File: Mathlib/CategoryTheory/Sites/Pretopology.lean

full_name: CategoryTheory.Pretopology.transitive
code: /--
A (Grothendieck) pretopology on `C` consists of a collection of families of morphisms with a fixed
target `X` for every object `X` in `C`, called "coverings" of `X`, which satisfies the following
three axioms:
1. Every family consisting of a single isomorphism is a covering family.
2. The collection of covering families is stable under pullback.
3. Given a covering family, and a covering family on each domain of the former, the composition
   is a covering family.

In some sense, a pretopology can be seen as Grothendieck topology with weaker saturation conditions,
in that each covering is not necessarily downward closed.

See: https://ncatlab.org/nlab/show/Grothendieck+pretopology, or
https://stacks.math.columbia.edu/tag/00VH, or [MM92] Chapter III, Section 2, Definition 2.
Note that Stacks calls a category together with a pretopology a site, and [MM92] calls this
a basis for a topology.
-/
@[ext]
structure Pretopology where
  coverings : ∀ X : C, Set (Presieve X)
  has_isos : ∀ ⦃X Y⦄ (f : Y ⟶ X) [IsIso f], Presieve.singleton f ∈ coverings X
  pullbacks : ∀ ⦃X Y⦄ (f : Y ⟶ X) (S), S ∈ coverings X → pullbackArrows f S ∈ coverings Y
  transitive :
    ∀ ⦃X : C⦄ (S : Presieve X) (Ti : ∀ ⦃Y⦄ (f : Y ⟶ X), S f → Presieve Y),
      S ∈ coverings X → (∀ ⦃Y⦄ (f) (H : S f), Ti f H ∈ coverings Y) → S.bind Ti ∈ coverings X
goal: S ∈ self.coverings X → (∀ ⦃Y : C⦄ (f : Y ⟶ X) (H : S f), Ti f H ∈ self.coverings Y) → S.bind Ti ∈ self.coverings X

full_name: CategoryTheory.Pretopology.coverings
code: /--
A (Grothendieck) pretopology on `C` consists of a collection of families of morphisms with a fixed
target `X` for every object `X` in `C`, called "coverings" of `X`, which satisfies the following
three axioms:
1. Every family consisting of a single isomorphism is a covering family.
2. The collection of covering families is stable under pullback.
3. Given a covering family, and a covering family on each domain of the former, the composition
   is a covering family.

In some sense, a pretopology can be seen as Grothendieck topology with weaker saturation conditions,
in that each covering is not necessarily downward closed.

See: https://ncatlab.org/nlab/show/Grothendieck+pretopology, or
https://stacks.math.columbia.edu/tag/00VH, or [MM92] Chapter III, Section 2, Definition 2.
Note that Stacks calls a category together with a pretopology a site, and [MM92] calls this
a basis for a topology.
-/
@[ext]
structure Pretopology where
  coverings : ∀ X : C, Set (Presieve X)
  has_isos : ∀ ⦃X Y⦄ (f : Y ⟶ X) [IsIso f], Presieve.singleton f ∈ coverings X
  pullbacks : ∀ ⦃X Y⦄ (f : Y ⟶ X) (S), S ∈ coverings X → pullbackArrows f S ∈ coverings Y
  transitive :
    ∀ ⦃X : C⦄ (S : Presieve X) (Ti : ∀ ⦃Y⦄ (f : Y ⟶ X), S f → Presieve Y),
      S ∈ coverings X → (∀ ⦃Y⦄ (f) (H : S f), Ti f H ∈ coverings Y) → S.bind Ti ∈ coverings X
goal: Set (CategoryTheory.Presieve X)

full_name: CategoryTheory.Pretopology.has_isos
code: /--
A (Grothendieck) pretopology on `C` consists of a collection of families of morphisms with a fixed
target `X` for every object `X` in `C`, called "coverings" of `X`, which satisfies the following
three axioms:
1. Every family consisting of a single isomorphism is a covering family.
2. The collection of covering families is stable under pullback.
3. Given a covering family, and a covering family on each domain of the former, the composition
   is a covering family.

In some sense, a pretopology can be seen as Grothendieck topology with weaker saturation conditions,
in that each covering is not necessarily downward closed.

