mathlib documentation

category_theory.limits.has_limits

Existence of limits and colimits #

In category_theory.limits.is_limit we defined is_limit c, the data showing that a cone c is a limit cone.

The two main structures defined in this file are:

has_limit is a propositional typeclass (it's important that it is a proposition merely asserting the existence of a limit, as otherwise we would have non-defeq problems from incompatible instances).

While has_limit only asserts the existence of a limit cone, we happily use the axiom of choice in mathlib, so there are convenience functions all depending on has_limit F:

Key to using the has_limit interface is that there is an @[ext] lemma stating that to check f = g, for f g : Z ⟶ limit F, it suffices to check f ≫ limit.π F j = g ≫ limit.π F j for every j. This, combined with @[simp] lemmas, makes it possible to prove many easy facts about limits using automation (e.g. tidy).

There are abbreviations has_limits_of_shape J C and has_limits C asserting the existence of classes of limits. Later more are introduced, for finite limits, special shapes of limits, etc.

Ideally, many results about limits should be stated first in terms of is_limit, and then a result in terms of has_limit derived from this. At this point, however, this is far from uniformly achieved in mathlib --- often statements are only written in terms of has_limit.

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.limit_cone {J : Type u₁} [category_theory.category J] {C : Type u} [category_theory.category C] (F : J C) :
Type (max u u₁ v)

limit_cone F contains a cone over F together with the information that it is a limit.

Instances for category_theory.limits.limit_cone
  • category_theory.limits.limit_cone.has_sizeof_inst
@[class]
structure category_theory.limits.has_limit {J : Type u₁} [category_theory.category J] {C : Type u} [category_theory.category C] (F : J C) :
Prop

has_limit F represents the mere existence of a limit for F.

Instances of this typeclass

Use the axiom of choice to extract explicit limit_cone F from has_limit F.

Equations
@[class]

C has limits of shape J if there exists a limit for every functor F : J ⥤ C.

Instances of this typeclass
@[class]

C has all limits of size v₁ u₁ (has_limits_of_size.{v₁ u₁} C) if it has limits of every shape J : Type u₁ with [category.{v₁} J].

Instances of this typeclass

C has all (small) limits if it has limits of every shape that is as big as its hom-sets.

An arbitrary choice of limit cone for a functor.

Equations

The projection from the limit object to a value of the functor.

Equations
Instances for category_theory.limits.limit.π
@[simp]
theorem category_theory.limits.limit.w_assoc {J : Type u₁} [category_theory.category J] {C : Type u} [category_theory.category C] (F : J C) [category_theory.limits.has_limit F] {j j' : J} (f : j j') {X' : C} (f' : F.obj j' X') :

Evidence that the arbitrary choice of cone provied by limit.cone F is a limit cone.

Equations

Functoriality of limits.

Usually this morphism should be accessed through lim.map, but may be needed separately when you have specified limits for the source and target functors, but not necessarily for all functors of shape J.

Equations

The isomorphism (in Type) between morphisms from a specified object W to the limit object, and cones with cone point W.

Equations
noncomputable def category_theory.limits.limit.hom_iso' {J : Type u₁} [category_theory.category J] {C : Type u} [category_theory.category C] (F : J C) [category_theory.limits.has_limit F] (W : C) :
ulift (W category_theory.limits.limit F) {p // ∀ {j j' : J} (f : j j'), p j F.map f = p j'}

The isomorphism (in Type) between morphisms from a specified object W to the limit object, and an explicit componentwise description of cones with cone point W.

Equations

If a functor F has a limit, so does any naturally isomorphic functor.

If a functor G has the same collection of cones as a functor F which has a limit, then G also has a limit.

The limits of F : J ⥤ C and G : K ⥤ C are isomorphic, if there is an equivalence e : J ≌ K making the triangle commute up to natural isomorphism.

Equations

The canonical morphism from G applied to the limit of F to the limit of F ⋙ G.

