mathlib documentation


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.


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


coverage, pretopology, site


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.


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.

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).


The largest pretopology generating the given Grothendieck topology.

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


The trivial pretopology, in which the coverings are exactly singleton isomorphisms. This topology is also known as the indiscrete, coarse, or chaotic topology.



The trivial pretopology induces the trivial grothendieck topology.