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feat(category_theory/limits): the isomorphism expressing preservation of chosen limits #2192

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10 changes: 10 additions & 0 deletions src/category_theory/limits/limits.lean
Original file line number Diff line number Diff line change
Expand Up @@ -70,6 +70,11 @@ def unique_up_to_iso {s t : cone F} (P : is_limit s) (Q : is_limit t) : s ≅ t
hom_inv_id' := P.uniq_cone_morphism,
inv_hom_id' := Q.uniq_cone_morphism }

/-- Limits of `F` are unique up to isomorphism. -/
-- We may later want to prove the coherence of these isomorphisms.
def cone_point_unique_up_to_iso {s t : cone F} (P : is_limit s) (Q : is_limit t) : s.X ≅ t.X :=
(cones.forget F).map_iso (unique_up_to_iso P Q)

/-- Transport evidence that a cone is a limit cone across an isomorphism of cones. -/
def of_iso_limit {r t : cone F} (P : is_limit r) (i : r ≅ t) : is_limit t :=
is_limit.mk_cone_morphism
Expand Down Expand Up @@ -274,6 +279,11 @@ def unique_up_to_iso {s t : cocone F} (P : is_colimit s) (Q : is_colimit t) : s
hom_inv_id' := P.uniq_cocone_morphism,
inv_hom_id' := Q.uniq_cocone_morphism }

/-- Colimits of `F` are unique up to isomorphism. -/
-- We may later want to prove the coherence of these isomorphisms.
def cone_point_unique_up_to_iso {s t : cocone F} (P : is_colimit s) (Q : is_colimit t) : s.X ≅ t.X :=
(cocones.forget F).map_iso (unique_up_to_iso P Q)

/-- Transport evidence that a cocone is a colimit cocone across an isomorphism of cocones. -/
def of_iso_colimit {r t : cocone F} (P : is_colimit r) (i : r ≅ t) : is_colimit t :=
is_colimit.mk_cocone_morphism
Expand Down
9 changes: 9 additions & 0 deletions src/category_theory/limits/preserves.lean
Original file line number Diff line number Diff line change
Expand Up @@ -49,6 +49,15 @@ class preserves_limit (K : J ⥤ C) (F : C ⥤ D) : Type (max u₁ u₂ v) :=
class preserves_colimit (K : J ⥤ C) (F : C ⥤ D) : Type (max u₁ u₂ v) :=
(preserves : Π {c : cocone K}, is_colimit c → is_colimit (F.map_cocone c))

/-- A functor which preserves limits preserves chosen limits up to isomorphism. -/
def preserves_limit_iso (K : J ⥤ C) [has_limit.{v} K] (F : C ⥤ D) [has_limit.{v} (K ⋙ F)] [preserves_limit K F] :
F.obj (limit K) ≅ limit (K ⋙ F) :=
is_limit.cone_point_unique_up_to_iso (preserves_limit.preserves F (limit.is_limit K)) (limit.is_limit (K ⋙ F))
/-- A functor which preserves colimits preserves chosen colimits up to isomorphism. -/
def preserves_colimit_iso (K : J ⥤ C) [has_colimit.{v} K] (F : C ⥤ D) [has_colimit.{v} (K ⋙ F)] [preserves_colimit K F] :
F.obj (colimit K) ≅ colimit (K ⋙ F) :=
is_colimit.cone_point_unique_up_to_iso (preserves_colimit.preserves F (colimit.is_colimit K)) (colimit.is_colimit (K ⋙ F))

class preserves_limits_of_shape (J : Type v) [small_category J] (F : C ⥤ D) : Type (max u₁ u₂ v) :=
(preserves_limit : Π {K : J ⥤ C}, preserves_limit K F)
class preserves_colimits_of_shape (J : Type v) [small_category J] (F : C ⥤ D) : Type (max u₁ u₂ v) :=
Expand Down