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[Merged by Bors] - feat(category_theory/limits/types): is_limit versions of limits in type #4130

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[Merged by Bors] - feat(category_theory/limits/types): is_limit versions of limits in type #4130

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28 changes: 23 additions & 5 deletions src/category_theory/limits/types.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -40,13 +40,34 @@ instance : has_limits (Type u) :=
{ has_limit := λ F, has_limit.mk
{ cone := limit_cone F, is_limit := limit_cone_is_limit F } } }

/--
The equivalence between a limiting cone of `F` in `Type u` and the "concrete" definition as the
sections of `F`.
-/
def is_limit_equiv_sections {F : J ⥤ Type u} {c : cone F} (t : is_limit c) :
c.X ≃ F.sections :=
(is_limit.cone_point_unique_up_to_iso t (limit_cone_is_limit F)).to_equiv

@[simp]
lemma is_limit_equiv_sections_apply {F : J ⥤ Type u} {c : cone F} (t : is_limit c) (j : J) (x : c.X) :
(((is_limit_equiv_sections t) x) : Π j, F.obj j) j = c.π.app j x :=
rfl

@[simp]
lemma is_limit_equiv_sections_symm_apply {F : J ⥤ Type u} {c : cone F} (t : is_limit c) (x : F.sections) (j : J) :
c.π.app j ((is_limit_equiv_sections t).symm x) = (x : Π j, F.obj j) j :=
begin
equiv_rw (is_limit_equiv_sections t).symm at x,
simp,
end

/--
The equivalence between the abstract limit of `F` in `Type u`
and the "concrete" definition as the sections of `F`.
-/
noncomputable
def limit_equiv_sections (F : J ⥤ Type u) : (limit F : Type u) ≃ F.sections :=
(is_limit.cone_point_unique_up_to_iso (limit.is_limit F) (limit_cone_is_limit F)).to_equiv
is_limit_equiv_sections (limit.is_limit _)

@[simp]
lemma limit_equiv_sections_apply (F : J ⥤ Type u) (x : limit F) (j : J) :
Expand All @@ -56,10 +77,7 @@ rfl
@[simp]
lemma limit_equiv_sections_symm_apply (F : J ⥤ Type u) (x : F.sections) (j : J) :
limit.π F j ((limit_equiv_sections F).symm x) = (x : Π j, F.obj j) j :=
begin
equiv_rw (limit_equiv_sections F).symm at x,
simp,
end
is_limit_equiv_sections_symm_apply _ _ _

/--
Construct a term of `limit F : Type u` from a family of terms `x : Π j, F.obj j`
Expand Down