Equations

If a E ⋙ F has a limit, and E is an equivalence, we can construct a limit of F.

limit F is functorial in F, when C has all limits of shape J.

Equations

The isomorphism between morphisms from W to the cone point of the limit cone for F and cones over F with cone point W is natural in F.

Equations

We can transport limits of shape J along an equivalence J ≌ J'.

has_limits_of_size_shrink.{v u} C tries to obtain has_limits_of_size.{v u} C from some other has_limits_of_size C.

@[nolint]
structure category_theory.limits.colimit_cocone {J : Type u₁} [category_theory.category J] {C : Type u} [category_theory.category C] (F : J C) :
Type (max u u₁ v)

colimit_cocone F contains a cocone over F together with the information that it is a colimit.

Instances for category_theory.limits.colimit_cocone
  • category_theory.limits.colimit_cocone.has_sizeof_inst
@[class]
structure category_theory.limits.has_colimit {J : Type u₁} [category_theory.category J] {C : Type u} [category_theory.category C] (F : J C) :
Prop

has_colimit F represents the mere existence of a colimit for F.

Instances of this typeclass
@[class]

C has colimits of shape J if there exists a colimit for every functor F : J ⥤ C.

Instances of this typeclass
@[class]

C has all colimits of size v₁ u₁ (has_colimits_of_size.{v₁ u₁} C) if it has colimits of every shape J : Type u₁ with [category.{v₁} J].

Instances of this typeclass

C has all (small) colimits if it has colimits of every shape that is as big as its hom-sets.

The coprojection from a value of the functor to the colimit object.

Equations
Instances for category_theory.limits.colimit.ι
@[simp]

We have lots of lemmas describing how to simplify colimit.ι F j ≫ _, and combined with colimit.ext we rely on these lemmas for many calculations.

However, since category.assoc is a @[simp] lemma, often expressions are right associated, and it's hard to apply these lemmas about colimit.ι.

We thus use reassoc to define additional @[simp] lemmas, with an arbitrary extra morphism. (see tactic/reassoc_axiom.lean)

Functoriality of colimits.

Usually this morphism should be accessed through colim.map, but may be needed separately when you have specified colimits for the source and target functors, but not necessarily for all functors of shape J.

Equations

The isomorphism (in Type) between morphisms from the colimit object to a specified object W, and cocones with cone point W.

Equations
noncomputable def category_theory.limits.colimit.hom_iso' {J : Type u₁} [category_theory.category J] {C : Type u} [category_theory.category C] (F : J C) [category_theory.limits.has_colimit F] (W : C) :
ulift (category_theory.limits.colimit F W) {p // ∀ {j j' : J} (f : j j'), F.map f p j' = p j}

The isomorphism (in Type) between morphisms from the colimit object to a specified object W, and an explicit componentwise description of cocones with cone point W.

Equations

If F has a colimit, so does any naturally isomorphic functor.

If a functor G has the same collection of cocones as a functor F which has a colimit, then G also has a colimit.

The colimits of F : J ⥤ C and G : K ⥤ C are isomorphic, if there is an equivalence e : J ≌ K making the triangle commute up to natural isomorphism.

Equations

The canonical morphism from G applied to the colimit of F ⋙ G to G applied to the colimit of F.

Equations

If a E ⋙ F has a colimit, and E is an equivalence, we can construct a colimit of F.

The isomorphism between morphisms from the cone point of the colimit cocone for F to W and cocones over F with cone point W is natural in F.

Equations

We can transport colimits of shape J along an equivalence J ≌ J'.

has_colimits_of_size_shrink.{v u} C tries to obtain has_colimits_of_size.{v u} C from some other has_colimits_of_size C.

If t : cone F is a limit cone, then t.op : cocone F.op is a colimit cocone.

Equations

If t : cocone F is a colimit cocone, then t.op : cone F.op is a limit cone.

Equations

If t : cone F.op is a limit cone, then t.unop : cocone F is a colimit cocone.

Equations

If t : cocone F.op is a colimit cocone, then t.unop : cone F. is a limit cone.

Equations