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values | symbolic_name stringlengths 1 131 | library stringclasses 417
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coe_comp {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) :
(f ≫ g : X → Z) = g ∘ f | (forget _).map_comp f g | lemma | category_theory.coe_comp | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [
"map_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply {X : C} (x : X) : ((𝟙 X) : X → X) x = x | congr_fun ((forget _).map_id X) x | lemma | category_theory.id_apply | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [
"map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :
(f ≫ g) x = g (f x) | congr_fun ((forget _).map_comp _ _) x | lemma | category_theory.comp_apply | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [
"map_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete_category.congr_hom {X Y : C} {f g : X ⟶ Y} (h : f = g) (x : X) : f x = g x | congr_fun (congr_arg (λ f : X ⟶ Y, (f : X → Y)) h) x | lemma | category_theory.concrete_category.congr_hom | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete_category.congr_arg {X Y : C} (f : X ⟶ Y) {x x' : X} (h : x = x') : f x = f x' | congr_arg (f : X → Y) h | lemma | category_theory.concrete_category.congr_arg | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete_category.mono_of_injective {X Y : C} (f : X ⟶ Y) (i : function.injective f) :
mono f | (forget C).mono_of_mono_map ((mono_iff_injective f).2 i) | lemma | category_theory.concrete_category.mono_of_injective | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | In any concrete category, injective morphisms are monomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concrete_category.injective_of_mono_of_preserves_pullback {X Y : C} (f : X ⟶ Y) [mono f]
[preserves_limits_of_shape walking_cospan (forget C)] : function.injective f | (mono_iff_injective ((forget C).map f)).mp infer_instance | lemma | category_theory.concrete_category.injective_of_mono_of_preserves_pullback | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete_category.mono_iff_injective_of_preserves_pullback {X Y : C} (f : X ⟶ Y)
[preserves_limits_of_shape walking_cospan (forget C)] : mono f ↔ function.injective f | ((forget C).mono_map_iff_mono _).symm.trans (mono_iff_injective _) | lemma | category_theory.concrete_category.mono_iff_injective_of_preserves_pullback | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete_category.epi_of_surjective {X Y : C} (f : X ⟶ Y) (s : function.surjective f) :
epi f | (forget C).epi_of_epi_map ((epi_iff_surjective f).2 s) | lemma | category_theory.concrete_category.epi_of_surjective | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | In any concrete category, surjective morphisms are epimorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concrete_category.surjective_of_epi_of_preserves_pushout {X Y : C} (f : X ⟶ Y) [epi f]
[preserves_colimits_of_shape walking_span (forget C)] : function.surjective f | (epi_iff_surjective ((forget C).map f)).mp infer_instance | lemma | category_theory.concrete_category.surjective_of_epi_of_preserves_pushout | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete_category.epi_iff_surjective_of_preserves_pushout {X Y : C} (f : X ⟶ Y)
[preserves_colimits_of_shape walking_span (forget C)] : epi f ↔ function.surjective f | ((forget C).epi_map_iff_epi _).symm.trans (epi_iff_surjective _) | lemma | category_theory.concrete_category.epi_iff_surjective_of_preserves_pushout | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete_category.bijective_of_is_iso {X Y : C} (f : X ⟶ Y) [is_iso f] :
function.bijective ((forget C).map f) | by { rw ← is_iso_iff_bijective, apply_instance, } | lemma | category_theory.concrete_category.bijective_of_is_iso | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete_category.has_coe_to_fun_Type {X Y : Type u} (f : X ⟶ Y) :
coe_fn f = f | rfl | lemma | category_theory.concrete_category.has_coe_to_fun_Type | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_forget₂ (C : Type u) (D : Type u') [category.{v} C] [concrete_category.{w} C]
[category.{v'} D] [concrete_category.{w} D] | (forget₂ : C ⥤ D)
(forget_comp : forget₂ ⋙ (forget D) = forget C . obviously) | class | category_theory.has_forget₂ | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | `has_forget₂ C D`, where `C` and `D` are both concrete categories, provides a functor
`forget₂ C D : C ⥤ D` and a proof that `forget₂ ⋙ (forget D) = forget C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂ (C : Type u) (D : Type u') [category.{v} C] [concrete_category.{w} C]
[category.{v'} D] [concrete_category.{w} D] [has_forget₂ C D] : C ⥤ D | has_forget₂.forget₂ | def | category_theory.forget₂ | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | The forgetful functor `C ⥤ D` between concrete categories for which we have an instance
`has_forget₂ C `. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_faithful (C : Type u) (D : Type u') [category.{v} C] [concrete_category.{w} C]
[category.{v'} D] [concrete_category.{w} D] [has_forget₂ C D] : faithful (forget₂ C D) | has_forget₂.forget_comp.faithful_of_comp | instance | category_theory.forget₂_faithful | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget₂_preserves_monomorphisms (C : Type u) (D : Type u')
[category.{v} C] [concrete_category.{w} C]
[category.{v'} D] [concrete_category.{w} D] [has_forget₂ C D]
