# mathlibdocumentation

category_theory.limits.is_limit

# Limits and colimits #

We set up the general theory of limits and colimits in a category. In this introduction we only describe the setup for limits; it is repeated, with slightly different names, for colimits.

The main structures defined in this file is

• `is_limit c`, for `c : cone F`, `F : J ⥤ C`, expressing that `c` is a limit cone,

See also `category_theory.limits.has_limits` which further builds:

• `limit_cone F`, which consists of a choice of cone for `F` and the fact it is a limit cone, and
• `has_limit F`, asserting the mere existence of some limit cone for `F`.

## Implementation #

At present we simply say everything twice, in order to handle both limits and colimits. It would be highly desirable to have some automation support, e.g. a `@[dualize]` attribute that behaves similarly to `@[to_additive]`.

## References #

@[nolint]
structure category_theory.limits.is_limit {J : Type u₁} {C : Type u₃} {F : J C}  :
Type (max u₁ u₃ v₃)

A cone `t` on `F` is a limit cone if each cone on `F` admits a unique cone morphism to `t`.

Instances for `category_theory.limits.is_limit`
@[simp]
theorem category_theory.limits.is_limit.fac {J : Type u₁} {C : Type u₃} {F : J C} (self : category_theory.limits.is_limit t) (j : J) :
self.lift s t.π.app j = s.π.app j
@[simp]
theorem category_theory.limits.is_limit.fac_assoc {J : Type u₁} {C : Type u₃} {F : J C} (self : category_theory.limits.is_limit t) (j : J) {X' : C} (f' : F.obj j X') :
self.lift s t.π.app j f' = s.π.app j f'
theorem category_theory.limits.is_limit.uniq {J : Type u₁} {C : Type u₃} {F : J C} (self : category_theory.limits.is_limit t) (m : s.X t.X) (w : ∀ (j : J), m t.π.app j = s.π.app j) :
m = self.lift s
@[protected, instance]
def category_theory.limits.is_limit.subsingleton {J : Type u₁} {C : Type u₃} {F : J C}  :
def category_theory.limits.is_limit.map {J : Type u₁} {C : Type u₃} {F G : J C} (α : F G) :
s.X t.X

Given a natural transformation `α : F ⟶ G`, we give a morphism from the cone point of any cone over `F` to the cone point of a limit cone over `G`.

Equations
@[simp]
theorem category_theory.limits.is_limit.map_π {J : Type u₁} {C : Type u₃} {F G : J C} (α : F G) (j : J) :
d.π.app j = c.π.app j α.app j
@[simp]
theorem category_theory.limits.is_limit.map_π_assoc {J : Type u₁} {C : Type u₃} {F G : J C} (α : F G) (j : J) {X' : C} (f' : G.obj j X') :
d.π.app j f' = c.π.app j α.app j f'
theorem category_theory.limits.is_limit.lift_self {J : Type u₁} {C : Type u₃} {F : J C}  :
t.lift c = 𝟙 c.X
def category_theory.limits.is_limit.lift_cone_morphism {J : Type u₁} {C : Type u₃} {F : J C}  :
s t

The universal morphism from any other cone to a limit cone.

Equations
@[simp]
theorem category_theory.limits.is_limit.lift_cone_morphism_hom {J : Type u₁} {C : Type u₃} {F : J C}  :
theorem category_theory.limits.is_limit.uniq_cone_morphism {J : Type u₁} {C : Type u₃} {F : J C} {s t : category_theory.limits.cone F} {f f' : s t} :
f = f'
theorem category_theory.limits.is_limit.exists_unique {J : Type u₁} {C : Type u₃} {F : J C}  :
∃! (l : s.X t.X), ∀ (j : J), l t.π.app j = s.π.app j

Restating the definition of a limit cone in terms of the ∃! operator.

noncomputable def category_theory.limits.is_limit.of_exists_unique {J : Type u₁} {C : Type u₃} {F : J C} (ht : ∀ (s : , ∃! (l : s.X t.X), ∀ (j : J), l t.π.app j = s.π.app j) :

Noncomputably make a colimit cocone from the existence of unique factorizations.

Equations
@[simp]
theorem category_theory.limits.is_limit.mk_cone_morphism_lift {J : Type u₁} {C : Type u₃} {F : J C} (lift : Π (s : , s t) (uniq' : ∀ (s : (m : s t), m = lift s)  :
uniq').lift s = (lift s).hom
def category_theory.limits.is_limit.mk_cone_morphism {J : Type u₁} {C : Type u₃} {F : J C} (lift : Π (s : , s t) (uniq' : ∀ (s : (m : s t), m = lift s) :

Alternative constructor for `is_limit`, providing a morphism of cones rather than a morphism between the cone points and separately the factorisation condition.

Equations
@[simp]
theorem category_theory.limits.is_limit.unique_up_to_iso_inv {J : Type u₁} {C : Type u₃} {F : J C} {s t : category_theory.limits.cone F}  :
def category_theory.limits.is_limit.unique_up_to_iso {J : Type u₁} {C : Type u₃} {F : J C} {s t : category_theory.limits.cone F}  :
s t

Limit cones on `F` are unique up to isomorphism.

