# 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.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 v} {C : Type u} {F : J C}  :
Type (max u v)

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

@[simp]
theorem category_theory.limits.is_limit.fac {J : Type v} {C : Type u} {F : J C} (j : J) :
c.lift s t.π.app j = s.π.app j
@[simp]
theorem category_theory.limits.is_limit.fac_assoc {J : Type v} {C : Type u} {F : J C} (j : J) {X' : C} (f' : F.obj j X') :
c.lift s t.π.app j f' = s.π.app j f'
theorem category_theory.limits.is_limit.uniq {J : Type v} {C : Type u} {F : J C} (m : s.X t.X) (w : ∀ (j : J), m t.π.app j = s.π.app j) :
m = c.lift s
@[instance]
def category_theory.limits.is_limit.subsingleton {J : Type v} {C : Type u} {F : J C}  :
def category_theory.limits.is_limit.map {J : Type v} {C : Type u} {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 v} {C : Type u} {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 v} {C : Type u} {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 v} {C : Type u} {F : J C}  :
t.lift c = 𝟙 c.X
def category_theory.limits.is_limit.lift_cone_morphism {J : Type v} {C : Type u} {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 v} {C : Type u} {F : J C}  :
theorem category_theory.limits.is_limit.uniq_cone_morphism {J : Type v} {C : Type u} {F : J C} {s t : category_theory.limits.cone F} {f f' : s t} :
f = f'
@[simp]
theorem category_theory.limits.is_limit.mk_cone_morphism_lift {J : Type v} {C : Type u} {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 v} {C : Type u} {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 v} {C : Type u} {F : J C} {s t : category_theory.limits.cone F}  :
def category_theory.limits.is_limit.unique_up_to_iso {J : Type v} {C : Type u} {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 v} {C : Type u} {F : J C} {s t : category_theory.limits.cone F}  :
theorem category_theory.limits.is_limit.hom_is_iso {J : Type v} {C : Type u} {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 v} {C : Type u} {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 v} {C : Type u} {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 v} {C : Type u} {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 v} {C : Type u} {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 v} {C : Type u} {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 v} {C : Type u} {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]
@[simp]
theorem category_theory.limits.is_limit.lift_comp_cone_point_unique_up_to_iso_inv_assoc {J : Type v} {C : Type u} {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]
def category_theory.limits.is_limit.of_iso_limit {J : Type v} {C : Type u} {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 v} {C : Type u} {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 v} {C : Type u} {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 v} {C : Type u} {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 v} {C : Type u} {F : J C} {r t : category_theory.limits.cone F} (i : r t)  :
def category_theory.limits.is_limit.of_point_iso {J : Type v} {C : Type u} {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 v} {C : Type u} {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 v} {C : Type u} {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 K : Type v} {C : Type u} {F : J C} {D : Type u'} {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 K : Type v} {C : Type u} {F : J C} {D : Type u'} {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 K : Type v} {C : Type u} {F : J C} {D : Type u'} {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 K : Type v} {C : Type u} {F : J C} {D : Type u'} {G : K D}  :
def category_theory.limits.is_limit.postcompose_hom_equiv {J : Type v} {C : Type u} {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 v} {C : Type u} {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.cone_points_iso_of_nat_iso {J : Type v} {C : Type u} {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 v} {C : Type u} {F G : J C} (w : F G) :
@[simp]
theorem category_theory.limits.is_limit.cone_points_iso_of_nat_iso_inv {J : Type v} {C : Type u} {F G : J C} (w : F G) :
theorem category_theory.limits.is_limit.cone_points_iso_of_nat_iso_hom_comp_assoc {J : Type v} {C : Type u} {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 v} {C : Type u} {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 v} {C : Type u} {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 v} {C : Type u} {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 v} {C : Type u} {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 v} {C : Type u} {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' =
def category_theory.limits.is_limit.whisker_equivalence {J K : Type v} {C : Type u} {F : J C} (e : K J) :