[(forget C).preserves_monomorphisms] : (forget₂ C D).preserves_monomorphisms | have (forget₂ C D ⋙ forget D).preserves_monomorphisms,
by { simp only [has_forget₂.forget_comp], apply_instance },
by exactI functor.preserves_monomorphisms_of_preserves_of_reflects _ (forget D) | instance | category_theory.forget₂_preserves_monomorphisms | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget₂_preserves_epimorphisms (C : Type u) (D : Type u')
[category.{v} C] [concrete_category.{w} C]
[category.{v'} D] [concrete_category.{w} D] [has_forget₂ C D]
[(forget C).preserves_epimorphisms] : (forget₂ C D).preserves_epimorphisms | have (forget₂ C D ⋙ forget D).preserves_epimorphisms,
by { simp only [has_forget₂.forget_comp], apply_instance },
by exactI functor.preserves_epimorphisms_of_preserves_of_reflects _ (forget D) | instance | category_theory.forget₂_preserves_epimorphisms | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_category.concrete_category {C : Type u} {D : Type u'} [category.{v'} D]
[concrete_category.{w} D] (f : C → D) :
concrete_category.{w} (induced_category D f) | { forget := induced_functor f ⋙ forget D } | instance | category_theory.induced_category.concrete_category | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_category.has_forget₂
{C : Type u} {D : Type u'} [category.{v'} D] [concrete_category.{w} D]
(f : C → D) :
has_forget₂ (induced_category D f) D | { forget₂ := induced_functor f,
forget_comp := rfl } | instance | category_theory.induced_category.has_forget₂ | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
full_subcategory.concrete_category {C : Type u} [category.{v} C] [concrete_category.{w} C]
(Z : C → Prop) : concrete_category (full_subcategory Z) | { forget := full_subcategory_inclusion Z ⋙ forget C } | instance | category_theory.full_subcategory.concrete_category | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
full_subcategory.has_forget₂ {C : Type u} [category.{v} C] [concrete_category.{w} C]
(Z : C → Prop) : has_forget₂ (full_subcategory Z) C | { forget₂ := full_subcategory_inclusion Z,
forget_comp := rfl } | instance | category_theory.full_subcategory.has_forget₂ | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_forget₂.mk' {C : Type u} {D : Type u'} [category.{v} C] [concrete_category.{w} C]
[category.{v'} D] [concrete_category.{w} D] (obj : C → D)
(h_obj : ∀ X, (forget D).obj (obj X) = (forget C).obj X)
(map : Π {X Y}, (X ⟶ Y) → (obj X ⟶ obj Y))
(h_map : ∀ {X Y} {f : X ⟶ Y}, (forget D).map (map f) == (forget C).m... | { forget₂ := faithful.div _ _ _ @h_obj _ @h_map,
forget_comp := by apply faithful.div_comp } | def | category_theory.has_forget₂.mk' | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | In order to construct a “partially forgetting” functor, we do not need to verify functor laws;
it suffices to ensure that compositions agree with `forget₂ C D ⋙ forget D = forget C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_forget_to_Type (C : Type u) [category.{v} C] [concrete_category.{w} C] :
has_forget₂ C (Type w) | { forget₂ := forget C,
forget_comp := functor.comp_id _ } | def | category_theory.has_forget_to_Type | category_theory.concrete_category | src/category_theory/concrete_category/basic.lean | [
"category_theory.types",
"category_theory.functor.epi_mono",
"category_theory.limits.constructions.epi_mono"
] | [] | Every forgetful functor factors through the identity functor. This is not a global instance as
it is prone to creating type class resolution loops. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bundled (c : Type u → Type v) : Type (max (u+1) v) | (α : Type u)
(str : c α . tactic.apply_instance) | structure | category_theory.bundled | category_theory.concrete_category | src/category_theory/concrete_category/bundled.lean | [
"tactic.lint"
] | [] | `bundled` is a type bundled with a type class instance for that type. Only
the type class is exposed as a parameter. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of {c : Type u → Type v} (α : Type u) [str : c α] : bundled c | ⟨α, str⟩ | def | category_theory.bundled.of | category_theory.concrete_category | src/category_theory/concrete_category/bundled.lean | [
"tactic.lint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (α) (str) : (@bundled.mk c α str : Type u) = α | rfl | lemma | category_theory.bundled.coe_mk | category_theory.concrete_category | src/category_theory/concrete_category/bundled.lean | [
"tactic.lint"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (f : Π {α}, c α → d α) (b : bundled c) : bundled d | ⟨b, f b.str⟩ | def | category_theory.bundled.map | category_theory.concrete_category | src/category_theory/concrete_category/bundled.lean | [
"tactic.lint"
] | [] | Map over the bundled structure | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bundled_hom | (to_fun : Π {α β : Type u} (Iα : c α) (Iβ : c β), hom Iα Iβ → α → β)
(id : Π {α : Type u} (I : c α), hom I I)
(comp : Π {α β γ : Type u} (Iα : c α) (Iβ : c β) (Iγ : c γ),
hom Iβ Iγ → hom Iα Iβ → hom Iα Iγ)
(hom_ext : ∀ {α β : Type u} (Iα : c α) (Iβ : c β), function.injective (to_fun Iα Iβ) . obviously)
(id_to_fun : ∀... | structure | category_theory.bundled_hom | category_theory.concrete_category | src/category_theory/concrete_category/bundled_hom.lean | [
"category_theory.concrete_category.basic",
"category_theory.concrete_category.bundled"
] | [
"hom_ext"
] | Class for bundled homs. Note that the arguments order follows that of lemmas for `monoid_hom`.
