is_bilimit_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"
] | [] | A functor `F` preserves biproducts of `f` if `F` maps every bilimit bicone over `f` to a
bilimit bicone over `F.obj ∘ f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_biproducts_of_shape (F : C ⥤ D) [preserves_zero_morphisms F] | (preserves : Π {f : J → C}, preserves_biproduct f F) | class | category_theory.limits.preserves_biproducts_of_shape | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | A functor `F` preserves biproducts of shape `J` if it preserves biproducts of `f` for every
`f : J → C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_finite_biproducts (F : C ⥤ D) [preserves_zero_morphisms F] | (preserves : Π {J : Type} [fintype J], preserves_biproducts_of_shape J F) | class | category_theory.limits.preserves_finite_biproducts | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [
"fintype"
] | A functor `F` preserves finite biproducts if it preserves biproducts of shape `J` whenever
`J` is a fintype. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_biproducts (F : C ⥤ D) [preserves_zero_morphisms F] | (preserves : Π {J : Type w₁}, preserves_biproducts_of_shape J F) | class | category_theory.limits.preserves_biproducts | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | A functor `F` preserves biproducts if it preserves biproducts of any shape `J` of size `w`.
The usual notion of preservation of biproducts is recovered by choosing `w` to be the universe
of the morphisms of `C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_biproducts_shrink (F : C ⥤ D) [preserves_zero_morphisms F]
[hp : preserves_biproducts.{max w₁ w₂} F] : preserves_biproducts.{w₁} F | ⟨λ J, ⟨λ f, ⟨λ b ib, ((F.map_bicone b).whisker_is_bilimit_iff _).to_fun
(is_bilimit_of_preserves F ((b.whisker_is_bilimit_iff equiv.ulift.{w₂}).inv_fun ib))⟩⟩⟩ | def | category_theory.limits.preserves_biproducts_shrink | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [
"inv_fun"
] | Preserving biproducts at a bigger universe level implies preserving biproducts at a
smaller universe level. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_finite_biproducts_of_preserves_biproducts (F : C ⥤ D)
[preserves_zero_morphisms F] [preserves_biproducts.{w₁} F] : preserves_finite_biproducts F | { preserves := λ J _, by letI := preserves_biproducts_shrink.{0} F; apply_instance } | instance | category_theory.limits.preserves_finite_biproducts_of_preserves_biproducts | 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 | |
preserves_binary_biproduct (X Y : C) (F : C ⥤ D) [preserves_zero_morphisms F] | (preserves : Π {b : binary_bicone X Y}, b.is_bilimit → (F.map_binary_bicone b).is_bilimit) | class | category_theory.limits.preserves_binary_biproduct | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | A functor `F` preserves binary biproducts of `X` and `Y` if `F` maps every bilimit bicone over
`X` and `Y` to a bilimit bicone over `F.obj X` and `F.obj Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_binary_bilimit_of_preserves {X Y : C} (F : C ⥤ D) [preserves_zero_morphisms F]
[preserves_binary_biproduct X Y F] {b : binary_bicone X Y} (hb : b.is_bilimit) :
(F.map_binary_bicone b).is_bilimit | preserves_binary_biproduct.preserves hb | def | category_theory.limits.is_binary_bilimit_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"
] | [] | A functor `F` preserves binary biproducts of `X` and `Y` if `F` maps every bilimit bicone over
`X` and `Y` to a bilimit bicone over `F.obj X` and `F.obj Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_binary_biproducts (F : C ⥤ D) [preserves_zero_morphisms F] | (preserves : Π {X Y : C}, preserves_binary_biproduct X Y F . tactic.apply_instance) | class | category_theory.limits.preserves_binary_biproducts | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | A functor `F` preserves binary biproducts if it preserves the binary biproduct of `X` and `Y`
for all `X` and `Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_binary_biproduct_of_preserves_biproduct (F : C ⥤ D) [preserves_zero_morphisms F]
