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.

Tags #

coverage, pretopology, site

References #

theorem CategoryTheory.Pretopology.ext {C : Type u} :
∀ {inst : CategoryTheory.Category.{v, u} C} {inst_1 : CategoryTheory.Limits.HasPullbacks C} (x y : CategoryTheory.Pretopology C), x.coverings = y.coveringsx = y

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:, or, 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, 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.


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