This way we can use `⟨@monoid_hom.to_fun, @monoid_hom.id ...⟩` in an instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
category : category (bundled c) | by refine
{ hom := λ X Y, @hom X Y X.str Y.str,
id := λ X, @bundled_hom.id c hom 𝒞 X X.str,
comp := λ X Y Z f g, @bundled_hom.comp c hom 𝒞 X Y Z X.str Y.str Z.str g f,
comp_id' := _,
id_comp' := _,
assoc' := _};
intros; apply 𝒞.hom_ext;
simp only [𝒞.id_to_fun, 𝒞.comp_to_fun, function.left_id, function.... | instance | category_theory.bundled_hom.category | category_theory.concrete_category | src/category_theory/concrete_category/bundled_hom.lean | [
"category_theory.concrete_category.basic",
"category_theory.concrete_category.bundled"
] | [] | Every `@bundled_hom c _` defines a category with objects in `bundled c`.
This instance generates the type-class problem `bundled_hom ?m` (which is why this is marked as
`[nolint]`). Currently that is not a problem, as there are almost no instances of `bundled_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concrete_category : concrete_category.{u} (bundled c) | { forget := { obj := λ X, X,
map := λ X Y f, 𝒞.to_fun X.str Y.str f,
map_id' := λ X, 𝒞.id_to_fun X.str,
map_comp' := by intros; erw 𝒞.comp_to_fun; refl },
forget_faithful := { map_injective' := by intros; apply 𝒞.hom_ext } } | instance | category_theory.bundled_hom.concrete_category | category_theory.concrete_category | src/category_theory/concrete_category/bundled_hom.lean | [
"category_theory.concrete_category.basic",
"category_theory.concrete_category.bundled"
] | [] | A category given by `bundled_hom` is a concrete category.
This instance generates the type-class problem `bundled_hom ?m` (which is why this is marked as
`[nolint]`). Currently that is not a problem, as there are almost no instances of `bundled_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_has_forget₂ {d : Type u → Type u} {hom_d : Π ⦃α β : Type u⦄ (Iα : d α) (Iβ : d β), Type u}
[bundled_hom hom_d] (obj : Π ⦃α⦄, c α → d α)
(map : Π {X Y : bundled c}, (X ⟶ Y) → ((bundled.map obj X) ⟶ (bundled.map obj Y)))
(h_map : ∀ {X Y : bundled c} (f : X ⟶ Y), (map f : X → Y) = f)
: has_forget₂ (bundled c) (... | has_forget₂.mk'
(bundled.map @obj)
(λ _, rfl)
@map
(by intros; apply heq_of_eq; apply h_map) | def | category_theory.bundled_hom.mk_has_forget₂ | category_theory.concrete_category | src/category_theory/concrete_category/bundled_hom.lean | [
"category_theory.concrete_category.basic",
"category_theory.concrete_category.bundled"
] | [] | A version of `has_forget₂.mk'` for categories defined using `@bundled_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_hom (F : Π {α}, d α → c α) : Π ⦃α β : Type u⦄ (Iα : d α) (Iβ : d β), Type u | λ α β iα iβ, hom (F iα) (F iβ) | def | category_theory.bundled_hom.map_hom | category_theory.concrete_category | src/category_theory/concrete_category/bundled_hom.lean | [
"category_theory.concrete_category.basic",
"category_theory.concrete_category.bundled"
] | [] | The `hom` corresponding to first forgetting along `F`, then taking the `hom` associated to `c`.
For typical usage, see the construction of `CommMon` from `Mon`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map (F : Π {α}, d α → c α) : bundled_hom (map_hom hom @F) | { to_fun := λ α β iα iβ f, 𝒞.to_fun (F iα) (F iβ) f,
id := λ α iα, 𝒞.id (F iα),
comp := λ α β γ iα iβ iγ f g, 𝒞.comp (F iα) (F iβ) (F iγ) f g,
hom_ext := λ α β iα iβ f g h, 𝒞.hom_ext (F iα) (F iβ) h } | def | category_theory.bundled_hom.map | category_theory.concrete_category | src/category_theory/concrete_category/bundled_hom.lean | [
"category_theory.concrete_category.basic",
"category_theory.concrete_category.bundled"
] | [
"hom_ext"
] | Construct the `bundled_hom` induced by a map between type classes.
This is useful for building categories such as `CommMon` from `Mon`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
parent_projection (F : Π {α}, d α → c α) | class | category_theory.bundled_hom.parent_projection | category_theory.concrete_category | src/category_theory/concrete_category/bundled_hom.lean | [
"category_theory.concrete_category.basic",
"category_theory.concrete_category.bundled"
] | [] | We use the empty `parent_projection` class to label functions like `comm_monoid.to_monoid`,
which we would like to use to automatically construct `bundled_hom` instances from.