Equations
@[simp]
theorem category_theory.limits.is_limit.unique_up_to_iso_hom {J : Type u₁} {C : Type u₃} {F : J C} {s t : category_theory.limits.cone F}  :
theorem category_theory.limits.is_limit.hom_is_iso {J : Type u₁} {C : Type u₃} {F : J C} {s t : category_theory.limits.cone F} (f : s t) :

Any cone morphism between limit cones is an isomorphism.

def category_theory.limits.is_limit.cone_point_unique_up_to_iso {J : Type u₁} {C : Type u₃} {F : J C} {s t : category_theory.limits.cone F}  :
s.X t.X

Limits of `F` are unique up to isomorphism.

Equations
@[simp]
theorem category_theory.limits.is_limit.cone_point_unique_up_to_iso_hom_comp_assoc {J : Type u₁} {C : Type u₃} {F : J C} {s t : category_theory.limits.cone F} (j : J) {X' : C} (f' : F.obj j X') :
t.π.app j f' = s.π.app j f'
@[simp]
theorem category_theory.limits.is_limit.cone_point_unique_up_to_iso_hom_comp {J : Type u₁} {C : Type u₃} {F : J C} {s t : category_theory.limits.cone F} (j : J) :
t.π.app j = s.π.app j
@[simp]
theorem category_theory.limits.is_limit.cone_point_unique_up_to_iso_inv_comp {J : Type u₁} {C : Type u₃} {F : J C} {s t : category_theory.limits.cone F} (j : J) :
s.π.app j = t.π.app j
@[simp]
theorem category_theory.limits.is_limit.cone_point_unique_up_to_iso_inv_comp_assoc {J : Type u₁} {C : Type u₃} {F : J C} {s t : category_theory.limits.cone F} (j : J) {X' : C} (f' : F.obj j X') :
s.π.app j f' = t.π.app j f'
@[simp]
theorem category_theory.limits.is_limit.lift_comp_cone_point_unique_up_to_iso_hom_assoc {J : Type u₁} {C : Type u₃} {F : J C} {r s t : category_theory.limits.cone F} {X' : C} (f' : t.X X') :
P.lift r f' = Q.lift r f'
@[simp]
theorem category_theory.limits.is_limit.lift_comp_cone_point_unique_up_to_iso_hom {J : Type u₁} {C : Type u₃} {F : J C} {r s t : category_theory.limits.cone F}  :
P.lift r = Q.lift r
@[simp]
theorem category_theory.limits.is_limit.lift_comp_cone_point_unique_up_to_iso_inv_assoc {J : Type u₁} {C : Type u₃} {F : J C} {r s t : category_theory.limits.cone F} {X' : C} (f' : s.X X') :
Q.lift r f' = P.lift r f'
@[simp]
theorem category_theory.limits.is_limit.lift_comp_cone_point_unique_up_to_iso_inv {J : Type u₁} {C : Type u₃} {F : J C} {r s t : category_theory.limits.cone F}  :
Q.lift r = P.lift r
def category_theory.limits.is_limit.of_iso_limit {J : Type u₁} {C : Type u₃} {F : J C} {r t : category_theory.limits.cone F} (i : r t) :

Transport evidence that a cone is a limit cone across an isomorphism of cones.

Equations
@[simp]
theorem category_theory.limits.is_limit.of_iso_limit_lift {J : Type u₁} {C : Type u₃} {F : J C} {r t : category_theory.limits.cone F} (i : r t)  :
(P.of_iso_limit i).lift s = P.lift s i.hom.hom
def category_theory.limits.is_limit.equiv_iso_limit {J : Type u₁} {C : Type u₃} {F : J C} {r t : category_theory.limits.cone F} (i : r t) :

Isomorphism of cones preserves whether or not they are limiting cones.

Equations
@[simp]
theorem category_theory.limits.is_limit.equiv_iso_limit_apply {J : Type u₁} {C : Type u₃} {F : J C} {r t : category_theory.limits.cone F} (i : r t)  :
@[simp]
theorem category_theory.limits.is_limit.equiv_iso_limit_symm_apply {J : Type u₁} {C : Type u₃} {F : J C} {r t : category_theory.limits.cone F} (i : r t)  :
noncomputable def category_theory.limits.is_limit.of_point_iso {J : Type u₁} {C : Type u₃} {F : J C} {r t : category_theory.limits.cone F} [i : category_theory.is_iso (P.lift t)] :

If the canonical morphism from a cone point to a limiting cone point is an iso, then the first cone was limiting also.

Equations
theorem category_theory.limits.is_limit.hom_lift {J : Type u₁} {C : Type u₃} {F : J C} {W : C} (m : W t.X) :
m = h.lift {X := W, π := {app := λ (b : J), m t.π.app b, naturality' := _}}
theorem category_theory.limits.is_limit.hom_ext {J : Type u₁} {C : Type u₃} {F : J C} {W : C} {f f' : W t.X} (w : ∀ (j : J), f t.π.app j = f' t.π.app j) :
f = f'

Two morphisms into a limit are equal if their compositions with each cone morphism are equal.

def category_theory.limits.is_limit.of_right_adjoint {J : Type u₁} {K : Type u₂} {C : Type u₃} {F : J C} {D : Type u₄} {G : K D}  :

Given a right adjoint functor between categories of cones, the image of a limit cone is a limit cone.

