mathlib documentation

category_theory.limits.types

(internal implementation) the limit cone of a functor, implemented as flat sections of a pi type

Equations

(internal implementation) the fact that the proposed limit cone is the limit

Equations
@[instance]

The category of types has all limits.

See https://stacks.math.columbia.edu/tag/002U.

The equivalence between a limiting cone of F in Type u and the "concrete" definition as the sections of F.

Equations

The equivalence between the abstract limit of F in Type u and the "concrete" definition as the sections of F.

Equations
@[ext]
def category_theory.limits.types.limit.mk {J : Type u} [category_theory.small_category J] (F : J Type u) (x : Π (j : J), F.obj j) :
(∀ (j j' : J) (f : j j'), F.map f (x j) = x j')category_theory.limits.limit F

Construct a term of limit F : Type u from a family of terms x : Π j, F.obj j which are "coherent": ∀ (j j') (f : j ⟶ j'), F.map f (x j) = x j'.

Equations
@[simp]
theorem category_theory.limits.types.limit.π_mk {J : Type u} [category_theory.small_category J] (F : J Type u) (x : Π (j : J), F.obj j) (h : ∀ (j j' : J) (f : j j'), F.map f (x j) = x j') (j : J) :

def category_theory.limits.types.quot.rel {J : Type u} [category_theory.small_category J] (F : J Type u) :
(Σ (j : J), F.obj j)(Σ (j : J), F.obj j) → Prop

The relation defining the quotient type which implements the colimit of a functor F : J ⥤ Type u. See category_theory.limits.types.quot.

Equations
@[nolint]
def category_theory.limits.types.quot {J : Type u} [category_theory.small_category J] :
J Type uType u

A quotient type implementing the colimit of a functor F : J ⥤ Type u, as pairs ⟨j, x⟩ where x : F.obj j, modulo the equivalence relation generated by ⟨j, x⟩ ~ ⟨j', x'⟩ whenever there is a morphism f : j ⟶ j' so F.map f x = x'.

Equations

(internal implementation) the colimit cocone of a functor, implemented as a quotient of a sigma type

Equations

(internal implementation) the fact that the proposed colimit cocone is the colimit

Equations
@[instance]

The category of types has all colimits.

See https://stacks.math.columbia.edu/tag/002U.

@[simp]
theorem category_theory.limits.types.colimit.w_apply {J : Type u} [category_theory.small_category J] {F : J Type u} {j j' : J} {x : F.obj j} (f : j j') :

theorem category_theory.limits.types.colimit_sound {J : Type u} [category_theory.small_category J] {F : J Type u} {j j' : J} {x : F.obj j} {x' : F.obj j'} (f : j j') :

theorem category_theory.limits.types.colimit_sound' {J : Type u} [category_theory.small_category J] {F : J Type u} {j j' : J} {x : F.obj j} {x' : F.obj j'} {j'' : J} (f : j j'') (f' : j' j'') :

theorem category_theory.limits.types.jointly_surjective {J : Type u} [category_theory.small_category J] (F : J Type u) {t : category_theory.limits.cocone F} (h : category_theory.limits.is_colimit t) (x : t.X) :
∃ (j : J) (y : F.obj j), t.ι.app j y = x

A variant of jointly_surjective for x : colimit F.

def category_theory.limits.types.filtered_colimit.r {J : Type u} [category_theory.small_category J] (F : J Type u) :
(Σ (j : J), F.obj j)(Σ (j : J), F.obj j) → Prop

An alternative relation on Σ j, F.obj j, which generates the same equivalence relation as we use to define the colimit in Type above, but that is more convenient when working with filtered colimits.

Elements in F.obj j and F.obj j' are equivalent if there is some k : J to the right where their images are equal.

Equations
theorem category_theory.limits.types.filtered_colimit.r_ge {J : Type u} [category_theory.small_category J] (F : J Type u) (x y : Σ (j : J), F.obj j) :

def category_theory.limits.types.filtered_colimit.is_colimit_of {J : Type u} [category_theory.small_category J] (F : J Type u) (t : category_theory.limits.cocone F) :
(∀ (x : t.X), ∃ (i : J) (xi : F.obj i), x = t.ι.app i xi)(∀ (i j : J) (xi : F.obj i) (xj : F.obj j), t.ι.app i xi = t.ι.app j xj(∃ (k : J) (f : i k) (g : j k), F.map f xi = F.map g xj))category_theory.limits.is_colimit t

Recognizing filtered colimits of types.

Equations
theorem category_theory.limits.types.filtered_colimit.is_colimit_eq_iff {J : Type u} [category_theory.small_category J] (F : J Type u) {t : category_theory.limits.cocone F} [category_theory.is_filtered_or_empty J] (ht : category_theory.limits.is_colimit t) {i j : J} {xi : F.obj i} {xj : F.obj j} :
t.ι.app i xi = t.ι.app j xj ∃ (k : J) (f : i k) (g : j k), F.map f xi = F.map g xj

theorem category_theory.limits.types.filtered_colimit.colimit_eq_iff {J : Type u} [category_theory.small_category J] (F : J Type u) [category_theory.is_filtered_or_empty J] {i j : J} {xi : F.obj i} {xj : F.obj j} :
category_theory.limits.colimit.ι F i xi = category_theory.limits.colimit.ι F j xj ∃ (k : J) (f : i k) (g : j k), F.map f xi = F.map g xj

def category_theory.limits.types.image {α β : Type u} :
β)Type u

the image of a morphism in Type is just set.range f

Equations

the inclusion of image f into the target

Equations

the universal property for the image factorisation

Equations

the facorisation through a mono has the universal property of the image.

Equations
@[instance]

Equations