nLab
Grothendieck pretopology

Note: “pretopology” redirects here. For a generalization of topological spaces based on neighborhoods, see pretopological space.

A Grothendieck pretopology is a collection of families of maps in a category which can be considered as covers. It is sometimes known as a ”basis for a Grothendieck topology”, as a pretopology generates a Grothendieck topology. Note that different pretopologies can generate the same Grothendieck topology.

An even weaker notion than a Grothendieck pretopology, which also generates a Grothendieck toplogy, is a coverage. A Grothendieck pretopology can be defined as a coverage that also satisfies a couple of extra saturation conditions.

Definition

Let $S$ be a category. A Grothendieck pretopology is a collection of families $\{\{U_i\to A{\}}_{i\in I}\}$, called covering families, for each object $A$, satisfying the following conditions:

• If $A\prime \to A$ is an isomorphism, then $\{A\prime \to A\}$ is a covering family,

• Given a covering family $\{U_i\to A{\}}_{i\in I}$ and a map $B\to A$, the pullbacks $B{\times }_A{U}_i$ exist and $\{B{\times }_A{U}_i\to B{\}}_{i\in I}$ is a covering family,

• Given a covering family $\{U_i\to A{\}}_{i\in I}$ and for each $i\in I$ a covering family $\{V_{ij}\to U_i{\}}_{j\in J_i}$, then $\{V_{ij}\to A{\}}_{j\in J_i,i\in I}$ is a covering family.

If we drop the first and third conditions, we obtain the notion of a coverage; conversely every coverage generates a Grothendieck pretopology by an evident closure process. However, many coverages that arise in practice are actually already Grothendieck pretopologies.

Examples

The prototype is the pretopology consisting of open covers of a topological space/manifold. Other 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 submersion?s. All of these have covering families consisting of single arrows. Such a pretopology is called a singleton pretopology (or, if you prefer the name coverage, a singleton coverage).

Most of the examples of coverages are in fact Grothendieck pretopologies.

(Other examples ..)