Once we've set up `Mon` as the category of bundled monoids,
this allows us to set up `CommMon` by defining an instance
```instance : parent_pro... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bundled_hom_of_parent_projection (F : Π {α}, d α → c α) [parent_projection @F] :
bundled_hom (map_hom hom @F) | map hom @F | instance | category_theory.bundled_hom.bundled_hom_of_parent_projection | category_theory.concrete_category | src/category_theory/concrete_category/bundled_hom.lean | [
"category_theory.concrete_category.basic",
"category_theory.concrete_category.bundled"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget₂ (F : Π {α}, d α → c α) [parent_projection @F] :
has_forget₂ (bundled d) (bundled c) | { forget₂ :=
{ obj := λ X, ⟨X, F X.2⟩,
map := λ X Y f, f } } | instance | category_theory.bundled_hom.forget₂ | category_theory.concrete_category | src/category_theory/concrete_category/bundled_hom.lean | [
"category_theory.concrete_category.basic",
"category_theory.concrete_category.bundled"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget₂_full (F : Π {α}, d α → c α) [parent_projection @F] :
full (forget₂ (bundled d) (bundled c)) | { preimage := λ X Y f, f } | instance | category_theory.bundled_hom.forget₂_full | category_theory.concrete_category | src/category_theory/concrete_category/bundled_hom.lean | [
"category_theory.concrete_category.basic",
"category_theory.concrete_category.bundled"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_isomorphisms_forget₂ [has_forget₂ C D] [reflects_isomorphisms (forget C)] :
reflects_isomorphisms (forget₂ C D) | { reflects := λ X Y f i,
begin
resetI,
haveI i' : is_iso ((forget D).map ((forget₂ C D).map f)) := functor.map_is_iso (forget D) _,
haveI : is_iso ((forget C).map f) :=
begin
have := has_forget₂.forget_comp,
dsimp at this,
rw ←this,
exact i',
end,
apply is_iso_of_reflec... | lemma | category_theory.reflects_isomorphisms_forget₂ | category_theory.concrete_category | src/category_theory/concrete_category/reflects_isomorphisms.lean | [
"category_theory.concrete_category.basic",
"category_theory.functor.reflects_isomorphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unbundled_hom {c : Type u → Type u} (hom : Π {α β}, c α → c β → (α → β) → Prop) | (hom_id [] : ∀ {α} (ia : c α), hom ia ia id)
(hom_comp [] : ∀ {α β γ} {Iα : c α} {Iβ : c β} {Iγ : c γ} {g : β → γ} {f : α → β}
(hg : hom Iβ Iγ g) (hf : hom Iα Iβ f), hom Iα Iγ (g ∘ f)) | class | category_theory.unbundled_hom | category_theory.concrete_category | src/category_theory/concrete_category/unbundled_hom.lean | [
"category_theory.concrete_category.bundled_hom"
] | [] | A class for unbundled homs used to define a category. `hom` must
take two types `α`, `β` and instances of the corresponding structures,
and return a predicate on `α → β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bundled_hom : bundled_hom (λ α β (Iα : c α) (Iβ : c β), subtype (hom Iα Iβ)) | { to_fun := λ _ _ _ _, subtype.val,
id := λ α Iα, ⟨id, hom_id hom Iα⟩,
id_to_fun := by intros; refl,
comp := λ _ _ _ _ _ _ g f, ⟨g.1 ∘ f.1, hom_comp c g.2 f.2⟩,
comp_to_fun := by intros; refl,
hom_ext := by intros; apply subtype.eq } | instance | category_theory.unbundled_hom.bundled_hom | category_theory.concrete_category | src/category_theory/concrete_category/unbundled_hom.lean | [
"category_theory.concrete_category.bundled_hom"
] | [
"hom_ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_has_forget₂ : has_forget₂ (bundled c) (bundled c') | bundled_hom.mk_has_forget₂ obj (λ X Y f, ⟨f.val, map f.property⟩) (λ _ _ _, rfl) | def | category_theory.unbundled_hom.mk_has_forget₂ | category_theory.concrete_category | src/category_theory/concrete_category/unbundled_hom.lean | [
"category_theory.concrete_category.bundled_hom"
] | [] | A custom constructor for forgetful functor
between concrete categories defined using `unbundled_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra (F : C ⥤ C) | (A : C)
(str : F.obj A ⟶ A) | structure | category_theory.endofunctor.algebra | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [
"algebra"
] | An algebra of an endofunctor; `str` stands for "structure morphism" | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom (A₀ A₁ : algebra F) | (f : A₀.1 ⟶ A₁.1)
(h' : F.map f ≫ A₁.str = A₀.str ≫ f . obviously) | structure | category_theory.endofunctor.algebra.hom | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [
"algebra"
] | A morphism between algebras of endofunctor `F` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id : hom A A | { f := 𝟙 _ } | def | category_theory.endofunctor.algebra.hom.id | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | The identity morphism of an algebra of endofunctor `F` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp (f : hom A₀ A₁) (g : hom A₁ A₂) : hom A₀ A₂ | { f := f.1 ≫ g.1 } | def | category_theory.endofunctor.algebra.hom.comp | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | The composition of morphisms between algebras of endofunctor `F` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_eq_id : algebra.hom.id A = 𝟙 A | rfl | lemma | category_theory.endofunctor.algebra.id_eq_id | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_f : (𝟙 _ : A ⟶ A).1 = 𝟙 A.1 | rfl | lemma | category_theory.endofunctor.algebra.id_f | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_eq_comp : algebra.hom.comp f g = f ≫ g | rfl | lemma | category_theory.endofunctor.algebra.comp_eq_comp | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_f : (f ≫ g).1 = f.1 ≫ g.1 | rfl | lemma | category_theory.endofunctor.algebra.comp_f | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_mk (h : A₀.1 ≅ A₁.1) (w : F.map h.hom ≫ A₁.str = A₀.str ≫ h.hom) : A₀ ≅ A₁ | { hom := { f := h.hom },