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

Equations
@[simp]
theorem category_theory.limits.is_limit.cone_points_iso_of_equivalence_hom {J K : Type v} {C : Type u} {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 K : Type v} {C : Type u} {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 K : Type v} {C : Type u} {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 F).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 v} {C : Type u} {F : J C} (W : C) :
(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 v} {C : Type u} {F : J C} {W : C} (f : W t.X) :
(h.hom_iso W).hom f = (t.extend f).π
def category_theory.limits.is_limit.nat_iso {J : Type v} {C : Type u} {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 v} {C : Type u} {F : J C} (W : C) :
(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 v} {C : Type u} {F : J C} {D : Type u'} (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 v} {C : Type u} {D : Type u'} {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 v} {C : Type u} {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 v} {C : Type u} {F : J C} {X : C} (h : F.cones) {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 v} {C : Type u} {F : J C} {X : C} (h : F.cones)  :
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 v} {C : Type u} {F : J C} {X : C} (h : F.cones)  :
@[simp]
theorem category_theory.limits.is_limit.of_nat_iso.hom_of_cone_of_hom {J : Type v} {C : Type u} {F : J C} {X : C} (h : F.cones) {Y : C} (f : Y X) :
def category_theory.limits.is_limit.of_nat_iso.limit_cone {J : Type v} {C : Type u} {F : J C} {X : C} (h : F.cones) :

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 v} {C : Type u} {F : J C} {X : C} (h : F.cones) {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 v} {C : Type u} {F : J C} {X : C} (h : F.cones)  :

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 v} {C : Type u} {F : J C} {X : C} (h : F.cones) :

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 v} {C : Type u} {F : J C}  :
Type (max u v)

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

@[simp]
theorem category_theory.limits.is_colimit.fac {J : Type v} {C : Type u} {F : J C} (j : J) :
t.ι.app j c.desc s = s.ι.app j
@[simp]
theorem category_theory.limits.is_colimit.fac_assoc {J : Type v} {C : Type u} {F : J C} (j : J) {X' : C} (f' : s.X X') :
t.ι.app j c.desc s f' = s.ι.app j f'
theorem category_theory.limits.is_colimit.uniq {J : Type v} {C : Type u} {F : J C} (m : t.X s.X) (w : ∀ (j : J), t.ι.app j m = s.ι.app j) :
m = c.desc s
@[instance]
def category_theory.limits.is_colimit.subsingleton {J : Type v} {C : Type u} {F : J C}  :
def category_theory.limits.is_colimit.map {J : Type v} {C : Type u} {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 v} {C : Type u} {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 v} {C : Type u} {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 v} {C : Type u} {F : J C}  :
h.desc t = 𝟙 t.X
def category_theory.limits.is_colimit.desc_cocone_morphism {J : Type v} {C : Type u} {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 v} {C : Type u} {F : J C}  :
theorem category_theory.limits.is_colimit.uniq_cocone_morphism {J : Type v} {C : Type u} {F : J C} {s t : category_theory.limits.cocone F} {f f' : t s} :
f = f'
def category_theory.limits.is_colimit.mk_cocone_morphism {J : Type v} {C : Type u} {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 v} {C : Type u} {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 v} {C : Type u} {F : J C} {s t : category_theory.limits.cocone F}  :
def category_theory.limits.is_colimit.unique_up_to_iso {J : Type v} {C : Type u} {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 v} {C : Type u} {F : J C} {s t : category_theory.limits.cocone F}  :
theorem category_theory.limits.is_colimit.hom_is_iso {J : Type v} {C : Type u} {F : J C} {s t : category_theory.limits.cocone F} (f : s t) :

Any cocone morphism between colimit cocones is an isomorphism.

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 v} {C : Type u} {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 v} {C : Type u} {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 v} {C : Type u} {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 v} {C : Type u} {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]
@[simp]
theorem category_theory.limits.is_colimit.cocone_point_unique_up_to_iso_hom_desc_assoc {J : Type v} {C : Type u} {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 v} {C : Type u} {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]
def category_theory.limits.is_colimit.of_iso_colimit {J : Type v} {C : Type u} {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 v} {C : Type u} {F : J C} {r t : category_theory.limits.cocone F} (i : r t)  :
def category_theory.limits.is_colimit.equiv_iso_colimit {J : Type v} {C : Type u} {F : J C} {r t : category_theory.limits.cocone F} (i : r t) :

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

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@[simp]
theorem category_theory.limits.is_colimit.equiv_iso_colimit_apply {J : Type v} {C : Type u} {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 v} {C : Type u} {F : J C} {r t : category_theory.limits.cocone F} (i : r t)  :
def category_theory.limits.is_colimit.of_point_iso {J : Type v} {C : Type u} {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.