See: https://ncatlab.org/nlab/show/Grothendieck+pretopology, or
https://stacks.math.columbia.edu/tag/00VH, or [MM92] Chapter III, Section 2, Definition 2.
Note that Stacks calls a category together with a pretopology a site, and [MM92] calls this
a basis for a topology.
-/
@[ext]
structure Pretopology where
  coverings : ∀ X : C, Set (Presieve X)
  has_isos : ∀ ⦃X Y⦄ (f : Y ⟶ X) [IsIso f], Presieve.singleton f ∈ coverings X
  pullbacks : ∀ ⦃X Y⦄ (f : Y ⟶ X) (S), S ∈ coverings X → pullbackArrows f S ∈ coverings Y
  transitive :
    ∀ ⦃X : C⦄ (S : Presieve X) (Ti : ∀ ⦃Y⦄ (f : Y ⟶ X), S f → Presieve Y),
      S ∈ coverings X → (∀ ⦃Y⦄ (f) (H : S f), Ti f H ∈ coverings Y) → S.bind Ti ∈ coverings X
goal: CategoryTheory.Presieve.singleton f ∈ self.coverings X

full_name: CategoryTheory.Pretopology
code: /--
A (Grothendieck) pretopology on `C` consists of a collection of families of morphisms with a fixed
target `X` for every object `X` in `C`, called "coverings" of `X`, which satisfies the following
three axioms:
1. Every family consisting of a single isomorphism is a covering family.
2. The collection of covering families is stable under pullback.
3. Given a covering family, and a covering family on each domain of the former, the composition
   is a covering family.

In some sense, a pretopology can be seen as Grothendieck topology with weaker saturation conditions,
in that each covering is not necessarily downward closed.

See: https://ncatlab.org/nlab/show/Grothendieck+pretopology, or
https://stacks.math.columbia.edu/tag/00VH, or [MM92] Chapter III, Section 2, Definition 2.
Note that Stacks calls a category together with a pretopology a site, and [MM92] calls this
a basis for a topology.
-/
@[ext]
structure Pretopology where
  coverings : ∀ X : C, Set (Presieve X)
  has_isos : ∀ ⦃X Y⦄ (f : Y ⟶ X) [IsIso f], Presieve.singleton f ∈ coverings X
  pullbacks : ∀ ⦃X Y⦄ (f : Y ⟶ X) (S), S ∈ coverings X → pullbackArrows f S ∈ coverings Y
  transitive :
    ∀ ⦃X : C⦄ (S : Presieve X) (Ti : ∀ ⦃Y⦄ (f : Y ⟶ X), S f → Presieve Y),
      S ∈ coverings X → (∀ ⦃Y⦄ (f) (H : S f), Ti f H ∈ coverings Y) → S.bind Ti ∈ coverings X
goal: Type (max u v)

full_name: CategoryTheory.Pretopology.pullbacks
code: /--
A (Grothendieck) pretopology on `C` consists of a collection of families of morphisms with a fixed
target `X` for every object `X` in `C`, called "coverings" of `X`, which satisfies the following
three axioms:
1. Every family consisting of a single isomorphism is a covering family.
2. The collection of covering families is stable under pullback.
3. Given a covering family, and a covering family on each domain of the former, the composition
   is a covering family.

In some sense, a pretopology can be seen as Grothendieck topology with weaker saturation conditions,
in that each covering is not necessarily downward closed.

See: https://ncatlab.org/nlab/show/Grothendieck+pretopology, or
https://stacks.math.columbia.edu/tag/00VH, or [MM92] Chapter III, Section 2, Definition 2.
Note that Stacks calls a category together with a pretopology a site, and [MM92] calls this
a basis for a topology.
-/
@[ext]
structure Pretopology where
  coverings : ∀ X : C, Set (Presieve X)
  has_isos : ∀ ⦃X Y⦄ (f : Y ⟶ X) [IsIso f], Presieve.singleton f ∈ coverings X
  pullbacks : ∀ ⦃X Y⦄ (f : Y ⟶ X) (S), S ∈ coverings X → pullbackArrows f S ∈ coverings Y
  transitive :
    ∀ ⦃X : C⦄ (S : Presieve X) (Ti : ∀ ⦃Y⦄ (f : Y ⟶ X), S f → Presieve Y),
      S ∈ coverings X → (∀ ⦃Y⦄ (f) (H : S f), Ti f H ∈ coverings Y) → S.bind Ti ∈ coverings X
goal: S ∈ self.coverings X → CategoryTheory.Presieve.pullbackArrows f S ∈ self.coverings Y