(X Y : C) [preserves_biproduct (pair_function X Y) F] : preserves_binary_biproduct X Y F | { preserves := λ b hb,
{ is_limit := is_limit.of_iso_limit
((is_limit.postcompose_hom_equiv (by exact diagram_iso_pair _) _).symm
((is_bilimit_of_preserves F (b.to_bicone_is_bilimit.symm hb)).is_limit)) $
cones.ext (iso.refl _) (λ j, by { rcases j with ⟨⟨⟩⟩, tidy, }),
is_colimit := is_colimit.... | def | category_theory.limits.preserves_binary_biproduct_of_preserves_biproduct | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | A functor that preserves biproducts of a pair preserves binary biproducts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_binary_biproducts_of_preserves_biproducts (F : C ⥤ D)
[preserves_zero_morphisms F] [preserves_biproducts_of_shape walking_pair F] :
preserves_binary_biproducts F | { preserves := λ X Y, preserves_binary_biproduct_of_preserves_biproduct F X Y } | def | category_theory.limits.preserves_binary_biproducts_of_preserves_biproducts | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/biproducts.lean | [
"category_theory.limits.shapes.biproducts",
"category_theory.limits.preserves.shapes.zero"
] | [] | A functor that preserves biproducts of a pair preserves binary biproducts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct_comparison : F.obj (⨁ f) ⟶ ⨁ (F.obj ∘ f) | biproduct.lift (λ j, F.map (biproduct.π f j)) | def | category_theory.functor.biproduct_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 biproducts gives rise to a morphism
`F.obj (⨁ f) ⟶ ⨁ (F.obj ∘ f)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct_comparison_π (j : J) :
biproduct_comparison F f ≫ biproduct.π _ j = F.map (biproduct.π f j) | biproduct.lift_π _ _ | lemma | category_theory.functor.biproduct_comparison_π | 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_comparison' : ⨁ (F.obj ∘ f) ⟶ F.obj (⨁ f) | biproduct.desc (λ j, F.map (biproduct.ι f j)) | def | category_theory.functor.biproduct_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 biproducts gives rise to a morphism
`⨁ (F.obj ∘ f) ⟶ F.obj (⨁ f)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_biproduct_comparison' (j : J) :
biproduct.ι _ j ≫ biproduct_comparison' F f = F.map (biproduct.ι f j) | biproduct.ι_desc _ _ | lemma | category_theory.functor.ι_biproduct_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 | |
biproduct_comparison'_comp_biproduct_comparison :
biproduct_comparison' F f ≫ biproduct_comparison F f = 𝟙 (⨁ (F.obj ∘ f)) | by { classical, ext, simp [biproduct.ι_π, ← functor.map_comp, eq_to_hom_map] } | lemma | category_theory.functor.biproduct_comparison'_comp_biproduct_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_biproduct_of_mono_biproduct_comparison`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
split_epi_biproduct_comparison : split_epi (biproduct_comparison F f) | ⟨biproduct_comparison' F f⟩ | def | category_theory.functor.split_epi_biproduct_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"
] | [] | `biproduct_comparison F f` is a split epimorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
split_mono_biproduct_comparison' : split_mono (biproduct_comparison' F f) | ⟨biproduct_comparison F f⟩ | def | category_theory.functor.split_mono_biproduct_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"
] | [] | `biproduct_comparison' F f` is a split monomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_biproduct_of_preserves : has_biproduct (F.obj ∘ f) | has_biproduct.mk
{ bicone := F.map_bicone (biproduct.bicone f),
is_bilimit := preserves_biproduct.preserves (biproduct.is_bilimit _) } | instance | category_theory.functor.has_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_biproduct : F.obj (⨁ f) ≅ ⨁ (F.obj ∘ f) | biproduct.unique_up_to_iso _ (preserves_biproduct.preserves (biproduct.is_bilimit _)) | def | category_theory.functor.map_biproduct | 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 biproduct, we get a definitionally nice isomorphism
`F.obj (⨁ f) ≅ ⨁ (F.obj ∘ f)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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