Equations
def category_theory.limits.is_limit.of_cone_equiv {J : Type u₁} {K : Type u₂} {C : Type u₃} {F : J C} {D : Type u₄} {G : K D}  :

Given two functors which have equivalent categories of cones, we can transport a limiting cone across the equivalence.

Equations
@[simp]
theorem category_theory.limits.is_limit.of_cone_equiv_apply_desc {J : Type u₁} {K : Type u₂} {C : Type u₃} {F : J C} {D : Type u₄} {G : K D} (P : category_theory.limits.is_limit (h.functor.obj c))  :
@[simp]
theorem category_theory.limits.is_limit.of_cone_equiv_symm_apply_desc {J : Type u₁} {K : Type u₂} {C : Type u₃} {F : J C} {D : Type u₄} {G : K D}  :
def category_theory.limits.is_limit.postcompose_hom_equiv {J : Type u₁} {C : Type u₃} {F G : J C} (α : F G)  :

A cone postcomposed with a natural isomorphism is a limit cone if and only if the original cone is.

Equations
def category_theory.limits.is_limit.postcompose_inv_equiv {J : Type u₁} {C : Type u₃} {F G : J C} (α : F G)  :

A cone postcomposed with the inverse of a natural isomorphism is a limit cone if and only if the original cone is.

Equations
def category_theory.limits.is_limit.equiv_of_nat_iso_of_iso {J : Type u₁} {C : Type u₃} {F G : J C} (α : F G) (w : d) :

Constructing an equivalence `is_limit c ≃ is_limit d` from a natural isomorphism between the underlying functors, and then an isomorphism between `c` transported along this and `d`.

Equations
def category_theory.limits.is_limit.cone_points_iso_of_nat_iso {J : Type u₁} {C : Type u₃} {F G : J C} (w : F G) :
s.X t.X

The cone points of two limit cones for naturally isomorphic functors are themselves isomorphic.

Equations
@[simp]
theorem category_theory.limits.is_limit.cone_points_iso_of_nat_iso_hom {J : Type u₁} {C : Type u₃} {F G : J C} (w : F G) :
@[simp]
theorem category_theory.limits.is_limit.cone_points_iso_of_nat_iso_inv {J : Type u₁} {C : Type u₃} {F G : J C} (w : F G) :
theorem category_theory.limits.is_limit.cone_points_iso_of_nat_iso_hom_comp_assoc {J : Type u₁} {C : Type u₃} {F G : J C} (w : F G) (j : J) {X' : C} (f' : G.obj j X') :
w).hom t.π.app j f' = s.π.app j w.hom.app j f'
theorem category_theory.limits.is_limit.cone_points_iso_of_nat_iso_hom_comp {J : Type u₁} {C : Type u₃} {F G : J C} (w : F G) (j : J) :
w).hom t.π.app j = s.π.app j w.hom.app j
theorem category_theory.limits.is_limit.cone_points_iso_of_nat_iso_inv_comp {J : Type u₁} {C : Type u₃} {F G : J C} (w : F G) (j : J) :
w).inv s.π.app j = t.π.app j w.inv.app j
theorem category_theory.limits.is_limit.cone_points_iso_of_nat_iso_inv_comp_assoc {J : Type u₁} {C : Type u₃} {F G : J C} (w : F G) (j : J) {X' : C} (f' : F.obj j X') :
w).inv s.π.app j f' = t.π.app j w.inv.app j f'
theorem category_theory.limits.is_limit.lift_comp_cone_points_iso_of_nat_iso_hom {J : Type u₁} {C : Type u₃} {F G : J C} {r s : category_theory.limits.cone F} (w : F G) :
P.lift r w).hom =
theorem category_theory.limits.is_limit.lift_comp_cone_points_iso_of_nat_iso_hom_assoc {J : Type u₁} {C : Type u₃} {F G : J C} {r s : category_theory.limits.cone F} (w : F G) {X' : C} (f' : t.X X') :
P.lift r w).hom f' =
theorem category_theory.limits.is_limit.lift_comp_cone_points_iso_of_nat_iso_inv_assoc {J : Type u₁} {C : Type u₃} {F G : J C} {r s : category_theory.limits.cone G} (w : F G) {X' : C} (f' : t.X X') :
Q.lift r w).inv f' =
theorem category_theory.limits.is_limit.lift_comp_cone_points_iso_of_nat_iso_inv {J : Type u₁} {C : Type u₃} {F G : J C} {r s : category_theory.limits.cone G} (w : F G) :
Q.lift r w).inv =
def category_theory.limits.is_limit.whisker_equivalence {J : Type u₁} {K : Type u₂} {C : Type u₃} {F : J C} (e : K J) :

If `s : cone F` is a limit cone, so is `s` whiskered by an equivalence `e`.

Equations
def category_theory.limits.is_limit.of_whisker_equivalence {J : Type u₁} {K : Type u₂} {C : Type u₃} {F : J C} (e : K J)  :

If `s : cone F` whiskered by an equivalence `e` is a limit cone, so is `s`.

Equations
def category_theory.limits.is_limit.whisker_equivalence_equiv {J : Type u₁} {K : Type u₂} {C : Type u₃} {F : J C} (e : K J) :

Given an equivalence of diagrams `e`, `s` is a limit cone iff `s.whisker e.functor` is.

