I've made this a new question, rather than expanding the first one.
Torsten gives a good answer, and it partially illustrates in practice the 'second approach' I outlined in my other question. (You don't need to know the fiddly details of my other question, but you are welcome to look at it).
I'm well aware of the machinery of stacks of categories, a la Giraud, but what I want to know is: why can't these stacks be given some geometry? And not just 'because we don't'. Here's a sketch of how it might work:
Consider the 2-category of stacks of categories over some site $S$, considered as functors $X \to S$. Call a stack representable if it is representable in the usual way (equivalent to $S/a$ for some $a\in S$). Call a map of stacks $p:X\to Y$ lax-representable if for every representable stack $a$ and map $f:a \to Y$ the comma object $(f/p)$ is representable (comma objects coincide the the usual 2-pullback one uses with stacks in groupoids when all the categories involved are groupoids).
A comma object fits into a 2-commuting square, but the 2-arrow filling the square isn't necessarily invertible. We then say as per usual that a lax-representable map has property P if all the projections $(f/p) \to a$ have property P for representable $a$. Then one says a stack in categories $X$ is geometric if given a notion of cover (like smooth maps of schemes) if it admits a lax-representable cover $j:u\to X$ by a representable stack $u$ (and the diagonal $X\to X\times X$ should be lax-representable as well). Then the comma object $(j/j)$ together with $u$ should form an internal category $(j/j) \rightrightarrows u$ in $S$.
This is all well and good, but what stops this happening in practice? Perhaps lax-representable maps just do not exist? Or not enough of them? One potential barrier, in the algebraic case is that the collection of not-necessarily-invertible maps between two objects might be too big to the algebraic. Consider the automorphisms of your favourite algebro-geometric object, such as a vector bundle, or algebraic curve. Such things can again be formed into geometric objects in nice cases. But the categories of such things may not be cartesian closed. I haven't thought too much about this, but it seems to be a possible barrier.
To be concrete, my question is this:
What goes wrong/can go wrong with the above recipe? If you like, consider a concrete example, like the stack of quasicoherent sheaves, or the stack of vector bundles (with all morphisms of vector bundles).
One more thing… to make this theory analogous to that of ordinary geometric stacks (in groupoids), one thing that is lacking is the notion of stackification. The stackification $st(G)$ of a algebraic groupoid $G$ is essentially the category of torsors for that groupoid. Then $G_0$ (the objects of $G$) comes with a (fairly) canonical map $G_0 \to st(G)$ which is the presentation described above.
The problem is how one wants to stackify. What is a torsor for an algebraic category $C$? If one is thinking in terms of descent, it might be a torsor for the maximal algebraic groupoid contained in $C$. The hint I am getting is that this might be the right choice, but it really depends on coming up with working examples, as in my question.
Best Answer
I think the right thing to do with category-valued stacks is to keep the notion of "representability" the same (that is, use (pseudo) 2-pullbacks rather than comma objects), but to replace representability of the diagonal by representability of $X^2 \to X\times X$, where $X^2$ is the power (cotensor) of X by the free-living arrow (which is equivalent to X in case X is groupoid-valued). This does imply that $(j/j)$ is representable whenever the domain of $j$ is so, since $(j/j)$ is the pullback of $X^2 \to X\times X$ along $j\times j$.
One reason I think this is the right thing is that in the well-developed theory of "indexed categories" over a topos S regarded as "large categories relative to S as a universe of sets", representability of $X^2 \to X\times X$ is equivalent to the standard notion of "local smallness". (More generally, various kinds of "comprehensibility" for indexed categories can be rephrased as the representability of certain functors in the above sense.) Therefore, the resulting notion of "geometricity" would coincide with "essential smallness", as one would expect.