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category_theory.sites.grothendieck

Grothendieck topologies

Definition and lemmas about Grothendieck topologies. A Grothendieck topology for a category C is a set of sieves on each object X satisfying certain closure conditions.

Alternate versions of the axioms (in arrow form) are also described. Two explicit examples of Grothendieck topologies are given:

A pretopology, or a basis for a topology is defined in pretopology.lean. The topology associated to a topological space is defined in spaces.lean.

Todo

Define sheaves on a topology, and the topos of sheaves.

Tags

Grothendieck topology, coverage, pretopology, site

References

Implementation notes

We use the definition of [nlab] and MM91, where Grothendieck topologies are saturated collections of morphisms, rather than the notions of the Stacks project (00VG) and the Elephant, in which topologies are allowed to be unsaturated, and are then completed. TODO (BM): Add the definition from Stacks, as a pretopology, and complete to a topology.

This is so that we can produce a bijective correspondence between Grothendieck topologies on a small category and Lawvere-Tierney topologies on its presheaf topos, as well as the equivalence between Grothendieck topoi and left exact reflective subcategories of presheaf toposes.

structure category_theory.grothendieck_topology (C : Type u) [category_theory.category C] :
Type (max u v)

The definition of a Grothendieck topology: a set of sieves J X on each object X satisfying three axioms:

  1. For every object X, the maximal sieve is in J X.
  2. If S ∈ J X then its pullback along any h : Y ⟶ X is in J Y.
  3. If S ∈ J X and R is a sieve on X, then provided that the pullback of R along any arrow f : Y ⟶ X in S is in J Y, we have that R itself is in J X.

A sieve S on X is referred to as J-covering, (or just covering), if S ∈ J X.

See https://stacks.math.columbia.edu/tag/00Z4, or [nlab], or [MM92] Chapter III, Section 2, Definition 1.

@[ext]
theorem category_theory.grothendieck_topology.ext {C : Type u} [category_theory.category C] {J₁ J₂ : category_theory.grothendieck_topology C} (h : J₁ = J₂) :
J₁ = J₂

An extensionality lemma in terms of the coercion to a pi-type. We prove this explicitly rather than deriving it so that it is in terms of the coercion rather than the projection .sieves.

theorem category_theory.grothendieck_topology.transitive {C : Type u} [category_theory.category C] {X : C} {S : category_theory.sieve X} (J : category_theory.grothendieck_topology C) (hS : S J X) (R : category_theory.sieve X) (h : ∀ ⦃Y : C⦄ ⦃f : Y X⦄, S fcategory_theory.sieve.pullback f R J Y) :
R J X

If S is a subset of R, and S is covering, then R is covering as well.

See https://stacks.math.columbia.edu/tag/00Z5 (2), or discussion after [MM92] Chapter III, Section 2, Definition 1.

The intersection of two covering sieves is covering.

See https://stacks.math.columbia.edu/tag/00Z5 (1), or [MM92] Chapter III, Section 2, Definition 1 (iv).

theorem category_theory.grothendieck_topology.bind_covering {C : Type u} [category_theory.category C] {X : C} (J : category_theory.grothendieck_topology C) {S : category_theory.sieve X} {R : Π ⦃Y : C⦄ ⦃f : Y X⦄, S fcategory_theory.sieve Y} (hS : S J X) (hR : ∀ ⦃Y : C⦄ ⦃f : Y X⦄ (H : S f), R H J Y) :

The sieve S on X J-covers an arrow f to X if S.pullback f ∈ J Y. This definition is an alternate way of presenting a Grothendieck topology.

Equations

The maximality axiom in 'arrow' form: Any arrow f in S is covered by S.

theorem category_theory.grothendieck_topology.arrow_stable {C : Type u} [category_theory.category C] {X Y : C} (J : category_theory.grothendieck_topology C) (f : Y X) (S : category_theory.sieve X) (h : J.covers S f) {Z : C} (g : Z Y) :
J.covers S (g f)

The stability axiom in 'arrow' form: If S covers f then S covers g ≫ f for any g.

theorem category_theory.grothendieck_topology.arrow_trans {C : Type u} [category_theory.category C] {X Y : C} (J : category_theory.grothendieck_topology C) (f : Y X) (S R : category_theory.sieve X) (h : J.covers S f) :
(∀ {Z : C} (g : Z X), S gJ.covers R g)J.covers R f

The transitivity axiom in 'arrow' form: If S covers f and every arrow in S is covered by R, then R covers f.

theorem category_theory.grothendieck_topology.arrow_intersect {C : Type u} [category_theory.category C] {X Y : C} (J : category_theory.grothendieck_topology C) (f : Y X) (S R : category_theory.sieve X) (hS : J.covers S f) (hR : J.covers R f) :
J.covers (S R) f

The trivial Grothendieck topology, in which only the maximal sieve is covering. This topology is also known as the indiscrete, coarse, or chaotic topology.

See [MM92] Chapter III, Section 2, example (a), or https://en.wikipedia.org/wiki/Grothendieck_topology#The_discrete_and_indiscrete_topologies

Equations

The discrete Grothendieck topology, in which every sieve is covering.

See https://en.wikipedia.org/wiki/Grothendieck_topology#The_discrete_and_indiscrete_topologies.

Equations
@[instance]

See https://stacks.math.columbia.edu/tag/00Z6

Equations

See https://stacks.math.columbia.edu/tag/00Z7

@[instance]

Construct a complete lattice from the Inf, but make the trivial and discrete topologies definitionally equal to the bottom and top respectively.

Equations
@[simp]

The dense Grothendieck topology.

See https://ncatlab.org/nlab/show/dense+topology, or [MM92] Chapter III, Section 2, example (e).

Equations
theorem category_theory.grothendieck_topology.dense_covering {C : Type u} [category_theory.category C] {X : C} {S : category_theory.sieve X} :
S category_theory.grothendieck_topology.dense X ∀ {Y : C} (f : Y X), ∃ (Z : C) (g : Z Y), S (g f)

A category satisfies the right Ore condition if any span can be completed to a commutative square. NB. Any category with pullbacks obviously satisfies the right Ore condition, see right_ore_of_pullbacks.

Equations

The atomic Grothendieck topology: a sieve is covering iff it is nonempty. For the pullback stability condition, we need the right Ore condition to hold.

See https://ncatlab.org/nlab/show/atomic+site, or [MM92] Chapter III, Section 2, example (f).

Equations