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theorem category_theory.limits.is_colimit.hom_desc {J : Type v} {C : Type u} {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 v} {C : Type u} {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 K : Type v} {C : Type u} {F : J C} {D : Type u'} {G : K D}  :

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

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def category_theory.limits.is_colimit.of_cocone_equiv {J K : Type v} {C : Type u} {F : J C} {D : Type u'} {G : K D}  :

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

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@[simp]
theorem category_theory.limits.is_colimit.of_cocone_equiv_apply_desc {J K : Type v} {C : Type u} {F : J C} {D : Type u'} {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 K : Type v} {C : Type u} {F : J C} {D : Type u'} {G : K D}  :
def category_theory.limits.is_colimit.precompose_hom_equiv {J : Type v} {C : Type u} {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.

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def category_theory.limits.is_colimit.precompose_inv_equiv {J : Type v} {C : Type u} {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.

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def category_theory.limits.is_colimit.cocone_points_iso_of_nat_iso {J : Type v} {C : Type u} {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.

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@[simp]
theorem category_theory.limits.is_colimit.cocone_points_iso_of_nat_iso_inv {J : Type v} {C : Type u} {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 v} {C : Type u} {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 v} {C : Type u} {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 v} {C : Type u} {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 v} {C : Type u} {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 v} {C : Type u} {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 v} {C : Type u} {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 v} {C : Type u} {F G : J C} {r t : category_theory.limits.cocone G} (w : F G) :
w).hom Q.desc r = P.map r w.hom
def category_theory.limits.is_colimit.whisker_equivalence {J K : Type v} {C : Type u} {F : J C} (e : K J) :

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

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@[simp]
theorem category_theory.limits.is_colimit.cocone_points_iso_of_equivalence_hom {J K : Type v} {C : Type u} {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 K : Type v} {C : Type u} {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 K : Type v} {C : Type u} {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 F).X` are isomorphic if

• both cocones are colimit ccoones
• 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).

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def category_theory.limits.is_colimit.hom_iso {J : Type v} {C : Type u} {F : J C} (W : C) :
(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`.

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

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

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def category_theory.limits.is_colimit.hom_iso' {J : Type v} {C : Type u} {F : J C} (W : C) :
(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`.

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def category_theory.limits.is_colimit.of_faithful {J : Type v} {C : Type u} {F : J C} {D : Type u'} (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.

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def category_theory.limits.is_colimit.map_cocone_equiv {J : Type v} {C : Type u} {D : Type u'} {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.

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def category_theory.limits.is_colimit.iso_unique_cocone_morphism {J : Type v} {C : Type u} {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.

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def category_theory.limits.is_colimit.of_nat_iso.cocone_of_hom {J : Type v} {C : Type u} {F : J C} {X : C} (h : F.cocones) {Y : C} (f : X Y) :

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

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def category_theory.limits.is_colimit.of_nat_iso.hom_of_cocone {J : Type v} {C : Type u} {F : J C} {X : C} (h : F.cocones)  :
X s.X

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

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@[simp]
theorem category_theory.limits.is_colimit.of_nat_iso.cocone_of_hom_of_cocone {J : Type v} {C : Type u} {F : J C} {X : C} (h : F.cocones)  :
@[simp]
theorem category_theory.limits.is_colimit.of_nat_iso.hom_of_cocone_of_hom {J : Type v} {C : Type u} {F : J C} {X : C} (h : F.cocones) {Y : C} (f : X Y) :
def category_theory.limits.is_colimit.of_nat_iso.colimit_cocone {J : Type v} {C : Type u} {F : J C} {X : C} (h : F.cocones) :

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

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theorem category_theory.limits.is_colimit.of_nat_iso.cocone_of_hom_fac {J : Type v} {C : Type u} {F : J C} {X : C} (h : F.cocones) {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 v} {C : Type u} {F : J C} {X : C} (h : F.cocones)  :

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 v} {C : Type u} {F : J C} {X : C} (h : F.cocones) :

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

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