Equations
@[simp]
theorem category_theory.limits.is_limit.cone_points_iso_of_equivalence_hom {J : Type u₁} {K : Type u₂} {C : Type u₃} {F : J C} {G : K C} (e : J K) (w : e.functor G F) :
w).hom =
@[simp]
theorem category_theory.limits.is_limit.cone_points_iso_of_equivalence_inv {J : Type u₁} {K : Type u₂} {C : Type u₃} {F : J C} {G : K C} (e : J K) (w : e.functor G F) :
w).inv =
def category_theory.limits.is_limit.cone_points_iso_of_equivalence {J : Type u₁} {K : Type u₂} {C : Type u₃} {F : J C} {G : K C} (e : J K) (w : e.functor G F) :
s.X t.X

We can prove two cone points `(s : cone F).X` and `(t.cone G).X` are isomorphic if

• both cones are limit cones
• their indexing categories are equivalent via some `e : J ≌ K`,
• the triangle of functors commutes up to a natural isomorphism: `e.functor ⋙ G ≅ F`.

This is the most general form of uniqueness of cone points, allowing relabelling of both the indexing category (up to equivalence) and the functor (up to natural isomorphism).

Equations
def category_theory.limits.is_limit.hom_iso {J : Type u₁} {C : Type u₃} {F : J C} (W : C) :
ulift (W t.X)

The universal property of a limit cone: a map `W ⟶ X` is the same as a cone on `F` with vertex `W`.

Equations
@[simp]
theorem category_theory.limits.is_limit.hom_iso_hom {J : Type u₁} {C : Type u₃} {F : J C} {W : C} (f : ulift (W t.X)) :
(h.hom_iso W).hom f = (t.extend f.down).π
def category_theory.limits.is_limit.nat_iso {J : Type u₁} {C : Type u₃} {F : J C}  :

The limit of `F` represents the functor taking `W` to the set of cones on `F` with vertex `W`.

Equations
def category_theory.limits.is_limit.hom_iso' {J : Type u₁} {C : Type u₃} {F : J C} (W : C) :
ulift (W t.X) {p // ∀ {j j' : J} (f : j j'), p j F.map f = p j'}

Another, more explicit, formulation of the universal property of a limit cone. See also `hom_iso`.

Equations
def category_theory.limits.is_limit.of_faithful {J : Type u₁} {C : Type u₃} {F : J C} {D : Type u₄} (G : C D) (ht : category_theory.limits.is_limit (G.map_cone t)) (lift : Π (s : , s.X t.X) (h : ∀ (s : , G.map (lift s) = ht.lift (G.map_cone s)) :

If G : C → D is a faithful functor which sends t to a limit cone, then it suffices to check that the induced maps for the image of t can be lifted to maps of C.

Equations
def category_theory.limits.is_limit.map_cone_equiv {J : Type u₁} {C : Type u₃} {D : Type u₄} {K : J C} {F G : C D} (h : F G) (t : category_theory.limits.is_limit (F.map_cone c)) :

If `F` and `G` are naturally isomorphic, then `F.map_cone c` being a limit implies `G.map_cone c` is also a limit.

Equations
def category_theory.limits.is_limit.iso_unique_cone_morphism {J : Type u₁} {C : Type u₃} {F : J C}  :
Π (s : , unique (s t)

A cone is a limit cone exactly if there is a unique cone morphism from any other cone.

Equations
def category_theory.limits.is_limit.of_nat_iso.cone_of_hom {J : Type u₁} {C : Type u₃} {F : J C} {X : C} {Y : C} (f : Y X) :

If `F.cones` is represented by `X`, each morphism `f : Y ⟶ X` gives a cone with cone point `Y`.

Equations
def category_theory.limits.is_limit.of_nat_iso.hom_of_cone {J : Type u₁} {C : Type u₃} {F : J C} {X : C}  :
s.X X

If `F.cones` is represented by `X`, each cone `s` gives a morphism `s.X ⟶ X`.

Equations
@[simp]
theorem category_theory.limits.is_limit.of_nat_iso.cone_of_hom_of_cone {J : Type u₁} {C : Type u₃} {F : J C} {X : C}  :
@[simp]
theorem category_theory.limits.is_limit.of_nat_iso.hom_of_cone_of_hom {J : Type u₁} {C : Type u₃} {F : J C} {X : C} {Y : C} (f : Y X) :
def category_theory.limits.is_limit.of_nat_iso.limit_cone {J : Type u₁} {C : Type u₃} {F : J C} {X : C}  :

If `F.cones` is represented by `X`, the cone corresponding to the identity morphism on `X` will be a limit cone.

Equations
theorem category_theory.limits.is_limit.of_nat_iso.cone_of_hom_fac {J : Type u₁} {C : Type u₃} {F : J C} {X : C} {Y : C} (f : Y X) :

If `F.cones` is represented by `X`, the cone corresponding to a morphism `f : Y ⟶ X` is the limit cone extended by `f`.

theorem category_theory.limits.is_limit.of_nat_iso.cone_fac {J : Type u₁} {C : Type u₃} {F : J C} {X : C}  :

If `F.cones` is represented by `X`, any cone is the extension of the limit cone by the corresponding morphism.

def category_theory.limits.is_limit.of_nat_iso {J : Type u₁} {C : Type u₃} {F : J C} {X : C}  :

If `F.cones` is representable, then the cone corresponding to the identity morphism on the representing object is a limit cone.