inv :=
{ f := h.inv,
h' := by { rw [h.eq_comp_inv, category.assoc, ←w, ←functor.map_comp_assoc], simp } } } | def | category_theory.endofunctor.algebra.iso_mk | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | To construct an isomorphism of algebras, it suffices to give an isomorphism of the As which
commutes with the structure morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget (F : C ⥤ C) : algebra F ⥤ C | { obj := λ A, A.1,
map := λ A B f, f.1 } | def | category_theory.endofunctor.algebra.forget | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [
"algebra"
] | The forgetful functor from the category of algebras, forgetting the algebraic structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_of_iso (f : A₀ ⟶ A₁) [is_iso f.1] : is_iso f | ⟨⟨{ f := inv f.1,
h' := by { rw [is_iso.eq_comp_inv f.1, category.assoc, ← f.h], simp } }, by tidy⟩⟩ | lemma | category_theory.endofunctor.algebra.iso_of_iso | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | An algebra morphism with an underlying isomorphism hom in `C` is an algebra isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_reflects_iso : reflects_isomorphisms (forget F) | { reflects := λ A B, iso_of_iso } | instance | category_theory.endofunctor.algebra.forget_reflects_iso | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_faithful : faithful (forget F) | {} | instance | category_theory.endofunctor.algebra.forget_faithful | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_of_epi {X Y : algebra F} (f : X ⟶ Y) [h : epi f.1] : epi f | (forget F).epi_of_epi_map h | lemma | category_theory.endofunctor.algebra.epi_of_epi | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [
"algebra"
] | An algebra morphism with an underlying epimorphism hom in `C` is an algebra epimorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono_of_mono {X Y : algebra F} (f : X ⟶ Y) [h : mono f.1] : mono f | (forget F).mono_of_mono_map h | lemma | category_theory.endofunctor.algebra.mono_of_mono | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [
"algebra"
] | An algebra morphism with an underlying monomorphism hom in `C` is an algebra monomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor_of_nat_trans {F G : C ⥤ C} (α : G ⟶ F) : algebra F ⥤ algebra G | { obj := λ A,
{ A := A.1,
str := α.app A.1 ≫ A.str },
map := λ A₀ A₁ f,
{ f := f.1 } } | def | category_theory.endofunctor.algebra.functor_of_nat_trans | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [
"algebra"
] | From a natural transformation `α : G → F` we get a functor from
algebras of `F` to algebras of `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor_of_nat_trans_id :
functor_of_nat_trans (𝟙 F) ≅ 𝟭 _ | nat_iso.of_components
(λ X, iso_mk (iso.refl _) (by { dsimp, simp, }))
(λ X Y f, by { ext, dsimp, simp }) | def | category_theory.endofunctor.algebra.functor_of_nat_trans_id | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | The identity transformation induces the identity endofunctor on the category of algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor_of_nat_trans_comp {F₀ F₁ F₂ : C ⥤ C} (α : F₀ ⟶ F₁) (β : F₁ ⟶ F₂) :
functor_of_nat_trans (α ≫ β) ≅
functor_of_nat_trans β ⋙ functor_of_nat_trans α | nat_iso.of_components
(λ X, iso_mk (iso.refl _) (by { dsimp, simp }))
(λ X Y f, by { ext, dsimp, simp }) | def | category_theory.endofunctor.algebra.functor_of_nat_trans_comp | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | A composition of natural transformations gives the composition of corresponding functors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor_of_nat_trans_eq {F G : C ⥤ C} {α β : F ⟶ G} (h : α = β) :
functor_of_nat_trans α ≅ functor_of_nat_trans β | nat_iso.of_components
(λ X, iso_mk (iso.refl _) (by { dsimp, simp [h] }))
(λ X Y f, by { ext, dsimp, simp }) | def | category_theory.endofunctor.algebra.functor_of_nat_trans_eq | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | If `α` and `β` are two equal natural transformations, then the functors of algebras induced by them
are isomorphic.
We define it like this as opposed to using `eq_to_iso` so that the components are nicer to prove
lemmas about. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_of_nat_iso {F G : C ⥤ C} (α : F ≅ G) :
algebra F ≌ algebra G | { functor := functor_of_nat_trans α.inv,
inverse := functor_of_nat_trans α.hom,
unit_iso :=
functor_of_nat_trans_id.symm ≪≫
functor_of_nat_trans_eq (by simp) ≪≫
functor_of_nat_trans_comp _ _,
counit_iso :=
(functor_of_nat_trans_comp _ _).symm ≪≫
functor_of_nat_trans_eq (by simp) ≪≫
functor... | def | category_theory.endofunctor.algebra.equiv_of_nat_iso | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [
"algebra"
] | Naturally isomorphic endofunctors give equivalent categories of algebras.
