Products in the over category #

Shows that products in the over category can be derived from wide pullbacks in the base category. The main result is over_product_of_widePullback, which says that if C has J-indexed wide pullbacks, then Over B has J-indexed products.

source

@[reducible, inline]

abbrev CategoryTheory.Over.ConstructProducts.widePullbackDiagramOfDiagramOver {C : Type u} [Category.{v, u} C] (B : C) {J : Type w} (F : Functor (Discrete J) (Over B)) :

Functor (Limits.WidePullbackShape J) C

(Implementation) Given a product diagram in C/B, construct the corresponding wide pullback diagram in C.

Equations

One or more equations did not get rendered due to their size.

Instances For

source

def CategoryTheory.Over.ConstructProducts.conesEquivInverseObj {C : Type u} [Category.{v, u} C] (B : C) {J : Type w} (F : Functor (Discrete J) (Over B)) (c : Limits.Cone F) :

Limits.Cone (widePullbackDiagramOfDiagramOver B F)

(Impl) A preliminary definition to avoid timeouts.

Equations

One or more equations did not get rendered due to their size.

Instances For

source

@[simp]

theorem CategoryTheory.Over.ConstructProducts.conesEquivInverseObj_pt {C : Type u} [Category.{v, u} C] (B : C) {J : Type w} (F : Functor (Discrete J) (Over B)) (c : Limits.Cone F) :

(conesEquivInverseObj B F c).pt = c.pt.left

source

@[simp]

theorem CategoryTheory.Over.ConstructProducts.conesEquivInverseObj_π_app {C : Type u} [Category.{v, u} C] (B : C) {J : Type w} (F : Functor (Discrete J) (Over B)) (c : Limits.Cone F) (X : Limits.WidePullbackShape J) :

(conesEquivInverseObj B F c).π.app X = Option.casesOn X c.pt.hom fun (j : J) => (c.π.app { as := j }).left

source

def CategoryTheory.Over.ConstructProducts.conesEquivInverse {C : Type u} [Category.{v, u} C] (B : C) {J : Type w} (F : Functor (Discrete J) (Over B)) :

Functor (Limits.Cone F) (Limits.Cone (widePullbackDiagramOfDiagramOver B F))

(Impl) A preliminary definition to avoid timeouts.

Equations

One or more equations did not get rendered due to their size.

Instances For

source

@[simp]

theorem CategoryTheory.Over.ConstructProducts.conesEquivInverse_obj {C : Type u} [Category.{v, u} C] (B : C) {J : Type w} (F : Functor (Discrete J) (Over B)) (c : Limits.Cone F) :

(conesEquivInverse B F).obj c = conesEquivInverseObj B F c

source

@[simp]

theorem CategoryTheory.Over.ConstructProducts.conesEquivInverse_map_hom {C : Type u} [Category.{v, u} C] (B : C) {J : Type w} (F : Functor (Discrete J) (Over B)) {X✝ Y✝ : Limits.Cone F} (f : X✝ ⟶ Y✝) :

((conesEquivInverse B F).map f).hom = f.hom.left

source

def CategoryTheory.Over.ConstructProducts.conesEquivFunctor {C : Type u} [Category.{v, u} C] (B : C) {J : Type w} (F : Functor (Discrete J) (Over B)) :

Functor (Limits.Cone (widePullbackDiagramOfDiagramOver B F)) (Limits.Cone F)

(Impl) A preliminary definition to avoid timeouts.

Equations

One or more equations did not get rendered due to their size.

Instances For

source

@[simp]

theorem CategoryTheory.Over.ConstructProducts.conesEquivFunctor_obj_π_app {C : Type u} [Category.{v, u} C] (B : C) {J : Type w} (F : Functor (Discrete J) (Over B)) (c : Limits.Cone (widePullbackDiagramOfDiagramOver B F)) (x✝ : Discrete J) :

((conesEquivFunctor B F).obj c).π.app x✝ = match x✝ with | { as := j } => homMk (c.π.app (some j)) ⋯

source

@[simp]

theorem CategoryTheory.Over.ConstructProducts.conesEquivFunctor_obj_pt {C : Type u} [Category.{v, u} C] (B : C) {J : Type w} (F : Functor (Discrete J) (Over B)) (c : Limits.Cone (widePullbackDiagramOfDiagramOver B F)) :

((conesEquivFunctor B F).obj c).pt = mk (c.π.app none)

source

@[simp]

theorem CategoryTheory.Over.ConstructProducts.conesEquivFunctor_map_hom {C : Type u} [Category.{v, u} C] (B : C) {J : Type w} (F : Functor (Discrete J) (Over B)) {X✝ Y✝ : Limits.Cone (widePullbackDiagramOfDiagramOver B F)} (f : X✝ ⟶ Y✝) :

((conesEquivFunctor B F).map f).hom = homMk f.hom ⋯

source

def CategoryTheory.Over.ConstructProducts.conesEquivUnitIso {J : Type w} {C : Type u} [Category.{v, u} C] (B : C) (F : Functor (Discrete J) (Over B)) :

Functor.id (Limits.Cone (widePullbackDiagramOfDiagramOver B F)) ≅ (conesEquivFunctor B F).comp (conesEquivInverse B F)

(Impl) A preliminary definition to avoid timeouts.

Equations

One or more equations did not get rendered due to their size.