Equations
@[nolint]
structure category_theory.limits.is_colimit {J : Type u₁} {C : Type u₃} {F : J C}  :
Type (max u₁ u₃ v₃)

A cocone `t` on `F` is a colimit cocone if each cocone on `F` admits a unique cocone morphism from `t`.

Instances for `category_theory.limits.is_colimit`
@[simp]
theorem category_theory.limits.is_colimit.fac {J : Type u₁} {C : Type u₃} {F : J C} (self : category_theory.limits.is_colimit t) (j : J) :
t.ι.app j self.desc s = s.ι.app j
@[simp]
theorem category_theory.limits.is_colimit.fac_assoc {J : Type u₁} {C : Type u₃} {F : J C} (self : category_theory.limits.is_colimit t) (j : J) {X' : C} (f' : s.X X') :
t.ι.app j self.desc s f' = s.ι.app j f'
theorem category_theory.limits.is_colimit.uniq {J : Type u₁} {C : Type u₃} {F : J C} (self : category_theory.limits.is_colimit t) (m : t.X s.X) (w : ∀ (j : J), t.ι.app j m = s.ι.app j) :
m = self.desc s
@[protected, instance]
def category_theory.limits.is_colimit.subsingleton {J : Type u₁} {C : Type u₃} {F : J C}  :
def category_theory.limits.is_colimit.map {J : Type u₁} {C : Type u₃} {F G : J C} (α : F G) :
s.X t.X

Given a natural transformation `α : F ⟶ G`, we give a morphism from the cocone point of a colimit cocone over `F` to the cocone point of any cocone over `G`.

Equations
@[simp]
theorem category_theory.limits.is_colimit.ι_map_assoc {J : Type u₁} {C : Type u₃} {F G : J C} (α : F G) (j : J) {X' : C} (f' : d.X X') :
c.ι.app j hc.map d α f' = α.app j d.ι.app j f'
@[simp]
theorem category_theory.limits.is_colimit.ι_map {J : Type u₁} {C : Type u₃} {F G : J C} (α : F G) (j : J) :
c.ι.app j hc.map d α = α.app j d.ι.app j
@[simp]
theorem category_theory.limits.is_colimit.desc_self {J : Type u₁} {C : Type u₃} {F : J C}  :
h.desc t = 𝟙 t.X
def category_theory.limits.is_colimit.desc_cocone_morphism {J : Type u₁} {C : Type u₃} {F : J C}  :
t s

The universal morphism from a colimit cocone to any other cocone.

Equations
@[simp]
theorem category_theory.limits.is_colimit.desc_cocone_morphism_hom {J : Type u₁} {C : Type u₃} {F : J C}  :
theorem category_theory.limits.is_colimit.uniq_cocone_morphism {J : Type u₁} {C : Type u₃} {F : J C} {s t : category_theory.limits.cocone F} {f f' : t s} :
f = f'
theorem category_theory.limits.is_colimit.exists_unique {J : Type u₁} {C : Type u₃} {F : J C}  :
∃! (d : t.X s.X), ∀ (j : J), t.ι.app j d = s.ι.app j

Restating the definition of a colimit cocone in terms of the ∃! operator.

noncomputable def category_theory.limits.is_colimit.of_exists_unique {J : Type u₁} {C : Type u₃} {F : J C} (ht : ∀ (s : , ∃! (d : t.X s.X), ∀ (j : J), t.ι.app j d = s.ι.app j) :

Noncomputably make a colimit cocone from the existence of unique factorizations.

Equations
def category_theory.limits.is_colimit.mk_cocone_morphism {J : Type u₁} {C : Type u₃} {F : J C} (desc : Π (s : , t s) (uniq' : ∀ (s : (m : t s), m = desc s) :

Alternative constructor for `is_colimit`, providing a morphism of cocones rather than a morphism between the cocone points and separately the factorisation condition.

Equations
@[simp]
theorem category_theory.limits.is_colimit.mk_cocone_morphism_desc {J : Type u₁} {C : Type u₃} {F : J C} (desc : Π (s : , t s) (uniq' : ∀ (s : (m : t s), m = desc s)  :
.desc s = (desc s).hom
@[simp]
theorem category_theory.limits.is_colimit.unique_up_to_iso_hom {J : Type u₁} {C : Type u₃} {F : J C} {s t : category_theory.limits.cocone F}  :
def category_theory.limits.is_colimit.unique_up_to_iso {J : Type u₁} {C : Type u₃} {F : J C} {s t : category_theory.limits.cocone F}  :
s t

Colimit cocones on `F` are unique up to isomorphism.

Equations
@[simp]
theorem category_theory.limits.is_colimit.unique_up_to_iso_inv {J : Type u₁} {C : Type u₃} {F : J C} {s t : category_theory.limits.cocone F}  :
theorem category_theory.limits.is_colimit.hom_is_iso {J : Type u₁} {C : Type u₃} {F : J C} {s t : category_theory.limits.cocone F} (f : s t) :

Any cocone morphism between colimit cocones is an isomorphism.

def category_theory.limits.is_colimit.cocone_point_unique_up_to_iso {J : Type u₁} {C : Type u₃} {F : J C} {s t : category_theory.limits.cocone F}  :
s.X t.X

Colimits of `F` are unique up to isomorphism.