Furthermore, they are equivalent as categories over `C`, that is,
we have `equiv_of_nat_iso h ⋙ forget = forget`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
str_inv : A.1 ⟶ F.obj A.1 | (h.to ⟨ F.obj A.1 , F.map A.str ⟩).1 | def | category_theory.endofunctor.algebra.initial.str_inv | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | The inverse of the structure map of an initial algebra | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_inv' : (⟨str_inv h ≫ A.str⟩ : A ⟶ A) = 𝟙 A | limits.is_initial.hom_ext h _ (𝟙 A) | lemma | category_theory.endofunctor.algebra.initial.left_inv' | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_inv : str_inv h ≫ A.str = 𝟙 _ | congr_arg hom.f (left_inv' h) | lemma | category_theory.endofunctor.algebra.initial.left_inv | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_inv : A.str ≫ str_inv h = 𝟙 _ | by { rw [str_inv, ← (h.to ⟨ F.obj A.1 , F.map A.str ⟩).h,
← F.map_id, ← F.map_comp], congr, exact (left_inv h) } | lemma | category_theory.endofunctor.algebra.initial.right_inv | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
str_is_iso (h : limits.is_initial A) : is_iso A.str | { out := ⟨ str_inv h, right_inv _ , left_inv _ ⟩ } | lemma | category_theory.endofunctor.algebra.initial.str_is_iso | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | The structure map of the inital algebra is an isomorphism,
hence endofunctors preserve their initial algebras | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coalgebra (F : C ⥤ C) | (V : C)
(str : V ⟶ F.obj V) | structure | category_theory.endofunctor.coalgebra | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | A coalgebra of an endofunctor; `str` stands for "structure morphism" | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom (V₀ V₁ : coalgebra F) | (f : V₀.1 ⟶ V₁.1)
(h' : V₀.str ≫ F.map f = f ≫ V₁.str . obviously) | structure | category_theory.endofunctor.coalgebra.hom | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | A morphism between coalgebras of an endofunctor `F` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id : hom V V | { f := 𝟙 _ } | def | category_theory.endofunctor.coalgebra.hom.id | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | The identity morphism of an algebra of endofunctor `F` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp (f : hom V₀ V₁) (g : hom V₁ V₂) : hom V₀ V₂ | { f := f.1 ≫ g.1 } | def | category_theory.endofunctor.coalgebra.hom.comp | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | The composition of morphisms between algebras of endofunctor `F` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_eq_id : coalgebra.hom.id V = 𝟙 V | rfl | lemma | category_theory.endofunctor.coalgebra.id_eq_id | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_f : (𝟙 _ : V ⟶ V).1 = 𝟙 V.1 | rfl | lemma | category_theory.endofunctor.coalgebra.id_f | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_eq_comp : coalgebra.hom.comp f g = f ≫ g | rfl | lemma | category_theory.endofunctor.coalgebra.comp_eq_comp | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_mk (h : V₀.1 ≅ V₁.1) (w : V₀.str ≫ F.map h.hom = h.hom ≫ V₁.str ) : V₀ ≅ V₁ | { hom := { f := h.hom },
inv :=
{ f := h.inv,
h' := by { rw [h.eq_inv_comp, ← category.assoc, ←w, category.assoc, ← functor.map_comp],
simp only [iso.hom_inv_id, functor.map_id, category.comp_id] } } } | def | category_theory.endofunctor.coalgebra.iso_mk | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [
"functor.map_id"
] | To construct an isomorphism of coalgebras, it suffices to give an isomorphism of the Vs which
commutes with the structure morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget (F : C ⥤ C) : coalgebra F ⥤ C | { obj := λ A, A.1,
map := λ A B f, f.1 } | def | category_theory.endofunctor.coalgebra.forget | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | The forgetful functor from the category of coalgebras, forgetting the coalgebraic structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_of_iso (f : V₀ ⟶ V₁) [is_iso f.1] : is_iso f | ⟨⟨{ f := inv f.1,
h' := by { rw [is_iso.eq_inv_comp f.1, ← category.assoc, ← f.h, category.assoc], simp } },
by tidy⟩⟩ | lemma | category_theory.endofunctor.coalgebra.iso_of_iso | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | A coalgebra morphism with an underlying isomorphism hom in `C` is a coalgebra isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
epi_of_epi {X Y : coalgebra F} (f : X ⟶ Y) [h : epi f.1] : epi f | (forget F).epi_of_epi_map h | lemma | category_theory.endofunctor.coalgebra.epi_of_epi | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | An algebra morphism with an underlying epimorphism hom in `C` is an algebra epimorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono_of_mono {X Y : coalgebra F} (f : X ⟶ Y) [h : mono f.1] : mono f | (forget F).mono_of_mono_map h | lemma | category_theory.endofunctor.coalgebra.mono_of_mono | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | An algebra morphism with an underlying monomorphism hom in `C` is an algebra monomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor_of_nat_trans {F G : C ⥤ C} (α : F ⟶ G) : coalgebra F ⥤ coalgebra G | { obj := λ V,
{ V := V.1,
str := V.str ≫ α.app V.1 },
map := λ V₀ V₁ f, { f := f.1,
h' := by rw [category.assoc, ← α.naturality, ← category.assoc, f.h, category.assoc] } } | def | category_theory.endofunctor.coalgebra.functor_of_nat_trans | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | From a natural transformation `α : F → G` we get a functor from
coalgebras of `F` to coalgebras of `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor_of_nat_trans_comp {F₀ F₁ F₂ : C ⥤ C} (α : F₀ ⟶ F₁) (β : F₁ ⟶ F₂) :
functor_of_nat_trans (α ≫ β) ≅
functor_of_nat_trans α ⋙ functor_of_nat_trans β | nat_iso.of_components
(λ X, iso_mk (iso.refl _) (by { dsimp, simp }))
(λ X Y f, by { ext, dsimp, simp }) | def | category_theory.endofunctor.coalgebra.functor_of_nat_trans_comp | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | A composition of natural transformations gives the composition of corresponding functors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_of_nat_iso {F G : C ⥤ C} (α : F ≅ G) :
coalgebra F ≌ coalgebra G | { functor := functor_of_nat_trans α.hom,
inverse := functor_of_nat_trans α.inv,
unit_iso :=
functor_of_nat_trans_id.symm ≪≫
functor_of_nat_trans_eq (by simp) ≪≫
functor_of_nat_trans_comp _ _,
counit_iso :=
(functor_of_nat_trans_comp _ _).symm ≪≫
functor_of_nat_trans_eq (by simp) ≪≫
functor... | def | category_theory.endofunctor.coalgebra.equiv_of_nat_iso | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [] | Naturally isomorphic endofunctors give equivalent categories of coalgebras.
