Grothendieck topology

An accessible Grothendieck topology on a topos \(T\) is a class \(\tau\) of monomorphisms in \(T\), called the covering monomorphisms, such that:

1. The class \(\tau\) contains all isomorphisms, and its saturation \(\tau^s\) is of small generation;

2. The class \(\tau\) is a local class, in the sense of Definition 2.46;

3. Covering monomorphisms are closed under composition;

4. Given monomorphisms \(f\colon X \hookrightarrow Y\) and \(g\colon Y \hookrightarrow Z\) in \(T\), if \(g \circ f\) is a covering monomorphism, then so is \(g\).

We refer to a \(\tau\)-local object as a \(\tau\)-sheaf and denote the full subcategory of \(\tau\)-sheaves by