nLab Grothendieck pretopology

Contents

This entry is about pretopologies on sites. For a similarly-named generalization of topological spaces based on neighborhoods, see pretopological space.


Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Idea

A Grothendieck pretopology or basis for a Grothendieck topology is a collection of families of morphisms in a category which can be considered as covers.

Every Grothendieck pretopology generates a genuine Grothendieck topology. Different pretopologies may give rise to the same topology.

An even weaker notion than a Grothendieck pretopology, which also generates a Grothendieck topology, is a coverage. A Grothendieck pretopology can be defined as a coverage that also satisfies a couple of extra saturation conditions. (Note that it is coverages, not pretopologies, that most directly corresponds to bases of topological spaces.

Definition

See Definition II.1.3 in SGA 4.

Definition

Let CC be a category. A Grothendieck pretopology or basis (for a Grothendieck topology) on CC is an assignment to each object UU of CC of a collection of families {U iU}\{U_i \to U\} of morphisms, called covering families such that

  1. (Stability under base changes.) The collection of covering families is stable under pullback: if {U iU}\{U_i \to U\} is a covering family and f:VUf : V \to U is any morphism in CC, then {f *U iV}\{f^* U_i \to V\} exists and is a covering family;

  2. (Stability under composition.) If {U iU} iI\{U_i \to U\}_{i \in I} is a covering family and for each ii also {U i,jU i} jJ i\{U_{i,j} \to U_i\}_{j \in J_i} is a covering family, then also the family of composites {U i,jU iU} iI,jJ i\{U_{i,j} \to U_i \to U\}_{i\in I, j \in J_i} is a covering family.

  3. (Isomorphisms cover.) Every family consisting of a single isomorphism {VU}\{V \stackrel{\cong}{\to}U\} is a covering family;

If we drop the second and third conditions, we obtain something a bit stronger than a coverage; at the page coverage this notion is called a cartesian coverage. Conversely every coverage on a category with pullbacks generates a Grothendieck pretopology by an evident closure process. However, many coverages that arise in practice are actually already Grothendieck pretopologies. On the other hand, for some analogues in noncommutative algebraic geometry, rather the stability axiom fails.

Definition

The Grothendieck topology on CC generated from a basis of covering families is that for which a sieve {S iU}\{S_i \to U\} is covering precisely if it contains a covering family of morphisms.

Properties

Given any Grothendieck topology on CC, there is a maximal basis which generates it: this has as covering families precisely thoses families of morphisms that generate a covering sieve under completion under precomposition.

Examples

The prototype is the pretopology on the category of open subsets Op(X)Op(X) of a topological space XX, consisting of open covers of XX.

Given a base for the topology on XX, we can construct a pretopology on Op(X)Op(X) by declaring that a family {U iV} iI\{U_i\to V\}_{i\in I} is a covering family if it is an open cover and for every iIi\in I the open set U iU_i is the intersection of VV and an element of the base.

If instead we impose a stronger requirement that U iU_i belongs to the base, then the resulting coverage is not a pretopology, since the intersection of such a covering family with an arbitrary open subset of XX is an open cover whose elements need not belong to the base.

Grothendieck pretopologies on Top include:

An example for the category Diff of manifolds is the pretopology of surjective submersions. All of these have covering families consisting of single morphisms. Such a pretopology is called a singleton pretopology (and, in particular, it is a singleton coverage).

An example of a coverage that is not a pretopology is the coverage of good open covers, say on Diff. In general the pullback of a good open cover is just an open cover, not necessarily still one where all finite non-empty intersections are contractible.

References

The definition appears for instance as definition 2 on page 111 of

Last revised on May 10, 2022 at 13:23:00. See the history of this page for a list of all contributions to it.