Instances For

source

@[simp]

theorem CategoryTheory.Over.ConstructProducts.conesEquivUnitIso_inv_app_hom {J : Type w} {C : Type u} [Category.{v, u} C] (B : C) (F : Functor (Discrete J) (Over B)) (X : Limits.Cone (widePullbackDiagramOfDiagramOver B F)) :

((conesEquivUnitIso B F).inv.app X).hom = CategoryStruct.id X.pt

source

@[simp]

theorem CategoryTheory.Over.ConstructProducts.conesEquivUnitIso_hom_app_hom {J : Type w} {C : Type u} [Category.{v, u} C] (B : C) (F : Functor (Discrete J) (Over B)) (X : Limits.Cone (widePullbackDiagramOfDiagramOver B F)) :

((conesEquivUnitIso B F).hom.app X).hom = CategoryStruct.id X.pt

source

def CategoryTheory.Over.ConstructProducts.conesEquivCounitIso {J : Type w} {C : Type u} [Category.{v, u} C] (B : C) (F : Functor (Discrete J) (Over B)) :

(conesEquivInverse B F).comp (conesEquivFunctor B F) ≅ Functor.id (Limits.Cone F)

(Impl) A preliminary definition to avoid timeouts.

Equations

One or more equations did not get rendered due to their size.

Instances For

source

@[simp]

theorem CategoryTheory.Over.ConstructProducts.conesEquivCounitIso_inv_app_hom_left {J : Type w} {C : Type u} [Category.{v, u} C] (B : C) (F : Functor (Discrete J) (Over B)) (X : Limits.Cone F) :

((conesEquivCounitIso B F).inv.app X).hom.left = CategoryStruct.id X.pt.left

source

@[simp]

theorem CategoryTheory.Over.ConstructProducts.conesEquivCounitIso_hom_app_hom_left {J : Type w} {C : Type u} [Category.{v, u} C] (B : C) (F : Functor (Discrete J) (Over B)) (X : Limits.Cone F) :

((conesEquivCounitIso B F).hom.app X).hom.left = CategoryStruct.id X.pt.left

source

def CategoryTheory.Over.ConstructProducts.conesEquiv {J : Type w} {C : Type u} [Category.{v, u} C] (B : C) (F : Functor (Discrete J) (Over B)) :

Limits.Cone (widePullbackDiagramOfDiagramOver B F) ≌ Limits.Cone F

(Impl) Establish an equivalence between the category of cones for F and for the "grown" F.

Equations

One or more equations did not get rendered due to their size.

Instances For

source

@[simp]

theorem CategoryTheory.Over.ConstructProducts.conesEquiv_functor {J : Type w} {C : Type u} [Category.{v, u} C] (B : C) (F : Functor (Discrete J) (Over B)) :

(conesEquiv B F).functor = conesEquivFunctor B F

source

@[simp]

theorem CategoryTheory.Over.ConstructProducts.conesEquiv_counitIso {J : Type w} {C : Type u} [Category.{v, u} C] (B : C) (F : Functor (Discrete J) (Over B)) :

(conesEquiv B F).counitIso = conesEquivCounitIso B F

source

@[simp]

theorem CategoryTheory.Over.ConstructProducts.conesEquiv_inverse {J : Type w} {C : Type u} [Category.{v, u} C] (B : C) (F : Functor (Discrete J) (Over B)) :

(conesEquiv B F).inverse = conesEquivInverse B F

source

@[simp]

theorem CategoryTheory.Over.ConstructProducts.conesEquiv_unitIso {J : Type w} {C : Type u} [Category.{v, u} C] (B : C) (F : Functor (Discrete J) (Over B)) :

(conesEquiv B F).unitIso = conesEquivUnitIso B F

source

theorem CategoryTheory.Over.ConstructProducts.has_over_limit_discrete_of_widePullback_limit {J : Type w} {C : Type u} [Category.{v, u} C] {B : C} (F : Functor (Discrete J) (Over B)) [Limits.HasLimit (widePullbackDiagramOfDiagramOver B F)] :

Limits.HasLimit F

Use the above equivalence to prove we have a limit.

source

theorem CategoryTheory.Over.ConstructProducts.over_product_of_widePullback {J : Type w} {C : Type u} [Category.{v, u} C] [Limits.HasLimitsOfShape (Limits.WidePullbackShape J) C] {B : C} :

Limits.HasLimitsOfShape (Discrete J) (Over B)

Given a wide pullback in C, construct a product in C/B.

source

theorem CategoryTheory.Over.ConstructProducts.over_binaryProduct_of_pullback {C : Type u} [Category.{v, u} C] [Limits.HasPullbacks C] {B : C} :

Limits.HasBinaryProducts (Over B)

Given a pullback in C, construct a binary product in C/B.

source

theorem CategoryTheory.Over.ConstructProducts.over_products_of_widePullbacks {C : Type u} [Category.{v, u} C] [Limits.HasWidePullbacks C] {B : C} :

Limits.HasProducts (Over B)

Given all wide pullbacks in C, construct products in C/B.

source

theorem CategoryTheory.Over.ConstructProducts.over_finiteProducts_of_finiteWidePullbacks {C : Type u} [Category.{v, u} C] [Limits.HasFiniteWidePullbacks C] {B : C} :

Limits.HasFiniteProducts (Over B)

Given all finite wide pullbacks in C, construct finite products in C/B.

source

theorem CategoryTheory.Over.over_hasTerminal {C : Type u} [Category.{v, u} C] (B : C) :

Limits.HasTerminal (Over B)

Construct terminal object in the over category. This isn't an instance as it's not typically the way we want to define terminal objects. (For instance, this gives a terminal object which is different from the generic one given by over_product_of_widePullback above.)

source