so... projective varieties (for example...) are "pseudo-stacky" in a certain sense...in the sense that, being non-affine, they're inadequately represented by the commutative rings of functions over them; instead you can represent them by the "categorified cpmmutative rings" of coherent sheaves over them... does this idea maybe repeat at a higher level in a simple-minded way? there are some specific examples of stacks that i've wondered whether they're such that they're inadequately represented by the coherent sheaves over them, but adequately represented by "the next thing up"; without however being actually "2-stacky" (in some sense...), and thus just "pseudo-2-stacky" or something like that... i think that i have some fairly specific idea of what "the next thing up" here (and "2-stacky" and so forth) is, though i'm not sure how systematically it forms part of a pattern...