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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

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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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.

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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The largest pretopology generating the given Grothendieck topology.

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

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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.