nLab Grothendieck pretopology

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

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

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

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

3. (Isomorphisms cover.) Every family consisting of a single isomorphism $\{V \stackrel{\cong}{\to}U\}$ 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 $C$ generated from a basis of covering families is that for which a sieve $\{S_i \to U\}$ is covering precisely if it contains a covering family of morphisms.

Properties

Given any Grothendieck topology on $C$, 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 $Op(X)$ of a topological space $X$, consisting of open covers of $X$.

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

If instead we impose a stronger requirement that $U_i$ 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 $X$ is an open cover whose elements need not belong to the base.

Grothendieck pretopologies on Top include:

• Local-section-admitting surjections
• Surjective open maps
• Surjective topological submersions
• Surjective local homeomorphisms

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

• Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic
• MO questions:
  • example-of-a-grothendieck-pretopology-satisfying-a-weak-saturation-condition;
  • colimits-of-covers;
  • When is a basis of a topological space a Grothendieck pretopology?