# mathlibdocumentation

category_theory.sites.pretopology

# Grothendieck pretopologies

Definition and lemmas about Grothendieck pretopologies. A Grothendieck pretopology for a category C is a set of families of morphisms with fixed codomain, satisfying certain closure conditions.

We show that a pretopology generates a genuine Grothendieck topology, and every topology has a maximal pretopology which generates it.

The pretopology associated to a topological space is defined in spaces.lean.

## Todo

Define sheaves on a pretopology, and show they are the same as the sheaves for the topology generated by the pretopology.

## Tags

coverage, pretopology, site

## References

• [https://ncatlab.org/nlab/show/Grothendieck+pretopology][nlab]
• [S. MacLane, I. Moerdijk, Sheaves in Geometry and Logic][MM92]
• [https://stacks.math.columbia.edu/tag/00VG][Stacks]
def category_theory.pullback_arrows {C : Type u} {X Y : C} (f : Y X) (S : category_theory.presieve X) :

Pullback a set of arrows with given codomain along a fixed map, by taking the pullback in the category. This is not the same as the arrow set of sieve.pullback, but there is a relation between them in pullback_arrows_comm.

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• = λ (Z : C) (g : Z Y), ∃ (Z' : C) (h : Z' X), S h
theorem category_theory.pullback_arrows_comm {C : Type u} {X Y : C} (f : Y X) (R : category_theory.presieve X) :

theorem category_theory.pullback_singleton {C : Type u} {X Y Z : C} (f : Y X) (g : Z X) :
∃ (W : C) (k : W Y),

@[ext]
structure category_theory.pretopology (C : Type u)  :
Type (max u v)
• coverings : Π (X : C),
• has_isos : ∀ ⦃X Y : C⦄ (f : Y X) [_inst_3 : ,
• pullbacks : ∀ ⦃X Y : C⦄ (f : Y X) (S : , S c.coverings X
• transitive : ∀ ⦃X : C⦄ (S : (Ti : Π ⦃Y : C⦄ (f : Y X), S f, S c.coverings X(∀ ⦃Y : C⦄ (f : Y X) (H : S f), Ti f H c.coverings Y)S.bind Ti c.coverings X

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.

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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).

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theorem category_theory.pretopology.mem_to_grothendieck (C : Type u) (X : C) (S : category_theory.sieve X) :
∃ (R : (H : R K X), R S

The largest pretopology generating the given Grothendieck topology.

See [MM92] Chapter III, Section 2, Equations (3,4).

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We have a galois insertion from pretopologies to Grothendieck topologies.

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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

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The trivial pretopology induces the trivial grothendieck topology.