full_name: CategoryTheory.Pretopology.instCoeFunForallSetPresieve
code: instance : CoeFun (Pretopology C) fun _ => ∀ X : C, Set (Presieve X) 
goal: CoeFun (CategoryTheory.Pretopology C) fun x => (X : C) → Set (CategoryTheory.Presieve X)

full_name: CategoryTheory.Pretopology.LE
code: instance LE : LE (Pretopology C) 
goal: LE (CategoryTheory.Pretopology C)

full_name: CategoryTheory.Pretopology.le_def
code: theorem le_def {K₁ K₂ : Pretopology C} : K₁ ≤ K₂ ↔ (K₁ : ∀ X : C, Set (Presieve X)) ≤ K₂ 
goal: K₁ ≤ K₂ ↔ K₁.coverings ≤ K₂.coverings

full_name: CategoryTheory.Pretopology.instPartialOrder
code: instance : PartialOrder (Pretopology C) 
goal: PartialOrder (CategoryTheory.Pretopology C)

full_name: CategoryTheory.Pretopology.instOrderTop
code: instance : OrderTop (Pretopology C) 
goal: OrderTop (CategoryTheory.Pretopology C)

full_name: CategoryTheory.Pretopology.instInhabited
code: instance : Inhabited (Pretopology C) 
goal: Inhabited (CategoryTheory.Pretopology C)

full_name: CategoryTheory.Pretopology.toGrothendieck
code: /-- A pretopology `K` can be completed to a Grothendieck topology `J` by declaring a sieve to be
`J`-covering if it contains a family in `K`.

See <https://stacks.math.columbia.edu/tag/00ZC>, or [MM92] Chapter III, Section 2, Equation (2).
-/
def toGrothendieck (K : Pretopology C) : GrothendieckTopology C 
goal: CategoryTheory.GrothendieckTopology C

full_name: CategoryTheory.Pretopology.mem_toGrothendieck
code: theorem mem_toGrothendieck (K : Pretopology C) (X S) :
    S ∈ toGrothendieck C K X ↔ ∃ R ∈ K X, R ≤ (S : Presieve X) 
goal: S ∈ (CategoryTheory.Pretopology.toGrothendieck C K).sieves X ↔ ∃ R ∈ K.coverings X, R ≤ S.arrows

full_name: CategoryTheory.Pretopology.ofGrothendieck
code: /-- The largest pretopology generating the given Grothendieck topology.

See [MM92] Chapter III, Section 2, Equations (3,4).
-/
def ofGrothendieck (J : GrothendieckTopology C) : Pretopology C 
goal: CategoryTheory.Pretopology C

full_name: CategoryTheory.Pretopology.gi
code: /-- We have a galois insertion from pretopologies to Grothendieck topologies. -/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) 
goal: GaloisInsertion (CategoryTheory.Pretopology.toGrothendieck C) (CategoryTheory.Pretopology.ofGrothendieck C)

full_name: CategoryTheory.Pretopology.trivial
code: /--
The trivial pretopology, in which the coverings are exactly singleton isomorphisms. This topology is
also known as the indiscrete, coarse, or chaotic topology.

See <https://stacks.math.columbia.edu/tag/07GE>
-/
def trivial : Pretopology C 
goal: CategoryTheory.Pretopology C

full_name: CategoryTheory.Pretopology.instOrderBot
code: instance : OrderBot (Pretopology C) 
goal: OrderBot (CategoryTheory.Pretopology C)

full_name: CategoryTheory.Pretopology.toGrothendieck_bot
code: /-- The trivial pretopology induces the trivial grothendieck topology. -/
theorem toGrothendieck_bot : toGrothendieck C ⊥ = ⊥ 
goal: CategoryTheory.Pretopology.toGrothendieck C ⊥ = ⊥