Equations
@[simp]
theorem category_theory.limits.is_colimit.comp_cocone_point_unique_up_to_iso_hom_assoc {J : Type u₁} {C : Type u₃} {F : J C} {s t : category_theory.limits.cocone F} (j : J) {X' : C} (f' : t.X X') :
s.ι.app j f' = t.ι.app j f'
@[simp]
theorem category_theory.limits.is_colimit.comp_cocone_point_unique_up_to_iso_hom {J : Type u₁} {C : Type u₃} {F : J C} {s t : category_theory.limits.cocone F} (j : J) :
s.ι.app j = t.ι.app j
@[simp]
theorem category_theory.limits.is_colimit.comp_cocone_point_unique_up_to_iso_inv {J : Type u₁} {C : Type u₃} {F : J C} {s t : category_theory.limits.cocone F} (j : J) :
t.ι.app j = s.ι.app j
@[simp]
theorem category_theory.limits.is_colimit.comp_cocone_point_unique_up_to_iso_inv_assoc {J : Type u₁} {C : Type u₃} {F : J C} {s t : category_theory.limits.cocone F} (j : J) {X' : C} (f' : s.X X') :
t.ι.app j f' = s.ι.app j f'
@[simp]
theorem category_theory.limits.is_colimit.cocone_point_unique_up_to_iso_hom_desc {J : Type u₁} {C : Type u₃} {F : J C} {r s t : category_theory.limits.cocone F}  :
Q.desc r = P.desc r
@[simp]
theorem category_theory.limits.is_colimit.cocone_point_unique_up_to_iso_hom_desc_assoc {J : Type u₁} {C : Type u₃} {F : J C} {r s t : category_theory.limits.cocone F} {X' : C} (f' : r.X X') :
Q.desc r f' = P.desc r f'
@[simp]
theorem category_theory.limits.is_colimit.cocone_point_unique_up_to_iso_inv_desc_assoc {J : Type u₁} {C : Type u₃} {F : J C} {r s t : category_theory.limits.cocone F} {X' : C} (f' : r.X X') :
P.desc r f' = Q.desc r f'
@[simp]
theorem category_theory.limits.is_colimit.cocone_point_unique_up_to_iso_inv_desc {J : Type u₁} {C : Type u₃} {F : J C} {r s t : category_theory.limits.cocone F}  :
P.desc r = Q.desc r
def category_theory.limits.is_colimit.of_iso_colimit {J : Type u₁} {C : Type u₃} {F : J C} {r t : category_theory.limits.cocone F} (i : r t) :

Transport evidence that a cocone is a colimit cocone across an isomorphism of cocones.

Equations
@[simp]
theorem category_theory.limits.is_colimit.of_iso_colimit_desc {J : Type u₁} {C : Type u₃} {F : J C} {r t : category_theory.limits.cocone F} (i : r t)  :
def category_theory.limits.is_colimit.equiv_iso_colimit {J : Type u₁} {C : Type u₃} {F : J C} {r t : category_theory.limits.cocone F} (i : r t) :

Isomorphism of cocones preserves whether or not they are colimiting cocones.

Equations
@[simp]
theorem category_theory.limits.is_colimit.equiv_iso_colimit_apply {J : Type u₁} {C : Type u₃} {F : J C} {r t : category_theory.limits.cocone F} (i : r t)  :
@[simp]
theorem category_theory.limits.is_colimit.equiv_iso_colimit_symm_apply {J : Type u₁} {C : Type u₃} {F : J C} {r t : category_theory.limits.cocone F} (i : r t)  :
noncomputable def category_theory.limits.is_colimit.of_point_iso {J : Type u₁} {C : Type u₃} {F : J C} {r t : category_theory.limits.cocone F} [i : category_theory.is_iso (P.desc t)] :

If the canonical morphism to a cocone point from a colimiting cocone point is an iso, then the first cocone was colimiting also.

Equations
theorem category_theory.limits.is_colimit.hom_desc {J : Type u₁} {C : Type u₃} {F : J C} {W : C} (m : t.X W) :
m = h.desc {X := W, ι := {app := λ (b : J), t.ι.app b m, naturality' := _}}
theorem category_theory.limits.is_colimit.hom_ext {J : Type u₁} {C : Type u₃} {F : J C} {W : C} {f f' : t.X W} (w : ∀ (j : J), t.ι.app j f = t.ι.app j f') :
f = f'

Two morphisms out of a colimit are equal if their compositions with each cocone morphism are equal.

def category_theory.limits.is_colimit.of_left_adjoint {J : Type u₁} {K : Type u₂} {C : Type u₃} {F : J C} {D : Type u₄} {G : K D}  :

Given a left adjoint functor between categories of cocones, the image of a colimit cocone is a colimit cocone.

Equations
def category_theory.limits.is_colimit.of_cocone_equiv {J : Type u₁} {K : Type u₂} {C : Type u₃} {F : J C} {D : Type u₄} {G : K D}  :

Given two functors which have equivalent categories of cocones, we can transport a colimiting cocone across the equivalence.