Furthermore, they are equivalent as categories over `C`, that is,
we have `equiv_of_nat_iso h ⋙ forget = forget`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra.hom_equiv_naturality_str (adj : F ⊣ G) (A₁ A₂ : algebra F)
(f : A₁ ⟶ A₂) : (adj.hom_equiv A₁.A A₁.A) A₁.str ≫ G.map f.f =
f.f ≫ (adj.hom_equiv A₂.A A₂.A) A₂.str | by { rw [← adjunction.hom_equiv_naturality_right, ← adjunction.hom_equiv_naturality_left, f.h] } | lemma | category_theory.endofunctor.adjunction.algebra.hom_equiv_naturality_str | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [
"adj",
"algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coalgebra.hom_equiv_naturality_str_symm (adj : F ⊣ G) (V₁ V₂ : coalgebra G)
(f : V₁ ⟶ V₂) : F.map f.f ≫ ((adj.hom_equiv V₂.V V₂.V).symm) V₂.str =
((adj.hom_equiv V₁.V V₁.V).symm) V₁.str ≫ f.f | by { rw [← adjunction.hom_equiv_naturality_left_symm, ← adjunction.hom_equiv_naturality_right_symm,
f.h] } | lemma | category_theory.endofunctor.adjunction.coalgebra.hom_equiv_naturality_str_symm | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [
"adj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra.to_coalgebra_of (adj : F ⊣ G) : algebra F ⥤ coalgebra G | { obj := λ A, { V := A.1,
str := (adj.hom_equiv A.1 A.1).to_fun A.2 },
map := λ A₁ A₂ f, { f := f.1,
h' := (algebra.hom_equiv_naturality_str adj A₁ A₂ f) } } | def | category_theory.endofunctor.adjunction.algebra.to_coalgebra_of | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [
"adj",
"algebra"
] | Given an adjunction `F ⊣ G`, the functor that associates to an algebra over `F` a
coalgebra over `G` defined via adjunction applied to the structure map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coalgebra.to_algebra_of (adj : F ⊣ G) : coalgebra G ⥤ algebra F | { obj := λ V, { A := V.1,
str := (adj.hom_equiv V.1 V.1).inv_fun V.2 },
map := λ V₁ V₂ f, { f := f.1,
h' := (coalgebra.hom_equiv_naturality_str_symm adj V₁ V₂ f) } } | def | category_theory.endofunctor.adjunction.coalgebra.to_algebra_of | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [
"adj",
"algebra",
"inv_fun"
] | Given an adjunction `F ⊣ G`, the functor that associates to a coalgebra over `G` an algebra over
`F` defined via adjunction applied to the structure map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
alg_coalg_equiv.unit_iso (adj : F ⊣ G) :
𝟭 (algebra F) ≅ (algebra.to_coalgebra_of adj) ⋙ (coalgebra.to_algebra_of adj) | { hom :=
{ app := λ A,
{ f := (𝟙 A.1),
h' := by { erw [F.map_id, category.id_comp, category.comp_id],
apply (adj.hom_equiv _ _).left_inv A.str } },
naturality' := λ A₁ A₂ f, by { ext1, dsimp, erw [category.id_comp, category.comp_id], refl } },
inv :=
{ app := λ A,
{ f := (𝟙 A.... | def | category_theory.endofunctor.adjunction.alg_coalg_equiv.unit_iso | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [
"adj",
"algebra"
] | Given an adjunction, assigning to an algebra over the left adjoint a coalgebra over its right
adjoint and going back is isomorphic to the identity functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
alg_coalg_equiv.counit_iso (adj : F ⊣ G) :
(coalgebra.to_algebra_of adj) ⋙ (algebra.to_coalgebra_of adj) ≅ 𝟭 (coalgebra G) | { hom :=
{ app := λ V,
{ f := (𝟙 V.1),
h' := by { dsimp, erw [G.map_id, category.id_comp, category.comp_id],
apply (adj.hom_equiv _ _).right_inv V.str } },
naturality' := λ V₁ V₂ f,
by { ext1, dsimp, erw [category.comp_id, category.id_comp], refl, } },
inv :=
{ app := λ V,
... | def | category_theory.endofunctor.adjunction.alg_coalg_equiv.counit_iso | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [
"adj"
] | Given an adjunction, assigning to a coalgebra over the right adjoint an algebra over the left
adjoint and going back is isomorphic to the identity functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra_coalgebra_equiv (adj : F ⊣ G) : algebra F ≌ coalgebra G | { functor := algebra.to_coalgebra_of adj,
inverse := coalgebra.to_algebra_of adj,
unit_iso := alg_coalg_equiv.unit_iso adj,
counit_iso := alg_coalg_equiv.counit_iso adj,
functor_unit_iso_comp' := λ A, by { ext, exact category.comp_id _ } } | def | category_theory.endofunctor.adjunction.algebra_coalgebra_equiv | category_theory.endofunctor | src/category_theory/endofunctor/algebra.lean | [
"category_theory.functor.reflects_isomorphisms",
"category_theory.limits.shapes.terminal"
] | [
"adj",
"algebra"
] | If `F` is left adjoint to `G`, then the category of algebras over `F` is equivalent to the
category of coalgebras over `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
enriched_category (C : Type u₁) | (hom : C → C → V)
(notation X ` ⟶[] ` Y:10 := hom X Y)
(id : Π X, 𝟙_ V ⟶ (X ⟶[] X))
(comp : Π X Y Z, (X ⟶[] Y) ⊗ (Y ⟶[] Z) ⟶ (X ⟶[] Z))
(id_comp : Π X Y, (λ_ (X ⟶[] Y)).inv ≫ (id X ⊗ 𝟙 _) ≫ comp X X Y = 𝟙 _ . obviously)
(comp_id : Π X Y, (ρ_ (X ⟶[] Y)).inv ≫ (𝟙 _ ⊗ id Y) ≫ comp X Y Y = 𝟙 _ . obviously)
(assoc :
... | class | category_theory.enriched_category | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | A `V`-category is a category enriched in a monoidal category `V`.
