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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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.
Let $C$ be a category with pullbacks. 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
isomorphisms cover – every family consisting of a single isomorphism $\{V \stackrel{\cong}{\to}U\}$ is a covering family;
stability axiom – 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 $\{f^* U_i \to V\}$ is a covering family;
transitivity axiom – 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.
If we drop the first and third conditions, we obtain the notion of a coverage on a category with pullbacks; 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.
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.
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.
The prototype is the pretopology on the category of open subsets $Op(X)$ of a topological space $X$, consisting of open covers of $X$.
Notice that a basis for the topology of $X$ is not a Grothendieck pretopology on $Op(X)$ (since it is in general not closed under pullback, which here is restriction) but is a coverage on $Op(X)$.
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.
The definition appears for instance as definition 2 on page 111 of