Equations
@[simp]
theorem category_theory.limits.is_colimit.of_cocone_equiv_apply_desc {J : Type u₁} {K : Type u₂} {C : Type u₃} {F : J C} {D : Type u₄} {G : K D} (P : category_theory.limits.is_colimit (h.functor.obj c))  :
@[simp]
theorem category_theory.limits.is_colimit.of_cocone_equiv_symm_apply_desc {J : Type u₁} {K : Type u₂} {C : Type u₃} {F : J C} {D : Type u₄} {G : K D}  :
def category_theory.limits.is_colimit.precompose_hom_equiv {J : Type u₁} {C : Type u₃} {F G : J C} (α : F G)  :

A cocone precomposed with a natural isomorphism is a colimit cocone if and only if the original cocone is.

Equations
def category_theory.limits.is_colimit.precompose_inv_equiv {J : Type u₁} {C : Type u₃} {F G : J C} (α : F G)  :

A cocone precomposed with the inverse of a natural isomorphism is a colimit cocone if and only if the original cocone is.

Equations
def category_theory.limits.is_colimit.equiv_of_nat_iso_of_iso {J : Type u₁} {C : Type u₃} {F G : J C} (α : F G) (w : d) :

Constructing an equivalence `is_colimit c ≃ is_colimit d` from a natural isomorphism between the underlying functors, and then an isomorphism between `c` transported along this and `d`.

Equations
def category_theory.limits.is_colimit.cocone_points_iso_of_nat_iso {J : Type u₁} {C : Type u₃} {F G : J C} (w : F G) :
s.X t.X

The cocone points of two colimit cocones for naturally isomorphic functors are themselves isomorphic.

Equations
@[simp]
theorem category_theory.limits.is_colimit.cocone_points_iso_of_nat_iso_inv {J : Type u₁} {C : Type u₃} {F G : J C} (w : F G) :
w).inv = Q.map s w.inv
@[simp]
theorem category_theory.limits.is_colimit.cocone_points_iso_of_nat_iso_hom {J : Type u₁} {C : Type u₃} {F G : J C} (w : F G) :
w).hom = P.map t w.hom
theorem category_theory.limits.is_colimit.comp_cocone_points_iso_of_nat_iso_hom {J : Type u₁} {C : Type u₃} {F G : J C} (w : F G) (j : J) :
s.ι.app j w).hom = w.hom.app j t.ι.app j
theorem category_theory.limits.is_colimit.comp_cocone_points_iso_of_nat_iso_hom_assoc {J : Type u₁} {C : Type u₃} {F G : J C} (w : F G) (j : J) {X' : C} (f' : t.X X') :
s.ι.app j w).hom f' = w.hom.app j t.ι.app j f'
theorem category_theory.limits.is_colimit.comp_cocone_points_iso_of_nat_iso_inv_assoc {J : Type u₁} {C : Type u₃} {F G : J C} (w : F G) (j : J) {X' : C} (f' : s.X X') :
t.ι.app j w).inv f' = w.inv.app j s.ι.app j f'
theorem category_theory.limits.is_colimit.comp_cocone_points_iso_of_nat_iso_inv {J : Type u₁} {C : Type u₃} {F G : J C} (w : F G) (j : J) :
t.ι.app j w).inv = w.inv.app j s.ι.app j
theorem category_theory.limits.is_colimit.cocone_points_iso_of_nat_iso_hom_desc_assoc {J : Type u₁} {C : Type u₃} {F G : J C} {r t : category_theory.limits.cocone G} (w : F G) {X' : C} (f' : r.X X') :
w).hom Q.desc r f' = P.map r w.hom f'
theorem category_theory.limits.is_colimit.cocone_points_iso_of_nat_iso_hom_desc {J : Type u₁} {C : Type u₃} {F G : J C} {r t : category_theory.limits.cocone G} (w : F G) :
w).hom Q.desc r = P.map r w.hom
theorem category_theory.limits.is_colimit.cocone_points_iso_of_nat_iso_inv_desc_assoc {J : Type u₁} {C : Type u₃} {F G : J C} {r t : category_theory.limits.cocone F} (w : F G) {X' : C} (f' : r.X X') :
w).inv P.desc r f' = Q.map r w.inv f'
theorem category_theory.limits.is_colimit.cocone_points_iso_of_nat_iso_inv_desc {J : Type u₁} {C : Type u₃} {F G : J C} {r t : category_theory.limits.cocone F} (w : F G) :
w).inv P.desc r = Q.map r w.inv
def category_theory.limits.is_colimit.whisker_equivalence {J : Type u₁} {K : Type u₂} {C : Type u₃} {F : J C} (e : K J) :

If `s : cocone F` is a colimit cocone, so is `s` whiskered by an equivalence `e`.

Equations
def category_theory.limits.is_colimit.of_whisker_equivalence {J : Type u₁} {K : Type u₂} {C : Type u₃} {F : J C} (e : K J)  :

If `s : cocone F` whiskered by an equivalence `e` is a colimit cocone, so is `s`.

Equations
def category_theory.limits.is_colimit.whisker_equivalence_equiv {J : Type u₁} {K : Type u₂} {C : Type u₃} {F : J C} (e : K J) :

Given an equivalence of diagrams `e`, `s` is a colimit cocone iff `s.whisker e.functor` is.

