Ross Street has written the articles
where the stacks are considered on a 2-site. A 2-site is a 2-category with a Grothendieck 2-topology (compare Grothendieck topology), which is in turn defined in terms of 2-sieves (compare sieve). There is a Giraud-type theorem proved in this context. In a later article there were some errata mentioned.
Zoran: Could one define -sieves somehow as subobjects (in quasi-category sense) of representables in enriched quasi-category setup ?
So if -stacks are really -sheaves, and stacks are really -sheaves, then these are the real -sheaves, that is -sheaves? (with notation following that of -category). —Toby