Note that we do not assume that `V` is a concrete category,
so there may not be an "honest" underlying category at all! | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
e_id (X : C) : 𝟙_ V ⟶ (X ⟶[V] X) | enriched_category.id X | def | category_theory.e_id | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | The `𝟙_ V`-shaped generalized element giving the identity in a `V`-enriched category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
e_comp (X Y Z : C) : (X ⟶[V] Y) ⊗ (Y ⟶[V] Z) ⟶ (X ⟶[V] Z) | enriched_category.comp X Y Z | def | category_theory.e_comp | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | The composition `V`-morphism for a `V`-enriched category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
e_id_comp (X Y : C) :
(λ_ (X ⟶[V] Y)).inv ≫ (e_id V X ⊗ 𝟙 _) ≫ e_comp V X X Y = 𝟙 (X ⟶[V] Y) | enriched_category.id_comp X Y | lemma | category_theory.e_id_comp | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
e_comp_id (X Y : C) :
(ρ_ (X ⟶[V] Y)).inv ≫ (𝟙 _ ⊗ e_id V Y) ≫ e_comp V X Y Y = 𝟙 (X ⟶[V] Y) | enriched_category.comp_id X Y | lemma | category_theory.e_comp_id | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
e_assoc (W X Y Z : C) :
(α_ _ _ _).inv ≫ (e_comp V W X Y ⊗ 𝟙 _) ≫ e_comp V W Y Z =
(𝟙 _ ⊗ e_comp V X Y Z) ≫ e_comp V W X Z | enriched_category.assoc W X Y Z | lemma | category_theory.e_assoc | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
transport_enrichment (F : lax_monoidal_functor V W) (C : Type u₁) | C | def | category_theory.transport_enrichment | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | A type synonym for `C`, which should come equipped with a `V`-enriched category structure.
In a moment we will equip this with the `W`-enriched category structure
obtained by applying the functor `F : lax_monoidal_functor V W` to each hom object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
category_of_enriched_category_Type (C : Type u₁) [𝒞 : enriched_category (Type v) C] :
category.{v} C | { hom := 𝒞.hom,
id := λ X, e_id (Type v) X punit.star,
comp := λ X Y Z f g, e_comp (Type v) X Y Z ⟨f, g⟩,
id_comp' := λ X Y f, congr_fun (e_id_comp (Type v) X Y) f,
comp_id' := λ X Y f, congr_fun (e_comp_id (Type v) X Y) f,
assoc' := λ W X Y Z f g h, (congr_fun (e_assoc (Type v) W X Y Z) ⟨f, g, h⟩ : _), } | def | category_theory.category_of_enriched_category_Type | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | Construct an honest category from a `Type v`-enriched category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
enriched_category_Type_of_category (C : Type u₁) [𝒞 : category.{v} C] :
enriched_category (Type v) C | { hom := 𝒞.hom,
id := λ X p, 𝟙 X,
comp := λ X Y Z p, p.1 ≫ p.2,
id_comp := λ X Y, by { ext, simp, },
comp_id := λ X Y, by { ext, simp, },
assoc := λ W X Y Z, by { ext ⟨f, g, h⟩, simp, }, } | def | category_theory.enriched_category_Type_of_category | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | Construct a `Type v`-enriched category from an honest category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
enriched_category_Type_equiv_category (C : Type u₁) :
(enriched_category (Type v) C) ≃ category.{v} C | { to_fun := λ 𝒞, by exactI category_of_enriched_category_Type C,
inv_fun := λ 𝒞, by exactI enriched_category_Type_of_category C,
left_inv := λ 𝒞, begin
cases 𝒞,
dsimp [enriched_category_Type_of_category],
congr,
{ ext X ⟨⟩, refl, },
{ ext X Y Z ⟨f, g⟩, refl, }
end,
right_inv := λ 𝒞, by ... | def | category_theory.enriched_category_Type_equiv_category | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [
"inv_fun"
] | We verify that an enriched category in `Type u` is just the same thing as an honest category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_enrichment
(W : Type (v+1)) [category.{v} W] [monoidal_category W] (C : Type u₁) [enriched_category W C] | C | def | category_theory.forget_enrichment | category_theory.enriched | src/category_theory/enriched/basic.lean | [
"category_theory.monoidal.types.symmetric",
"category_theory.monoidal.types.coyoneda",
"category_theory.monoidal.center",
"tactic.apply_fun"
] | [] | A type synonym for `C`, which should come equipped with a `V`-enriched category structure.
In a moment we will equip this with the (honest) category structure
so that `X ⟶ Y` is `(𝟙_ W) ⟶ (X ⟶[W] Y)`.
We obtain this category by
transporting the enrichment in `V` along the lax monoidal functor `coyoneda_tensor_unit`,
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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