Equations
@[simp]
theorem category_theory.limits.is_colimit.cocone_points_iso_of_equivalence_hom {J : Type u₁} {K : Type u₂} {C : Type u₃} {F : J C} {G : K C} (e : J K) (w : e.functor G F) :
@[simp]
theorem category_theory.limits.is_colimit.cocone_points_iso_of_equivalence_inv {J : Type u₁} {K : Type u₂} {C : Type u₃} {F : J C} {G : K C} (e : J K) (w : e.functor G F) :
def category_theory.limits.is_colimit.cocone_points_iso_of_equivalence {J : Type u₁} {K : Type u₂} {C : Type u₃} {F : J C} {G : K C} (e : J K) (w : e.functor G F) :
s.X t.X

We can prove two cocone points `(s : cocone F).X` and `(t.cocone G).X` are isomorphic if

• both cocones are colimit cocones
• their indexing categories are equivalent via some `e : J ≌ K`,
• the triangle of functors commutes up to a natural isomorphism: `e.functor ⋙ G ≅ F`.

This is the most general form of uniqueness of cocone points, allowing relabelling of both the indexing category (up to equivalence) and the functor (up to natural isomorphism).

Equations
def category_theory.limits.is_colimit.hom_iso {J : Type u₁} {C : Type u₃} {F : J C} (W : C) :
ulift (t.X W)

The universal property of a colimit cocone: a map `X ⟶ W` is the same as a cocone on `F` with vertex `W`.

Equations
@[simp]
theorem category_theory.limits.is_colimit.hom_iso_hom {J : Type u₁} {C : Type u₃} {F : J C} {W : C} (f : ulift (t.X W)) :
(h.hom_iso W).hom f = (t.extend f.down).ι
def category_theory.limits.is_colimit.nat_iso {J : Type u₁} {C : Type u₃} {F : J C}  :

The colimit of `F` represents the functor taking `W` to the set of cocones on `F` with vertex `W`.

Equations
def category_theory.limits.is_colimit.hom_iso' {J : Type u₁} {C : Type u₃} {F : J C} (W : C) :
ulift (t.X W) {p // ∀ {j j' : J} (f : j j'), F.map f p j' = p j}

Another, more explicit, formulation of the universal property of a colimit cocone. See also `hom_iso`.

Equations
def category_theory.limits.is_colimit.of_faithful {J : Type u₁} {C : Type u₃} {F : J C} {D : Type u₄} (G : C D) (ht : category_theory.limits.is_colimit (G.map_cocone t)) (desc : Π (s : , t.X s.X) (h : ∀ (s : , G.map (desc s) = ht.desc (G.map_cocone s)) :

If G : C → D is a faithful functor which sends t to a colimit cocone, then it suffices to check that the induced maps for the image of t can be lifted to maps of C.

Equations
def category_theory.limits.is_colimit.map_cocone_equiv {J : Type u₁} {C : Type u₃} {D : Type u₄} {K : J C} {F G : C D} (h : F G) (t : category_theory.limits.is_colimit (F.map_cocone c)) :

If `F` and `G` are naturally isomorphic, then `F.map_cone c` being a colimit implies `G.map_cone c` is also a colimit.

Equations
def category_theory.limits.is_colimit.iso_unique_cocone_morphism {J : Type u₁} {C : Type u₃} {F : J C}  :
Π (s : , unique (t s)

A cocone is a colimit cocone exactly if there is a unique cocone morphism from any other cocone.

Equations
def category_theory.limits.is_colimit.of_nat_iso.cocone_of_hom {J : Type u₁} {C : Type u₃} {F : J C} {X : C} {Y : C} (f : X Y) :

If `F.cocones` is corepresented by `X`, each morphism `f : X ⟶ Y` gives a cocone with cone point `Y`.

Equations
def category_theory.limits.is_colimit.of_nat_iso.hom_of_cocone {J : Type u₁} {C : Type u₃} {F : J C} {X : C}  :
X s.X

If `F.cocones` is corepresented by `X`, each cocone `s` gives a morphism `X ⟶ s.X`.

Equations
@[simp]
theorem category_theory.limits.is_colimit.of_nat_iso.cocone_of_hom_of_cocone {J : Type u₁} {C : Type u₃} {F : J C} {X : C}  :
@[simp]
theorem category_theory.limits.is_colimit.of_nat_iso.hom_of_cocone_of_hom {J : Type u₁} {C : Type u₃} {F : J C} {X : C} {Y : C} (f : X Y) :
def category_theory.limits.is_colimit.of_nat_iso.colimit_cocone {J : Type u₁} {C : Type u₃} {F : J C} {X : C}  :

If `F.cocones` is corepresented by `X`, the cocone corresponding to the identity morphism on `X` will be a colimit cocone.

Equations
theorem category_theory.limits.is_colimit.of_nat_iso.cocone_of_hom_fac {J : Type u₁} {C : Type u₃} {F : J C} {X : C} {Y : C} (f : X Y) :

If `F.cocones` is corepresented by `X`, the cocone corresponding to a morphism `f : Y ⟶ X` is the colimit cocone extended by `f`.

theorem category_theory.limits.is_colimit.of_nat_iso.cocone_fac {J : Type u₁} {C : Type u₃} {F : J C} {X : C}  :

If `F.cocones` is corepresented by `X`, any cocone is the extension of the colimit cocone by the corresponding morphism.

def category_theory.limits.is_colimit.of_nat_iso {J : Type u₁} {C : Type u₃} {F : J C} {X : C}  :

If `F.cocones` is corepresentable, then the cocone corresponding to the identity morphism on the representing object is a colimit cocone.

Equations