nLab two dimensional sheaf theory

Ross Street has written the articles

• Two dimensional sheaf theory, J. Pure Appl. Algebra 23 (1982) 251-270
• Characterization of bicategories of stacks, in: Category theory (Gummersbach 1981) Springer Lecture Notes in Mathematics 962, 1982, pp 282-291 transcript

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.

This should be related to the “$\infty$-dimensional sheaf theory” described at (infinity,1)-category of (infinity,1)-sheaves, somehow. Compare also derived stack.

Zoran: Could one define $(\infty,1)$-sieves somehow as subobjects (in quasi-category sense) of representables in enriched quasi-category setup ?

So if $\infty$-stacks are really $(\infty,1)$-sheaves, and stacks are really $(2,1)$-sheaves, then these are the real $2$-sheaves, that is $(2,2)$-sheaves? (with notation following that of $(n,r)$-category). —Toby

category: sheaf theory