Definition 6.126.
A category \(C\) is a local pretopos if the following conditions are satisfied:
\(C\) admits fiber products;
\(C\) admits finite coproducts, and they are van Kampen;
\(C\) admits colimits of groupoid objects, which are effective and universal (or equivalently van Kampen).
A pretopos is a local pretopos with a final object.